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}, - "distinct": true, - "id": "a1f2fa7ad06eb85536a1b0852542a815b0f80b3c", - "message": "Stark: Stone prover compatibility end to end for Fibonacci AIR (#596)\n\n* add test\n\n* make trace commitment SHARP compatible\n\n* wip\n\n* use powers of a single challenge for the boundary and transition coefficients\n\n* add permutation to match sharp compatible commitments on the trace\n\n* change trait bound from ByteConversion to Serializable\n\n* minor refactor\n\n* fmt, clippy\n\n* move std feature to inner trait function in Serializable\n\n* add IsStarkProver and IsStarkVerifier traits\n\n* proof of concept\n\n* composition poly breaker\n\n* WIP: commitment composition poly works. Opens are broken.\n\n* WIP Refactor open_trace_polys and open_composition_poly\n\n* Refactor sample iotas\n\n* Refactor sample iotas\n\n* make fri a trait\n\n* change trace ood evaluations in transcript\n\n* wip\n\n* sample gammas as power of a single challenge\n\n* fix z fri sampling\n\n* wip\n\n* wip\n\n* wip, broken\n\n* Compiles but fibonacci_5 does not work\n\n* Opens of query phase and OOD broken. Commit phase of FRI works.\n\n* Dont append to the transcript when grinding factor is zero\n\n* skip grinding factor when security bits is zero\n\n* remove permutation function\n\n* fmt\n\n* fix standard verifier\n\n* removes deep consistency check and openings of the first layer of fri for each query\n\n* SHARP computes the trace and composition polynomial openings and their symmetric elements consistently\n\n* Test symmetric elements in trace openings to compute deep composition polynomial\n\n* Composition polynomial opening evaluations are splitted between symmetric and not. The authentication paths remain equal\n\n* check openings in symmetric elements\n\n* make verifier sharp compatible\n\n* compute number of parts\n\n* fix verify fri for original prover\n\n* fix verify sym in stone prover\n\n* rename\n\n* rename file\n\n* wip\n\n* remove unnecessary variable\n\n* wip\n\n* move verifier\n\n* move fri\n\n* fix open\n\n* move stone to prover\n\n* remove file\n\n* fmt\n\n* clippy\n\n* clippy\n\n* remove redundant trait bounds\n\n* remove custom serialization/deserialization and replace it with serde_cbor\n\n* fmt\n\n* clippy\n\n* remove old files after merge from main\n\n* fmt\n\n* make field a type of IsStarkVerifier\n\n* remove frame serialization\n\n* separate compatibility test into individual tests\n\n* remove redundant test\n\n* add test case 2\n\n* minor refactor. add docs\n\n* minor refactor\n\n* remove unnecessary method\n\n* revert unintended changes to exercises\n\n* clippy\n\n* remove isFri trait\n\n* move Prover definition to the top of the file\n\n* update docs and add unit test\n\n* minor refactors. clippy\n\n* remove unused trait method\n\n* Move function only used for tests, to tests\n\n---------\n\nCo-authored-by: Agustin \nCo-authored-by: MauroFab ", - "timestamp": "2023-10-17T18:39:36Z", - "tree_id": "8a3bbe933499f1fa222b16676afc8b61010f40a6", - "url": "https://github.com/lambdaclass/lambdaworks/commit/a1f2fa7ad06eb85536a1b0852542a815b0f80b3c" - }, - "date": 1697569107765, - "tool": "cargo", - "benches": [ - { - "name": "Ordered FFT/Parallel (Metal)", - "value": 96709291, - "range": "± 10922609", - "unit": "ns/iter" - }, - { - "name": "Ordered FFT/Parallel (Metal) #2", - "value": 159255263, - "range": "± 4032066", - "unit": "ns/iter" - }, - { - "name": "Ordered FFT/Parallel (Metal) #3", - "value": 305754666, - "range": "± 5205265", - "unit": "ns/iter" - }, - { - "name": "Ordered FFT/Parallel (Metal) #4", - "value": 607407604, - "range": "± 17691302", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/Parallel (Metal)", - "value": 34510858, - "range": "± 586791", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/Parallel (Metal) #2", - "value": 67232865, - "range": "± 574259", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/Parallel (Metal) #3", - "value": 132588954, - "range": "± 362436", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/Parallel (Metal) #4", - "value": 274776510, - "range": "± 1258164", - "unit": "ns/iter" - }, - { - "name": "Bit-reverse permutation/Parallel (Metal)", - "value": 27819497, - "range": "± 893058", - "unit": "ns/iter" - }, - { - "name": "Bit-reverse permutation/Parallel (Metal) #2", - "value": 53507935, - "range": "± 2954437", - "unit": "ns/iter" - }, - { - "name": "Bit-reverse permutation/Parallel (Metal) #3", - "value": 106999948, - "range": "± 6934415", - "unit": "ns/iter" - }, - { - "name": "Bit-reverse permutation/Parallel (Metal) #4", - "value": 241193298, - "range": "± 11175116", - "unit": "ns/iter" - }, - { - "name": "Polynomial/evaluate_fft_metal", - "value": 118520853, - "range": "± 1154948", - "unit": "ns/iter" - }, - { - "name": "Polynomial/evaluate_fft_metal #2", - "value": 238856604, - "range": "± 1230388", - "unit": "ns/iter" - }, - { - "name": "Polynomial/evaluate_fft_metal #3", - "value": 491925437, - "range": "± 5151128", - "unit": "ns/iter" - }, - { - "name": "Polynomial/evaluate_fft_metal #4", - "value": 982654583, - "range": "± 14631880", - "unit": "ns/iter" - }, - { - "name": "Polynomial/interpolate_fft_metal", - "value": 407424333, - "range": "± 2710792", - "unit": "ns/iter" - }, - { - "name": "Polynomial/interpolate_fft_metal #2", - "value": 794036854, - "range": "± 6510413", - "unit": "ns/iter" - }, - { - "name": "Polynomial/interpolate_fft_metal #3", - "value": 1567799749, - "range": "± 14919128", - "unit": "ns/iter" - }, - { - "name": "Polynomial/interpolate_fft_metal #4", - "value": 3125214353, - "range": "± 22831261", - "unit": "ns/iter" - } - ] - }, - { - "commit": { - "author": { - "email": "41742639+schouhy@users.noreply.github.com", - "name": "Sergio Chouhy", - "username": "schouhy" - }, - "committer": { - "email": "noreply@github.com", - "name": "GitHub", - "username": "web-flow" - }, - "distinct": true, - "id": "a1f2fa7ad06eb85536a1b0852542a815b0f80b3c", - "message": "Stark: Stone prover compatibility end to end for Fibonacci AIR (#596)\n\n* add test\n\n* make trace commitment SHARP compatible\n\n* wip\n\n* use powers of a single challenge for the boundary and transition coefficients\n\n* add permutation to match sharp compatible commitments on the trace\n\n* change trait bound from ByteConversion to Serializable\n\n* minor refactor\n\n* fmt, clippy\n\n* move std feature to inner trait function in Serializable\n\n* add IsStarkProver and IsStarkVerifier traits\n\n* proof of concept\n\n* composition poly breaker\n\n* WIP: commitment composition poly works. Opens are broken.\n\n* WIP Refactor open_trace_polys and open_composition_poly\n\n* Refactor sample iotas\n\n* Refactor sample iotas\n\n* make fri a trait\n\n* change trace ood evaluations in transcript\n\n* wip\n\n* sample gammas as power of a single challenge\n\n* fix z fri sampling\n\n* wip\n\n* wip\n\n* wip, broken\n\n* Compiles but fibonacci_5 does not work\n\n* Opens of query phase and OOD broken. Commit phase of FRI works.\n\n* Dont append to the transcript when grinding factor is zero\n\n* skip grinding factor when security bits is zero\n\n* remove permutation function\n\n* fmt\n\n* fix standard verifier\n\n* removes deep consistency check and openings of the first layer of fri for each query\n\n* SHARP computes the trace and composition polynomial openings and their symmetric elements consistently\n\n* Test symmetric elements in trace openings to compute deep composition polynomial\n\n* Composition polynomial opening evaluations are splitted between symmetric and not. The authentication paths remain equal\n\n* check openings in symmetric elements\n\n* make verifier sharp compatible\n\n* compute number of parts\n\n* fix verify fri for original prover\n\n* fix verify sym in stone prover\n\n* rename\n\n* rename file\n\n* wip\n\n* remove unnecessary variable\n\n* wip\n\n* move verifier\n\n* move fri\n\n* fix open\n\n* move stone to prover\n\n* remove file\n\n* fmt\n\n* clippy\n\n* clippy\n\n* remove redundant trait bounds\n\n* remove custom serialization/deserialization and replace it with serde_cbor\n\n* fmt\n\n* clippy\n\n* remove old files after merge from main\n\n* fmt\n\n* make field a type of IsStarkVerifier\n\n* remove frame serialization\n\n* separate compatibility test into individual tests\n\n* remove redundant test\n\n* add test case 2\n\n* minor refactor. add docs\n\n* minor refactor\n\n* remove unnecessary method\n\n* revert unintended changes to exercises\n\n* clippy\n\n* remove isFri trait\n\n* move Prover definition to the top of the file\n\n* update docs and add unit test\n\n* minor refactors. clippy\n\n* remove unused trait method\n\n* Move function only used for tests, to tests\n\n---------\n\nCo-authored-by: Agustin \nCo-authored-by: MauroFab ", - 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Nicolini \n\n* Update provers/cairo/README.md\n\nCo-authored-by: Mariano A. Nicolini \n\n* Fix cargo\n\n* Re organized\n\n* Remove proof\n\n* Update readme\n\n* Fmt\n\n* CLI feature\n\n* CLI Fix\n\n* Fmt\n\n---------\n\nCo-authored-by: Jmunoz \nCo-authored-by: Mauro Toscano <12560266+MauroToscano@users.noreply.github.com>\nCo-authored-by: Mariano A. Nicolini \nCo-authored-by: MauroFab ", - "timestamp": "2023-10-18T19:14:15Z", - "tree_id": "e130fbbd180d9040ab5537d06cd7968eec722e64", - "url": "https://github.com/lambdaclass/lambdaworks/commit/8f7b415f56292e153cd7764a3212c14cad9aa2d2" - }, - "date": 1697657697567, - "tool": "cargo", - "benches": [ - { - "name": "Ordered FFT/Parallel (Metal)", - "value": 100379826, - "range": "± 6450454", - "unit": "ns/iter" - }, - { - "name": "Ordered FFT/Parallel (Metal) #2", - "value": 163892595, - "range": "± 13906378", - "unit": "ns/iter" - }, - { - "name": "Ordered FFT/Parallel (Metal) #3", - "value": 313616479, - "range": "± 1849445", - "unit": "ns/iter" - }, - { - "name": "Ordered FFT/Parallel (Metal) #4", - "value": 628623020, - "range": "± 28487626", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/Parallel (Metal)", - "value": 34463282, - "range": "± 666482", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/Parallel (Metal) #2", - "value": 69188242, - "range": "± 817067", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/Parallel (Metal) #3", - "value": 135843128, - "range": "± 1612443", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/Parallel (Metal) #4", - "value": 284956948, - "range": "± 2599386", - "unit": "ns/iter" - }, - { - "name": "Bit-reverse permutation/Parallel (Metal)", - "value": 27031847, - "range": "± 591414", - "unit": "ns/iter" - }, - { - "name": "Bit-reverse permutation/Parallel (Metal) #2", - "value": 56812024, - "range": "± 2967330", - "unit": "ns/iter" - }, - { - "name": "Bit-reverse permutation/Parallel (Metal) #3", - "value": 107609481, - "range": "± 2536657", - "unit": "ns/iter" - }, - { - "name": "Bit-reverse permutation/Parallel (Metal) #4", - "value": 238237111, - "range": "± 8088735", - "unit": "ns/iter" - }, - { - "name": "Polynomial/evaluate_fft_metal", - "value": 121057666, - "range": "± 893948", - "unit": "ns/iter" - }, - { - "name": "Polynomial/evaluate_fft_metal #2", - "value": 242204041, - "range": "± 1155622", - "unit": "ns/iter" - }, - { - "name": "Polynomial/evaluate_fft_metal #3", - "value": 493782687, - "range": "± 8468116", - "unit": "ns/iter" - }, - { - "name": "Polynomial/evaluate_fft_metal #4", - "value": 988587854, - "range": "± 15511036", - "unit": "ns/iter" - }, - { - "name": "Polynomial/interpolate_fft_metal", - "value": 407255562, - "range": "± 2217412", - "unit": "ns/iter" - }, - { - "name": "Polynomial/interpolate_fft_metal #2", - "value": 799572187, - "range": "± 4528942", - "unit": "ns/iter" - }, - { - "name": "Polynomial/interpolate_fft_metal #3", - "value": 1572104187, - "range": "± 16821363", - "unit": "ns/iter" - }, - { - "name": "Polynomial/interpolate_fft_metal #4", - "value": 3147655791, - "range": "± 18371234", - "unit": "ns/iter" - } - ] - }, - { - "commit": { - "author": { - "email": "62400508+juan518munoz@users.noreply.github.com", - "name": "juan518munoz", - "username": "juan518munoz" - }, - "committer": { - "email": "noreply@github.com", - "name": "GitHub", - "username": "web-flow" - }, - "distinct": true, - "id": "8f7b415f56292e153cd7764a3212c14cad9aa2d2", - "message": "Cairo reorganized (#607)\n\n* initial commit\n\n* modified README\n\n* fix cairo programs dir\n\n* Update provers/cairo/README.md\n\nCo-authored-by: Mariano A. 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Nicolini \n\n* stark curve, pedersen hash, precalculated points (#597)\n\n* stark curve, pedersen hash, precalculated points\n\n* lint\n\n* lint + array index fixes\n\n* cargo fmt\n\n* clippy\n\n* fmt\n\n* ref to starknet-rs\n\n* reviews\n\n* constant points moved to another file\n\n* move 'from_affine_hex_string' to starkcurve\n\n* make add_points a private fn\n\n* docs for pedersen\n\n* rename add_points -> lookup_and_accumulate\n\n---------\n\nCo-authored-by: Mariano A. Nicolini \n\n* Addition fuzzer for Stark Field (#601)\n\n* Add fuzzer\n\n* Remove artifacts\n\n---------\n\nCo-authored-by: Mariano A. Nicolini \n\n* update README\n\n* Update Makefile\n\n* update README bin\n\n* changed println for eprintln\n\n* fix README\n\n* rmv fuzz and wasm from README\n\n---------\n\nCo-authored-by: Jmunoz \nCo-authored-by: Mariano A. Nicolini \nCo-authored-by: MauroFab \nCo-authored-by: irfan \nCo-authored-by: Mauro Toscano <12560266+MauroToscano@users.noreply.github.com>", - "timestamp": "2023-10-20T18:06:45Z", - "tree_id": "a9b32edd6c63bf0926a468d79e3d39453c5ebbe8", - "url": "https://github.com/lambdaclass/lambdaworks/commit/6d686d3005c025a9a2da47c90ef4dc75228967ac" - }, - "date": 1697826448129, - "tool": "cargo", - "benches": [ - { - "name": "Ordered FFT/Parallel (Metal)", - "value": 98238472, - "range": "± 4349828", - "unit": "ns/iter" - }, - { - "name": "Ordered FFT/Parallel (Metal) #2", - "value": 159132724, - "range": "± 14049041", - "unit": "ns/iter" - }, - { - "name": "Ordered FFT/Parallel (Metal) #3", - "value": 306286145, - "range": "± 2845789", - "unit": "ns/iter" - }, - { - "name": "Ordered FFT/Parallel (Metal) #4", - "value": 612408792, - "range": "± 31916216", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/Parallel (Metal)", - "value": 34367410, - "range": "± 384259", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/Parallel (Metal) #2", - "value": 68075305, - "range": "± 911721", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/Parallel (Metal) #3", - "value": 133483552, - "range": "± 591638", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/Parallel (Metal) #4", - "value": 286020593, - "range": "± 2101313", - "unit": "ns/iter" - }, - { - "name": "Bit-reverse permutation/Parallel (Metal)", - "value": 29647960, - "range": "± 310623", - "unit": "ns/iter" - }, - { - "name": "Bit-reverse permutation/Parallel (Metal) #2", - "value": 55705983, - "range": "± 2205231", - "unit": "ns/iter" - }, - { - "name": "Bit-reverse permutation/Parallel (Metal) #3", - "value": 112749535, - "range": "± 2919139", - "unit": "ns/iter" - }, - { - "name": "Bit-reverse permutation/Parallel (Metal) #4", - "value": 241977388, - "range": "± 7888407", - "unit": "ns/iter" - }, - { - "name": "Polynomial/evaluate_fft_metal", - "value": 116717875, - "range": "± 569680", - "unit": "ns/iter" - }, - { - "name": "Polynomial/evaluate_fft_metal #2", - "value": 237875013, - "range": "± 1608643", - "unit": "ns/iter" - }, - { - "name": "Polynomial/evaluate_fft_metal #3", - "value": 490296520, - "range": "± 5223011", - "unit": "ns/iter" - }, - { - "name": "Polynomial/evaluate_fft_metal #4", - "value": 967902416, - "range": "± 13679593", - "unit": "ns/iter" - }, - { - "name": "Polynomial/interpolate_fft_metal", - "value": 404326281, - "range": "± 4578206", - "unit": "ns/iter" - }, - { - "name": "Polynomial/interpolate_fft_metal #2", - "value": 789941166, - "range": "± 7339159", - "unit": "ns/iter" - }, - { - "name": "Polynomial/interpolate_fft_metal #3", - "value": 1579340083, - "range": "± 25654235", - "unit": "ns/iter" - }, - { - "name": "Polynomial/interpolate_fft_metal #4", - "value": 3129161124, - "range": "± 35265682", - "unit": "ns/iter" - } - ] - }, - { - "commit": { - "author": { - "email": "62400508+juan518munoz@users.noreply.github.com", - "name": "juan518munoz", - "username": "juan518munoz" - }, - "committer": { - "email": "noreply@github.com", - "name": "GitHub", - "username": "web-flow" - }, - "distinct": true, - "id": "6d686d3005c025a9a2da47c90ef4dc75228967ac", - "message": "CLI improvements - followup (#603)\n\n* migrate cli to extern crate\n\n* add clap4 for cli handling\n\n* cargo fmt\n\n* readme changes\n\n* Add disclaimer readme\n\n* Remove bit security text from cairo prover crate\n\n* add compile option to makefile\n\n* add compile and run all to makefile\n\n* add example to readme\n\n* cargo fmt\n\n* add compile and prove to makefile\n\n* better readme\n\n* add quiet flag to makefile binaries\n\n* reorganized readme\n\n* initial commit\n\n* add remove after compile to docker run\n\n* externalize commands\n\n* Update README\n\n* cargo fmt\n\n* update README\n\n* err handling for non existing files\n\n* moved prints to err handler\n\n* Update provers/cairo-prover-cli/src/main.rs\n\nCo-authored-by: Mariano A. Nicolini \n\n* stark curve, pedersen hash, precalculated points (#597)\n\n* stark curve, pedersen hash, precalculated points\n\n* lint\n\n* lint + array index fixes\n\n* cargo fmt\n\n* clippy\n\n* fmt\n\n* ref to starknet-rs\n\n* reviews\n\n* constant points moved to another file\n\n* move 'from_affine_hex_string' to starkcurve\n\n* make add_points a private fn\n\n* docs for pedersen\n\n* rename add_points -> lookup_and_accumulate\n\n---------\n\nCo-authored-by: Mariano A. Nicolini \n\n* Addition fuzzer for Stark Field (#601)\n\n* Add fuzzer\n\n* Remove artifacts\n\n---------\n\nCo-authored-by: Mariano A. Nicolini \n\n* update README\n\n* Update Makefile\n\n* update README bin\n\n* changed println for eprintln\n\n* fix README\n\n* rmv fuzz and wasm from README\n\n---------\n\nCo-authored-by: Jmunoz \nCo-authored-by: Mariano A. Nicolini \nCo-authored-by: MauroFab \nCo-authored-by: irfan \nCo-authored-by: Mauro Toscano <12560266+MauroToscano@users.noreply.github.com>", - "timestamp": "2023-10-20T18:06:45Z", - "tree_id": "a9b32edd6c63bf0926a468d79e3d39453c5ebbe8", - "url": "https://github.com/lambdaclass/lambdaworks/commit/6d686d3005c025a9a2da47c90ef4dc75228967ac" - }, - "date": 1697827990305, - "tool": "cargo", - "benches": [ - { - "name": "Ordered FFT/Sequential from NR radix2", - "value": 1004128292, - "range": "± 1405401", - "unit": "ns/iter" - }, - { - "name": "Ordered FFT/Sequential from RN radix2", - "value": 3242638094, - "range": "± 44680735", - "unit": "ns/iter" - }, - { - "name": "Ordered FFT/Sequential from NR radix2 #2", - "value": 2108698236, - "range": "± 3245321", - "unit": "ns/iter" - }, - { - "name": "Ordered FFT/Sequential from RN radix2 #2", - "value": 7020801216, - "range": "± 103514760", - "unit": "ns/iter" - }, - { - "name": "Ordered FFT/Sequential from NR radix2 #3", - "value": 4394335381, - "range": "± 4925617", - "unit": "ns/iter" - }, - { - "name": "Ordered FFT/Sequential from RN radix2 #3", - "value": 15204910306, - "range": "± 285114687", - "unit": "ns/iter" - }, - { - "name": "Ordered FFT/Sequential from NR radix2 #4", - "value": 9167250028, - "range": "± 11922382", - "unit": "ns/iter" - }, - { - "name": "Ordered FFT/Sequential from RN radix2 #4", - "value": 32240140942, - "range": "± 182080515", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/natural", - "value": 36320726, - "range": "± 368973", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/natural inversed", - "value": 36311106, - "range": "± 186025", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/bit-reversed", - "value": 64408076, - "range": "± 591544", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/bit-reversed inversed", - "value": 64479583, - "range": "± 964920", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/natural #2", - "value": 73131392, - "range": "± 264529", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/natural inversed #2", - "value": 72475846, - "range": "± 300255", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/bit-reversed #2", - "value": 130276485, - "range": "± 2096882", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/bit-reversed inversed #2", - "value": 130668883, - "range": "± 927928", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/natural #3", - "value": 145937621, - "range": "± 397982", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/natural inversed #3", - "value": 147191964, - "range": "± 1296450", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/bit-reversed #3", - "value": 266507858, - "range": "± 1281898", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/bit-reversed inversed #3", - "value": 266866468, - "range": "± 1214920", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/natural #4", - "value": 289490519, - "range": "± 336427", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/natural inversed #4", - "value": 292586205, - "range": "± 418351", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/bit-reversed #4", - "value": 528857650, - "range": "± 1569586", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/bit-reversed inversed #4", - "value": 526135034, - "range": "± 1722263", - "unit": "ns/iter" - }, - { - "name": "Bit-reverse permutation/Sequential", - "value": 58122793, - "range": "± 199213", - "unit": "ns/iter" - }, - { - "name": "Bit-reverse permutation/Sequential #2", - "value": 119530051, - "range": "± 341616", - "unit": "ns/iter" - }, - { - "name": "Bit-reverse permutation/Sequential #3", - "value": 248811420, - "range": "± 905277", - "unit": "ns/iter" - }, - { - "name": "Bit-reverse permutation/Sequential #4", - "value": 512786997, - "range": "± 2205869", - "unit": "ns/iter" - }, - { - "name": "Polynomial evaluation/Sequential FFT", - "value": 1197412768, - "range": "± 7554820", - "unit": "ns/iter" - }, - { - "name": "Polynomial evaluation/Sequential FFT #2", - "value": 2492932248, - "range": "± 1872923", - "unit": "ns/iter" - }, - { - "name": "Polynomial evaluation/Sequential FFT #3", - "value": 5174859890, - "range": "± 8852740", - "unit": "ns/iter" - }, - { - "name": "Polynomial evaluation/Sequential FFT #4", - "value": 10681259750, - "range": "± 15414629", - "unit": "ns/iter" - }, - { - "name": "Polynomial interpolation/Sequential FFT", - "value": 1272181905, - "range": "± 3801252", - "unit": "ns/iter" - }, - { - "name": "Polynomial interpolation/Sequential FFT #2", - "value": 2642301224, - "range": "± 3853416", - "unit": "ns/iter" - }, - { - "name": "Polynomial interpolation/Sequential FFT #3", - "value": 5476179368, - "range": "± 8750121", - "unit": "ns/iter" - }, - { - "name": "Polynomial interpolation/Sequential FFT #4", - "value": 11311998618, - "range": "± 61240453", - "unit": "ns/iter" - }, - { - "name": "Polynomial/evaluate", - 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Nicolini", - "username": "entropidelic" - }, - "committer": { - "email": "noreply@github.com", - "name": "GitHub", - "username": "web-flow" - }, - "distinct": true, - "id": "94790afba6421a09ce5c90b09b6e794dfb994f53", - "message": "fix: polish some CLI details (#620)\n\n* Polish CLI\n\n* Fix README\n\n* Polish some README details\n\n* Change compile-and-run-all to compile-prove-and-verify", - "timestamp": "2023-10-23T14:25:02Z", - "tree_id": "745a4c9d9b815a844c7a5498172767746d9c7321", - "url": "https://github.com/lambdaclass/lambdaworks/commit/94790afba6421a09ce5c90b09b6e794dfb994f53" - }, - "date": 1698072501171, - "tool": "cargo", - "benches": [ - { - "name": "Ordered FFT/Parallel (Metal)", - "value": 127061771, - "range": "± 11517567", - "unit": "ns/iter" - }, - { - "name": "Ordered FFT/Parallel (Metal) #2", - "value": 197581527, - "range": "± 20399295", - "unit": "ns/iter" - }, - { - "name": "Ordered FFT/Parallel (Metal) #3", - "value": 416631937, - "range": "± 34550915", - "unit": "ns/iter" - }, - { - "name": "Ordered FFT/Parallel (Metal) #4", - "value": 868288937, - "range": "± 106873676", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/Parallel (Metal)", - "value": 55388726, - "range": "± 5114172", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/Parallel (Metal) #2", - "value": 101607262, - "range": "± 13409342", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/Parallel (Metal) #3", - "value": 183382432, - "range": "± 11738458", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/Parallel (Metal) #4", - "value": 345815906, - "range": "± 7256705", - "unit": "ns/iter" - }, - { - "name": "Bit-reverse permutation/Parallel (Metal)", - "value": 51961151, - "range": "± 3739910", - "unit": "ns/iter" - }, - { - "name": "Bit-reverse permutation/Parallel (Metal) #2", - "value": 60531389, - "range": "± 7595220", - "unit": "ns/iter" - }, - { - "name": "Bit-reverse permutation/Parallel (Metal) #3", - "value": 139539945, - "range": "± 13850489", - "unit": "ns/iter" - }, - { - "name": "Bit-reverse permutation/Parallel (Metal) #4", - "value": 234472968, - "range": "± 15092400", - "unit": "ns/iter" - }, - { - "name": "Polynomial/evaluate_fft_metal", - "value": 138204019, - "range": "± 3388004", - "unit": "ns/iter" - }, - { - "name": "Polynomial/evaluate_fft_metal #2", - "value": 274848646, - "range": "± 8643331", - "unit": "ns/iter" - }, - { - "name": "Polynomial/evaluate_fft_metal #3", - "value": 547513625, - "range": "± 9993694", - "unit": "ns/iter" - }, - { - "name": "Polynomial/evaluate_fft_metal #4", - "value": 1088816104, - "range": "± 13484309", - "unit": "ns/iter" - }, - { - "name": "Polynomial/interpolate_fft_metal", - "value": 462348302, - "range": "± 10988572", - "unit": "ns/iter" - }, - { - "name": "Polynomial/interpolate_fft_metal #2", - "value": 904413646, - "range": "± 22255961", - "unit": "ns/iter" - }, - { - "name": "Polynomial/interpolate_fft_metal #3", - "value": 1768836604, - "range": "± 48618707", - "unit": "ns/iter" - }, - { - "name": "Polynomial/interpolate_fft_metal #4", - "value": 3389915749, - "range": "± 158497650", - "unit": "ns/iter" - } - ] - }, - { - "commit": { - "author": { - "email": "mariano.nicolini.91@gmail.com", - "name": "Mariano A. Nicolini", - "username": "entropidelic" - }, - "committer": { - "email": "noreply@github.com", - "name": "GitHub", - "username": "web-flow" - }, - "distinct": true, - "id": "94790afba6421a09ce5c90b09b6e794dfb994f53", - "message": "fix: polish some CLI details (#620)\n\n* Polish CLI\n\n* Fix README\n\n* Polish some README details\n\n* Change compile-and-run-all to compile-prove-and-verify", - "timestamp": "2023-10-23T14:25:02Z", - "tree_id": "745a4c9d9b815a844c7a5498172767746d9c7321", - "url": "https://github.com/lambdaclass/lambdaworks/commit/94790afba6421a09ce5c90b09b6e794dfb994f53" - }, - "date": 1698074046253, - "tool": "cargo", - "benches": [ - { - "name": "Ordered FFT/Sequential from NR radix2", - "value": 1061230685, - "range": "± 47346069", - "unit": "ns/iter" - }, - { - "name": "Ordered FFT/Sequential from RN radix2", - "value": 3492273094, - "range": "± 29148892", - "unit": "ns/iter" - }, - { - "name": "Ordered FFT/Sequential from NR radix2 #2", - "value": 2099247882, - "range": "± 38553663", - "unit": "ns/iter" - }, - { - "name": "Ordered FFT/Sequential from RN radix2 #2", - "value": 7643146468, - "range": "± 129888098", - "unit": "ns/iter" - }, - { - "name": "Ordered FFT/Sequential from NR radix2 #3", - "value": 4491744400, - "range": "± 152673223", - "unit": "ns/iter" - }, - { - "name": "Ordered FFT/Sequential from RN radix2 #3", - "value": 16366095744, - "range": "± 91348632", - "unit": "ns/iter" - }, - { - "name": "Ordered FFT/Sequential from NR radix2 #4", - "value": 8914804420, - "range": "± 335507621", - "unit": "ns/iter" - }, - { - "name": "Ordered FFT/Sequential from RN radix2 #4", - "value": 34952237206, - "range": "± 117356871", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/natural", - "value": 32372040, - "range": "± 1046057", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/natural inversed", - "value": 35227043, - "range": "± 1351963", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/bit-reversed", - "value": 58358887, - "range": "± 5695272", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/bit-reversed inversed", - "value": 61622801, - "range": "± 1365673", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/natural #2", - "value": 70754515, - "range": "± 1763557", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/natural inversed #2", - "value": 73337478, - "range": "± 1195081", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/bit-reversed #2", - "value": 125733429, - "range": "± 2135079", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/bit-reversed inversed #2", - "value": 131762344, - "range": "± 1993851", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/natural #3", - "value": 143235203, - "range": "± 3703858", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/natural inversed #3", - "value": 148636447, - "range": "± 10663861", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/bit-reversed #3", - "value": 282453106, - "range": "± 16257430", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/bit-reversed inversed #3", - "value": 270269382, - "range": "± 6725530", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/natural #4", - "value": 283748068, - "range": "± 3478368", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/natural inversed #4", - "value": 291804148, - "range": "± 7776686", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/bit-reversed #4", - "value": 548043727, - "range": "± 5389195", - "unit": "ns/iter" - }, - { - "name": "FFT twiddles generation/bit-reversed inversed #4", - "value": 547595583, - "range": "± 8577704", - "unit": "ns/iter" - }, - { - "name": "Bit-reverse permutation/Sequential", - "value": 58351099, - "range": "± 1716289", - "unit": "ns/iter" - }, - { - "name": "Bit-reverse permutation/Sequential #2", - "value": 124822549, - "range": "± 2001061", - "unit": "ns/iter" - }, - { - "name": "Bit-reverse permutation/Sequential #3", - 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b' with Ruffini", - "value": 200, - "range": "± 8", - "unit": "ns/iter" - } - ] - } - ] - } -} \ No newline at end of file diff --git a/bench/index.html b/bench/index.html deleted file mode 100644 index 6c887805e..000000000 --- a/bench/index.html +++ /dev/null @@ -1,281 +0,0 @@ - - - - - - - Benchmarks - - - - -
- - - - - - - diff --git a/print.html b/print.html index b70eeb82e..22df3d51b 100644 --- a/print.html +++ b/print.html @@ -1185,12 +1185,12 @@

Protocol

In summary, at a very high level, the STARK protocol can be organized into three major parts:

  • Arithmetization and commitment of execution trace.
    • -
  • Construction of composition polynomial .
    • +
  • Construction and commitment of composition polynomial .
  • Opening of polynomials at random .

Arithmetization

As the Recap mentions, the trace is a table containing the system's state at every step. In this section, we will denote the trace as . A trace can have several columns to store different aspects or features of a particular state at a specific moment. We will refer to the -th column as . You can think of a trace as a matrix where the entry is the -th element of the -th state.

-

Most proving systems' primary tool is polynomials over a finite field . Each column of the trace will be interpreted as evaluations of such a polynomial . Consequently, any information about the states must be encoded somehow as an element in .

+

Most proving systems' primary tool is polynomials over a finite field . Each column of the trace will be interpreted as evaluations of such a polynomial . Consequently, any information about the states must be encoded somehow as an element in .

To ease notation, we will assume here and in the protocol that the constraints encoding transition rules depend only on a state and the previous one. Everything can be easily generalized to transitions that depend on many preceding states. Then, constraints can be expressed as multivariate polynomials in variables A transition from state to state will be valid if and only if when we plug row of in the first variables and row in the second variables of , we get for all . In mathematical notation, this is @@ -1202,73 +1202,68 @@

Vector commitments

Given a vector , commiting to means the following. The prover builds a Merkle tree out of it and sends its root to the verifier. The verifier can then ask the prover to reveal, or open, the value of the vector at some index . The prover won't have any choice except to send the correct value. The verifier will expect the corresponding value and the authentication path to the tree's root to check its authenticity. The authentication path also encodes the vector's position and its length .

-

In STARKs, all commited vectors are of the form for some polynomial and some domain fixed domain . The domain is always known to the prover and the verifier.

+

The root of the Merkle tree is said to be the commitment of , and we denote it here by .

FRI

-

The FRI protocol is a potent tool to prove that the commitment of a vector corresponds to the evaluations of a polynomial of a certain degree. The degree is a configuration of the protocol.

+

In STARKs, all commited vectors are of the form for some polynomial and some fixed domain . The domain is always known to the prover and the verifier. It can be proved, as long as is less than the total number of field elements, that every vector is equal to for a unique polynomial of degree at most . This is called the Lagrange interpolation theorem. It means, there is a unique polynomial of degree at most such that for all . And is an upper bound to the degree of . It could be less. For example, the vector of all ones is the evaluation of the constant polynomial , which has degree .

+

Suppose the vector is the vector of evaluations of a polynomial of degree strictly less than . Suppose one party holds the vector and another party holds only the commitment of it. The FRI protocol is an efficient interactive protocol with which the former can convince the latter that the commitment they hold corresponds to the vector of evaluations of a polynomial of degree strictly less than .

+

More precisely, the protocol depends on the following parameters

+
    +
  • Powers of two and with .
    • +
  • A vector , with , with a nonzero value in and a primitive -root of unity
    • +
+

A prover holds a vector , and the verifier holds the commitment of it. The result of the FRI protocol will be Accept if the unique polynomial of degree less than such that has degree less than . Even more precisely, the protocol proves that is very close to a vector with of degree less than , but it may differ in negligible proportion of the coordinates.

+

The number is called the blowup factor and the security of the protocol depends in part on this parameter. The specific shape of the domain set has some symmetric properties important for the inner workings of FRI, such as for all .

+

Variant useful for STARKs

+

FRI is usually described as above. In STARK, FRI is used as a building block for the polynomial commitment scheme of the next section. For that, a small variant of FRI is needed.

+

Suppose the prover holds a vector and the verifier holds its commitment as before. Suppose further that both parties know a function that takes two field elements and outputs another field element. For example could be the function . More precisely, the kind of functions we need are .

+

The protocol can be used to prove that the transformed vector is the vector of evaluations of a polynomial of degree at most . Note that in this variant, the verifier holds originally the commitment of the vector and not the commitment of the transformed vector. In the example, the verifier holds the commitment and FRI will return Accept if is the vector of evaluations of a polynomial of degree at most .

Polynomial commitments

STARK uses a univariate polynomial commitment scheme. The following is what is expected from the commit and open protocols:

  • Commit: given a polynomial , the prover produces a sort of hash of it. We denote it here by , called the commitment of . This hash is unique to . The prover usually sends to the verifier.
  • Open: this is an interactive protocol between the prover and the verifier. The prover holds the polynomial . The verifier only has the commitment . The verifier sends a value to the prover at which he wants to know the value . The prover sends a value to the verifier, and then they engage in the Open protocol. As a result, the verifier gets convinced that the polynomial corresponding to the hash evaluates to at .
-

Let's see how both of these protocols work in detail. Some configuration parameters are needed:

+

Let's see how both of these protocols work in detail. The same configuration parameters of FRI are needed:

  • Powers of two and with .
    • -
  • A vector , with in for all and for all .
    • -
  • An integer .
    • +
  • A vector , with , with a nonzero value in and a primitive -root of unity
-

The commitment scheme will only work for polynomials of degree at most . This means anyone can commit to any polynomial, but the Open protocol will pass only for polynomials satisfying that degree bound.

+

The commitment scheme will only work for polynomials of degree at most (polynomials of degree are allowed). This means: anyone can commit to any polynomial, but the Open protocol will pass only for polynomials satisfying that degree bound.

Commit

Given a polynomial , the commitment is just the commitment of the vector . That is, is the root of the Merkle tree of the vector of evaluations of at .

Open

It is an interactive protocol. So assume there is a prover and a verifier. We describe the process considering an honest prover. In the next section, we analyze what happens for malicious provers.

-

The prover holds the polynomial , and the verifier only the commitment of it. There is also an element . The prover evaluates and sends the result to the verifier. As we mentioned, the goal is to generate proof of the validity of the evaluation. Let us denote . This is a value now both the prover and verifier have.

-

The protocol has three steps:

-
    -
  • -

    FRI on : First, the prover and the verifier engage in a FRI protocol for polynomials of degree at most to check that is the commitment of such a polynomial. We will assume from now on that the degree of is at most . There is an optimization that completely removes this step—more on that later.

    -
    • -
  • -

    FRI on : Since , the polynomial can be written as for some polynomial . The prover computes the commitment and sends it to the verifier. Now they engage in a FRI protocol for polynomials of degree at most , which convinces the verifier that is the commitment of a polynomial of degree at most .

    -
    • -
  • -

    Point checks: From the point of view of the verifier the commitments and are still potentially unrelated. Therefore, there is a check to ensure that was computed properly from following the formula and that is its commitment. To do this, the verifier challenges the prover to open and as vectors. They use the open protocol of the vector commitment scheme to reveal the values and for some random point chosen by the verifier. Next he checks that . They repeat this last part times and, as we'll analyze in the next section, this whole thing will convince the verifier that as polynomials with overwhelming probability (about ). Finally, the verifier deduces that from this equality.

    -
    • -
+

The prover holds the polynomial , and the verifier only the commitment of it. There is also an element chosen by the verifier. The prover evaluates and sends the result back. As we mentioned, the goal is to generate proof of the validity of the evaluation. Let us denote the value received by the verifier.

+

Now they engage in the variant of the FRI protocol for the function . The verifier accepts the value if and only if the result of FRI is Accept.

+

Let's see why this makes sense.

+

Completeness

+

If the prover is honest, is of degree at most and equals . That means that + +for some polynomial . Since is of degree at most , then is of degree at most . The vector is then a vector of evaluations of a polynomial of degree at most . And it is equal to . So the FRI protocol will succeed.

Soundness

-

Let's analyze why the open protocol is sound. Keep in mind that the prover always has to provide a commitment of a polynomial of degree at most that satisfies for the chosen values of the verifier. That's the goal. An essential but subtle part of the soundness is that is a vector of length . To understand why let's see what would happen if .

-

Case

-

Suppose the prover tries to cheat the verifier by sending a value different from . This means that is not divisible by as polynomials. But as long as is not in the prover can interpolate the values -at and obtain some polynomial of degree at most . This polynomial satisfies for all , but the equality of polynomials does not hold. Mainly because that would imply that , and we are assuming the opposite case. Nevertheless, if they continue with the open protocol, FRI will pass since is a polynomial of degree at most . All subsequent point checks will pass since was crafted especially for that.

-

So how come is different from but has the property that for all ? FRI guarantees is of degree at most , which implies that is of degree at most . Then, since is of degree at most , the difference is again of degree at most . A polynomial of degree at most is zero if and only if we can prove that for distinct points. But the construction of only shows that for all , a set of points. It's not enough. One extra point and the equality would hold. Of course, that point doesn't exist, again, because that would imply that , which we assume is not true.

-

Case

-

Let's see one more example where we double the de size of but keep the same bound for the degree of the polynomials. So polynomials are of degree at most , and the length of is .

-

Again let's assume the prover wants to cheat the verifier similarly and claims that is but, in reality, . Inevitably, for the Open protocol to succeed, the prover has to send a commitment of some polynomial of degree at most . The strategy of the prover is pretty much constrained to be the same. But now he can't interpolate as before in every element of because that would produce a polynomial of degree at most . This most likely won't pass the FRI check.

-

Alternatively, he can choose some subset of of size and interpolate only there to get a polynomial of degree at most . That will succeed in the FRI phase, but what happens with the point checks? The check will pass if the verifier chooses a challenge that belongs to . If , then the check will inevitably fail. This is because, as we saw in the previous case, one extra successful point to the already points where was interpolated was enough to prove that , which we assume now not to hold. This means, if , then does not coincide with and the point check fails. So if the verifier chooses the challenges following a uniform distribution, the chances of the prover being caught cheating are for only one challenge. If the verifier performs challenges, then the chance of the prover not getting caught is .

-

General case: the blowup factor

-

If the ratio between and is , then challenges give of probability for a malicious prover to succeed. If the ratio is , that is, if , then that probability is for the same number of point checks. This provides another way of improving the soundness error. The larger the ratio, the more complex it is to cheat in the open protocol. This ratio is what's called the blowup factor. It is a security parameter, and finding a balance between the number of challenges and the size of is part of the configuration of the protocol. Increasing the size of makes committing an expensive operation since it involves building a Merkle tree with a vector of the size of . But increasing the number of challenges makes the size of the final proof larger.

-

We denote the blowup factor by , and it's always assumed to be a power of .

+

Let's sketch an idea of the soundness. Note that the value is chosen by the verifier after receiving the commitment of . So the prover does not know in advance, at the moment of sending , what will be.

+

Suppose the prover is trying to cheat and sends the commitment of a vector that's not the vector of evaluations of a polynomial of degree at most . Then the coordinates of the transformed vector are . Since was chosen by the verifier, dividing by shuffles all the elements in a very unpredictable way for the prover. So it is extremely unlikely that the cheating prover can craft an invalid vector such that the transformed vector turns out to be of degree at most . The expected degree of the polynomial associated with a random vector is .

Batch

During proof generation, polynomials are committed and opened several times. Computing these for each polynomial independently is costly. In this section, we'll see how batching polynomials can reduce the amount of computation. Let be a set of polynomials. We will commit and open as a whole. We note this batch commitment as .

-

We need the same configuration parameters as before: , with , a vector and .

-

As described earlier, to commit to a single polynomial , a Merkle tree is built over the vector . When committing to a batch of polynomials , the leaves of the Merkle tree are instead the concatenation of the polynomial evaluations. That is, in the batch setting, the Merkle tree is built for the vector . The commitment is the root of this Merkle tree. This reduces the proof size: we only need one Merkle tree for polynomials. The verifier can then only ask for values in batches. When the verifier chooses an index , the prover sends along with one authentication path. The verifier on his side computes the concatenation and validates it with the authentication path and . This also reduces the computational time. By traversing the Merkle tree one time, it can reveal several components simultaneously.

-

The batch open protocol proceeds similarly to the case of a single polynomial. The prover will try to convince the verifier that the committed polynomials at evaluate to some values . In the batch case, the prover will construct the following polynomial:

-

-

Where are challenges provided by the verifier. The prover commits to and sends to the verifier. Then the prover and verifier continue similarly to the normal open protocol for only. This means they engage in a FRI protocol for polynomials of degree at most for . Then they engage in the point checks for . Here, for each challenge , the prover uses one authentication path for to reveal and use one authentication path for to batch reveal values . Successful point checks here mean that .

+

We need the same configuration parameters as before: , with , a vector .

+

As described earlier, to commit to a single polynomial , a Merkle tree is built over the vector . When committing to a batch of polynomials , the leaves of the Merkle tree are instead the concatenation of the polynomial evaluations. That is, in the batch setting, the Merkle tree is built for the vector + +The commitment is the root of this Merkle tree. This reduces the proof size: we only need one Merkle tree for polynomials. The verifier can then only ask for values in batches. When the verifier chooses an index , the prover sends along with one authentication path. The verifier on his side computes the concatenation and validates it with the authentication path and . This also reduces the computational time. By traversing the Merkle tree one time, it can reveal several components simultaneously.

+

The batch open protocol proceeds similarly to the case of a single polynomial. The verifier sends evaluations points to the prover at which they wish to know the value of . The prover will try to convince the verifier that the committed polynomials , evaluate to some values . There is a generalization of the variant of FRI where the function takes more parameters, and in this case is +Where are challenges provided by the verifier. Then FRI return Accept if and only if the vector +is close to the vector of evaluations of a polynomial of degree at most . If this is the case, the verifier accepts the openings. In the context of STARKs, the polynomial is called the DEEP composition polynomial.

This is equivalent to running the open protocol times, one for each term and . Note that this optimization makes a huge difference, as we only need to run the FRI protocol once instead of running it once for each polynomial.

-

Optimize the open protocol reusing FRI internal challenges

-

There is an optimization for the open protocol to avoid running FRI to check that is of degree at most . The idea is as follows. Part of FRI protocol for , to check that it is of degree at most , involves revealing values of at other random points also chosen by the verifier. These are part of the internal workings of FRI. These challenges are unrelated to what we mentioned before. So if one removes the FRI check for , the point checks of the open protocol need to be performed on these challenges of the FRI protocol for . This optimization is included in the formal description of the protocol.

References

High-level description of the protocol

-

The protocol is split into rounds. Each round more or less represents an interaction with the verifier. Each round will generally start by getting a challenge from the verifier.

-

The prover will need to interpolate polynomials, and he will always do it over the set , where is a root of unity in . Also, the vector commitments will be performed over the set where is a root of unity and is some field element. This is the set we denoted in the commitment scheme section. The specific choices for the shapes of these sets are motivated by optimizations at a code level.

+

The protocol is split into rounds. Each round more or less represents an interaction with the verifier. Each round will generally start by getting a challenge from the verifier.

+

The prover will need to interpolate polynomials, and he will always do it over the set , where is a root of unity in . Also, the vector commitments will be performed over the set where is a root of unity and is some field element. This is the set we denoted in the commitment scheme section.

Round 1: Arithmetization and commitment of the execution trace

In round 1, the prover commits to the columns of the trace . He does so by interpolating each column and obtaining univariate polynomials . Then the prover commits to over . In this way, we have . @@ -1278,9 +1273,9 @@

. This function will have the property that it is a polynomial if and only if the trace that the prover committed to at round 1 is valid and satisfies the agreed polynomial constraints. That is, will be a polynomial if and only if is a trace that satisfies all the transition and boundary constraints.

Note that we can compose the polynomials , the ones that interpolate the columns of the trace , with the multivariate constraint polynomials as follows. -These result in univariate polynomials. And the same can be done for the boundary constraints. Since , these univariate polynomials vanish at every element of if and only if the trace is valid.

+These result in univariate polynomials. The same can be done for the boundary constraints. Since , these univariate polynomials vanish at every element of if and only if the trace is valid.

As we already mentioned, this is assuming that transitions only depend on the current and previous state. But it can be generalized to include frames with three or more rows or more context for each constraint. For example, in the Fibonacci case, the most natural way is to encode it as one transition constraint that depends on a row and the two preceding it, as we already did in the Recap section. The STARK protocol checks whether the function is a polynomial instead of checking that the polynomial is zero over the domain . The two statements are equivalent.

-

The verifier could check that all are polynomials one by one, and the same for the polynomials coming from the boundary constraints. But this is inefficient; the same can be obtained with a single polynomial. To do this, the prover samples challenges and obtains a random linear combination of these polynomials. The result of this is denoted by and is called the composition polynomial. It integrates all the constraints by adding them up. So after computing , the prover commits to it and sends the commitment to the verifier. The rest of the protocol aims to prove that was constructed correctly and is a polynomial, which can only be true if the prover has a valid extension of the original trace.

+

The verifier could check that all are polynomials one by one, and the same for the polynomials coming from the boundary constraints. However, this is inefficient; the same can be obtained with a single polynomial. To do this, the prover samples challenges and obtains a random linear combination of these polynomials. The result of this is denoted by and is called the composition polynomial. It integrates all the constraints by adding them up. So after computing , the prover commits to it and sends the commitment to the verifier. The rest of the protocol aims to prove that was constructed correctly and is a polynomial, which can only be true if the prover has a valid extension of the original trace.

Round 3: Evaluation of polynomials at

The verifier must check that was constructed according to the protocol rules. That is, has to be a linear combination of all the functions and similar terms for the boundary constraints. To do so, in round 3 the verifier chooses a random point and the prover computes , and for all . With all these, the verifier can check that and the expected linear combination coincide, at least when evaluated at . Since was chosen randomly, this proves with overwhelming probability that was properly constructed.

Round 4: Run batch open protocol

@@ -1289,11 +1284,11 @@

STARKs protocol

In this section we describe precisely the STARKs protocol used in Lambdaworks.

We begin with some additional considerations and notation for most of the relevant objects and values to refer to them later on.

-

Grinding

-

This is a technique to increase the soundness of the protocol by adding proof of work. It works as follows. At some fixed point in the protocol, the verifier sends a challenge to the prover. The prover needs to find a string such that begins with a predefined number of zeroes. Here denotes the concatenation of and , seen as bit strings. -The number of zeroes is called the grinding factor. The hash function can be any hash function, independent of other hash functions used in the rest of the protocol. In Lambdaworks we use Keccak256.

Transcript

The Fiat-Shamir heuristic is used to make the protocol noninteractive. We assume there is a transcript object to which values can be added and from which challenges can be sampled.

+

Grinding

+

This is a technique to increase the soundness of the protocol by adding proof of work. It works as follows. At some fixed point in the protocol, a value is derived in a deterministic way from all the interactions between the prover and the verifier up to that point (the state of the transcript). The prover needs to find a string such that begins with a predefined number of zeroes. Here denotes the concatenation of and , seen as bit strings. +The number of zeroes is called the grinding factor. The hash function can be any hash function, independent of other hash functions used in the rest of the protocol. In Lambdaworks we use Keccak256.

General notation

  • denotes a finite field.
    • @@ -1312,7 +1307,7 @@

    is the total number of columns: .
  • denote the transition constraint polynomials for . We are assuming these are of degree at most 2.
  • denote the transition constraint zerofiers for .
    • -
  • is the blowup factor.
    • +
  • is the blowup factor.
  • is the grinding factor.
  • is number of FRI queries.
  • We assume there is a fixed hash function from to binary strings. We also assume all Merkle trees are constructed using this hash function.
    • @@ -1336,9 +1331,11 @@

    Derived values<

    Notation of important operations

    Vector commitment scheme

    Given a vector . The operation returns the root of the Merkle tree that has the hash of the elements of as leaves.

    -

    For , the operation returns and the authentication path to the Merkle tree root.

    +

    For , the operation returns the pair , where is the authentication path to the Merkle tree root.

    The operation returns Accept or Reject depending on whether the -th element of is . It checks whether the authentication path is compatible with , and the Merkle tree root .

    In our cases the sets will be of the form for some elements . It will be convenient to use the following abuse of notation. We will write to mean . Similarly, we will write instead of . Note that this is only notation and is only checking that the is the -th element of the commited vector.

    +
    Batch
    +

    As we mentioned in the protocol overview. When committing to multiple vectors , where one can build a single Merkle tree. Its -th leaf is the concatenation of all the -th coordinates of all vectors, that is, . The commitment to this batch of vectors is the root of this Merkle tree.

    Protocol

    Prover

    Round 0: Transcript initialization

    @@ -1363,15 +1360,15 @@
    Round 2: Construction of composition polynomial
      -
    • Sample and in from the transcript.
      • -
    • Sample and in from the transcript.
      • +
    • Sample in from the transcript.
      • +
    • Sample in from the transcript.
    • Compute .
    • Compute .
    • Compute the composition polynomial
    • Decompose as
      • -
    • Compute commitments and .
      • +
    • Compute commitments and (Batch commitment optimization applies here).
    • Add and to the transcript.

    Round 3: Evaluation of polynomials at

    @@ -1387,8 +1384,7 @@

    Round 4.1.k: FRI commit phase

      -
    • Let , and .
      • -
    • Add to the transcript.
      • +
    • Let .
    • For do the following: -
      Round 4.4: Open deep composition polynomial components
      -
        -
      • For do the following: -
          -
        • Compute , .
          • -
        • Compute for all .
          • +
        • Sample random index from the transcript and let .
          • +
        • Compute and for all .
          • +
        • Compute and .
          • +
        • Compute and .
          • +
        • Compute and for all .

      Build proof

      • Send the proof to the verifier: -
        • +

      Verifier

      -

      Notation

      +

      From the point of view of the verifier, the proof they receive is a bunch of values that may or may not be what they claim to be. To make this explicit, we avoid denoting values like as such, because that implicitly assumes that the value was obtained after evaluating a polynomial at . And that's something the verifier can't assume. We use the following convention.

        -
      • Bold capital letters refer to commitments. For example is the claimed commitment .
        • -
      • Greek letters with superscripts refer to claimed function evaluations. For example is the claimed evaluation .
        • -
      • Gothic letters refer to authentication paths (e.g. is the authentication path of the opening of ).
        • +
      • Bold capital letters refer to commitments. For example is the claimed commitment .
        • +
      • Greek letters with superscripts refer to claimed function evaluations. For example is the claimed evaluation and is the claimed evaluation of . Note that field elements in superscripts never indicate powers. They are just notation.
        • +
      • Gothic letters refer to authentication paths. For example is the authentication path of a opening of .
        • +
      • Recall that every opening is a pair , where is the claimed value at index and is the authentication path. So for example, is denoted as from the verifier's end.

      Input

      -

      +

      This is the proof using the notation described above. The elements appear in the same exact order as they are in the Prover section, serving also as a complete reference of the meaning of each value.

      +

      Step 1: Replay interactions and recover challenges

      • Start a transcript
        • -
      • (Strong Fiat Shamir) Commit to the set of coefficients of the transition and boundary polynomials, and add the commitments to the transcript.
        • +
      • (Strong Fiat Shamir) Add all public values to the transcript.
      • Add to the transcript for all .
      • Sample random values from the transcript.
      • Add to the transcript for .
        • @@ -1457,7 +1448,6 @@

        Sample from the transcript.
      • Add , , and to the transcript.
      • Sample , , and from the transcript.
        • -
      • Add to the transcript
      • For do the following: @@ -1485,40 +1474,56 @@

        Step 3: Verify FRI

          -
        • Check that the following are all Accept: +
        • Reconstruct the deep composition polynomial values at and . That is, define +
          • +
        • For all : +
            +
          • For all :
              -
            • for all .
              • -
            • for all , .
              • +
            • Check that and are Accept.
              • +
            • Solve the following system of equations on the variables +
              • +
            • If , check that equals
              • +
            • If , check that equals .
            • -
          • For all : - For all : - Solve the following system of equations on the variables - -- Define -- Check that is equal to .
          -

          Step 4: Verify deep composition polynomial is FRI first layer

          +
          • +
        +

        Step 4: Verify trace and composition polynomials openings

        • For do the following:
          • Check that the following are all Accept:
              +
            • for all .
            • .
            • .
              • -
            • for all .
              • +
            • for all .
              • +
            • .
              • +
            • .
            • -
          • Check that is equal to the following: -
        -

        Other

        -

        Notes on Optimizations

        -
          -
        • Inversions of finite field elements are slow. There is a very well known trick to batch invert many elements at once replacing inversions by multiplications. See here for the algorithm.
          • -
        • One of the most computationally intensive operations performed is polynomial division. These can be optimized by utilizing Fast Fourier Transform (FFT) to divide each field element in Lagrange form.
          • -
        • In specific scenarios, such as dividing by a polynomial of the form (x-a), Ruffini's rule can be employed to further enhance performance.
          • -
        +

        Notes on Optimizations and variants

        +

        Sampling of challenges variant

        +

        To build the composition the prover samples challenges and for and . A variant of this is sampling a single challenge and defining and as powers of . That is, define for and for .

        +

        The same variant applies for the challenges for used to build the deep composition polynomial. In this case the variant samples a single challenge and defines , for all , and .

        +

        Batch inversion

        +

        Inversions of finite field elements are slow. There is a very well known trick to batch invert many elements at once replacing inversions by multiplications. See here for the algorithm.

        +

        FFT

        +

        One of the most computationally intensive operations performed is polynomial division. These can be optimized by utilizing Fast Fourier Transform (FFT) to divide each field element in Lagrange form.

        +

        Ruffini's rule

        +

        In specific scenarios, such as dividing by a polynomial of the form , for example when building the deep composition polynomial, Ruffini's rule can be employed to further enhance performance.

        +

        Bit-reversal ordering of Merkle tree leaves

        +

        As one can see from inspecting the protocol, there are multiple times where, for a polynomial , the prover sends both openings and . This implies, a priori, sending two authentication paths. Domains can be indexed using bit-reverse ordering to reduce this to a single authentication path for both openings, as follows.

        +

        The natural way of building a Merkle tree to commit to a vector , is assigning the value to leaf . If this is the case, the value is at position and the value is at position . This is because equals for the value used in the protocol.

        +

        Instead of this naive approach, a better solution is to assign the value to leaf , where is the bit-reversal permutation. This is the permutation that maps to the index whose binary representation (padded to bits), is the binary representation of but in reverse order. For example, if and , then its binary representation is , which reversed is . Therefore . In the same way and . Check out the wikipedia article. With this ordering of the leaves, if is even, element is at index and is at index . Which means that a single authentication path serves to validate both points simultaneously.

        +

        Redundant values in the proof

        +

        The prover opens the polynomials of the FRI layers at and for all . Later on, the verifier uses each of those pairs to reconstruct one of the values of the next layer, namely . So there's no need to add the value to the proof, as the verifier reconstructs them. The prover only needs to send the authentication paths for them.

        +

        The protocol is only modified at Step 3 of the verifier as follows. Checking that is skipped. After computing , the verifier uses it to check that is Accept, which proves that is actually , and continues to the next iteration of the loop.

        STARKs Prover Lambdaworks Implementation

        The goal of this section will be to go over the details of the implementation of the proving system. To this end, we will follow the flow the example in the recap chapter, diving deeper into the code when necessary and explaining how it fits into a more general case.

        @@ -1842,7 +1847,7 @@

        FFT evaluation and interpolation

        When evaluating or interpolating a polynomial, if the input (be it coefficients or evaluations) size isn't a power of two then the FFT API will extend it with zero padding until this requirement is met. This is because the library currently only uses a radix-2 FFT algorithm.

        Also, right now FFT only supports inputs with a size up to elements.

        -

        Other

        +

        Other

        Why use roots of unity?

        Whenever we interpolate or evaluate trace, boundary and constraint polynomials, we use some -th roots of unity. There are a few reasons for this:

          diff --git a/searchindex.js b/searchindex.js index a9ffaa7bb..78ae00468 100644 --- a/searchindex.js +++ b/searchindex.js @@ -1 +1 @@ -Object.assign(window.search, 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site hosts the main documentation for Lambdaworks as a whole. It is still a work in progress.","breadcrumbs":"Introduction » Introduction","id":"0","title":"Introduction"},"1":{"body":"","breadcrumbs":"FFT Library » Benchmarks » FFT Benchmarks","id":"1","title":"FFT Benchmarks"},"10":{"body":"The claim in the previous section is clearly not an \"if and only if\" statement because the following trace columns do satisfy the equations but do not correspond to a valid execution: A B C 2 3 6 0 0 0 20 - 19 The V matrix encodes the carry of the results from one gate to the right or left operand of a subsequent one. These are called wirings . Like the Q matrix, it's independent of the particular evaluation. It consists of indices for all input and intermediate variables. In this case that matrix is: L R O 0 1 2 2 1 3 3 - 4 Here 0 is the index of e, 1 is the index of x, 2 is the index of u, 3 is the index of v and 4 is the index of the output w. Now we can update the claim to have an \"if and only if\" statement. Claim: Let T be a matrix with columns A,B,C. It correspond to a valid evaluation of the circuit if and only if a) for all i the following equality holds Ai​(QL​)i​+Bi​(QR​)i​+Ai​Bi​QM​+Ci​(QO​)i​+(QC​)i​=0, b) for all i,j,k,l such that Vi,j​=Vk,l​ we have Ti,j​=Tk,l​. So now our malformed example does not pass the second check.","breadcrumbs":"Plonk » Recap » The V matrix","id":"10","title":"The V matrix"},"100":{"body":"The way we specify our fibonacci transition constraint looks like this: fn compute_transition( &self, frame: &air::frame::Frame, _rap_challenges: &Self::RAPChallenges,\n) -> Vec> { let first_row = frame.get_row(0); let second_row = frame.get_row(1); let third_row = frame.get_row(2); vec![third_row[0] - second_row[0] - first_row[0]]\n} It's not completely obvious why this is how we chose to express transition constraints, so let's talk a little about it. What we need to specify in this method is the relationship that has to hold between the current step of computation and the previous ones. For this, we get a Frame as an argument. This is a struct holding the current step (i.e. the current row of the trace) and all previous ones needed to encode our constraint. In our case, this is the current row and the two previous ones. To access rows we use the get_row method. The current step is always the last row (in our case 2), with the others coming before it. In our compute_transition method we get the three rows we need and return third_row[0] - second_row[0] - first_row[0] which is the value that needs to be zero for our constraint to hold. Because we support multiple transition constraints, we actually return a vector with one value per constraint, so the first element holds the first constraint value and so on.","breadcrumbs":"STARKs » Implementation » High Level API » Transition Constraints","id":"100","title":"Transition Constraints"},"101":{"body":"After defining our AIR, we create our specific trace to prove against it. let trace = fibonacci_trace([FE17::new(1), FE17::new(1)], 4); let trace_table = TraceTable { table: trace.clone(), num_cols: 1,\n}; TraceTable is the struct holding execution traces; the num_cols says how many columns the trace has, the table field is a vec holding the actual values of the trace in row-major form, meaning if the trace looks like this | 1 | 2 |\n| 3 | 4 |\n| 5 | 6 | then its corresponding TraceTable is let trace_table = TraceTable { table: vec![1, 2, 3, 4, 5, 6], num_cols: 2,\n}; In our example, fibonacci_trace is just a helper function we use to generate the fibonacci trace with 4 rows and [1, 1] as the first two values.","breadcrumbs":"STARKs » Implementation » High Level API » TraceTable","id":"101","title":"TraceTable"},"102":{"body":"After specifying our constraints and trace, the only thing left to do is provide a few parameters related to the STARK protocol and our AIR. These specify things such as the number of columns of the trace and proof configuration, among others. They are all encapsulated in the AirContext struct, which in our example we instantiate like this: let context = AirContext { options: ProofOptions { blowup_factor: 2, fri_number_of_queries: 1, coset_offset: 3, }, trace_columns: trace_table.n_cols, transition_degrees: vec![1], transition_exemptions: vec![2], transition_offsets: vec![0, 1, 2], num_transition_constraints: 1,\n}; Let's go over each of them: options requires a ProofOptions struct holding specific parameters related to the STARK protocol to be used when proving. They are: The blowup_factor used for the trace LDE extension, a parameter related to the security of the protocol. The number of queries performed by the verifier when doing FRI, also related to security. The offset used for the LDE coset. This depends on the field being used for the STARK proof. trace_columns are the number of columns of the trace, respectively. transition_degrees holds the degree of each transition constraint. transition_exemptions is a Vec which tells us, for each column, the number of rows the transition constraints should not apply, starting from the end of the trace. In the example, the transition constraints won't apply on the last two rows of the trace. transition_offsets holds the indexes that define a frame for our AIR. In our fibonacci case, these are [0, 1, 2] because we need the current row and the two previous one to define our transition constraint. num_transition_constraints simply says how many transition constraints our AIR has.","breadcrumbs":"STARKs » Implementation » High Level API » AIR Context","id":"102","title":"AIR Context"},"103":{"body":"Having defined all of the above, proving our fibonacci example amounts to instantiating the necessary structs and then calling prove passing the trace, public inputs and proof options. We use a simple implementation of a hasher called TestHasher to handle merkle proof building. let proof = prove(&trace_table, &pub_inputs, &proof_options); Verifying is then done by passing the proof of execution along with the same AIR to the verify function. assert!(verify(&proof, &pub_inputs, &proof_options));","breadcrumbs":"STARKs » Implementation » High Level API » Proving execution","id":"103","title":"Proving execution"},"104":{"body":"In this section we go over how a few things in the prove and verify functions are implemented. If you just need to use the prover, then you probably don't need to read this. If you're going through the code to try to understand it, read on. We will once again use the fibonacci example as an ilustration. Recall from the recap that the main steps for the prover and verifier are the following:","breadcrumbs":"STARKs » Implementation » Under the hood » How this works under the hood","id":"104","title":"How this works under the hood"},"105":{"body":"Compute the trace polynomial t by interpolating the trace column over a set of 2n-th roots of unity {gi:0≤i<2n}. Compute the boundary polynomial B. Compute the transition constraint polynomial C. Construct the composition polynomial H from B and C. Sample an out of domain point z and provide the evaluation H(z) and all the necessary trace evaluations to reconstruct it. In the fibonacci case, these are t(z), t(zg), and t(zg2). Sample a domain point x_0 and provide the evaluation H(x0​) and t(x0​). Construct the deep composition polynomial Deep(x) from H, t, and the evaluations from the item above. Do FRI on Deep(x) and provide the resulting FRI commitment to the verifier. Provide the merkle root of t and the merkle proof of t(x0​).","breadcrumbs":"STARKs » Implementation » Under the hood » Prover side","id":"105","title":"Prover side"},"106":{"body":"Take the evaluation H(z) along with the trace evaluations the prover provided. Reconstruct the evaluations B(z) and C(z) from the trace evaluations. Check that the claimed evaluation H(z) the prover gave us actually satisfies H(z)=β1​B(z)+β2​C(z) Take the evaluations H(x0​) and t(x0​). Check that the claimed evaluation Deep(x0​) the prover gave us actually satisfies Deep(x0​)=γ1​x0​−zH(x0​)−H(z)​+γ2​x0​−zt(x0​)−t(z)​+γ3​x0​−zgt(x0​)−t(zg)​+γ4​x0​−zg2t(x0​)−t(zg2)​ Take the provided FRI commitment and check that it verifies. Using the merkle root and the merkle proof the prover provided, check that t(x0​) belongs to the trace. Following along the code in the prove and verify functions, most of it maps pretty well to the steps above. The main things that are not immediately clear are: How we take the constraints defined in the AIR through the compute_transition method and map them to transition constraint polynomials. How we then construct H from them and the boundary constraint polynomials. What the composition polynomial even/odd decomposition is. What an ood frame is. What the transcript is.","breadcrumbs":"STARKs » Implementation » Under the hood » Verifier side","id":"106","title":"Verifier side"},"107":{"body":"This is possibly the most complex part of the code, so what follows is a long explanation for it. In our fibonacci example, after obtaining the trace polynomial t by interpolating, the transition constraint polynomial is C(x)=∏i=05​(x−gi)t(xg2)−t(xg)−t(x)​ On our prove code, if someone passes us a fibonacci AIR like the one we showed above used in one of our tests, we somehow need to construct C(x). However, what we are given is not a polynomial, but rather this method fn compute_transition( &self, frame: &air::frame::Frame, ) -> Vec> { let first_row = frame.get_row(0); let second_row = frame.get_row(1); let third_row = frame.get_row(2); vec![third_row[0] - second_row[0] - first_row[0]]\n} So how do we get to C(x) from this? The answer is interpolation. What the method above is doing is the following: if you pass it a frame that looks like this ​t(x0​)t(x0​g)t(x0​g2)​​ for any given point x0​, it will return the value t(x0​g2)−t(x0​g)−t(x0​) which is the numerator in C(x0​). Using the transition_exemptions field we defined in our AIR, we can also compute evaluations in the denominator, i.e. the zerofier evaluations. This is done under the hood by the transition_divisors() method. The above means that even though we don't explicitly have the polynomial C(x), we can evaluate it on points given an appropriate frame. If we can evaluate it on enough points, we can then interpolate them to recover C(x). This is exactly how we construct both transition constraint polynomials and subsequently the composition polynomial H. The job of evaluating H on enough points so we can then interpolate it is done by the ConstraintEvaluator struct. You'll notice prove does the following let constraint_evaluations = evaluator.evaluate( &lde_trace, &lde_roots_of_unity_coset, &alpha_and_beta_transition_coefficients, &alpha_and_beta_boundary_coefficients,\n); This function call will return the evaluations of the boundary terms βiB​Bi​(x) and constraint terms βiT​Ci​(x) for every i. The constraint_evaluations value returned is a ConstraintEvaluationTable struct, which is nothing more than a big list of evaluations of each polynomial required to construct H. With this in hand, we just call let composition_poly = constraint_evaluations.compute_composition_poly(& lde_roots_of_unity_coset); which simply interpolates the sum of all evaluations to obtain H. Let's go into more detail on how the evaluate method reconstructs C(x) in our fibonacci example. It receives the lde_trace as an argument, which is this: ​t(ω0)t(ω1)…t(ω15)​​ where ω is the primitive root of unity used for the LDE, that is, ω satisfies ω2=g. We need to recover C(x), a polynomial whose degree can't be more than t(x)'s. Because t was built by interpolating 8 points (the trace), we know we can recover C(x) by interpolating it on 16 points. We choose these points to be the LDE roots of unity {ω0,ω,ω2,…,ω15} Remember that to evaluate C(x) on these points, all we need are the evaluations of the polynomial t(xg2)−t(xg)−t(x) as the zerofier ones we can compute easily. These become: t(ω0g2)−t(ω0g)−t(ω0)t(ωg2)−t(ωg)−t(ω)t(ω2g2)−t(ω2g)−t(ω2)⋮t(ω15g2)−t(ω15g)−t(ω15) If we remember that ω2=g, this is t(ω4)−t(ω2)−t(ω0)t(ω5)−t(ω3)−t(ω)t(ω6)−t(ω4)−t(ω2)⋮t(ω3)−t(ω)−t(ω15) and we can compute each evaluation here by calling compute_transition on the appropriate frame built from the lde_trace. Specifically, for the first evaluation we can build the frame: ​t(ω0)t(ω2)t(ω4)​​ Calling compute_transition on this frame gives us the first evaluation. We can get the rest in a similar fashion, which is what this piece of code in the evaluate method does: for (i, d) in lde_domain.iter().enumerate() { let frame = Frame::read_from_trace( lde_trace, i, blowup_factor, &self.air.context().transition_offsets, ) let mut evaluations = self.air.compute_transition(&frame); ...\n} Each iteration builds a frame as above and computes one of the evaluations needed. The rest of the code just adds the zerofier evaluations, along with the alphas and betas. It then also computes boundary polynomial evaluations by explicitly constructing them.","breadcrumbs":"STARKs » Implementation » Under the hood » Reconstructing the transition constraint polynomials","id":"107","title":"Reconstructing the transition constraint polynomials"},"108":{"body":"The verifier employs the same trick to reconstruct the evaluations on the out of domain point Ci​(z) for the consistency check.","breadcrumbs":"STARKs » Implementation » Under the hood » Verifier","id":"108","title":"Verifier"},"109":{"body":"At the end of the recap we talked about how in our code we don't actually commit to H, but rather an even/odd decomposition for it. These are two polynomials H_1 and H_2 that satisfy H(x)=H1​(x2)+xH2​(x2) This all happens on this piece of code let composition_poly = constraint_evaluations.compute_composition_poly(&lde_roots_of_unity_coset); let (composition_poly_even, composition_poly_odd) = composition_poly.even_odd_decomposition(); // Evaluate H_1 and H_2 in z^2.\nlet composition_poly_evaluations = vec![ composition_poly_even.evaluate(&z_squared), composition_poly_odd.evaluate(&z_squared),\n]; After this, we don't really use H anymore, but rather H_1 and H_2. There's not that much to say other than that.","breadcrumbs":"STARKs » Implementation » Under the hood » Even/odd decomposition for H","id":"109","title":"Even/odd decomposition for H"},"11":{"body":"Our matrices are fine now. But they can be optimized. Let's do that to showcase this flexibility of PLONK and also reduce the size of our example. PLONK has the flexibility to construct more sophisticated gates as combinations of the five columns. And therefore the same program can be expressed in multiple ways. In our case all three gates can actually be merged into a single custom gate. The Q matrix ends up being a single row. QL​ QR​ QM​ QO​ QC​ 1 1 1 -1 1 and also the V matrix L R O 0 1 2 The trace matrix for this representation is just A B C 2 3 8 And we check that it satisfies the equation 2×1+3×1+2×3×1+8×(−1)+(−1)=0 Of course, we can't always squash an entire program into a single gate.","breadcrumbs":"Plonk » Recap » Custom gates","id":"11","title":"Custom gates"},"110":{"body":"As part of the consistency check, the prover needs to provide evaluations of the trace polynomials in all the points needed by the verifier to check that H was constructed correctly. In the fibonacci example, these are t(z), t(zg), and t(zg2). In code, the prover passes these evaluations as a Frame, which we call the out of domain (ood) frame. The reason we do this is simple: with the frame in hand, the verifier can reconstruct the evaluations of the constraint polynomials Ci​(z) by calling the compute_transition method on the ood frame and then adding the alphas, betas, and so on, just like we explained in the section above.","breadcrumbs":"STARKs » Implementation » Under the hood » Out of Domain Frame","id":"110","title":"Out of Domain Frame"},"111":{"body":"Throughout the protocol, there are a number of times where the verifier randomly samples some values that the prover needs to use (think of the alphas and betas used when constructing H). Because we don't actually have an interaction between prover and verifier, we emulate it by using a hash function, which we assume is a source of randomness the prover can't control. The job of providing these samples for both prover and verifier is done by the Transcript struct, which you can think of as a stateful rng; whenever you call challenge() on a transcript you get a random value and the internal state gets mutated, so the next time you call challenge() you get a different one. You can also call append on it to mutate its internal state yourself. This is done a number of times throughout the protocol to keep the prover honest so it can't predict or manipulate the outcome of challenge(). Notice that to sample the same values, both prover and verifier need to call challenge and append in the same order (and with the same values in the case of append) and the same number of times. The idea explained above is called the Fiat-Shamir heuristic or just Fiat-Shamir, and is more generally used throughout proving systems to remove interaction between prover and verifier. Though the concept is very simple, getting it right so the prover can't cheat is not, but we won't go into that here.","breadcrumbs":"STARKs » Implementation » Under the hood » Transcript","id":"111","title":"Transcript"},"112":{"body":"The generated proof has got all the information needed for the verifier to verify it: Trace length: The number of rows of the trace table, needed to know the max degree of the polynomials that appear in the system. LDE trace commitments. DEEP composition polynomial out of domain even and odd evaluations. DEEP composition polynomial root. FRI layers merkle roots. FRI last layer value. Query list. DEEP composition poly openings. Nonce: Proof of work setting used to generate the proof.","breadcrumbs":"STARKs » Implementation » Under the hood » Proof","id":"112","title":"Proof"},"113":{"body":"","breadcrumbs":"STARKs » Implementation » Under the hood » Special considerations","id":"113","title":"Special considerations"},"114":{"body":"When evaluating or interpolating a polynomial, if the input (be it coefficients or evaluations) size isn't a power of two then the FFT API will extend it with zero padding until this requirement is met. This is because the library currently only uses a radix-2 FFT algorithm. Also, right now FFT only supports inputs with a size up to 2264 elements.","breadcrumbs":"STARKs » Implementation » Under the hood » FFT evaluation and interpolation","id":"114","title":"FFT evaluation and interpolation"},"115":{"body":"","breadcrumbs":"STARKs » Implementation » Under the hood » Other","id":"115","title":"Other"},"116":{"body":"Whenever we interpolate or evaluate trace, boundary and constraint polynomials, we use some 2n-th roots of unity. There are a few reasons for this: Using roots of unity means we can use the Fast Fourier Transform and its inverse to evaluate and interpolate polynomials. This method is much faster than the naive Lagrange interpolation one. Since a huge part of the STARK protocol involves both evaluating and interpolating, this is a huge performance improvement. When computing boundary and constraint polynomials, we divide them by their zerofiers, polynomials that vanish on a few points (the trace elements where the constraints do not hold). These polynomials take the form Z(X)=∏(X−xi​) where the xi​ are the points where we want it to vanish. When implementing this, evaluating this polynomial can be very expensive as it involves a huge product. However, if we are using roots of unity, we can use the following trick. The vanishing polynomial for all the 2n roots of unity is X2n−1 Instead of expressing the zerofier as a product of the places where it should vanish, we express it as the vanishing polynomial above divided by the exemptions polynomial; the polynomial whose roots are the places where constraints don't need to hold. Z(X)=∏(X−ei​)X2n−1​ where the ei​ are now the points where we don't want it to vanish. This exemptions polynomial in the denominator is usually much smaller, and because the vanishing polynomial in the numerator is only two terms, evaluating it is really fast.","breadcrumbs":"STARKs » Implementation » Under the hood » Why use roots of unity?","id":"116","title":"Why use roots of unity?"},"117":{"body":"The n-th roots of unity are the numbers x that satisfy xn=1 There are n such numbers, because they are the roots of the polynomial Xn−1. The set of n-th roots of unity always has a generator, a root g that can be used to obtain every other root of unity by exponentiating. What this means is that the set of n-th roots of unity is {gi:0≤i1 and 0≤j≤J and I=∣Lj′′​∣. Extend M with additional L′′ rows to obtain a matrix M∈F(L+L′+L′′)×33 by appending copies of R′′ at the bottom. Pad M with copies of its last row until it has a power of two number of rows. As a result we obtain a matrix T∈F2n×33.","breadcrumbs":"Cairo » Trace Formal Description » Construction of execution trace T:","id":"121","title":"Construction of execution trace T:"},"122":{"body":"","breadcrumbs":"Cairo » Trace Detailed Description » Cairo execution trace","id":"122","title":"Cairo execution trace"},"123":{"body":"After the execution of a Cairo program in the Cairo VM, three files are generated that are the core components for the construction of the execution trace, needed for the proving system: trace file : Has the information on the state of the three Cairo VM registers ap, fp, and pc at every cycle of the execution of the program. To reduce ambiguity in terms, we should call these the register states of the Cairo VM, and leave the term trace to the final product that is passed to the prover to generate a proof. memory file : A file with the information of the VM's memory at the end of the program run, after the memory has been relocated. public inputs : A file with all the information that must be publicly available to the prover and verifier, such as the total number of execution steps, public memory, used builtins and their respective addresses range in memory. The next section will explain in detail how these elements are used to build the final execution trace.","breadcrumbs":"Cairo » Trace Detailed Description » Raw materials","id":"123","title":"Raw materials"},"124":{"body":"The execution trace is built in two stages. In the first one, the information on the files described in the previous section is aggregated to build a main trace table. In the second stage, there is an interaction with the verifier to add some extension columns to the main trace.","breadcrumbs":"Cairo » Trace Detailed Description » Construction details","id":"124","title":"Construction details"},"125":{"body":"The layout of the main execution trace is as follows: A. flags (16): Decoded instruction flags B. res (1): Res value C. pointers (2): Temporary memory pointers (ap and fp) D. mem_a (4): Memory addresses (pc, dst_addr, op0_addr, op1_addr) E. mem_v (4): Memory values (inst, dst, op0, op1) F. offsets (3) : (off_dst, off_op0, off_op1) G. derived (3) : (t0, t1, mul) A B C D E F G\n|xxxxxxxxxxxxxxxx|x|xx|xxxx|xxxx|xxx|xxx| Each letter from A to G represents some subsection of columns, and the number specifies how many columns correspond to that subsection. Cairo instructions It is important to have in mind the information that each executed Cairo instruction holds, since it is a key component of the construction of the execution trace. For a detailed explanation of how the building components of the instruction interact to change the VM state, refer to the Cairo whitepaper, sections 4.4 and 4.5. Structure of the 63-bit that forms the first word of each instruction: ┌─────────────────────────────────────────────────────────────────────────┐ │ off_dst (biased representation) │ ├─────────────────────────────────────────────────────────────────────────┤ │ off_op0 (biased representation) │ ├─────────────────────────────────────────────────────────────────────────┤ │ off_op1 (biased representation) │ ├─────┬─────┬───────┬───────┬───────────┬────────┬───────────────────┬────┤ │ dst │ op0 │ op1 │ res │ pc │ ap │ opcode │ 0 │ │ reg │ reg │ src │ logic │ update │ update │ │ │ ├─────┼─────┼───┬───┼───┬───┼───┬───┬───┼───┬────┼────┬────┬────┬────┼────┤ │ 0 │ 1 │ 2 │ 3 │ 4 │ 5 │ 6 │ 7 │ 8 │ 9 │ 10 │ 11 │ 12 │ 13 │ 14 │ 15 │ └─────┴─────┴───┴───┴───┴───┴───┴───┴───┴───┴────┴────┴────┴────┴────┴────┘ Columns The construction of the following columns corresponds to a colloquial explanation of what is done in the build_cairo_execution_trace function. Section A - Flags The flags section A corresponds to the 16 bits that represent the configuration of the dst_reg, op0_reg, op1_src, res_logic, pc_update, ap_update and opcode flags, as well as the zero flag. So there is one column for each bit of the flags decomposition. Section C - Temporary memory pointers The two columns in this section, as well as the pc column from section D , are the most trivial. For each step of the register states, the corresponding values are added to the columns, which are pointers to some memory cell in the VM's memory. Section D - Memory addresses As already mentioned, the first column of this section, pc, is trivially obtained from the register states for each cycle. Columns dst_addr, op0_addr, and op1_addr from section D are addresses constructed from pointers stored at ap or fp, and their respective offsets off_dst, off_op0 and off_op1. The exact way these are computed depends on the particular values of the flags for each instruction. Section E - Memory values The inst column, is obtained by fetching in memory the value stored at pointer pc, which corresponds to a 63-bit Cairo instruction. Columns dst, op0, and op1 are computed by fetching in memory by their respective addresses. Section F - Offsets/Range-checked values These columns represent integer values that are used to construct addresses dst_addr, op0_addr and op1_addr and are decoded directly from the instruction. These values have the property to be numbered in the range from 0 to 2^16. Section B - Res This column is computed depending on the decoded opcode and res_logic of every instruction. In some cases, res is unused in the instruction, and the value for (dst)^(-1) is used in that place as an optimization. Section G - Derived To have constraints of max degree two, some more columns are derived from the already calculated, t0, t1, and mul: t0 is the product of the values of DST and the PC_JNZ flag for each step. t1 is the product of t0 and res for each step. mul is the product of op0 and op1 for each step. Range check and Memory holes For the values constrained between ranges 0 and 216, the offsets, the prover uses a permutation argument to optimize enforcing this. In particular, it checks an ordered list with the offsets are the same as the original one, is continuous, the first value is rcmin​, and the last one is less than rcmax​. Since not all values are used, there may be unused values, and so the ordered offset may not be continuous. These unused values are called holes, and they need to be filled with the missing values, so the checks can be done. This is explained in section 9.9 of the Cairo Paper In the case of memory, something similar happens, where the values should be continuous, but if there are built-ins, this may not be the case. For example, the built-in may be using addresses in ranges higher than the ones used by the program. To fix this, holes in the memory cells are filled, just like the ones of the RC. It's something important to note that when filling the holes, we can't use dedicated columns like op0_addr, since this would break the constraints. For this to work, we either need new columns for holes, or make use of subcolumns, which are explained in their dedicated section. No matter which approach is used , either by subcolumns or columns, we will need cells where the constraints of the range check and memory are applied, but not the specific ones related to the instructions. Finally, using these columns, we can fill the holes without breaking the constraint system. Dummy memory accesses As part of proving the execution of a Cairo program, we need to prove that memory used by the program extends the public memory. This is important since public memory contains for example the bytecode that was executed and the outputs. The bytecode is something critical for the verifier to not only know that something was executed correctly but to know what was executed. To do this, we the permutation check, which that proves the memory is continuous and single-valued To do this, the permutation check of the memory is modified. Initially, we had M, V and M′, V′ pairs of memory addresses and values, where M and V are the values as used and without order, and M′, V′ the pairs ordered by address. For the public memory, we will add it directly to M′, V′. We also need to add dummy accesses to the pairs M V. These dummy accesses are just pairs of (M,V)=(0,0). This change makes the statement that (M′,V′) is a permutation of (M,V), which means that they are the same values in a different order, no longer true. This means that the two cumulative products used to check the statement are no longer equal, and their division Luckily, we can know the new expected value on the verifier, since we have access to the public memory. Even more, it's this fact that enables the verifier to check the memory is an extension of the public memory. The math is quite straightforward. When the memory was the same, we expected a final value pn−1​=1. This came from two cumulative products that should equal, one from the unordered pairs, and one from the ordered ones. Now, adding zeros to one side and the real values to the other unbalances the cumulative products, so the verifier will need to balance it by dividing pn−1​ by the extra factors that appeared with the addition of the public memory. Doing so will make the final value 1 again. Since they only depend on the public memory, the verifier has enough data to recalculate them and use them. Even more, if the prover lies, the equality that the verifier is expecting won't hold. In reality, instead of dividing and expecting the result to equal to 1, we can just check the equality against the new expected value, and avoid doing that inversion. All of this is explained in section 9.8 of the Cairo paper. Trace extension / Padding The last step is padding the trace to a power of two for efficiency. We may also need to pad the trace if for some reason some unbalance is given by the layout. For this, we will copy the last executed instruction until reaching the desired length. But there's a trick. If the last executed instruction is any instruction, and it's copied, the transition constraints won't be satisfied. To be able to do this, we need to use something called \"proof mode\". In proof mode, the main function of the program is wrapped in another one, which calls it and returns to an infinite loop. This loop is a jump relative 0. Since this loop can be executed many times without changing the validity of the trace, it can be copied as many times as needed, solving the issues mentioned before. Summary To construct the execution trace, we augment the RegisterStates with the information obtained from the Memory. This includes decoding instruction of each steps, and writing all the data needed to check the execution is valid. Additionally, memory holes have to be filled, public memory added, and a final pad using an infinite loop is needed for everything to work properly. Adding all of that, we create an execution trace that's ready for the prover to generate a proof.","breadcrumbs":"Cairo » Trace Detailed Description » Main trace construction","id":"125","title":"Main trace construction"},"126":{"body":"The verifier sends challenges α,z∈F (or the prover samples them from the transcript). Additional columns are added to incorporate the memory constraints. To define them the prover follows these steps: Stack the rows of the submatrix of T defined by the columns pc, dst_addr, op0_addr, op1_addr into a vector a of length 2n+2 (this means that the first entries of a are pc[0], dst_addr[0], op0_addr[0], op1_addr[0], pc[1], dst_addr[1],...). Stack the the rows of the submatrix defined by the columns inst, dst, op0, op1 into a vector v of length 2n+2. Define MMem​∈F2n+2×2 to be the matrix with columns a, v. Define MMemRepl​∈F2n+2×2 to be the matrix that's equal to MMem​ in the first 2n+2−Lpub​ rows, and its last Lpub​ entries are the addresses and values of the actual public memory (program code). Sort MMemRepl​ by the first column in increasing order. The result is a matrix MMemReplSorted​ of size 2n+2×2. Denote its columns by a′ and v′. Compute the vector p of size 2n+2 with entries pi​:=j=0∏i​z−(ai​+αvi​)z−(ai′​+αvi′​)​ Reshape the matrix MMemReplSorted​ into a 2n×8 in row-major. Reshape the vector p into a 2n×4 matrix in row-major. Concatenate these 12 rows. The result is a matrix MMemRAP2​ of size 2n×12 The verifier sends challenge z′∈F. Further columns are added to incorporate the range check constraints following these steps: Stack the rows of the submatrix of T defined by the columns in the group offsets into a vector b of length 3⋅2n. Sort the values of b in increasing order. Let b′ be the result. Compute the vector p′ of size 3⋅2n with entries pi′​:=j=0∏i​z′−bi​z′−bi′​​ Reshape b′ and p′ into matrices of size 2n×3 each and concatenate them into a matrix MRangeCheckRAP2​ of size 2n×6. Concatenate MMemRAP2​ and MRangeCheckRAP2​ into a matrix MRAP2​ of size 2n×18. Using the notation described at the beginning, m′=33, m′′=18 and m=52. They are respectively the columns of the first and second part of the rap, and the total number of columns. Putting all together, the final layout of the trace is the following A. flags (16) : Decoded instruction flags B. res (1) : Res value C. pointers (2) : Temporary memory pointers (ap and fp) D. mem_a (4) : Memory addresses (pc, dst_addr, op0_addr, op1_addr) E. mem_v (4) : Memory values (inst, dst, op0, op1) F. offsets (3) : (off_dst, off_op0, off_op1) G. derived (3) : (t0, t1, mul) H. mem_a' (4) : Sorted memory addresses I. mem_v' (4) : Sorted memory values J. mem_p (4) : Memory permutation argument columns K. offsets_b' (3) : Sorted offset columns L. offsets_p' (3) : Range check permutation argument columns A B C D E F G H I J K L\n|xxxxxxxxxxxxxxxx|x|xx|xxxx|xxxx|xxx|xxx|xxxx|xxxx|xxxx|xxx|xxx|","breadcrumbs":"Cairo » RAP » Extended columns","id":"126","title":"Extended columns"},"127":{"body":"","breadcrumbs":"Cairo » Virtual Columns » Virtual columns and Subcolumns","id":"127","title":"Virtual columns and Subcolumns"},"128":{"body":"In previous chapters, we have seen how the registers states and the memory are augmented to generate a provable trace. While we have shown a way of doing that, there isn't only one possible provable trace. In fact, there are multiple configurations possible. For example, in the Cairo VM, we have 15 flags. These flags include \"DstReg\", \"Op0Reg\", \"OpCode\" and others. For simplification, let's imagine we have 3 flags with letters from \"A\" to \"C\", where \"A\" is the first flag. Now, let's assume we have 4 steps in our trace. If we were to only use plain columns, the layout would look like this: FlagA FlagB FlagB A0 B0 C0 A1 B1 C1 A2 B2 C2 A3 B3 C3 But, we could also organize them like this Flags A0 B0 C0 A1 B1 C1 A2 B2 C2 A3 B3 C3 The only problem is that now the constraints for each transition of the rows are not the same. We will have to define then a concept called \"Virtual Column\". A Virtual Column is like a traditional column, which has its own set of constraints, but it exists interleaved with another one. In the previous example, each row is associated with a column, but in practice, we could have different ratios. We could have 3 rows corresponding to one Virtual Column, and the next one corresponding to another one. For the time being, let's focus on this simpler example. Each row corresponding to Flag A will have the constraints associated with its own Virtual Column, and the same will apply to Flag B and Flag C. Now, to do this, we will need to evaluate the multiple rows taking into account that they are part of the same step. For a real case, we will add a dummy flag D, whose purpose is to make the evaluation move in a number that is a power of 2. Let's see how it works. If we were evaluating the Frame where the constraints should give 0, the frame movement would look like this: + A0 | B0 | C0\n+ A1 | B1 | C1 A2 | B2 | C2 A3 | B3 | C3 A0 | B0 | C0\n+ A1 | B1 | C1\n+ A2 | B2 | C2 A3 | B3 | C3 A0 | B0 | C0 A1 | B1 | C1\n+ A2 | B2 | C2\n+ A3 | B3 | C3 In the second case, the evaluation would look like this: + A0 |\n+ B0 |\n+ C0 |\n+ D0 |\n+ A1 |\n+ B1 |\n+ C1 |\n+ D1 | A2 | B2 | C2 | D2 | A3 | B3 | C3 | D3 | A0 | B0 | C0 | D0 |\n+ A1 |\n+ B1 |\n+ C1 |\n+ D1 |\n+ A2 |\n+ B2 |\n+ C2 |\n+ D2 | A3 | B3 | C3 | D3 | A0 | B0 | C0 | D0 | A1 | B1 | C1 | D1 |\n+ A2 |\n+ B2 |\n+ C2 |\n+ D2 |\n+ A3 |\n+ B3 |\n+ C3 |\n+ D3 | When evaluating the composition polynomial, we will do it over the points on the LDE, where the constraints won't evaluate to 0, but we will use the same spacing. Assume we have three constraints for each flag, C0​, C1​, and C2​, and that they don't involve other trace cells. Let's call the index of the frame evaluation i, starting from 0. In the first case, the constraint C0​, C1​ and C2​ would be applied over the same rows, giving an equation that looks like this: ‘Ck​(wi,wi+1)‘ In the second case, the equations would look like: ‘C0​(wi∗4,wi∗4+4)‘ ‘C1​(wi∗4+1,wi∗4+5)‘ ‘C2​(wi∗4+2,wi∗4+6)‘","breadcrumbs":"Cairo » Virtual Columns » Virtual Columns","id":"128","title":"Virtual Columns"},"129":{"body":"Assume now we have 3 columns that share some constraints. For example, let's have three flags that can be either 0 or 1. Each flag will also have its own set of dedicated constraints. Let's denote the shared constraint B, and the independent constraints Ci​. What we can do is define a Column for the flags, where the binary constraint B is enforced. Additionally, we will define a subcolumn for each flag, which will enforce each Ci​. In summary, if we have a set of shared constraints to apply, we will be using a Column. If we want to mix or interleave Columns, we will define them as Virtual Columns. And if we want to apply more constraints to a subset of a Column of Virtual Columns, or share constraints between columns, we will define Virtual Subcolumns.","breadcrumbs":"Cairo » Virtual Columns » Virtual Subcolumns","id":"129","title":"Virtual Subcolumns"},"13":{"body":"In the previous section we showed how the arithmetization process works in PLONK. For a program with n public inputs and m gates, we constructed two matrices Q and V, of sizes (n+m+1)×5 and (n+m+1)×3 that satisfy the following. Let N=n+m+1. Claim: Let T be a N×3 matrix with columns A,B,C and PI a N×1 matrix. They correspond to a valid execution instance with public input given by PI if and only if a) for all i the following equality holds Ai​(QL​)i​+Bi​(QR​)i​+Ai​Bi​QM​+Ci​(QO​)i​+(QC​)i​+(PI)i​=0, b) for all i,j,k,l such that Vi,j​=Vk,l​ we have Ti,j​=Tk,l​, c) (PI)i​=0 for all i>n. Polynomials enter now to squash most of these equations. We will traduce the set of all equations in conditions (a) and (b) to just a few equations on polynomials. Let ω be a primitive N-th root of unity and let H=ωi:0≤in. Then we constructed polynomials qL​,qR​,qM​,qO​,qC​,Sσ1​,Sσ2​,Sσ3​, f, g off the matrices Q and V. They are the result of interpolating at a domain H={1,ω,ω2,…,ωN−1} for some N-th primitive root of unity and a few random values. And also constructed polynomials a,b,c,pi off the matrices T and PI. Loosely speaking, the above fact can be reformulated in terms of polynomial equations as follows. Fact: Let zH​=XN−1. Let T be a N×3 matrix with columns A,B,C and PI a N×1 matrix. They correspond to a valid execution instance with public input given by PI if and only if a) There is a polynomial t1​ such that the following equality holds aqL​+bqR​+abqM​+cqO​+qC​+pi=zH​t1​, b) There are polynomials t2​,t3​, z such that zf−gz′=zH​t2​ and (z−1)L1​=zH​t3​, where z′(X)=z(Xω) You might be wondering where the polynomials ti​ came from. Recall that for a polynomial F, we have F(h)=0 for all h∈H if and only if F=zH​t for some polynomial t. Finally both conditions (a) and (b) are equivalent to a single equation (c) if we let more randomness to come into play. This is: (c) Let α be a random field element. There is a polynomial t such that zH​t=​aqL​+bqR​+abqM​+cqO​+qC​+pi+α(gz′−fz)+α2(z−1)L1​​ This last step is not obvious. You can check the paper to see the proof. Anyway, this is the equation you'll recognize below in the description of the protocol. Randomness is a delicate matter and an important part of the protocol is where it comes from, who chooses it and when they choose it. Check out the protocol to see how it works.","breadcrumbs":"Plonk » Recap » Wrapping up the whole thing","id":"15","title":"Wrapping up the whole thing"},"16":{"body":"","breadcrumbs":"Plonk » Protocol » Protocol","id":"16","title":"Protocol"},"17":{"body":"","breadcrumbs":"Plonk » Protocol » Details and tricks","id":"17","title":"Details and tricks"},"18":{"body":"A polynomial commitment scheme (PCS) is a cryptographic tool that allows one party to commit to a polynomial, and later prove properties of that polynomial. This commitment polynomial hides the original polynomial's coefficients and can be publicly shared without revealing any information about the original polynomial. Later, the party can use the commitment to prove certain properties of the polynomial, such as that it satisfies certain constraints or that it evaluates to a certain value at a specific point. In the implementation section we'll explain the inner workings of the Kate-Zaverucha-Goldberg scheme, a popular PCS chosen in Lambdaworks for PLONK. For the moment we only need the following about it: It consists of a finite group G and the following algorithms: Commit(f) : This algorithm takes a polynomial f and produces an element of the group G. It is called the commitment of f and is denoted by [f]1​. It is homomorphic in the sense that [f+g]1​=[f]1​+[g]1​. The former sum being addition of polynomials. The latter is addition in the group G. Open(f,ζ) : It takes a polynomial f and a field element ζ and produces an element π of the group G. This element is called an opening proof for f(ζ). It is the proof that f evaluated at ζ gives f(ζ). Verify([f]1​, π, ζ, y) : It takes group elements [f]1​ and π, and also field elements ζ and y. With overwhelming probability it outputs Accept if f(z)=y and Reject otherwise.","breadcrumbs":"Plonk » Protocol » Polynomial commitment scheme","id":"18","title":"Polynomial commitment scheme"},"19":{"body":"As you will see in the protocol, the prover reveals the value taken by a bunch of the polynomials at a random ζ. In order for the protocol to be Honest Verifier Zero Knowledge , these polynomials need to be blinded . This is a process that makes the values of these polynomials at ζ seemingly random by forcing them to be of certain degree. Here's how it works. Let's take for example the polynomial a the prover constructs. This is the interpolation of the first column of the trace matrix T at the domain H. This matrix has all of the left operands of all the gates. The prover wishes to keep them secret. Say the trace matrix T has N rows. And so H is {1,ω,ω2,…,ωN−1}. The invariant that the prover cannot violate is that ablinded​(ωi) must take the value T0,i​, for all i. This is what the interpolation polynomial a satisfies. And is the unique such polynomial of degree at most N−1 with such property. But for higher degrees, there are many such polynomials. The blinding process takes a and a desired degree M≥N, and produces a new polynomial ablinded​ of degree exactly M. This new polynomial satisfies that ablinded​(ωi)=a(ωi) for all i. But outside H differs from a. This may seem hard but it's actually very simple. Let zH​ be the polynomial zH​=XN−1. If M=N+k, with k≥0, then sample random values b0​,…,bk​ and define ablinded​:=(b0​+b1​X+⋯+bk​Xk)zH​+a The reason why this does the job is that zH​(ωi)=0 for all i. Therefore the added term vanishes at H and leaves the values of a at H unchanged.","breadcrumbs":"Plonk » Protocol » Blindings","id":"19","title":"Blindings"},"2":{"body":"Three methods of polynomial interpolation were benchmarked, with different input sizes each time: CPU Lagrange: Finding the Lagrange polynomial of a set of random points via a naive algorithm (see math/src/polynomial.rs:interpolate()) CPU FFT: Finding the lowest degree polynomial that interpolates pairs of twiddle factors and Fourier coefficients (the results of applying the Fourier transform to the coefficients of a polynomial) (see math/src/polynomial.rs:interpolate_fft()). GPU FFT (Metal): Same as CPU FFT but the FFT algorithm is run on the GPU via the Metal API (Apple). these were run with criterion-rs in a MacBook Pro M1 (18.3), statistically measuring the total run time of one iteration. The field used was a 256 bit STARK-friendly prime field. All values of time are in milliseconds. Those cases which were greater than 30 seconds were marked respectively as they're too slow and weren't worth to be benchmarked. The input size refers to d + 1 where d is the polynomial's degree (so size is amount of coefficients). Input size CPU Lagrange CPU FFT GPU FFT (Metal) 2^4 2.2 ms 0.2 ms 2.5 ms 2^5 9.6 ms 0.4 ms 2.5 ms 2^6 42.6 ms 0.8 ms 2.5 ms 2^7 200.8 ms 1.7 ms 2.9 ms ... ... .. .. 2^21 >30000 ms 28745 ms 574.2 ms 2^22 >30000 ms >30000 ms 1144.9 ms 2^23 >30000 ms >30000 ms 2340.1 ms 2^24 >30000 ms >30000 ms 4652.9 ms NOTE: Metal FFT execution includes the Metal state setup, twiddle factors generation and a bit-reverse permutation.","breadcrumbs":"FFT Library » Benchmarks » Polynomial interpolation methods comparison","id":"2","title":"Polynomial interpolation methods comparison"},"20":{"body":"This is an optimization in PLONK to reduce the number of checks of the verifier. One of the main checks in PLONK boils down to check that p(ζ)=zH​(ζ)t(ζ), with p some polynomial that looks like p=aqL​+bqR​+abqM​+⋯, and so on. In particular the verifier needs to get the value p(ζ) from somewhere. For the sake of simplicity, in this section assume p is exactly aqL​+bqR​. Secret to the prover here are only a,b. The polynomials qL​ and qR​ are known also to the verifier. The verifier will already have the commitments [a]1​,[b]1​,[qL​]1​ and [qR​]1​. So the prover could send just a(ζ), b(ζ) along with their opening proofs and let the verifier compute by himself qL​(ζ) and qR​(ζ). Then with all these values the verifier could compute p(ζ)=a(ζ)qL​(ζ)+b(ζ)qR​(ζ). And also use his commitments to validate the opening proofs of a(ζ) and b(ζ). This has the problem that computing qL​(ζ) and qR​(ζ) is expensive. The prover can instead save the verifier this by sending also qL​(ζ),qR​(ζ) along with opening proofs. Since the verifier will have the commitments [qL​]1​ and [qR​]1​ beforehand, he can check that the prover is not cheating and cheaply be convinced that the claimed values are actually qL​(ζ) and qR​(ζ). This is much better. It involves the check of four opening proofs and the computation of p(ζ) off the values received from the prover. But it can be further improved as follows. As before, the prover sends a(ζ),b(ζ) along with their opening proofs. She constructs the polynomial f=a(ζ)qL​+b(ζ)qR​. She sends the value f(ζ) along with an opening proof of it. Notice that the value of f(ζ) is exactly p(ζ). The verifier can compute by himself [f]1​ as a(ζ)[qL​]1​+b(ζ)[qR​]1​. The verifier has everything to check all three openings and get convinced that the claimed value f(ζ) is true. And this value is actually p(ζ). So this means no more work for the verifier. And the whole thing got reduced to three openings. This is called the linearization trick. The polynomial f is called the linearization of p.","breadcrumbs":"Plonk » Protocol » Linearization trick","id":"20","title":"Linearization trick"},"21":{"body":"There's a one time setup phase to compute some values common to any execution and proof of the particular circuit. Precisely, the following commitments are computed and published. [qL​]1​,[qR​]1​,[qM​]1​,[qO​]1​,[qC​]1​,[Sσ1​]1​,[Sσ2​]1​,[Sσ3​]1​","breadcrumbs":"Plonk » Protocol » Setup","id":"21","title":"Setup"},"22":{"body":"Next we describe the proving algorithm for a program of size N. That includes public inputs. Let ω be a primitive N-th root of unity. Let H={1,ω,ω2,…,ωN−1}. Define ZH​:=XN−1. Assume the eight polynomials of common preprocessed input are already given. The prover computes the trace matrix T as described in the first sections. That means, with the first rows corresponding to the public inputs. It should be a N×3 matrix.","breadcrumbs":"Plonk » Protocol » Proving algorithm","id":"22","title":"Proving algorithm"},"23":{"body":"Add to the transcript the following: [Sσ1​]1​,[Sσ2​]1​,[Sσ3​]1​,[qL​]1​,[qR​]1​,[qM​]1​,[qO​]1​,[qC​]1​ Compute polynomials a′,b′,c′ as the interpolation polynomials of the columns of T at the domain H. Sample random b1​,b2​,b3​,b4​,b5​,b6​ Let a:=(b1​X+b2​)ZH​+a′ b:=(b3​X+b4​)ZH​+b′ c:=(b5​X+b6​)ZH​+c′ Compute [a]1​,[b]1​,[c]1​ and add them to the transcript.","breadcrumbs":"Plonk » Protocol » Round 1","id":"23","title":"Round 1"},"24":{"body":"Sample β,γ from the transcript. Let z0​=1 and define recursively for 0≤k { pub n: usize, pub domain: Vec>, pub omega: FieldElement, pub k1: FieldElement, pub ql: Polynomial>, pub qr: Polynomial>, pub qo: Polynomial>, pub qm: Polynomial>, pub qc: Polynomial>, pub s1: Polynomial>, pub s2: Polynomial>, pub s3: Polynomial>, pub s1_lagrange: Vec>, pub s2_lagrange: Vec>, pub s3_lagrange: Vec>,\n} Apart from the eight polynomials in the canonical basis, we store also here the number of constraints n, the domain H, the primitive n-th of unity ω and the element k1​. The element k2​ will be k12​. For convenience, we also store the polynomials Sσi​ in Lagrange form. The following lines define the particular values of the program input x and the private input e. // Input\nlet x = FieldElement::from(2_u64); // Private input\nlet e = FieldElement::from(3_u64);\nlet y, witness = test_witness_2(x, e); The function test_witness_2(x, e) returns an instance of the following struct, that holds the polynomials that interpolate the columns A,B,C of the trace matrix T. pub struct Witness { pub a: Vec>, pub b: Vec>, pub c: Vec>,\n} Next the commitment scheme KZG (Kate-Zaverucha-Goldberg) is instantiated. let srs = test_srs(common_preprocessed_input.n);\nlet kzg = KZG::new(srs); The setup function performs the setup phase. It only needs the common preprocessed input and the commitment scheme. let verifying_key = setup(&common_preprocessed_input, &kzg); It outputs an instance of the struct VerificationKey. pub struct VerificationKey { pub qm_1: G1Point, pub ql_1: G1Point, pub qr_1: G1Point, pub qo_1: G1Point, pub qc_1: G1Point, pub s1_1: G1Point, pub s2_1: G1Point, pub s3_1: G1Point,\n} It stores the commitments of the eight polynomials of the common preprocessed input. The suffix _1 means it is a commitment. It comes from the notation [f]1​, where f is a polynomial. Then a prover is instantiated let random_generator = TestRandomFieldGenerator {};\nlet prover = Prover::new(kzg.clone(), random_generator); The prover is an instance of the struct Prover: pub struct Prover\nwhere F: IsField, CS: IsCommitmentScheme, R: IsRandomFieldElementGenerator { commitment_scheme: CS, random_generator: R, phantom: PhantomData,\n} It stores an instance of a commitment scheme and a random field element generator needed for blinding polynomials. Then the public input is defined. As we mentioned in the recap, the public input contains the output of the program. let public_input = vec![x.clone(), y]; We then generate a proof using the prover's method prove let proof = prover.prove( &witness, &public_input, &common_preprocessed_input, &verifying_key,\n); The output is an instance of the struct Proof. pub struct Proof> { // Round 1. /// Commitment to the wire polynomial `a(x)` pub a_1: CS::Commitment, /// Commitment to the wire polynomial `b(x)` pub b_1: CS::Commitment, /// Commitment to the wire polynomial `c(x)` pub c_1: CS::Commitment, // Round 2. /// Commitment to the copy constraints polynomial `z(x)` pub z_1: CS::Commitment, // Round 3. /// Commitment to the low part of the quotient polynomial t(X) pub t_lo_1: CS::Commitment, /// Commitment to the middle part of the quotient polynomial t(X) pub t_mid_1: CS::Commitment, /// Commitment to the high part of the quotient polynomial t(X) pub t_hi_1: CS::Commitment, // Round 4. /// Value of `a(ζ)`. pub a_zeta: FieldElement, /// Value of `b(ζ)`. pub b_zeta: FieldElement, /// Value of `c(ζ)`. pub c_zeta: FieldElement, /// Value of `S_σ1(ζ)`. pub s1_zeta: FieldElement, /// Value of `S_σ2(ζ)`. pub s2_zeta: FieldElement, /// Value of `z(ζω)`. pub z_zeta_omega: FieldElement, // Round 5 /// Value of `p_non_constant(ζ)`. pub p_non_constant_zeta: FieldElement, /// Value of `t(ζ)`. pub t_zeta: FieldElement, /// Batch opening proof for all the evaluations at ζ pub w_zeta_1: CS::Commitment, /// Single opening proof for `z(ζω)`. pub w_zeta_omega_1: CS::Commitment,\n} Finally, we instantiate a verifier. let verifier = Verifier::new(kzg); It's an instance of Verifier: struct Verifier> { commitment_scheme: CS, phantom: PhantomData,\n} Finally, we call the verifier's method verify that outputs a bool. assert!(verifier.verify( &proof, &public_input, &common_preprocessed_input, &verifying_key\n));","breadcrumbs":"Plonk » Implementation » Usage","id":"34","title":"Usage"},"35":{"body":"All the matrices Q,V,T,PI are padded with dummy rows so that their length is a power of two. To be able to interpolate their columns, we need a primitive root of unity ω of that order. Given the particular field used in our implementation, that means that the maximum possible size for a circuit is 232. The entries of the dummy rows are filled in with zeroes in the Q, V and PI matrices. The T matrix needs to be consistent with the V matrix. Therefore it is filled with the value of the variable with index 0. Some other rows in the V matrix have also dummy values. These are the rows corresponding to the B and C columns of the public input rows. In the recap we denoted them with the empty - symbol. They are filled in with the same logic as the padding rows, as well as the corresponding values in the T matrix.","breadcrumbs":"Plonk » Implementation » Padding","id":"35","title":"Padding"},"36":{"body":"The implementation pretty much follows the rounds as are described in the protocol section. There are a few details that are worth mentioning.","breadcrumbs":"Plonk » Implementation » Implementation details","id":"36","title":"Implementation details"},"37":{"body":"The commitment scheme we use is the Kate-Zaverucha-Goldberg scheme with the BLS 12 381 curve and the ate pairing. It can be found in the commitments module of the lambdaworks_crypto package. The order r of the cyclic subgroup is 0x73eda753299d7d483339d80809a1d80553bda402fffe5bfeffffffff00000001 The maximum power of two that divides r−1 is 232. Therefore, that is the maximum possible order for a primitive root of unity in Fr​ with order a power of two.","breadcrumbs":"Plonk » Implementation » Commitment Scheme","id":"37","title":"Commitment Scheme"},"38":{"body":"","breadcrumbs":"Plonk » Implementation » Fiat-Shamir","id":"38","title":"Fiat-Shamir"},"39":{"body":"Here we describe our implementation of the transcript used for the Fiat-Shamir heuristic. A Transcript exposes two methods: append and challenge. The method append adds a message to the transcript by updating the internal state of the hasher with the raw bytes of the message. The method challenge returns the result of the hasher using the current internal state of the hasher. It subsequently resets the hasher and updates the internal state with the last result. Here is an example of this process: Start a fresh transcript. Call append and pass message_1. Call append and pass message_2. The internal state of the hasher at this point is message_2 || message_1. Call challenge. The output is Hash(message_2 || message_1). Call append and pass message_3. Call challenge. The output is Hash(message_3 || Hash(message_2 || message_1)). Call append and pass message_4. The internal state of the hasher at the end of this exercise is message_4 || Hash(message_3 || Hash(message_2 || message_1)) The underlying hasher function we use is h=sha3.","breadcrumbs":"Plonk » Implementation » Transcript strategy","id":"39","title":"Transcript strategy"},"4":{"body":"We use the following notation. The symbol F denotes a finite field. It is fixed all along. The symbol ω denotes a primitive root of unity in F. All polynomials have coefficients in F and the variable is usually denoted by X. We denote polynomials by single letters like p,a,b,z. We only denote them as z(X) when we want to emphasize the fact that it is a polynomial in X, or we need that to explicitly define a polynomial from another one. For example when composing a polynomial z with the polynomial ωX, the result being denoted by z′:=z(ωX). The symbol ′ is not used to denote derivatives. When interpolating at a domain H={h0​,…,hn​}⊂F, the symbols Li​ denote the Lagrange basis. That is Li​ is the polynomial such that Li​(hj​)=0 for all j=i, and that Li​(hi​)=1. If M is a matrix, then Mi,j​ denotes the value at the row i and column j.","breadcrumbs":"Plonk » Recap » Notation","id":"4","title":"Notation"},"40":{"body":"The result of every challenge is a 256-bit string, which is interpreted as an integer in big-endian order. A field element is constructed out of it by taking modulo the field order. The prime field used in this implementation has a 255-bit order. Therefore some field elements are more probable to occur than others because they have more representatives as 256-bit integers.","breadcrumbs":"Plonk » Implementation » Field elements","id":"40","title":"Field elements"},"41":{"body":"The first messages added to the transcript are all commitments of the polynomials of the common preprocessed input and the values of the public inputs. This prevents a known vulnerability called \"weak Fiat-Shamir\". Check out the following resources to learn more about it. What can go wrong (zkdocs) How not to Prove Yourself: Pitfalls of the Fiat-Shamir Heuristic and Applications to Helios Weak Fiat-Shamir Attacks on Modern Proof Systems","breadcrumbs":"Plonk » Implementation » Strong Fiat-Shamir","id":"41","title":"Strong Fiat-Shamir"},"42":{"body":"In this section, we'll discuss how to build your own constraint system to prove the execution of a particular program.","breadcrumbs":"Plonk » Circuit API » Circuit API","id":"42","title":"Circuit API"},"43":{"body":"Let's take the following simple program as an example. We have two public inputs: x and y. We want to prove to a verifier that we know a private input e such that x * e = y. You can achieve this by building the following constraint system: use lambdaworks_plonk::constraint_system::ConstraintSystem;\nuse lambdaworks_math::elliptic_curve::short_weierstrass::curves::bls12_381::default_types::FrField; fn main() { let system = &mut ConstraintSystem::::new(); let x = system.new_public_input(); let y = system.new_public_input(); let e = system.new_variable(); let z = system.mul(&x, &e); // This constraint system asserts that x * e == y system.assert_eq(&y, &z);\n} This code creates a constraint system over the field of the BLS12381 curve. Then, it creates three variables: two public inputs x and y, and a private variable e. Note that every variable is private except for the public inputs. Finally, it adds the constraints that represent a multiplication and an assertion. Before generating proofs for this system, we need to run a setup and obtain a verifying key: let common = CommonPreprocessedInput::from_constraint_system(&system, &ORDER_R_MINUS_1_ROOT_UNITY);\nlet srs = test_srs(common.n);\nlet kzg = KZG::new(srs); // The commitment scheme for plonk.\nlet vk = setup(&common, &kzg); Now we can generate proofs for our system. We just need to specify the public inputs and obtain a witness that is a solution for our constraint system: let inputs = HashMap::from([(x, FieldElement::from(4)), (e, FieldElement::from(3))]);\nlet assignments = system.solve(inputs).unwrap();\nlet witness = Witness::new(assignments, &system); Once you have all these ingredients, you can call the prover: let public_inputs = system.public_input_values(&assignments);\nlet prover = Prover::new(kzg.clone(), TestRandomFieldGenerator {});\nlet proof = prover.prove(&witness, &public_inputs, &common, &vk); and verify: let verifier = Verifier::new(kzg);\nassert!(verifier.verify(&proof, &public_inputs, &common, &vk));","breadcrumbs":"Plonk » Circuit API » Simple Example","id":"43","title":"Simple Example"},"44":{"body":"Some operations are common, and it makes sense to wrap the set of constraints that do these operations in a function and use it several times. Lambdaworks comes with a collection of functions to help you build your own constraint systems, such as conditionals, inverses, and hash functions. However, if you have an operation that does not come with Lambdaworks, you can easily extend Lambdaworks functionality. Suppose that the exponentiation operation is something common in your program. You can write the square and multiply algorithm and put it inside a function: pub fn pow( system: &mut ConstraintSystem, base: Variable, exponent: Variable,\n) -> Variable { let exponent_bits = system.new_u32(&exponent); let mut result = system.new_constant(FieldElement::one()); for i in 0..32 { if i != 0 { result = system.mul(&result, &result); } let result_times_base = system.mul(&result, &base); result = system.if_else(&exponent_bits[i], &result_times_base, &result); } result\n} This function can then be used to modify our simple program from the previous section. The following circuit checks that the prover knows e such that pow(x, e) = y: use lambdaworks_plonk::constraint_system::ConstraintSystem;\nuse lambdaworks_math::elliptic_curve::short_weierstrass::curves::bls12_381::default_types::FrField; fn main() { let system = &mut ConstraintSystem::::new(); let x = system.new_public_input(); let y = system.new_public_input(); let e = system.new_variable(); let z = pow(system, &x, &e); system.assert_eq(&y, &z);\n} You can keep composing these functions in order to create more complex systems.","breadcrumbs":"Plonk » Circuit API » Building Complex Systems","id":"44","title":"Building Complex Systems"},"45":{"body":"The goal of this document is to give a good a understanding of our stark prover code. To this end, in the first section we go through a recap of how the proving system works at a high level mathematically; then we dive into how that's actually implemented in our code.","breadcrumbs":"STARKs » STARK Prover","id":"45","title":"STARK Prover"},"46":{"body":"","breadcrumbs":"STARKs » Recap » STARKs Recap","id":"46","title":"STARKs Recap"},"47":{"body":"In general, we express computation in our proving system by providing an execution trace satisfying certain constraints . The execution trace is a table containing the state of the system at every step of computation. This computation needs to follow certain rules to be valid; these rules are our constraints . The constraints for our computation are expressed using an Algebraic Intermediate Representation or AIR. This representation uses polynomials to encode constraints, which is why sometimes they are called polynomial constraints. To make all this less abstract, let's go through two examples.","breadcrumbs":"STARKs » Recap » Verifying Computation through Polynomials","id":"47","title":"Verifying Computation through Polynomials"},"48":{"body":"Throughout this section and the following we will use this example extensively to have a concrete example. Even though it's a bit contrived (no one cares about computing fibonacci numbers), it's simple enough to be useful. STARKs and proving systems in general are very abstract things; having an example in mind is essential to not get lost. Let's say our computation consists of calculating the k-th number in the fibonacci sequence. This is just the sequence of numbers \\(a_n\\) satisfying \\[ a_0 = 1 \\] \\[ a_1 = 1 \\] \\[ a_{n+2} = a_{n + 1} + a_n \\] An execution trace for this just consists of a table with one column, where each row is the i-th number in the sequence: a_i 1 1 2 3 5 8 13 21 A valid trace for this computation is a table satisfying two things: The first two rows are 1. The value on any other row is the sum of the two preceding ones. The first item is called a boundary constraint, it just enforces specific values on the trace at certain points. The second one is a transition constraint; it tells you how to go from one step of computation to the next.","breadcrumbs":"STARKs » Recap » Fibonacci numbers","id":"48","title":"Fibonacci numbers"},"49":{"body":"The example above is extremely useful to have a mental model, but it's not really useful for anything else. The problem is it just works for the very narrow example of computing fibonacci numbers. If we wanted to prove execution of something else, we would have to write an AIR for it. What we're actually aiming for is an AIR for an entire general purpose Virtual Machine. This way, we can provide proofs of execution for any computation using just one AIR. This is what cairo as a programming language does. Cairo code compiles to the bytecode of a virtual machine with an already defined AIR. The general flow when using cairo is the following: User writes a cairo program. The program is compiled into Cairo's VM bytecode. The VM executes said code and provides an execution trace for it. The trace is passed on to a STARK prover, which creates a proof of correct execution according to Cairo's AIR. The proof is passed to a verifier, who checks that the proof is valid. Ultimately, our goal is to give the tools to write a STARK prover for the cairo VM and do so. However, this is not a good example to start out as it's incredibly complex. The execution trace of a cairo program has around 30 columns, some for general purpose registers, some for other reasons. Cairo's AIR contains a lot of different transition constraints, encoding all the different possible instructions (arithmetic operations, jumps, etc). Use the fibonacci example as your go-to for understanding all the moving parts; keep the Cairo example in mind as the thing we are actually building towards.","breadcrumbs":"STARKs » Recap » Cairo","id":"49","title":"Cairo"},"5":{"body":"","breadcrumbs":"Plonk » Recap » The ideas and components","id":"5","title":"The ideas and components"},"50":{"body":"Below we go through a step by step explanation of a STARK prover. We will assume the trace of the fibonacci sequence mentioned above; it consists of only one column of length \\(2^n\\). In this case, we'll take n=3. The trace looks like this a_i a_0 a_1 a_2 a_3 a_4 a_5 a_6 a_7","breadcrumbs":"STARKs » Recap » Fibonacci step by step walkthrough","id":"50","title":"Fibonacci step by step walkthrough"},"51":{"body":"The first step is to interpolate these values to generate the trace polynomial. This will be a polynomial encoding all the information about the trace. The way we do it is the following: in the finite field we are working in, we take an 8-th primitive root of unity, let's call it g. It being a primitive root means two things: g is an 8-th root of unity, i.e., \\(g^8 = 1\\). Every 8-th root of unity is of the form \\(g^i\\) for some \\(0 \\leq i \\leq 7\\). With g in hand, we take the trace polynomial t to be the one satisfying t(gi)=ai​ From here onwards, we will talk about the validity of the trace in terms of properties that this polynomial must satisfy. We will also implicitly identify a certain power of \\(g\\) with its corresponding trace element, so for example we sometimes think of \\(g^5\\) as \\(a_5\\), the fifth row in the trace, even though technically it's \\(t\\) evaluated in \\(g^5\\) that equals \\(a_5\\). We talked about two different types of constraints the trace must satisfy to be valid. They were: The first two rows are 1. The value on any other row is the sum of the two preceding ones. In terms of t, this translates to \\(t(g^0) = 1\\) and \\(t(g) = 1\\). \\(t(x g^2) - t(xg) - t(x) = 0\\) for all \\(x \\in {g^0, g^1, g^2, g^3, g^4, g^5}\\). This is because multiplying by g is the same as advancing a row in the trace.","breadcrumbs":"STARKs » Recap » Trace polynomial","id":"51","title":"Trace polynomial"},"52":{"body":"To convince the verifier that the trace polynomial satisfies the relationships above, the prover will construct another polynomial that shows that both the boundary and transition constraints are satisfied and commit to it. We call this polynomial the composition polynomial, and usually denote it with \\(H\\). Constructing it involves a lot of different things, so we'll go step by step introducing all the moving parts required.","breadcrumbs":"STARKs » Recap » Composition Polynomial","id":"52","title":"Composition Polynomial"},"53":{"body":"To show that the boundary constraints are satisfied, we construct the boundary polynomial. Recall that our boundary constraints are \\(t(g^0) = t(g) = 1\\). Let's call \\(P\\) the polynomial that interpolates these constraints, that is, \\(P\\) satisfies: P(1)=1P(g)=1 The boundary polynomial \\(B\\) is defined as follows: B(x)=(x−1)(x−g)t(x)−P(x)​ The denominator here is called the boundary zerofier, and it's the polynomial whose roots are the elements of the trace where the boundary constraints must hold. How does \\(B\\) encode the boundary constraints? The idea is that, if the trace satisfies said constraints, then t(1)−P(1)=1−1=0 t(g)−P(g)=1−1=0 so \\(t(x) - P(x)\\) has \\(1\\) and \\(g\\) as roots. Showing these values are roots is the same as showing that \\(B(x)\\) is a polynomial instead of a rational function, and that's why we construct \\(B\\) this way.","breadcrumbs":"STARKs » Recap » Boundary polynomial","id":"53","title":"Boundary polynomial"},"54":{"body":"To convince the verifier that the transition constraints are satisfied, we construct the transition constraint polynomial and call it \\(C(x)\\). It's defined as follows: C(x)=∏i=05​(x−gi)t(xg2)−t(xg)−t(x)​ How does \\(C\\) encode the transition constraints? We mentioned above that these are satisfied if the polynomial in the numerator vanishes in the elements \\({g^0, g^1, g^2, g^3, g^4, g^5}\\). As with \\(B\\), this is the same as showing that \\(C(x)\\) is a polynomial instead of a rational function.","breadcrumbs":"STARKs » Recap » Transition constraint polynomial","id":"54","title":"Transition constraint polynomial"},"55":{"body":"With the boundary and transition constraint polynomials in hand, we build the composition polynomial \\(H\\) as follows: The verifier will sample four numbers \\(\\beta_1, \\beta_2\\) and \\(H\\) will be H(x)=β1​B(x)+β2​C(x) Why not just take \\(H(x) = B(x) + C(x)\\)? The reason for the betas is to make the resulting \\(H\\) be always different and unpredictable for the prover, so they can't precompute stuff beforehand. With what we discussed above, showing that the constraints are satisfied is equivalent to saying that H is a polynomial and not a rational function (we are simplifying things a bit here, but it works for our purposes).","breadcrumbs":"STARKs » Recap » Constructing \\(H\\)","id":"55","title":"Constructing \\(H\\)"},"56":{"body":"To show \\(H\\) is a polynomial we are going to use the FRI protocol, which we treat as a black box. For all we care, a FRI proof will verify if what we committed to is indeed a polynomial. Thus, the prover will provide a FRI commitment to H, and if it passes, the verifier will be convinced that the constraints are satisfied. There is one catch here though: how does the verifier know that FRI was applied to H and not any other polynomial? For this we need to add an additional step to the protocol.","breadcrumbs":"STARKs » Recap » Commiting to \\(H\\)","id":"56","title":"Commiting to \\(H\\)"},"57":{"body":"After commiting to H, the prover needs to show that H was constructed correctly according to the formula above. To do this, it will ask the prover to provide an evaluation of H on some random point z and evaluations of the trace at the points \\(t(z), t(zg)\\) and \\(t(zg^2)\\). Because the boundary and transition constraints are a public part of the protocol, the verifier knows them, and thus the only thing it needs to compute the evaluation \\((z)\\) by itself are the three trace evaluations mentioned above. Because it asked the prover for them, it can check both sides of the equation: H(z)=β1​B(z)+β2​C(z) and be convinced that \\(H\\) was constructed correctly. We are still not done, however, as the prover could have now cheated on the values of the trace or composition polynomial evaluations.","breadcrumbs":"STARKs » Recap » Consistency check","id":"57","title":"Consistency check"},"58":{"body":"There are two things left the prover needs to show to complete the proof: That \\(H\\) effectively is a polynomial, i.e., that the constraints are satisfied. That the evaluations the prover provided on the consistency check were indeed evaluations of the trace polynomial and composition polynomial on the out of domain point z. Earlier we said we would use the FRI protocol to commit to H and show the first item in the list. However, we can slightly modify the polynomial we do FRI on to show both the first and second items at the same time. This new modified polynomial is called the DEEP composition polynomial. We define it as follows: Deep(x)=γ1​x−zH(x)−H(z)​+γ2​x−zt(x)−t(z)​+γ3​x−zgt(x)−t(zg)​+γ4​x−zg2t(x)−t(zg2)​ where the numbers \\(\\gamma_i\\) are randomly sampled by the verifier. The high level idea is the following: If we apply FRI to this polynomial and it verifies, we are simultaneously showing that \\(H\\) is a polynomial and the prover indeed provided H(z) as one of the out of domain evaluations. This is the first summand in Deep(x). The trace evaluations provided by the prover were the correct ones, i.e., they were \\(t(z)\\), \\(t(zg)\\), and \\(t(zg^2)\\). These are the remaining summands of the Deep(x).","breadcrumbs":"STARKs » Recap » Deep Composition Polynomial","id":"58","title":"Deep Composition Polynomial"},"59":{"body":"The prover needs to show that Deep was constructed correctly according to the formula above. To do this, the verifier will ask the prover to provide: An evaluation of H on z and x_0 Evaluations of the trace at the points \\(t(z)\\), \\(t(zg)\\), \\(t(zg^2)\\) and \\(t(x_0)\\) Where z is the same random, out of domain point used in the consistency check of the composition polynomial, and x_0 is a random point that belongs to the trace domain. With the values provided by the prover, the verifier can check both sides of the equation: Deep(x0​)=γ1​x0​−zH(x0​)−H(z)​+γ2​x0​−zt(x0​)−t(z)​+γ3​x0​−zgt(x0​)−t(zg)​+γ4​x0​−zg2t(x0​)−t(zg2)​ The prover also needs to show that the trace evaluation \\(t(x_0)\\) belongs to the trace. To achieve this, it needs to commit the merkle roots of t and the merkle proof of \\(t(x_0)\\).","breadcrumbs":"STARKs » Recap » Consistency check","id":"59","title":"Consistency check"},"6":{"body":"For better clarity, we'll be using the following toy program throughout this recap. INPUT: x PRIVATE INPUT: e OUTPUT: e * x + x - 1 The observer would have noticed that this program could also be written as (e+1)∗x−1, which is more sensible. But the way it is written now serves us to better explain the arithmetization of PLONK. So we'll stick to it. The idea here is that the verifier holds some value x, say x=3. He gives it to the prover. She executes the program using her own chosen value e, and sends the output value, say 8, along with a proof π demonstrating correct execution of the program and obtaining the correct output. In the context of PLONK, both the inputs and outputs of the program are considered public inputs . This may sound odd, but it is because these are the inputs to the verification algorithm. This is the algorithm that takes, in this case, the tuple (3,8,π) and outputs Accept if the toy program was executed with input x=3, some private value e not revealed to the verifier, and out came 8. Otherwise it outputs Reject . PLONK can be used to delegate program executions to untrusted parties, but it can also be used as a proof of knowledge. Our program could be used by a prover to demostrate that she knows the multiplicative inverse of some value x in the finite field without revealing it. She would do it by sending the verifier the tuple (x,0,π), where π is the proof of the execution of our toy program. In our toy example this is pointless because inverting field elements is easily performed by any verifier. But change our program to the following and you get proofs of knowledge of the preimage of SHA256 digests. PRIVATE INPUT: e OUTPUT: SHA256(e) Here there's no input aside from the prover's private input. As we mentioned, the output h of the program is then part of the inputs to the verification algorithm. Which in this case just takes (h,π).","breadcrumbs":"Plonk » Recap » Programs. Our toy example","id":"6","title":"Programs. Our toy example"},"60":{"body":"We summarize below the steps required in a STARK proof for both prover and verifier.","breadcrumbs":"STARKs » Recap » Summary","id":"60","title":"Summary"},"61":{"body":"Compute the trace polynomial t by interpolating the trace column over a set of \\(2^n\\)-th roots of unity \\({g^i : 0 \\leq i < 2^n}\\). Compute the boundary polynomial B. Compute the transition constraint polynomial C. Construct the composition polynomial H from B and C. Sample an out of domain point z and provide the evaluations \\(H(z)\\), \\(t(z)\\), \\(t(zg)\\), and \\(t(zg^2)\\) to the verifier. Sample a domain point x_0 and provide the evaluations \\(H(x_0)\\) and \\(t(x_0)\\) to the verifier. Construct the deep composition polynomial Deep(x) from H, t, and the evaluations from the item above. Do FRI on Deep(x) and provide the resulting FRI commitment to the verifier. Provide the merkle root of t and the merkle proof of \\(t(x_0)\\).","breadcrumbs":"STARKs » Recap » Prover side","id":"61","title":"Prover side"},"62":{"body":"Take the evaluations \\(H(z)\\), \\(H(x_0)\\), \\(t(z)\\), \\(t(zg)\\), \\(t(zg^2)\\) and \\(t(x_0)\\) the prover provided. Reconstruct the evaluations \\(B(z)\\) and \\(C(z)\\) from the trace evaluations we were given. Check that the claimed evaluation \\(H(z)\\) the prover gave us actually satisfies H(z)=β1​B(z)+β2​C(z) Check that the claimed evaluation \\(Deep(x_0)\\) the prover gave us actually satisfies Deep(x0​)=γ1​x0​−zH(x0​)−H(z)​+γ2​x0​−zt(x0​)−t(z)​+γ3​x0​−zgt(x0​)−t(zg)​+γ4​x0​−zg2t(x0​)−t(zg2)​ Using the merkle root and the merkle proof the prover provided, check that \\(t(x_0)\\) belongs to the trace. Take the provided FRI commitment and check that it verifies.","breadcrumbs":"STARKs » Recap » Verifier side","id":"62","title":"Verifier side"},"63":{"body":"The walkthrough above was for the fibonacci example which, because of its simplicity, allowed us to sweep under the rug a few more complexities that we'll have to tackle on the implementation side. They are:","breadcrumbs":"STARKs » Recap » Simplifications and Omissions","id":"63","title":"Simplifications and Omissions"},"64":{"body":"Our trace contained only one column, but in the general setting there can be multiple (the Cairo AIR has around 30). This means there isn't just one trace polynomial, but several; one for each column. This also means there are multiple boundary constraint polynomials. The general idea, however, remains the same. The deep composition polynomial H is now the sum of several terms containing the boundary constraint polynomials \\(B_1(x), \\dots, B_k(x)\\) (one per column), and each \\(B_i\\) is in turn constructed from the \\(i\\)-th trace polynomial \\(t_i(x)\\).","breadcrumbs":"STARKs » Recap » Multiple trace columns","id":"64","title":"Multiple trace columns"},"65":{"body":"Much in the same way, our fibonacci AIR had only one transition constraint, but there could be several. We will therefore have multiple transition constraint polynomials \\(C_1(x), \\dots, C_n(x)\\), each of which encodes a different relationship between rows that must be satisfied. Also, because there are multiple trace columns, a transition constraint can mix different trace polynomials. One such constraint could be C1​(x)=t1​(gx)−t2​(x) which means \"The first column on the next row has to be equal to the second column in the current row\". Again, even though this seems way more complex, the ideas remain the same. The composition polynomial H will now include a term for every \\(C_i(x)\\), and for each one the prover will have to provide out of domain evaluations of the trace polynomials at the appropriate values. In our example above, to perform the consistency check on \\(C_1(x)\\) the prover will have to provide the evaluations \\(t_1(zg)\\) and \\(t_2(z)\\).","breadcrumbs":"STARKs » Recap » Multiple transition constraints","id":"65","title":"Multiple transition constraints"},"66":{"body":"In the actual implementation, we won't commit to \\(H\\), but rather to a decomposition of \\(H\\) into an even term \\(H_1(x)\\) and an odd term \\(H_2(x)\\), which satisfy H(x)=H1​(x2)+xH2​(x2) This way, we don't commit to \\(H\\) but to \\(H_1\\) and \\(H_2\\). This is just an optimization at the code level; once again, the ideas remain exactly the same.","breadcrumbs":"STARKs » Recap » Composition polynomial decomposition","id":"66","title":"Composition polynomial decomposition"},"67":{"body":"We treated FRI as a black box entirely. However, there is one thing we do need to understand about it: low degree extensions. When applying FRI to a polynomial of degree \\(n\\), we need to provide evaluations of it over a domain with more than \\(n\\) points. In our case, the DEEP composition polynomial's degree is around the same as the trace's, which is, at most, \\(2^n - 1\\) (because it interpolates the trace containing \\(2^n\\) points). The domain we are going to choose to evaluate our DEEP polynomial on will be a set of higher roots of unity. In our fibonacci example, we will take a primitive \\(16\\)-th root of unity \\(\\omega\\). As a reminder, this means: \\(\\omega\\) is an \\(16\\)-th root of unity, i.e., \\(\\omega^{16} = 1\\). Every \\(16\\)-th root of unity is of the form \\(\\omega^i\\) for some \\(0 \\leq i \\leq 15\\). Additionally, we also take it so that \\(\\omega\\) satisfies \\(\\omega^2 = g\\) (\\(g\\) being the \\(8\\)-th primitive root of unity we used to construct t). The evaluation of \\(t\\) on the set \\({\\omega^i : 0 \\leq i \\leq 15}\\) is called a low degree extension (LDE) of \\(t\\). Notice this is not a new polynomial, they're evaluations of \\(t\\) on some set of points. Also note that, because \\(\\omega^2 = g\\), the LDE contains all the evaluations of \\(t\\) on the set of powers of \\(g\\). In fact, {t(ω2i):0≤i≤15}={t(gi):0≤i≤7} This will be extremely important when we get to implementation. For our LDE, we chose \\(16\\)-th roots of unity, but we could have chosen any other power of two greater than \\(8\\). In general, this choice is called the blowup factor, so that if the trace has \\(2^n\\) elements, a blowup factor of \\(b\\) means our LDE evaluates over the \\(2^{n} * b\\) roots of unity (\\(b\\) needs to be a power of two). The blowup factor is a parameter of the protocol related to its security.","breadcrumbs":"STARKs » Recap » FRI, low degree extensions and roots of unity","id":"67","title":"FRI, low degree extensions and roots of unity"},"68":{"body":"In this section, we start diving deeper before showing the formal protocol. If you haven't done so, we recommend reading the \"Recap\" section first. At a high level, the protocol works as follows. The starting point is a matrix T that encodes the trace of a valid execution of the program. This matrix needs to be in a particular format so that its correctness is equivalent to checking a finite number of polynomial equations on its rows. Transforming the execution to this matrix is what's called the arithmetization process. Then a single polynomial F is constructed that encodes the set of all the polynomial constraints. The satisfiability of all these constraints is equivalent to F being divisible by some public polynomial G. So the prover constructs H as the quotient F/G called the composition polynomial. Then the verifier chooses a random point z and challenges the prover to reveal the values F(z) and H(z). Then the verifier checks that H(z)=F(z)/G(z), which convinces him that the same relation holds at a level of polynomials and, in consequence, convinces the verifier that the private trace T of the prover is valid. In summary, at a very high level, the STARK protocol can be organized into three major parts: Arithmetization and commitment of execution trace. Construction of composition polynomial H. Opening of polynomials at random z.","breadcrumbs":"STARKs » Protocol overview » Protocol Overview","id":"68","title":"Protocol Overview"},"69":{"body":"As the Recap mentions, the trace is a table containing the system's state at every step. In this section, we will denote the trace as T. A trace can have several columns to store different aspects or features of a particular state at a specific moment. We will refer to the j-th column as Tj​. You can think of a trace as a matrix T where the entry Tij​ is the j-th element of the i-th state. Most proving systems' primary tool is polynomials over a finite field F. Each column Tj​ of the trace T will be interpreted as evaluations of such a polynomial tj​. Consequently, any information about the states must be encoded somehow as an element in F. To ease notation, we will assume here and in the protocol that the constraints encoding transition rules depend only on a state and the previous one. Everything can be easily generalized to transitions that depend on many preceding states. Then, constraints can be expressed as multivariate polynomials in 2m variables PkT​(X1​,…,Xm​,Y1​,…,Ym​) A transition from state i to state i+1 will be valid if and only if when we plug row i of T in the first m variables and row i+1 in the second m variables of PkT​, we get 0 for all k. In mathematical notation, this is PkT​(Ti,0​,…,Ti,m​,Ti+1,0​,…,Ti+1,m​)=0 for all k These are called transition constraints and check the trace's local properties, where local means relative to specific rows. There is another type of constraint, called boundary constraint , and denoted PjB​. These enforce parts of the trace to take particular values. It is helpful, for example, to verify the initial states. So far, these constraints can only express the local properties of the trace. There are situations where the global properties of the trace need to be checked for consistency. For example, a column may need to take all values in a range but not in any predefined way. Several methods exist to express these global properties as local by adding redundant columns. Usually, they need to involve randomness from the verifier to make sense, and they turn into an interactive protocol called Randomized AIR with Preprocessing .","breadcrumbs":"STARKs » Protocol overview » Arithmetization","id":"69","title":"Arithmetization"},"7":{"body":"This is the process that takes the circuit of a particular program and produces a set of mathematical tools that can be used to generate and verify proofs of execution. The end result will be a set of eight polynomials. To compute them we need first to define two matrices. We call them the Q matrix and the V matrix. The polynomials and the matrices depend only on the program and not on any particular execution of it. So they can be computed once and used for every execution instance. To understand what they are useful for, we need to start from execution traces .","breadcrumbs":"Plonk » Recap » PLONK Arithmetization","id":"7","title":"PLONK Arithmetization"},"70":{"body":"To make interactions possible, a crucial cryptographic primitive is the Polynomial Commitment Scheme. This prevents the prover from changing the polynomials to adjust them to what the verifier expects. Such a scheme consists of the commit and the open protocols. STARK uses a univariate polynomial commitment scheme that internally combines a vector commitment scheme and a protocol called FRI. Let's begin with these two components and see how they build up the polynomial commitment scheme.","breadcrumbs":"STARKs » Protocol overview » Polynomial commitment scheme","id":"70","title":"Polynomial commitment scheme"},"71":{"body":"Given a vector Y=(y0​,…,yM​), commiting to Y means the following. The prover builds a Merkle tree out of it and sends its root to the verifier. The verifier can then ask the prover to reveal, or open , the value of the vector Y at some index i. The prover won't have any choice except to send the correct value. The verifier will expect the corresponding value yi​ and the authentication path to the tree's root to check its authenticity. The authentication path also encodes the vector's position i and its length M. In STARKs, all commited vectors are of the form Y=(p(d1​),…,p(dM​)) for some polynomial p and some domain fixed domain D=(d1​,…,dM​). The domain is always known to the prover and the verifier.","breadcrumbs":"STARKs » Protocol overview » Vector commitments","id":"71","title":"Vector commitments"},"72":{"body":"The FRI protocol is a potent tool to prove that the commitment of a vector (p(d1​),…,p(dM​)) corresponds to the evaluations of a polynomial p of a certain degree. The degree is a configuration of the protocol.","breadcrumbs":"STARKs » Protocol overview » FRI","id":"72","title":"FRI"},"73":{"body":"STARK uses a univariate polynomial commitment scheme. The following is what is expected from the commit and open protocols: Commit : given a polynomial p, the prover produces a sort of hash of it. We denote it here by [p], called the commitment of p. This hash is unique to p. The prover usually sends [p] to the verifier. Open : this is an interactive protocol between the prover and the verifier. The prover holds the polynomial p. The verifier only has the commitment [p]. The verifier sends a value z to the prover at which he wants to know the value y=p(z). The prover sends a value y to the verifier, and then they engage in the Open protocol. As a result, the verifier gets convinced that the polynomial corresponding to the hash [p] evaluates to y at z. Let's see how both of these protocols work in detail. Some configuration parameters are needed: Powers of two N=2n and M=2m with n0. The commitment scheme will only work for polynomials of degree at most N. This means anyone can commit to any polynomial, but the Open protocol will pass only for polynomials satisfying that degree bound.","breadcrumbs":"STARKs » Protocol overview » Polynomial commitments","id":"73","title":"Polynomial commitments"},"74":{"body":"Given a polynomial p, the commitment [p] is just the commitment of the vector (p(d1​),…,p(dM​)). That is, [p] is the root of the Merkle tree of the vector of evaluations of p at D.","breadcrumbs":"STARKs » Protocol overview » Commit","id":"74","title":"Commit"},"75":{"body":"It is an interactive protocol. So assume there is a prover and a verifier. We describe the process considering an honest prover. In the next section, we analyze what happens for malicious provers. The prover holds the polynomial p, and the verifier only the commitment [p] of it. There is also an element z. The prover evaluates p(z) and sends the result to the verifier. As we mentioned, the goal is to generate proof of the validity of the evaluation. Let us denote y:=p(z). This is a value now both the prover and verifier have. The protocol has three steps: FRI on p : First, the prover and the verifier engage in a FRI protocol for polynomials of degree at most N to check that [p] is the commitment of such a polynomial. We will assume from now on that the degree of p is at most N. There is an optimization that completely removes this step—more on that later . FRI on (p−y)/(x−z) : Since p(z)=y, the polynomial p can be written as p=y+(x−z)q for some polynomial q. The prover computes the commitment [q] and sends it to the verifier. Now they engage in a FRI protocol for polynomials of degree at most N−1, which convinces the verifier that [q] is the commitment of a polynomial of degree at most N−1. Point checks : From the point of view of the verifier the commitments [p] and [q] are still potentially unrelated. Therefore, there is a check to ensure that q was computed properly from p following the formula q=(p−y)/(x−z) and that [q] is its commitment. To do this, the verifier challenges the prover to open [p] and [q] as vectors. They use the open protocol of the vector commitment scheme to reveal the values p(di​) and q(di​) for some random point di​∈D chosen by the verifier. Next he checks that p(di​)=y+(di​−z)q(di​). They repeat this last part k times and, as we'll analyze in the next section, this whole thing will convince the verifier that p=y+(x−z)q as polynomials with overwhelming probability (about (N/M)k). Finally, the verifier deduces that p(z)=y from this equality.","breadcrumbs":"STARKs » Protocol overview » Open","id":"75","title":"Open"},"76":{"body":"Let's analyze why the open protocol is sound. Keep in mind that the prover always has to provide a commitment [q] of a polynomial of degree at most N−1 that satisfies p(di​)=y+(di​−z)q(di​) for the chosen values of the verifier. That's the goal. An essential but subtle part of the soundness is that D is a vector of length M>N. To understand why let's see what would happen if N=M. Case M=N Suppose the prover tries to cheat the verifier by sending a value y different from p(z). This means that p−y is not divisible by X−z as polynomials. But as long as z is not in D the prover can interpolate the values d1​−zp(d1​)−y​,…,dN​−zp(dN​)−y​, at D and obtain some polynomial q of degree at most N−1. This polynomial satisfies p(di​)=y+(di​−z)q(di​) for all di​∈D, but the equality of polynomials p=y+(x−z)q does not hold. Mainly because that would imply that p(z)=y, and we are assuming the opposite case. Nevertheless, if they continue with the open protocol, FRI will pass since q is a polynomial of degree at most N−1. All subsequent point checks will pass since q was crafted especially for that. So how come p is different from y+(x−z)q but has the property that p(di​)=y+(di​−z)q(di​) for all di​∈D? FRI guarantees q is of degree at most N−1, which implies that y+(x−z)q is of degree at most N. Then, since p is of degree at most N, the difference f:=y+(X−z)q−p is again of degree at most N. A polynomial f of degree at most N is zero if and only if we can prove that f(d)=0 for N+1 distinct points. But the construction of q only shows that f(di​)=0 for all di​∈D, a set of N points. It's not enough. One extra point and the equality f=0 would hold. Of course, that point doesn't exist, again, because that would imply that p(z)=y, which we assume is not true. Case M=2N Let's see one more example where we double the de size of D but keep the same bound for the degree of the polynomials. So polynomials are of degree at most N, and the length of D is M=2N. Again let's assume the prover wants to cheat the verifier similarly and claims that y is p(z) but, in reality, y=p(z). Inevitably, for the Open protocol to succeed, the prover has to send a commitment [q] of some polynomial of degree at most N−1. The strategy of the prover is pretty much constrained to be the same. But now he can't interpolate q as before in every element of D because that would produce a polynomial of degree at most 2N−1. This most likely won't pass the FRI check. Alternatively, he can choose some subset of D′⊂D of size N and interpolate only there to get a polynomial q of degree at most N−1. That will succeed in the FRI phase, but what happens with the point checks? The check will pass if the verifier chooses a challenge di​ that belongs to D′. If di​∈/D′, then the check will inevitably fail. This is because, as we saw in the previous case, one extra successful point to the already N points where q was interpolated was enough to prove that p(z)=y, which we assume now not to hold. This means, if di​∈/D′, then p(di​) does not coincide with y+(di​−z)q(di​) and the point check fails. So if the verifier chooses the challenges following a uniform distribution, the chances of the prover being caught cheating are 1/2 for only one challenge. If the verifier performs k challenges, then the chance of the prover not getting caught is 1/2k. General case: the blowup factor If the ratio between M and N is 2, then k challenges give 1/2k of probability for a malicious prover to succeed. If the ratio is 4, that is, if M=4N, then that probability is 1/4k for the same number of point checks. This provides another way of improving the soundness error. The larger the ratio, the more complex it is to cheat in the open protocol. This ratio is what's called the blowup factor . It is a security parameter, and finding a balance between the number of challenges k and the size of D is part of the configuration of the protocol. Increasing the size of D makes committing an expensive operation since it involves building a Merkle tree with a vector of the size of D. But increasing the number of challenges makes the size of the final proof larger. We denote the blowup factor by b, and it's always assumed to be a power of 2.","breadcrumbs":"STARKs » Protocol overview » Soundness","id":"76","title":"Soundness"},"77":{"body":"During proof generation, polynomials are committed and opened several times. Computing these for each polynomial independently is costly. In this section, we'll see how batching polynomials can reduce the amount of computation. Let P={p1​,…,pL​} be a set of polynomials. We will commit and open P as a whole. We note this batch commitment as [P]. We need the same configuration parameters as before: N=2n, M=2m with N0. As described earlier, to commit to a single polynomial p, a Merkle tree is built over the vector (p(d1​),…,p(dm​)). When committing to a batch of polynomials P={p1​,…,pn​}, the leaves of the Merkle tree are instead the concatenation of the polynomial evaluations. That is, in the batch setting, the Merkle tree is built for the vector (p1​(d1​)∣∣…∣∣pL​(d1​),…,pL​(dm​)∣∣…∣∣pn​(dm​)). The commitment [P] is the root of this Merkle tree. This reduces the proof size: we only need one Merkle tree for L polynomials. The verifier can then only ask for values in batches. When the verifier chooses an index i, the prover sends p1​(di​),…,pL​(di​) along with one authentication path. The verifier on his side computes the concatenation p1​(di​)∣∣…∣∣pL​(di​) and validates it with the authentication path and [P]. This also reduces the computational time. By traversing the Merkle tree one time, it can reveal several components simultaneously. The batch open protocol proceeds similarly to the case of a single polynomial. The prover will try to convince the verifier that the committed polynomials P at z evaluate to some values yi​=pi​(z). In the batch case, the prover will construct the following polynomial: Q:=i=1∑L​γi​X−zpi​−yi​​ Where γi​ are challenges provided by the verifier. The prover commits to Q and sends [Q] to the verifier. Then the prover and verifier continue similarly to the normal open protocol for Q only. This means they engage in a FRI protocol for polynomials of degree at most N−1 for Q. Then they engage in the point checks for Q. Here, for each challenge di​, the prover uses one authentication path for [Q] to reveal Q(di​) and use one authentication path for [P] to batch reveal values p1​(di​),…,pL​(di​). Successful point checks here mean that Q(di​)=∑i​γi​(pi​(di​)−yi​)/(di​−z). This is equivalent to running the open protocol L times, one for each term pi​ and yi​. Note that this optimization makes a huge difference, as we only need to run the FRI protocol once instead of running it once for each polynomial.","breadcrumbs":"STARKs » Protocol overview » Batch","id":"77","title":"Batch"},"78":{"body":"There is an optimization for the open protocol to avoid running FRI to check that p is of degree at most N. The idea is as follows. Part of FRI protocol for [q], to check that it is of degree at most N−1, involves revealing values of q at other random points di​ also chosen by the verifier. These are part of the internal workings of FRI. These challenges are unrelated to what we mentioned before. So if one removes the FRI check for p, the point checks of the open protocol need to be performed on these challenges di​ of the FRI protocol for [q]. This optimization is included in the formal description of the protocol.","breadcrumbs":"STARKs » Protocol overview » Optimize the open protocol reusing FRI internal challenges","id":"78","title":"Optimize the open protocol reusing FRI internal challenges"},"79":{"body":"Transparent Polynomial Commitment Scheme with Polylogarithmic Communication Complexity Summary on FRI low degree test DEEP FRI Thank goodness it's FRIday Diving DEEP FRI","breadcrumbs":"STARKs » Protocol overview » References","id":"79","title":"References"},"8":{"body":"See the program as a sequence of gates that have left operand, a right operand and an output. The two most basic gates are multiplication and addition gates. In our example, one way of seeing our toy program is as a composition of three gates. Gate 1: left: e, right: x, output: u = e * x Gate 2: left: u, right: x, output: v = e + x Gate 3: left: v, right: 1, output: w = v - 1 On executing the circuit, all these variables will take a concrete value. All that information can be put in table form. It will be a matrix with all left, right and output values of all the gates. One row per gate. We call the columns of this matrix L,R,O. Let's build them for x=3 and e=2. We get u=6, v=9 and w=8. So the first matrix is: A B C 2 3 6 6 3 9 9 - 8 The last gate subtracts a constant value that is part of the program and is not a variable. So it actually has only one input instead of two. And the output is the result of subtracting 1 to it. That's why it is handled a bit different from the second gate. The symbol \"-\" in the R column is a consequence of that. With that we mean \"any value\" because it won't change the result. In the next section we'll see how we implement that. Here we'll use this notation when any value can be put there. In case we have to choose some, we'll default to 0. What we got is a valid execution trace. Not all matrices of that shape will be the trace of an execution. The matrices Q and V will be the tool we need to distinguish between valid and invalid execution traces.","breadcrumbs":"Plonk » Recap » Circuits and execution traces","id":"8","title":"Circuits and execution traces"},"80":{"body":"The protocol is split into rounds. Each round more or less represents an interaction with the verifier. Each round will generally start by getting a challenge from the verifier. The prover will need to interpolate polynomials, and he will always do it over the set DS​={gi}i=02n−1​⊆F, where g is a 2n root of unity in F. Also, the vector commitments will be performed over the set DLDE​=(h,hω,hω2,…,hω2n+l) where ω is a 2n+l root of unity and h is some field element. This is the set we denoted D in the commitment scheme section. The specific choices for the shapes of these sets are motivated by optimizations at a code level.","breadcrumbs":"STARKs » Protocol overview » High-level description of the protocol","id":"80","title":"High-level description of the protocol"},"81":{"body":"In round 1 , the prover commits to the columns of the trace T. He does so by interpolating each column j and obtaining univariate polynomials tj​. Then the prover commits to tj​ over DLDE​. In this way, we have Ti,j​=tj​(gi). From now on, the prover won't be able to change the trace values T. The verifier will leverage this and send challenges to the prover. The prover cannot know in advance what these challenges will be. Thus he cannot handcraft a trace to deceive the verifier. As mentioned before, if some constraints cannot be expressed locally, more columns can be added to make a constraint-friendly trace. This is done by committing to the first set of columns, then sampling challenges from the verifier and repeating round 1. The sampling of challenges serves to add new constraints. These constraints will ensure the new columns have some common structure with the original trace. In the protocol, extended columns are referred to as the RAP2 (Randomized AIR with Preprocessing). The matrix of the extended columns is denoted MRAP2​.","breadcrumbs":"STARKs » Protocol overview » Round 1: Arithmetization and commitment of the execution trace","id":"81","title":"Round 1: Arithmetization and commitment of the execution trace"},"82":{"body":"round 2 aims to build the composition polynomial H. This function will have the property that it is a polynomial if and only if the trace that the prover committed to at round 1 is valid and satisfies the agreed polynomial constraints. That is, H will be a polynomial if and only if T is a trace that satisfies all the transition and boundary constraints. Note that we can compose the polynomials tj​, the ones that interpolate the columns of the trace T, with the multivariate constraint polynomials as follows. QkT​(x)=PkT​(t1​(x),…,tm​(x),t1​(gx),…,tm​(ωx)) These result in univariate polynomials. And the same can be done for the boundary constraints. Since Ti,j​=tj​(gi), these univariate polynomials vanish at every element of D if and only if the trace T is valid. As we already mentioned, this is assuming that transitions only depend on the current and previous state. But it can be generalized to include frames with three or more rows or more context for each constraint. For example, in the Fibonacci case, the most natural way is to encode it as one transition constraint that depends on a row and the two preceding it, as we already did in the Recap section. The STARK protocol checks whether the function X2n−1QkT​​ is a polynomial instead of checking that the polynomial is zero over the domain D={gi​}i=02n−1​. The two statements are equivalent. The verifier could check that all X2n−1QkT​​ are polynomials one by one, and the same for the polynomials coming from the boundary constraints. But this is inefficient; the same can be obtained with a single polynomial. To do this, the prover samples challenges and obtains a random linear combination of these polynomials. The result of this is denoted by H and is called the composition polynomial. It integrates all the constraints by adding them up. So after computing H, the prover commits to it and sends the commitment to the verifier. The rest of the protocol aims to prove that H was constructed correctly and is a polynomial, which can only be true if the prover has a valid extension of the original trace.","breadcrumbs":"STARKs » Protocol overview » Round 2: Construction of composition polynomial H","id":"82","title":"Round 2: Construction of composition polynomial H"},"83":{"body":"The verifier must check that H was constructed according to the protocol rules. That is, H has to be a linear combination of all the functions X2n−1QkT​​ and similar terms for the boundary constraints. To do so, in round 3 the verifier chooses a random point z∈F and the prover computes H(z), tj​(z) and tj​(gz) for all j. With all these, the verifier can check that H and the expected linear combination coincide, at least when evaluated at z. Since z was chosen randomly, this proves with overwhelming probability that H was properly constructed.","breadcrumbs":"STARKs » Protocol overview » Round 3: Evaluation of polynomials at z","id":"83","title":"Round 3: Evaluation of polynomials at z"},"84":{"body":"In this round, the prover and verifier engage in the batch open protocol of the polynomial commitment scheme described above to validate all the evaluations at z from the previous round.","breadcrumbs":"STARKs » Protocol overview » Round 4: Run batch open protocol","id":"84","title":"Round 4: Run batch open protocol"},"85":{"body":"In this section we describe precisely the STARKs protocol used in Lambdaworks. We begin with some additional considerations and notation for most of the relevant objects and values to refer to them later on.","breadcrumbs":"STARKs » Protocol » STARKs protocol","id":"85","title":"STARKs protocol"},"86":{"body":"This is a technique to increase the soundness of the protocol by adding proof of work. It works as follows. At some fixed point in the protocol, the verifier sends a challenge x to the prover. The prover needs to find a string y such that H(x∣∣y) begins with a predefined number of zeroes. Here x∣∣y denotes the concatenation of x and y, seen as bit strings. The number of zeroes is called the grinding factor . The hash function H can be any hash function, independent of other hash functions used in the rest of the protocol. In Lambdaworks we use Keccak256.","breadcrumbs":"STARKs » Protocol » Grinding","id":"86","title":"Grinding"},"87":{"body":"The Fiat-Shamir heuristic is used to make the protocol noninteractive. We assume there is a transcript object to which values can be added and from which challenges can be sampled.","breadcrumbs":"STARKs » Protocol » Transcript","id":"87","title":"Transcript"},"88":{"body":"F denotes a finite field. Given a vector D=(y1​,…,yL​) and a function f:set(D)→F, denote by f(D) the vector (f(y1​),…,f(yL​)). Here set(D) denotes the underlying set of A. A polynomial p∈F[X] induces a function f:A→F for every subset A of F, where f(a):=p(a). Let p,q∈F[X] be two polynomials. A function f:A→F can be induced from them for every subset A disjoint from the set of roots of q, defined by f(a):=p(a)q(a)−1. We abuse notation and denote f by p/q.","breadcrumbs":"STARKs » Protocol » General notation","id":"88","title":"General notation"},"89":{"body":"We assume the prover has already obtained the trace of the execution of the program. This is a matrix T with entries in a finite field F. We assume the number of rows of T is 2n for some n in N. Values known by the prover and verifier prior to the interactions These values are determined the program, the specifications of the AIR being used and the security parameters chosen. m′ is the number of columns of the trace matrix T. r the number of RAP challenges. m′′ is the number of extended columns of the trace matrix in the (optional) second round of RAP. m is the total number of columns: m:=m′+m′′. PkT​ denote the transition constraint polynomials for k=1,…,nT​. We are assuming these are of degree at most 2. ZjT​ denote the transition constraint zerofiers for k=1,…,nT​. b=2l is the blowup factor . c is the grinding factor . Q is number of FRI queries. We assume there is a fixed hash function from F to binary strings. We also assume all Merkle trees are constructed using this hash function. Values computed by the prover These values are computed by the prover from the execution trace and are sent to the verifier along with the proof. 2n is the number of rows of the trace matrix after RAP. ω a primitive 2n+l-th root of unity. g=ωb. An element h∈F∖{ωi}i≥0​. This is called the coset factor . Boundary constraints polynomials PjB​ for j=1,…,m. Boundary constraint zerofiers ZjB​ for j=1,…,m.. Derived values Both prover and verifier compute the following. The interpolation domain: the vector DS​=(1,g,…,g2n−1). The Low Degree Extension DLDE​=(h,hω,hω2,…,hω2n+l−1). Recall 2l is the blowup factor.","breadcrumbs":"STARKs » Protocol » Definitions","id":"89","title":"Definitions"},"9":{"body":"As we said, it only depends on the program itself and not on any particular evaluation of it. It has one row for each gate and its columns are called QL​,QR​,QO​,QM​,QC​. They encode the type of gate of the rows and are designed to satisfy the following. Claim: if columns L,R,O correspond to a valid evaluation of the circuit then for all i the following equality holds Ai​(QL​)i​+Bi​(QR​)i​+Ai​Bi​QM​+Ci​(QO​)i​+(QC​)i​=0 This is better seen with examples. A multiplication gate is represented by the row: QL​ QR​ QM​ QO​ QC​ 0 0 1 -1 0 And the row in the trace matrix that corresponds to the execution of that gate is A B C 2 3 6 The equation in the claim for that row is that 2×0+3×0+2×3×1+6×(−1)+0, which equals 0. The next is an addition gate. This is represented by the row QL​ QR​ QM​ QO​ QC​ 1 1 0 -1 0 The corresponding row in the trace matrix its A B C 6 3 9 And the equation of the claim is 6×1+3×1+2×3×0+9×(−1)+0, which adds up to 0. Our last row is the gate that adds a constant. Addition by constant C can be represented by the row QL​ QR​ QM​ QO​ QC​ 1 0 0 -1 C In our case C=−1. The corresponding row in the execution trace is A B C 9 - 8 And the equation of the claim is 9×1+0×0+9×0×0+8×(−1)+C. This is also zero. Putting it altogether, the full Q matrix is QL​ QR​ QM​ QO​ QC​ 0 0 1 -1 0 1 1 0 -1 0 1 0 0 -1 -1 And we saw that the claim is true for our particular execution: 2×0+3×0+2×3×1+6×(−1)+0=0 6×1+3×1+6×3×0+9×(−1)+0=0 9×1+0×0+9×0×0+8×(−1)+(−1)=0 Not important to our example, but multiplication by constant C can be represented by: QL​ QR​ QM​ QO​ QC​ C 0 0 -1 0 As you might have already noticed, there are several ways of representing the same gate in some cases. We'll exploit this in a moment.","breadcrumbs":"Plonk » Recap » The Q matrix","id":"9","title":"The Q matrix"},"90":{"body":"Vector commitment scheme Given a vector A=(y0​,…,yL​). The operation Commit(A) returns the root r of the Merkle tree that has the hash of the elements of A as leaves. For i∈[0,2n+k), the operation open(A,i) returns yi​ and the authentication path s to the Merkle tree root. The operation Verify(i,y,r,s) returns Accept or Reject depending on whether the i-th element of A is y. It checks whether the authentication path s is compatible with i, y and the Merkle tree root r. In our cases the sets A will be of the form A=(f(a),f(ab),f(ab2),…,f(abL)) for some elements a,b∈F. It will be convenient to use the following abuse of notation. We will write Open(A,abi) to mean Open(A,i). Similarly, we will write Verify(abi,y,r,s) instead of Verify(i,y,r,s). Note that this is only notation and Verify(abi,y,r,s) is only checking that the y is the i-th element of the commited vector.","breadcrumbs":"STARKs » Protocol » Notation of important operations","id":"90","title":"Notation of important operations"},"91":{"body":"","breadcrumbs":"STARKs » Protocol » Protocol","id":"91","title":"Protocol"},"92":{"body":"Round 0: Transcript initialization Start a new transcript. (Strong Fiat Shamir) Add to it all the public values. Round 1: Arithmetization and commitment of the execution trace Round 1.1: Commit main trace For each column Mj​ of the execution trace matrix T, interpolate its values at the domain DS​ and obtain polynomials tj​ such that tj​(gi)=Ti,j​. Compute [tj​]:=Commit(tj​(DLED​)) for all j=1,…,m′ ( Batch commitment optimization applies here ). Add [tj​] to the transcript in increasing order. Round 1.2: Commit extended trace Sample random values a1​,…,al​ in F from the transcript. Use a1​,…,al​ to build MRAP2​∈F2n×m′′ following the specifications of the RAP process. For each column M^j​ of the matrix MRAP2​, interpolate its values at the domain DS​ and obtain polynomials tm′+1​,…,tm′+m′′​ such that tj​(gi)=M^i,j​. Compute [tj​]:=Commit(tj​(DLED​)) for all j=m′+1,…,m′+m′′ ( Batch commitment optimization applies here ). Add [tj​] to the transcript in increasing order for all j=m′+1,…,m′+m′′. Round 2: Construction of composition polynomial H Sample α1B​,…,αmB​ and β1B​,…,βmB​ in F from the transcript. Sample α1T​,…,αnT​T​ and β1T​,…,βnT​T​ in F from the transcript. Compute Bj​:=ZjB​tj​−PjB​​. Compute Ck​:=ZkT​PkT​(t1​,…,tm​,t1​(gX),…,tm​(gX))​. Compute the composition polynomial H:=k∑​βkT​Ck​+j∑​βjB​Bj​ Decompose H as H=H1​(X2)+XH2​(X2) Compute commitments [H1​] and [H2​]. Add [H1​] and [H2​] to the transcript. Round 3: Evaluation of polynomials at z Sample from the transcript until obtaining z∈F∖DLDE​. Compute H1​(z2), H2​(z2), and tj​(z) and tj​(gz) for all j. Add H1​(z2), H2​(z2), and tj​(z) and tj​(gz) for all j to the transcript. Round 4: Run batch open protocol Sample γ, γ′, and γ1​,…,γm​, γ1′​,…,γm′​ in F from the transcript. Compute p0​ as γX−z2H1​−H1​(z2)​+γ′X−z2H2​−H2​(z2)​+j∑​γj​X−ztj​−tj​(z)​+γj′​X−gztj​−tj​(gz)​ Round 4.1.k: FRI commit phase Let D0​:=DLDE​, and [p0​]:=Commit(p0​(D0​)). Add [p0​] to the transcript. For k=1,…,n do the following: Sample ζk−1​ from the transcript. Decompose pk−1​ into even and odd parts, that is, pk−1​=pk−1odd​(X2)+Xpk−1even​(X2). Define pk​:=pk−1odd​(X)+ζk−1​pk−1even​(X). If k>(&trace, &pub_inputs, &proof_options).unwrap(); assert!(verify::>(&proof, &pub_inputs, &proof_options));\n} The proving system revolves around the prove function, that takes a trace, public inputs and proof options as inputs to generate a proof, and a verify function that takes the generated proof, the public inputs and the proof options as inputs, outputting true when the proof is verified correctly and false otherwise. Note that the public inputs and proof options should be the same for both. Public inputs should be shared by the Cairo runner to prover and verifier, and the proof options should have been agreed on beforehand by the two entities beforehand. Below we go over the main things involved in this code.","breadcrumbs":"STARKs » Implementation » High Level API » High level API: Fibonacci example","id":"97","title":"High level API: Fibonacci example"},"98":{"body":"To prove the integrity of a fibonacci trace, we first need to define what it means for a trace to be valid. As we've talked about in the recap, this involves defining an AIR for our computation where we specify both the boundary and transition constraints for a fibonacci sequence. In code, this is done through the AIR trait. Implementing AIR requires defining a couple methods, but the two most important ones are boundary_constraints and compute_transition, which encode the boundary and transition constraints of our computation.","breadcrumbs":"STARKs » Implementation » High Level API » AIR","id":"98","title":"AIR"},"99":{"body":"For our Fibonacci AIR, boundary constraints look like this: fn boundary_constraints( &self, _rap_challenges: &Self::RAPChallenges,\n) -> BoundaryConstraints { let a0 = BoundaryConstraint::new_simple(0, self.pub_inputs.a0.clone()); let a1 = BoundaryConstraint::new_simple(1, self.pub_inputs.a1.clone()); BoundaryConstraints::from_constraints(vec![a0, a1])\n} The BoundaryConstraint struct represents a specific boundary constraint, meaning \"column i at row j should be equal to x\". In this case, because we have only one column, we are using the new_simple method to simply say Row 0 should equal the public input a0, which in the typical fibonacci is set to 1. 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Which means that a single authentication path serves to validate both points simultaneously.","breadcrumbs":"STARKs » Protocol » Bit-reversal ordering of Merkle tree leaves","id":"100","title":"Bit-reversal ordering of Merkle tree leaves"},"101":{"body":"The prover opens the polynomials pk​ of the FRI layers at υs2k​ and −υs2k​ for all k>1. Later on, the verifier uses each of those pairs to reconstruct one of the values of the next layer, namely pk+1​(υ2k+1). So there's no need to add the value pk​(υ2k+1) to the proof, as the verifier reconstructs them. The prover only needs to send the authentication paths Pk​ for them. The protocol is only modified at Step 3 of the verifier as follows. Checking that Verify((υs2k​,πkυs2k​​),Pk​,Pk​) is skipped. After computing x:=G+ζk​H, the verifier uses it to check that Verify((υs2k​,x),Pk​,Pk​) is Accept , which proves that x is actually πkυs2k​​, and continues to the next iteration of the loop.","breadcrumbs":"STARKs » Protocol » Redundant values in the proof","id":"101","title":"Redundant values in the proof"},"102":{"body":"The goal of this section will be to go over the details of the implementation of the proving system. To this end, we will follow the flow the example in the recap chapter, diving deeper into the code when necessary and explaining how it fits into a more general case. This implementation couldn't be done without checking Facebook's Winterfell and Max Gillett's Giza . We want to thank everyone involved in them, along with Shahar Papini and Lior Goldberg from Starkware who also provided us valuable insight.","breadcrumbs":"STARKs » Implementation » STARKs Prover Lambdaworks Implementation","id":"102","title":"STARKs Prover Lambdaworks Implementation"},"103":{"body":"Let's go over the main test we use for our prover, where we compute a STARK proof for a fibonacci trace with 4 rows and then verify it. fn test_prove_fib() { let trace = simple_fibonacci::fibonacci_trace([FE::from(1), FE::from(1)], 8); let proof_options = ProofOptions::default_test_options(); let pub_inputs = FibonacciPublicInputs { a0: FE::one(), a1: FE::one(), }; let proof = prove::>(&trace, &pub_inputs, &proof_options).unwrap(); assert!(verify::>(&proof, &pub_inputs, &proof_options));\n} The proving system revolves around the prove function, that takes a trace, public inputs and proof options as inputs to generate a proof, and a verify function that takes the generated proof, the public inputs and the proof options as inputs, outputting true when the proof is verified correctly and false otherwise. Note that the public inputs and proof options should be the same for both. Public inputs should be shared by the Cairo runner to prover and verifier, and the proof options should have been agreed on beforehand by the two entities beforehand. Below we go over the main things involved in this code.","breadcrumbs":"STARKs » Implementation » High Level API » High level API: Fibonacci example","id":"103","title":"High level API: Fibonacci example"},"104":{"body":"To prove the integrity of a fibonacci trace, we first need to define what it means for a trace to be valid. As we've talked about in the recap, this involves defining an AIR for our computation where we specify both the boundary and transition constraints for a fibonacci sequence. In code, this is done through the AIR trait. Implementing AIR requires defining a couple methods, but the two most important ones are boundary_constraints and compute_transition, which encode the boundary and transition constraints of our computation.","breadcrumbs":"STARKs » Implementation » High Level API » AIR","id":"104","title":"AIR"},"105":{"body":"For our Fibonacci AIR, boundary constraints look like this: fn boundary_constraints( &self, _rap_challenges: &Self::RAPChallenges,\n) -> BoundaryConstraints { let a0 = BoundaryConstraint::new_simple(0, self.pub_inputs.a0.clone()); let a1 = BoundaryConstraint::new_simple(1, self.pub_inputs.a1.clone()); BoundaryConstraints::from_constraints(vec![a0, a1])\n} The BoundaryConstraint struct represents a specific boundary constraint, meaning \"column i at row j should be equal to x\". In this case, because we have only one column, we are using the new_simple method to simply say Row 0 should equal the public input a0, which in the typical fibonacci is set to 1. Row 1 should equal the public input a1, which in the typical fibonacci is set to 1. In the case of multiple columns, the new method exists so you can also specify column number. After instantiating each of these constraints, we return all of them through the struct BoundaryConstraints.","breadcrumbs":"STARKs » Implementation » High Level API » Boundary Constraints","id":"105","title":"Boundary Constraints"},"106":{"body":"The way we specify our fibonacci transition constraint looks like this: fn compute_transition( &self, frame: &air::frame::Frame, _rap_challenges: &Self::RAPChallenges,\n) -> Vec> { let first_row = frame.get_row(0); let second_row = frame.get_row(1); let third_row = frame.get_row(2); vec![third_row[0] - second_row[0] - first_row[0]]\n} It's not completely obvious why this is how we chose to express transition constraints, so let's talk a little about it. What we need to specify in this method is the relationship that has to hold between the current step of computation and the previous ones. For this, we get a Frame as an argument. This is a struct holding the current step (i.e. the current row of the trace) and all previous ones needed to encode our constraint. In our case, this is the current row and the two previous ones. To access rows we use the get_row method. The current step is always the last row (in our case 2), with the others coming before it. In our compute_transition method we get the three rows we need and return third_row[0] - second_row[0] - first_row[0] which is the value that needs to be zero for our constraint to hold. Because we support multiple transition constraints, we actually return a vector with one value per constraint, so the first element holds the first constraint value and so on.","breadcrumbs":"STARKs » Implementation » High Level API » Transition Constraints","id":"106","title":"Transition Constraints"},"107":{"body":"After defining our AIR, we create our specific trace to prove against it. let trace = fibonacci_trace([FE17::new(1), FE17::new(1)], 4); let trace_table = TraceTable { table: trace.clone(), num_cols: 1,\n}; TraceTable is the struct holding execution traces; the num_cols says how many columns the trace has, the table field is a vec holding the actual values of the trace in row-major form, meaning if the trace looks like this | 1 | 2 |\n| 3 | 4 |\n| 5 | 6 | then its corresponding TraceTable is let trace_table = TraceTable { table: vec![1, 2, 3, 4, 5, 6], num_cols: 2,\n}; In our example, fibonacci_trace is just a helper function we use to generate the fibonacci trace with 4 rows and [1, 1] as the first two values.","breadcrumbs":"STARKs » Implementation » High Level API » TraceTable","id":"107","title":"TraceTable"},"108":{"body":"After specifying our constraints and trace, the only thing left to do is provide a few parameters related to the STARK protocol and our AIR. These specify things such as the number of columns of the trace and proof configuration, among others. They are all encapsulated in the AirContext struct, which in our example we instantiate like this: let context = AirContext { options: ProofOptions { blowup_factor: 2, fri_number_of_queries: 1, coset_offset: 3, }, trace_columns: trace_table.n_cols, transition_degrees: vec![1], transition_exemptions: vec![2], transition_offsets: vec![0, 1, 2], num_transition_constraints: 1,\n}; Let's go over each of them: options requires a ProofOptions struct holding specific parameters related to the STARK protocol to be used when proving. They are: The blowup_factor used for the trace LDE extension, a parameter related to the security of the protocol. The number of queries performed by the verifier when doing FRI, also related to security. The offset used for the LDE coset. This depends on the field being used for the STARK proof. trace_columns are the number of columns of the trace, respectively. transition_degrees holds the degree of each transition constraint. transition_exemptions is a Vec which tells us, for each column, the number of rows the transition constraints should not apply, starting from the end of the trace. In the example, the transition constraints won't apply on the last two rows of the trace. transition_offsets holds the indexes that define a frame for our AIR. In our fibonacci case, these are [0, 1, 2] because we need the current row and the two previous one to define our transition constraint. num_transition_constraints simply says how many transition constraints our AIR has.","breadcrumbs":"STARKs » Implementation » High Level API » AIR Context","id":"108","title":"AIR Context"},"109":{"body":"Having defined all of the above, proving our fibonacci example amounts to instantiating the necessary structs and then calling prove passing the trace, public inputs and proof options. We use a simple implementation of a hasher called TestHasher to handle merkle proof building. let proof = prove(&trace_table, &pub_inputs, &proof_options); Verifying is then done by passing the proof of execution along with the same AIR to the verify function. assert!(verify(&proof, &pub_inputs, &proof_options));","breadcrumbs":"STARKs » Implementation » High Level API » Proving execution","id":"109","title":"Proving execution"},"11":{"body":"Our matrices are fine now. But they can be optimized. Let's do that to showcase this flexibility of PLONK and also reduce the size of our example. PLONK has the flexibility to construct more sophisticated gates as combinations of the five columns. And therefore the same program can be expressed in multiple ways. In our case all three gates can actually be merged into a single custom gate. The Q matrix ends up being a single row. QL​ QR​ QM​ QO​ QC​ 1 1 1 -1 1 and also the V matrix L R O 0 1 2 The trace matrix for this representation is just A B C 2 3 8 And we check that it satisfies the equation 2×1+3×1+2×3×1+8×(−1)+(−1)=0 Of course, we can't always squash an entire program into a single gate.","breadcrumbs":"Plonk » Recap » Custom gates","id":"11","title":"Custom gates"},"110":{"body":"In this section we go over how a few things in the prove and verify functions are implemented. If you just need to use the prover, then you probably don't need to read this. If you're going through the code to try to understand it, read on. We will once again use the fibonacci example as an ilustration. Recall from the recap that the main steps for the prover and verifier are the following:","breadcrumbs":"STARKs » Implementation » Under the hood » How this works under the hood","id":"110","title":"How this works under the hood"},"111":{"body":"Compute the trace polynomial t by interpolating the trace column over a set of 2n-th roots of unity {gi:0≤i<2n}. Compute the boundary polynomial B. Compute the transition constraint polynomial C. Construct the composition polynomial H from B and C. Sample an out of domain point z and provide the evaluation H(z) and all the necessary trace evaluations to reconstruct it. In the fibonacci case, these are t(z), t(zg), and t(zg2). Sample a domain point x_0 and provide the evaluation H(x0​) and t(x0​). Construct the deep composition polynomial Deep(x) from H, t, and the evaluations from the item above. Do FRI on Deep(x) and provide the resulting FRI commitment to the verifier. Provide the merkle root of t and the merkle proof of t(x0​).","breadcrumbs":"STARKs » Implementation » Under the hood » Prover side","id":"111","title":"Prover side"},"112":{"body":"Take the evaluation H(z) along with the trace evaluations the prover provided. Reconstruct the evaluations B(z) and C(z) from the trace evaluations. Check that the claimed evaluation H(z) the prover gave us actually satisfies H(z)=β1​B(z)+β2​C(z) Take the evaluations H(x0​) and t(x0​). Check that the claimed evaluation Deep(x0​) the prover gave us actually satisfies Deep(x0​)=γ1​x0​−zH(x0​)−H(z)​+γ2​x0​−zt(x0​)−t(z)​+γ3​x0​−zgt(x0​)−t(zg)​+γ4​x0​−zg2t(x0​)−t(zg2)​ Take the provided FRI commitment and check that it verifies. Using the merkle root and the merkle proof the prover provided, check that t(x0​) belongs to the trace. Following along the code in the prove and verify functions, most of it maps pretty well to the steps above. The main things that are not immediately clear are: How we take the constraints defined in the AIR through the compute_transition method and map them to transition constraint polynomials. How we then construct H from them and the boundary constraint polynomials. What the composition polynomial even/odd decomposition is. What an ood frame is. What the transcript is.","breadcrumbs":"STARKs » Implementation » Under the hood » Verifier side","id":"112","title":"Verifier side"},"113":{"body":"This is possibly the most complex part of the code, so what follows is a long explanation for it. In our fibonacci example, after obtaining the trace polynomial t by interpolating, the transition constraint polynomial is C(x)=∏i=05​(x−gi)t(xg2)−t(xg)−t(x)​ On our prove code, if someone passes us a fibonacci AIR like the one we showed above used in one of our tests, we somehow need to construct C(x). However, what we are given is not a polynomial, but rather this method fn compute_transition( &self, frame: &air::frame::Frame, ) -> Vec> { let first_row = frame.get_row(0); let second_row = frame.get_row(1); let third_row = frame.get_row(2); vec![third_row[0] - second_row[0] - first_row[0]]\n} So how do we get to C(x) from this? The answer is interpolation. What the method above is doing is the following: if you pass it a frame that looks like this ​t(x0​)t(x0​g)t(x0​g2)​​ for any given point x0​, it will return the value t(x0​g2)−t(x0​g)−t(x0​) which is the numerator in C(x0​). Using the transition_exemptions field we defined in our AIR, we can also compute evaluations in the denominator, i.e. the zerofier evaluations. This is done under the hood by the transition_divisors() method. The above means that even though we don't explicitly have the polynomial C(x), we can evaluate it on points given an appropriate frame. If we can evaluate it on enough points, we can then interpolate them to recover C(x). This is exactly how we construct both transition constraint polynomials and subsequently the composition polynomial H. The job of evaluating H on enough points so we can then interpolate it is done by the ConstraintEvaluator struct. You'll notice prove does the following let constraint_evaluations = evaluator.evaluate( &lde_trace, &lde_roots_of_unity_coset, &alpha_and_beta_transition_coefficients, &alpha_and_beta_boundary_coefficients,\n); This function call will return the evaluations of the boundary terms βiB​Bi​(x) and constraint terms βiT​Ci​(x) for every i. The constraint_evaluations value returned is a ConstraintEvaluationTable struct, which is nothing more than a big list of evaluations of each polynomial required to construct H. With this in hand, we just call let composition_poly = constraint_evaluations.compute_composition_poly(& lde_roots_of_unity_coset); which simply interpolates the sum of all evaluations to obtain H. Let's go into more detail on how the evaluate method reconstructs C(x) in our fibonacci example. It receives the lde_trace as an argument, which is this: ​t(ω0)t(ω1)…t(ω15)​​ where ω is the primitive root of unity used for the LDE, that is, ω satisfies ω2=g. We need to recover C(x), a polynomial whose degree can't be more than t(x)'s. Because t was built by interpolating 8 points (the trace), we know we can recover C(x) by interpolating it on 16 points. We choose these points to be the LDE roots of unity {ω0,ω,ω2,…,ω15} Remember that to evaluate C(x) on these points, all we need are the evaluations of the polynomial t(xg2)−t(xg)−t(x) as the zerofier ones we can compute easily. These become: t(ω0g2)−t(ω0g)−t(ω0)t(ωg2)−t(ωg)−t(ω)t(ω2g2)−t(ω2g)−t(ω2)⋮t(ω15g2)−t(ω15g)−t(ω15) If we remember that ω2=g, this is t(ω4)−t(ω2)−t(ω0)t(ω5)−t(ω3)−t(ω)t(ω6)−t(ω4)−t(ω2)⋮t(ω3)−t(ω)−t(ω15) and we can compute each evaluation here by calling compute_transition on the appropriate frame built from the lde_trace. Specifically, for the first evaluation we can build the frame: ​t(ω0)t(ω2)t(ω4)​​ Calling compute_transition on this frame gives us the first evaluation. We can get the rest in a similar fashion, which is what this piece of code in the evaluate method does: for (i, d) in lde_domain.iter().enumerate() { let frame = Frame::read_from_trace( lde_trace, i, blowup_factor, &self.air.context().transition_offsets, ) let mut evaluations = self.air.compute_transition(&frame); ...\n} Each iteration builds a frame as above and computes one of the evaluations needed. The rest of the code just adds the zerofier evaluations, along with the alphas and betas. It then also computes boundary polynomial evaluations by explicitly constructing them.","breadcrumbs":"STARKs » Implementation » Under the hood » Reconstructing the transition constraint polynomials","id":"113","title":"Reconstructing the transition constraint polynomials"},"114":{"body":"The verifier employs the same trick to reconstruct the evaluations on the out of domain point Ci​(z) for the consistency check.","breadcrumbs":"STARKs » Implementation » Under the hood » Verifier","id":"114","title":"Verifier"},"115":{"body":"At the end of the recap we talked about how in our code we don't actually commit to H, but rather an even/odd decomposition for it. These are two polynomials H_1 and H_2 that satisfy H(x)=H1​(x2)+xH2​(x2) This all happens on this piece of code let composition_poly = constraint_evaluations.compute_composition_poly(&lde_roots_of_unity_coset); let (composition_poly_even, composition_poly_odd) = composition_poly.even_odd_decomposition(); // Evaluate H_1 and H_2 in z^2.\nlet composition_poly_evaluations = vec![ composition_poly_even.evaluate(&z_squared), composition_poly_odd.evaluate(&z_squared),\n]; After this, we don't really use H anymore, but rather H_1 and H_2. There's not that much to say other than that.","breadcrumbs":"STARKs » Implementation » Under the hood » Even/odd decomposition for H","id":"115","title":"Even/odd decomposition for H"},"116":{"body":"As part of the consistency check, the prover needs to provide evaluations of the trace polynomials in all the points needed by the verifier to check that H was constructed correctly. In the fibonacci example, these are t(z), t(zg), and t(zg2). In code, the prover passes these evaluations as a Frame, which we call the out of domain (ood) frame. The reason we do this is simple: with the frame in hand, the verifier can reconstruct the evaluations of the constraint polynomials Ci​(z) by calling the compute_transition method on the ood frame and then adding the alphas, betas, and so on, just like we explained in the section above.","breadcrumbs":"STARKs » Implementation » Under the hood » Out of Domain Frame","id":"116","title":"Out of Domain Frame"},"117":{"body":"Throughout the protocol, there are a number of times where the verifier randomly samples some values that the prover needs to use (think of the alphas and betas used when constructing H). Because we don't actually have an interaction between prover and verifier, we emulate it by using a hash function, which we assume is a source of randomness the prover can't control. The job of providing these samples for both prover and verifier is done by the Transcript struct, which you can think of as a stateful rng; whenever you call challenge() on a transcript you get a random value and the internal state gets mutated, so the next time you call challenge() you get a different one. You can also call append on it to mutate its internal state yourself. This is done a number of times throughout the protocol to keep the prover honest so it can't predict or manipulate the outcome of challenge(). Notice that to sample the same values, both prover and verifier need to call challenge and append in the same order (and with the same values in the case of append) and the same number of times. The idea explained above is called the Fiat-Shamir heuristic or just Fiat-Shamir, and is more generally used throughout proving systems to remove interaction between prover and verifier. Though the concept is very simple, getting it right so the prover can't cheat is not, but we won't go into that here.","breadcrumbs":"STARKs » Implementation » Under the hood » Transcript","id":"117","title":"Transcript"},"118":{"body":"The generated proof has got all the information needed for the verifier to verify it: Trace length: The number of rows of the trace table, needed to know the max degree of the polynomials that appear in the system. LDE trace commitments. DEEP composition polynomial out of domain even and odd evaluations. DEEP composition polynomial root. FRI layers merkle roots. FRI last layer value. Query list. DEEP composition poly openings. Nonce: Proof of work setting used to generate the proof.","breadcrumbs":"STARKs » Implementation » Under the hood » Proof","id":"118","title":"Proof"},"119":{"body":"","breadcrumbs":"STARKs » Implementation » Under the hood » Special considerations","id":"119","title":"Special considerations"},"12":{"body":"Aside from the gates that execute the program operations, additional rows must be incorporated into these matrices. This is due to the fact that the prover must demonstrate not only that she executed the program, but also that she used the appropriate inputs. Furthermore, the proof must include an assertion of the output value. As a result, a few extra rows are necessary. In our case these are the first two and the last one. The original one sits now in the third row. QL​ QR​ QM​ QO​ QC​ -1 0 0 0 3 -1 0 0 0 8 1 1 1 -1 1 1 -1 0 0 0 And this is the updated V matrix L R O 0 - - 1 - - 2 0 3 1 3 - The first row is there to force the variable with index 0 to take the value 3. Similarly the second row forces variable with index 1 to take the value 8. These two will be the public inputs of the verifier. The last row checks that the output of the program is the claimed one. And the trace matrix is now A B C 3 - - 8 - - 2 3 8 8 8 - With these extra rows, equations add up to zero only for valid executions of the program with input 3 and output 8. An astute observer would notice that by incorporating these new rows, the matrix Q is no longer independent of the specific evaluation. This is because the first two rows of the QC​ column contain concrete values that are specific to a particular execution instance. To maintain independence, we can remove these values and consider them as part of an extra one-column matrix called PI (stands for Public Input). This column has zeros in all rows not related to public inputs. We put zeros in the QC​ columns. The responsibility of filling in the PI matrix is of the prover and verifier. In our example it is PI 3 8 0 0 And the final Q matrix is QL​ QR​ QM​ QO​ QC​ -1 0 0 0 0 -1 0 0 0 0 1 1 1 -1 1 1 -1 0 0 0 We ended up with two matrices that depend only on the program, Q and V. And two matrices that depend on a particular evaluation, namely the ABC and PI matrices. The updated version of the claim is the following: Claim: Let T be a matrix with columns A,B,C. It corresponds to a evaluation of the circuit if and only if a) for all i the following equality holds Ai​(QL​)i​+Bi​(QR​)i​+Ai​Bi​QM​+Ci​(QO​)i​+(QC​)i​+(PI)i​=0, b) for all i,j,k,l such that Vi,j​=Vk,l​ we have Ti,j​=Tk,l​.","breadcrumbs":"Plonk » Recap » Public inputs","id":"12","title":"Public inputs"},"120":{"body":"When evaluating or interpolating a polynomial, if the input (be it coefficients or evaluations) size isn't a power of two then the FFT API will extend it with zero padding until this requirement is met. This is because the library currently only uses a radix-2 FFT algorithm. Also, right now FFT only supports inputs with a size up to 2264 elements.","breadcrumbs":"STARKs » Implementation » Under the hood » FFT evaluation and interpolation","id":"120","title":"FFT evaluation and interpolation"},"121":{"body":"","breadcrumbs":"STARKs » Implementation » Under the hood » Other","id":"121","title":"Other"},"122":{"body":"Whenever we interpolate or evaluate trace, boundary and constraint polynomials, we use some 2n-th roots of unity. There are a few reasons for this: Using roots of unity means we can use the Fast Fourier Transform and its inverse to evaluate and interpolate polynomials. This method is much faster than the naive Lagrange interpolation one. Since a huge part of the STARK protocol involves both evaluating and interpolating, this is a huge performance improvement. When computing boundary and constraint polynomials, we divide them by their zerofiers, polynomials that vanish on a few points (the trace elements where the constraints do not hold). These polynomials take the form Z(X)=∏(X−xi​) where the xi​ are the points where we want it to vanish. When implementing this, evaluating this polynomial can be very expensive as it involves a huge product. However, if we are using roots of unity, we can use the following trick. The vanishing polynomial for all the 2n roots of unity is X2n−1 Instead of expressing the zerofier as a product of the places where it should vanish, we express it as the vanishing polynomial above divided by the exemptions polynomial; the polynomial whose roots are the places where constraints don't need to hold. Z(X)=∏(X−ei​)X2n−1​ where the ei​ are now the points where we don't want it to vanish. This exemptions polynomial in the denominator is usually much smaller, and because the vanishing polynomial in the numerator is only two terms, evaluating it is really fast.","breadcrumbs":"STARKs » Implementation » Under the hood » Why use roots of unity?","id":"122","title":"Why use roots of unity?"},"123":{"body":"The n-th roots of unity are the numbers x that satisfy xn=1 There are n such numbers, because they are the roots of the polynomial Xn−1. The set of n-th roots of unity always has a generator, a root g that can be used to obtain every other root of unity by exponentiating. What this means is that the set of n-th roots of unity is {gi:0≤i1 and 0≤j≤J and I=∣Lj′′​∣. Extend M with additional L′′ rows to obtain a matrix M∈F(L+L′+L′′)×33 by appending copies of R′′ at the bottom. Pad M with copies of its last row until it has a power of two number of rows. As a result we obtain a matrix T∈F2n×33.","breadcrumbs":"Cairo » Trace Formal Description » Construction of execution trace T:","id":"127","title":"Construction of execution trace T:"},"128":{"body":"","breadcrumbs":"Cairo » Trace Detailed Description » Cairo execution trace","id":"128","title":"Cairo execution trace"},"129":{"body":"After the execution of a Cairo program in the Cairo VM, three files are generated that are the core components for the construction of the execution trace, needed for the proving system: trace file : Has the information on the state of the three Cairo VM registers ap, fp, and pc at every cycle of the execution of the program. To reduce ambiguity in terms, we should call these the register states of the Cairo VM, and leave the term trace to the final product that is passed to the prover to generate a proof. memory file : A file with the information of the VM's memory at the end of the program run, after the memory has been relocated. public inputs : A file with all the information that must be publicly available to the prover and verifier, such as the total number of execution steps, public memory, used builtins and their respective addresses range in memory. The next section will explain in detail how these elements are used to build the final execution trace.","breadcrumbs":"Cairo » Trace Detailed Description » Raw materials","id":"129","title":"Raw materials"},"13":{"body":"In the previous section we showed how the arithmetization process works in PLONK. For a program with n public inputs and m gates, we constructed two matrices Q and V, of sizes (n+m+1)×5 and (n+m+1)×3 that satisfy the following. Let N=n+m+1. Claim: Let T be a N×3 matrix with columns A,B,C and PI a N×1 matrix. They correspond to a valid execution instance with public input given by PI if and only if a) for all i the following equality holds Ai​(QL​)i​+Bi​(QR​)i​+Ai​Bi​QM​+Ci​(QO​)i​+(QC​)i​+(PI)i​=0, b) for all i,j,k,l such that Vi,j​=Vk,l​ we have Ti,j​=Tk,l​, c) (PI)i​=0 for all i>n. Polynomials enter now to squash most of these equations. We will traduce the set of all equations in conditions (a) and (b) to just a few equations on polynomials. Let ω be a primitive N-th root of unity and let H=ωi:0≤in. Then we constructed polynomials qL​,qR​,qM​,qO​,qC​,Sσ1​,Sσ2​,Sσ3​, f, g off the matrices Q and V. They are the result of interpolating at a domain H={1,ω,ω2,…,ωN−1} for some N-th primitive root of unity and a few random values. And also constructed polynomials a,b,c,pi off the matrices T and PI. Loosely speaking, the above fact can be reformulated in terms of polynomial equations as follows. Fact: Let zH​=XN−1. Let T be a N×3 matrix with columns A,B,C and PI a N×1 matrix. They correspond to a valid execution instance with public input given by PI if and only if a) There is a polynomial t1​ such that the following equality holds aqL​+bqR​+abqM​+cqO​+qC​+pi=zH​t1​, b) There are polynomials t2​,t3​, z such that zf−gz′=zH​t2​ and (z−1)L1​=zH​t3​, where z′(X)=z(Xω) You might be wondering where the polynomials ti​ came from. Recall that for a polynomial F, we have F(h)=0 for all h∈H if and only if F=zH​t for some polynomial t. Finally both conditions (a) and (b) are equivalent to a single equation (c) if we let more randomness to come into play. This is: (c) Let α be a random field element. There is a polynomial t such that zH​t=​aqL​+bqR​+abqM​+cqO​+qC​+pi+α(gz′−fz)+α2(z−1)L1​​ This last step is not obvious. You can check the paper to see the proof. Anyway, this is the equation you'll recognize below in the description of the protocol. Randomness is a delicate matter and an important part of the protocol is where it comes from, who chooses it and when they choose it. Check out the protocol to see how it works.","breadcrumbs":"Plonk » Recap » Wrapping up the whole thing","id":"15","title":"Wrapping up the whole thing"},"16":{"body":"","breadcrumbs":"Plonk » Protocol » Protocol","id":"16","title":"Protocol"},"17":{"body":"","breadcrumbs":"Plonk » Protocol » Details and tricks","id":"17","title":"Details and tricks"},"18":{"body":"A polynomial commitment scheme (PCS) is a cryptographic tool that allows one party to commit to a polynomial, and later prove properties of that polynomial. This commitment polynomial hides the original polynomial's coefficients and can be publicly shared without revealing any information about the original polynomial. Later, the party can use the commitment to prove certain properties of the polynomial, such as that it satisfies certain constraints or that it evaluates to a certain value at a specific point. In the implementation section we'll explain the inner workings of the Kate-Zaverucha-Goldberg scheme, a popular PCS chosen in Lambdaworks for PLONK. For the moment we only need the following about it: It consists of a finite group G and the following algorithms: Commit(f) : This algorithm takes a polynomial f and produces an element of the group G. It is called the commitment of f and is denoted by [f]1​. It is homomorphic in the sense that [f+g]1​=[f]1​+[g]1​. The former sum being addition of polynomials. The latter is addition in the group G. Open(f,ζ) : It takes a polynomial f and a field element ζ and produces an element π of the group G. This element is called an opening proof for f(ζ). It is the proof that f evaluated at ζ gives f(ζ). Verify([f]1​, π, ζ, y) : It takes group elements [f]1​ and π, and also field elements ζ and y. With overwhelming probability it outputs Accept if f(z)=y and Reject otherwise.","breadcrumbs":"Plonk » Protocol » Polynomial commitment scheme","id":"18","title":"Polynomial commitment scheme"},"19":{"body":"As you will see in the protocol, the prover reveals the value taken by a bunch of the polynomials at a random ζ. In order for the protocol to be Honest Verifier Zero Knowledge , these polynomials need to be blinded . This is a process that makes the values of these polynomials at ζ seemingly random by forcing them to be of certain degree. Here's how it works. Let's take for example the polynomial a the prover constructs. This is the interpolation of the first column of the trace matrix T at the domain H. This matrix has all of the left operands of all the gates. The prover wishes to keep them secret. Say the trace matrix T has N rows. And so H is {1,ω,ω2,…,ωN−1}. The invariant that the prover cannot violate is that ablinded​(ωi) must take the value T0,i​, for all i. This is what the interpolation polynomial a satisfies. And is the unique such polynomial of degree at most N−1 with such property. But for higher degrees, there are many such polynomials. The blinding process takes a and a desired degree M≥N, and produces a new polynomial ablinded​ of degree exactly M. This new polynomial satisfies that ablinded​(ωi)=a(ωi) for all i. But outside H differs from a. This may seem hard but it's actually very simple. Let zH​ be the polynomial zH​=XN−1. If M=N+k, with k≥0, then sample random values b0​,…,bk​ and define ablinded​:=(b0​+b1​X+⋯+bk​Xk)zH​+a The reason why this does the job is that zH​(ωi)=0 for all i. Therefore the added term vanishes at H and leaves the values of a at H unchanged.","breadcrumbs":"Plonk » Protocol » Blindings","id":"19","title":"Blindings"},"2":{"body":"Three methods of polynomial interpolation were benchmarked, with different input sizes each time: CPU Lagrange: Finding the Lagrange polynomial of a set of random points via a naive algorithm (see math/src/polynomial.rs:interpolate()) CPU FFT: Finding the lowest degree polynomial that interpolates pairs of twiddle factors and Fourier coefficients (the results of applying the Fourier transform to the coefficients of a polynomial) (see math/src/polynomial.rs:interpolate_fft()). GPU FFT (Metal): Same as CPU FFT but the FFT algorithm is run on the GPU via the Metal API (Apple). these were run with criterion-rs in a MacBook Pro M1 (18.3), statistically measuring the total run time of one iteration. The field used was a 256 bit STARK-friendly prime field. All values of time are in milliseconds. Those cases which were greater than 30 seconds were marked respectively as they're too slow and weren't worth to be benchmarked. The input size refers to d + 1 where d is the polynomial's degree (so size is amount of coefficients). Input size CPU Lagrange CPU FFT GPU FFT (Metal) 2^4 2.2 ms 0.2 ms 2.5 ms 2^5 9.6 ms 0.4 ms 2.5 ms 2^6 42.6 ms 0.8 ms 2.5 ms 2^7 200.8 ms 1.7 ms 2.9 ms ... ... .. .. 2^21 >30000 ms 28745 ms 574.2 ms 2^22 >30000 ms >30000 ms 1144.9 ms 2^23 >30000 ms >30000 ms 2340.1 ms 2^24 >30000 ms >30000 ms 4652.9 ms NOTE: Metal FFT execution includes the Metal state setup, twiddle factors generation and a bit-reverse permutation.","breadcrumbs":"FFT Library » Benchmarks » Polynomial interpolation methods comparison","id":"2","title":"Polynomial interpolation methods comparison"},"20":{"body":"This is an optimization in PLONK to reduce the number of checks of the verifier. One of the main checks in PLONK boils down to check that p(ζ)=zH​(ζ)t(ζ), with p some polynomial that looks like p=aqL​+bqR​+abqM​+⋯, and so on. In particular the verifier needs to get the value p(ζ) from somewhere. For the sake of simplicity, in this section assume p is exactly aqL​+bqR​. Secret to the prover here are only a,b. The polynomials qL​ and qR​ are known also to the verifier. The verifier will already have the commitments [a]1​,[b]1​,[qL​]1​ and [qR​]1​. So the prover could send just a(ζ), b(ζ) along with their opening proofs and let the verifier compute by himself qL​(ζ) and qR​(ζ). Then with all these values the verifier could compute p(ζ)=a(ζ)qL​(ζ)+b(ζ)qR​(ζ). And also use his commitments to validate the opening proofs of a(ζ) and b(ζ). This has the problem that computing qL​(ζ) and qR​(ζ) is expensive. The prover can instead save the verifier this by sending also qL​(ζ),qR​(ζ) along with opening proofs. Since the verifier will have the commitments [qL​]1​ and [qR​]1​ beforehand, he can check that the prover is not cheating and cheaply be convinced that the claimed values are actually qL​(ζ) and qR​(ζ). This is much better. It involves the check of four opening proofs and the computation of p(ζ) off the values received from the prover. But it can be further improved as follows. As before, the prover sends a(ζ),b(ζ) along with their opening proofs. She constructs the polynomial f=a(ζ)qL​+b(ζ)qR​. She sends the value f(ζ) along with an opening proof of it. Notice that the value of f(ζ) is exactly p(ζ). The verifier can compute by himself [f]1​ as a(ζ)[qL​]1​+b(ζ)[qR​]1​. The verifier has everything to check all three openings and get convinced that the claimed value f(ζ) is true. And this value is actually p(ζ). So this means no more work for the verifier. And the whole thing got reduced to three openings. This is called the linearization trick. The polynomial f is called the linearization of p.","breadcrumbs":"Plonk » Protocol » Linearization trick","id":"20","title":"Linearization trick"},"21":{"body":"There's a one time setup phase to compute some values common to any execution and proof of the particular circuit. Precisely, the following commitments are computed and published. [qL​]1​,[qR​]1​,[qM​]1​,[qO​]1​,[qC​]1​,[Sσ1​]1​,[Sσ2​]1​,[Sσ3​]1​","breadcrumbs":"Plonk » Protocol » Setup","id":"21","title":"Setup"},"22":{"body":"Next we describe the proving algorithm for a program of size N. That includes public inputs. Let ω be a primitive N-th root of unity. Let H={1,ω,ω2,…,ωN−1}. Define ZH​:=XN−1. Assume the eight polynomials of common preprocessed input are already given. The prover computes the trace matrix T as described in the first sections. That means, with the first rows corresponding to the public inputs. It should be a N×3 matrix.","breadcrumbs":"Plonk » Protocol » Proving algorithm","id":"22","title":"Proving algorithm"},"23":{"body":"Add to the transcript the following: [Sσ1​]1​,[Sσ2​]1​,[Sσ3​]1​,[qL​]1​,[qR​]1​,[qM​]1​,[qO​]1​,[qC​]1​ Compute polynomials a′,b′,c′ as the interpolation polynomials of the columns of T at the domain H. Sample random b1​,b2​,b3​,b4​,b5​,b6​ Let a:=(b1​X+b2​)ZH​+a′ b:=(b3​X+b4​)ZH​+b′ c:=(b5​X+b6​)ZH​+c′ Compute [a]1​,[b]1​,[c]1​ and add them to the transcript.","breadcrumbs":"Plonk » Protocol » Round 1","id":"23","title":"Round 1"},"24":{"body":"Sample β,γ from the transcript. Let z0​=1 and define recursively for 0≤k { pub n: usize, pub domain: Vec>, pub omega: FieldElement, pub k1: FieldElement, pub ql: Polynomial>, pub qr: Polynomial>, pub qo: Polynomial>, pub qm: Polynomial>, pub qc: Polynomial>, pub s1: Polynomial>, pub s2: Polynomial>, pub s3: Polynomial>, pub s1_lagrange: Vec>, pub s2_lagrange: Vec>, pub s3_lagrange: Vec>,\n} Apart from the eight polynomials in the canonical basis, we store also here the number of constraints n, the domain H, the primitive n-th of unity ω and the element k1​. The element k2​ will be k12​. For convenience, we also store the polynomials Sσi​ in Lagrange form. The following lines define the particular values of the program input x and the private input e. // Input\nlet x = FieldElement::from(2_u64); // Private input\nlet e = FieldElement::from(3_u64);\nlet y, witness = test_witness_2(x, e); The function test_witness_2(x, e) returns an instance of the following struct, that holds the polynomials that interpolate the columns A,B,C of the trace matrix T. pub struct Witness { pub a: Vec>, pub b: Vec>, pub c: Vec>,\n} Next the commitment scheme KZG (Kate-Zaverucha-Goldberg) is instantiated. let srs = test_srs(common_preprocessed_input.n);\nlet kzg = KZG::new(srs); The setup function performs the setup phase. It only needs the common preprocessed input and the commitment scheme. let verifying_key = setup(&common_preprocessed_input, &kzg); It outputs an instance of the struct VerificationKey. pub struct VerificationKey { pub qm_1: G1Point, pub ql_1: G1Point, pub qr_1: G1Point, pub qo_1: G1Point, pub qc_1: G1Point, pub s1_1: G1Point, pub s2_1: G1Point, pub s3_1: G1Point,\n} It stores the commitments of the eight polynomials of the common preprocessed input. The suffix _1 means it is a commitment. It comes from the notation [f]1​, where f is a polynomial. Then a prover is instantiated let random_generator = TestRandomFieldGenerator {};\nlet prover = Prover::new(kzg.clone(), random_generator); The prover is an instance of the struct Prover: pub struct Prover\nwhere F: IsField, CS: IsCommitmentScheme, R: IsRandomFieldElementGenerator { commitment_scheme: CS, random_generator: R, phantom: PhantomData,\n} It stores an instance of a commitment scheme and a random field element generator needed for blinding polynomials. Then the public input is defined. As we mentioned in the recap, the public input contains the output of the program. let public_input = vec![x.clone(), y]; We then generate a proof using the prover's method prove let proof = prover.prove( &witness, &public_input, &common_preprocessed_input, &verifying_key,\n); The output is an instance of the struct Proof. pub struct Proof> { // Round 1. /// Commitment to the wire polynomial `a(x)` pub a_1: CS::Commitment, /// Commitment to the wire polynomial `b(x)` pub b_1: CS::Commitment, /// Commitment to the wire polynomial `c(x)` pub c_1: CS::Commitment, // Round 2. /// Commitment to the copy constraints polynomial `z(x)` pub z_1: CS::Commitment, // Round 3. /// Commitment to the low part of the quotient polynomial t(X) pub t_lo_1: CS::Commitment, /// Commitment to the middle part of the quotient polynomial t(X) pub t_mid_1: CS::Commitment, /// Commitment to the high part of the quotient polynomial t(X) pub t_hi_1: CS::Commitment, // Round 4. /// Value of `a(ζ)`. pub a_zeta: FieldElement, /// Value of `b(ζ)`. pub b_zeta: FieldElement, /// Value of `c(ζ)`. pub c_zeta: FieldElement, /// Value of `S_σ1(ζ)`. pub s1_zeta: FieldElement, /// Value of `S_σ2(ζ)`. pub s2_zeta: FieldElement, /// Value of `z(ζω)`. pub z_zeta_omega: FieldElement, // Round 5 /// Value of `p_non_constant(ζ)`. pub p_non_constant_zeta: FieldElement, /// Value of `t(ζ)`. pub t_zeta: FieldElement, /// Batch opening proof for all the evaluations at ζ pub w_zeta_1: CS::Commitment, /// Single opening proof for `z(ζω)`. pub w_zeta_omega_1: CS::Commitment,\n} Finally, we instantiate a verifier. let verifier = Verifier::new(kzg); It's an instance of Verifier: struct Verifier> { commitment_scheme: CS, phantom: PhantomData,\n} Finally, we call the verifier's method verify that outputs a bool. assert!(verifier.verify( &proof, &public_input, &common_preprocessed_input, &verifying_key\n));","breadcrumbs":"Plonk » Implementation » Usage","id":"34","title":"Usage"},"35":{"body":"All the matrices Q,V,T,PI are padded with dummy rows so that their length is a power of two. To be able to interpolate their columns, we need a primitive root of unity ω of that order. Given the particular field used in our implementation, that means that the maximum possible size for a circuit is 232. The entries of the dummy rows are filled in with zeroes in the Q, V and PI matrices. The T matrix needs to be consistent with the V matrix. Therefore it is filled with the value of the variable with index 0. Some other rows in the V matrix have also dummy values. These are the rows corresponding to the B and C columns of the public input rows. In the recap we denoted them with the empty - symbol. They are filled in with the same logic as the padding rows, as well as the corresponding values in the T matrix.","breadcrumbs":"Plonk » Implementation » Padding","id":"35","title":"Padding"},"36":{"body":"The implementation pretty much follows the rounds as are described in the protocol section. There are a few details that are worth mentioning.","breadcrumbs":"Plonk » Implementation » Implementation details","id":"36","title":"Implementation details"},"37":{"body":"The commitment scheme we use is the Kate-Zaverucha-Goldberg scheme with the BLS 12 381 curve and the ate pairing. It can be found in the commitments module of the lambdaworks_crypto package. The order r of the cyclic subgroup is 0x73eda753299d7d483339d80809a1d80553bda402fffe5bfeffffffff00000001 The maximum power of two that divides r−1 is 232. Therefore, that is the maximum possible order for a primitive root of unity in Fr​ with order a power of two.","breadcrumbs":"Plonk » Implementation » Commitment Scheme","id":"37","title":"Commitment Scheme"},"38":{"body":"","breadcrumbs":"Plonk » Implementation » Fiat-Shamir","id":"38","title":"Fiat-Shamir"},"39":{"body":"Here we describe our implementation of the transcript used for the Fiat-Shamir heuristic. A Transcript exposes two methods: append and challenge. The method append adds a message to the transcript by updating the internal state of the hasher with the raw bytes of the message. The method challenge returns the result of the hasher using the current internal state of the hasher. It subsequently resets the hasher and updates the internal state with the last result. Here is an example of this process: Start a fresh transcript. Call append and pass message_1. Call append and pass message_2. The internal state of the hasher at this point is message_2 || message_1. Call challenge. The output is Hash(message_2 || message_1). Call append and pass message_3. Call challenge. The output is Hash(message_3 || Hash(message_2 || message_1)). Call append and pass message_4. The internal state of the hasher at the end of this exercise is message_4 || Hash(message_3 || Hash(message_2 || message_1)) The underlying hasher function we use is h=sha3.","breadcrumbs":"Plonk » Implementation » Transcript strategy","id":"39","title":"Transcript strategy"},"4":{"body":"We use the following notation. The symbol F denotes a finite field. It is fixed all along. The symbol ω denotes a primitive root of unity in F. All polynomials have coefficients in F and the variable is usually denoted by X. We denote polynomials by single letters like p,a,b,z. We only denote them as z(X) when we want to emphasize the fact that it is a polynomial in X, or we need that to explicitly define a polynomial from another one. For example when composing a polynomial z with the polynomial ωX, the result being denoted by z′:=z(ωX). The symbol ′ is not used to denote derivatives. When interpolating at a domain H={h0​,…,hn​}⊂F, the symbols Li​ denote the Lagrange basis. That is Li​ is the polynomial such that Li​(hj​)=0 for all j=i, and that Li​(hi​)=1. If M is a matrix, then Mi,j​ denotes the value at the row i and column j.","breadcrumbs":"Plonk » Recap » Notation","id":"4","title":"Notation"},"40":{"body":"The result of every challenge is a 256-bit string, which is interpreted as an integer in big-endian order. A field element is constructed out of it by taking modulo the field order. The prime field used in this implementation has a 255-bit order. Therefore some field elements are more probable to occur than others because they have more representatives as 256-bit integers.","breadcrumbs":"Plonk » Implementation » Field elements","id":"40","title":"Field elements"},"41":{"body":"The first messages added to the transcript are all commitments of the polynomials of the common preprocessed input and the values of the public inputs. This prevents a known vulnerability called \"weak Fiat-Shamir\". Check out the following resources to learn more about it. What can go wrong (zkdocs) How not to Prove Yourself: Pitfalls of the Fiat-Shamir Heuristic and Applications to Helios Weak Fiat-Shamir Attacks on Modern Proof Systems","breadcrumbs":"Plonk » Implementation » Strong Fiat-Shamir","id":"41","title":"Strong Fiat-Shamir"},"42":{"body":"In this section, we'll discuss how to build your own constraint system to prove the execution of a particular program.","breadcrumbs":"Plonk » Circuit API » Circuit API","id":"42","title":"Circuit API"},"43":{"body":"Let's take the following simple program as an example. We have two public inputs: x and y. We want to prove to a verifier that we know a private input e such that x * e = y. You can achieve this by building the following constraint system: use lambdaworks_plonk::constraint_system::ConstraintSystem;\nuse lambdaworks_math::elliptic_curve::short_weierstrass::curves::bls12_381::default_types::FrField; fn main() { let system = &mut ConstraintSystem::::new(); let x = system.new_public_input(); let y = system.new_public_input(); let e = system.new_variable(); let z = system.mul(&x, &e); // This constraint system asserts that x * e == y system.assert_eq(&y, &z);\n} This code creates a constraint system over the field of the BLS12381 curve. Then, it creates three variables: two public inputs x and y, and a private variable e. Note that every variable is private except for the public inputs. Finally, it adds the constraints that represent a multiplication and an assertion. Before generating proofs for this system, we need to run a setup and obtain a verifying key: let common = CommonPreprocessedInput::from_constraint_system(&system, &ORDER_R_MINUS_1_ROOT_UNITY);\nlet srs = test_srs(common.n);\nlet kzg = KZG::new(srs); // The commitment scheme for plonk.\nlet vk = setup(&common, &kzg); Now we can generate proofs for our system. We just need to specify the public inputs and obtain a witness that is a solution for our constraint system: let inputs = HashMap::from([(x, FieldElement::from(4)), (e, FieldElement::from(3))]);\nlet assignments = system.solve(inputs).unwrap();\nlet witness = Witness::new(assignments, &system); Once you have all these ingredients, you can call the prover: let public_inputs = system.public_input_values(&assignments);\nlet prover = Prover::new(kzg.clone(), TestRandomFieldGenerator {});\nlet proof = prover.prove(&witness, &public_inputs, &common, &vk); and verify: let verifier = Verifier::new(kzg);\nassert!(verifier.verify(&proof, &public_inputs, &common, &vk));","breadcrumbs":"Plonk » Circuit API » Simple Example","id":"43","title":"Simple Example"},"44":{"body":"Some operations are common, and it makes sense to wrap the set of constraints that do these operations in a function and use it several times. Lambdaworks comes with a collection of functions to help you build your own constraint systems, such as conditionals, inverses, and hash functions. However, if you have an operation that does not come with Lambdaworks, you can easily extend Lambdaworks functionality. Suppose that the exponentiation operation is something common in your program. You can write the square and multiply algorithm and put it inside a function: pub fn pow( system: &mut ConstraintSystem, base: Variable, exponent: Variable,\n) -> Variable { let exponent_bits = system.new_u32(&exponent); let mut result = system.new_constant(FieldElement::one()); for i in 0..32 { if i != 0 { result = system.mul(&result, &result); } let result_times_base = system.mul(&result, &base); result = system.if_else(&exponent_bits[i], &result_times_base, &result); } result\n} This function can then be used to modify our simple program from the previous section. The following circuit checks that the prover knows e such that pow(x, e) = y: use lambdaworks_plonk::constraint_system::ConstraintSystem;\nuse lambdaworks_math::elliptic_curve::short_weierstrass::curves::bls12_381::default_types::FrField; fn main() { let system = &mut ConstraintSystem::::new(); let x = system.new_public_input(); let y = system.new_public_input(); let e = system.new_variable(); let z = pow(system, &x, &e); system.assert_eq(&y, &z);\n} You can keep composing these functions in order to create more complex systems.","breadcrumbs":"Plonk » Circuit API » Building Complex Systems","id":"44","title":"Building Complex Systems"},"45":{"body":"The goal of this document is to give a good a understanding of our stark prover code. To this end, in the first section we go through a recap of how the proving system works at a high level mathematically; then we dive into how that's actually implemented in our code.","breadcrumbs":"STARKs » STARK Prover","id":"45","title":"STARK Prover"},"46":{"body":"","breadcrumbs":"STARKs » Recap » STARKs Recap","id":"46","title":"STARKs Recap"},"47":{"body":"In general, we express computation in our proving system by providing an execution trace satisfying certain constraints . The execution trace is a table containing the state of the system at every step of computation. This computation needs to follow certain rules to be valid; these rules are our constraints . The constraints for our computation are expressed using an Algebraic Intermediate Representation or AIR. This representation uses polynomials to encode constraints, which is why sometimes they are called polynomial constraints. To make all this less abstract, let's go through two examples.","breadcrumbs":"STARKs » Recap » Verifying Computation through Polynomials","id":"47","title":"Verifying Computation through Polynomials"},"48":{"body":"Throughout this section and the following we will use this example extensively to have a concrete example. Even though it's a bit contrived (no one cares about computing fibonacci numbers), it's simple enough to be useful. STARKs and proving systems in general are very abstract things; having an example in mind is essential to not get lost. Let's say our computation consists of calculating the k-th number in the fibonacci sequence. This is just the sequence of numbers \\(a_n\\) satisfying \\[ a_0 = 1 \\] \\[ a_1 = 1 \\] \\[ a_{n+2} = a_{n + 1} + a_n \\] An execution trace for this just consists of a table with one column, where each row is the i-th number in the sequence: a_i 1 1 2 3 5 8 13 21 A valid trace for this computation is a table satisfying two things: The first two rows are 1. The value on any other row is the sum of the two preceding ones. The first item is called a boundary constraint, it just enforces specific values on the trace at certain points. The second one is a transition constraint; it tells you how to go from one step of computation to the next.","breadcrumbs":"STARKs » Recap » Fibonacci numbers","id":"48","title":"Fibonacci numbers"},"49":{"body":"The example above is extremely useful to have a mental model, but it's not really useful for anything else. The problem is it just works for the very narrow example of computing fibonacci numbers. If we wanted to prove execution of something else, we would have to write an AIR for it. What we're actually aiming for is an AIR for an entire general purpose Virtual Machine. This way, we can provide proofs of execution for any computation using just one AIR. This is what cairo as a programming language does. Cairo code compiles to the bytecode of a virtual machine with an already defined AIR. The general flow when using cairo is the following: User writes a cairo program. The program is compiled into Cairo's VM bytecode. The VM executes said code and provides an execution trace for it. The trace is passed on to a STARK prover, which creates a proof of correct execution according to Cairo's AIR. The proof is passed to a verifier, who checks that the proof is valid. Ultimately, our goal is to give the tools to write a STARK prover for the cairo VM and do so. However, this is not a good example to start out as it's incredibly complex. The execution trace of a cairo program has around 30 columns, some for general purpose registers, some for other reasons. Cairo's AIR contains a lot of different transition constraints, encoding all the different possible instructions (arithmetic operations, jumps, etc). Use the fibonacci example as your go-to for understanding all the moving parts; keep the Cairo example in mind as the thing we are actually building towards.","breadcrumbs":"STARKs » Recap » Cairo","id":"49","title":"Cairo"},"5":{"body":"","breadcrumbs":"Plonk » Recap » The ideas and components","id":"5","title":"The ideas and components"},"50":{"body":"Below we go through a step by step explanation of a STARK prover. We will assume the trace of the fibonacci sequence mentioned above; it consists of only one column of length \\(2^n\\). In this case, we'll take n=3. The trace looks like this a_i a_0 a_1 a_2 a_3 a_4 a_5 a_6 a_7","breadcrumbs":"STARKs » Recap » Fibonacci step by step walkthrough","id":"50","title":"Fibonacci step by step walkthrough"},"51":{"body":"The first step is to interpolate these values to generate the trace polynomial. This will be a polynomial encoding all the information about the trace. The way we do it is the following: in the finite field we are working in, we take an 8-th primitive root of unity, let's call it g. It being a primitive root means two things: g is an 8-th root of unity, i.e., \\(g^8 = 1\\). Every 8-th root of unity is of the form \\(g^i\\) for some \\(0 \\leq i \\leq 7\\). With g in hand, we take the trace polynomial t to be the one satisfying t(gi)=ai​ From here onwards, we will talk about the validity of the trace in terms of properties that this polynomial must satisfy. We will also implicitly identify a certain power of \\(g\\) with its corresponding trace element, so for example we sometimes think of \\(g^5\\) as \\(a_5\\), the fifth row in the trace, even though technically it's \\(t\\) evaluated in \\(g^5\\) that equals \\(a_5\\). We talked about two different types of constraints the trace must satisfy to be valid. They were: The first two rows are 1. The value on any other row is the sum of the two preceding ones. In terms of t, this translates to \\(t(g^0) = 1\\) and \\(t(g) = 1\\). \\(t(x g^2) - t(xg) - t(x) = 0\\) for all \\(x \\in {g^0, g^1, g^2, g^3, g^4, g^5}\\). This is because multiplying by g is the same as advancing a row in the trace.","breadcrumbs":"STARKs » Recap » Trace polynomial","id":"51","title":"Trace polynomial"},"52":{"body":"To convince the verifier that the trace polynomial satisfies the relationships above, the prover will construct another polynomial that shows that both the boundary and transition constraints are satisfied and commit to it. We call this polynomial the composition polynomial, and usually denote it with \\(H\\). Constructing it involves a lot of different things, so we'll go step by step introducing all the moving parts required.","breadcrumbs":"STARKs » Recap » Composition Polynomial","id":"52","title":"Composition Polynomial"},"53":{"body":"To show that the boundary constraints are satisfied, we construct the boundary polynomial. Recall that our boundary constraints are \\(t(g^0) = t(g) = 1\\). Let's call \\(P\\) the polynomial that interpolates these constraints, that is, \\(P\\) satisfies: P(1)=1P(g)=1 The boundary polynomial \\(B\\) is defined as follows: B(x)=(x−1)(x−g)t(x)−P(x)​ The denominator here is called the boundary zerofier, and it's the polynomial whose roots are the elements of the trace where the boundary constraints must hold. How does \\(B\\) encode the boundary constraints? The idea is that, if the trace satisfies said constraints, then t(1)−P(1)=1−1=0 t(g)−P(g)=1−1=0 so \\(t(x) - P(x)\\) has \\(1\\) and \\(g\\) as roots. Showing these values are roots is the same as showing that \\(B(x)\\) is a polynomial instead of a rational function, and that's why we construct \\(B\\) this way.","breadcrumbs":"STARKs » Recap » Boundary polynomial","id":"53","title":"Boundary polynomial"},"54":{"body":"To convince the verifier that the transition constraints are satisfied, we construct the transition constraint polynomial and call it \\(C(x)\\). It's defined as follows: C(x)=∏i=05​(x−gi)t(xg2)−t(xg)−t(x)​ How does \\(C\\) encode the transition constraints? We mentioned above that these are satisfied if the polynomial in the numerator vanishes in the elements \\({g^0, g^1, g^2, g^3, g^4, g^5}\\). As with \\(B\\), this is the same as showing that \\(C(x)\\) is a polynomial instead of a rational function.","breadcrumbs":"STARKs » Recap » Transition constraint polynomial","id":"54","title":"Transition constraint polynomial"},"55":{"body":"With the boundary and transition constraint polynomials in hand, we build the composition polynomial \\(H\\) as follows: The verifier will sample four numbers \\(\\beta_1, \\beta_2\\) and \\(H\\) will be H(x)=β1​B(x)+β2​C(x) Why not just take \\(H(x) = B(x) + C(x)\\)? The reason for the betas is to make the resulting \\(H\\) be always different and unpredictable for the prover, so they can't precompute stuff beforehand. With what we discussed above, showing that the constraints are satisfied is equivalent to saying that H is a polynomial and not a rational function (we are simplifying things a bit here, but it works for our purposes).","breadcrumbs":"STARKs » Recap » Constructing \\(H\\)","id":"55","title":"Constructing \\(H\\)"},"56":{"body":"To show \\(H\\) is a polynomial we are going to use the FRI protocol, which we treat as a black box. For all we care, a FRI proof will verify if what we committed to is indeed a polynomial. Thus, the prover will provide a FRI commitment to H, and if it passes, the verifier will be convinced that the constraints are satisfied. There is one catch here though: how does the verifier know that FRI was applied to H and not any other polynomial? For this we need to add an additional step to the protocol.","breadcrumbs":"STARKs » Recap » Commiting to \\(H\\)","id":"56","title":"Commiting to \\(H\\)"},"57":{"body":"After commiting to H, the prover needs to show that H was constructed correctly according to the formula above. To do this, it will ask the prover to provide an evaluation of H on some random point z and evaluations of the trace at the points \\(t(z), t(zg)\\) and \\(t(zg^2)\\). Because the boundary and transition constraints are a public part of the protocol, the verifier knows them, and thus the only thing it needs to compute the evaluation \\((z)\\) by itself are the three trace evaluations mentioned above. Because it asked the prover for them, it can check both sides of the equation: H(z)=β1​B(z)+β2​C(z) and be convinced that \\(H\\) was constructed correctly. We are still not done, however, as the prover could have now cheated on the values of the trace or composition polynomial evaluations.","breadcrumbs":"STARKs » Recap » Consistency check","id":"57","title":"Consistency check"},"58":{"body":"There are two things left the prover needs to show to complete the proof: That \\(H\\) effectively is a polynomial, i.e., that the constraints are satisfied. That the evaluations the prover provided on the consistency check were indeed evaluations of the trace polynomial and composition polynomial on the out of domain point z. Earlier we said we would use the FRI protocol to commit to H and show the first item in the list. However, we can slightly modify the polynomial we do FRI on to show both the first and second items at the same time. This new modified polynomial is called the DEEP composition polynomial. We define it as follows: Deep(x)=γ1​x−zH(x)−H(z)​+γ2​x−zt(x)−t(z)​+γ3​x−zgt(x)−t(zg)​+γ4​x−zg2t(x)−t(zg2)​ where the numbers \\(\\gamma_i\\) are randomly sampled by the verifier. The high level idea is the following: If we apply FRI to this polynomial and it verifies, we are simultaneously showing that \\(H\\) is a polynomial and the prover indeed provided H(z) as one of the out of domain evaluations. This is the first summand in Deep(x). The trace evaluations provided by the prover were the correct ones, i.e., they were \\(t(z)\\), \\(t(zg)\\), and \\(t(zg^2)\\). These are the remaining summands of the Deep(x).","breadcrumbs":"STARKs » Recap » Deep Composition Polynomial","id":"58","title":"Deep Composition Polynomial"},"59":{"body":"The prover needs to show that Deep was constructed correctly according to the formula above. To do this, the verifier will ask the prover to provide: An evaluation of H on z and x_0 Evaluations of the trace at the points \\(t(z)\\), \\(t(zg)\\), \\(t(zg^2)\\) and \\(t(x_0)\\) Where z is the same random, out of domain point used in the consistency check of the composition polynomial, and x_0 is a random point that belongs to the trace domain. With the values provided by the prover, the verifier can check both sides of the equation: Deep(x0​)=γ1​x0​−zH(x0​)−H(z)​+γ2​x0​−zt(x0​)−t(z)​+γ3​x0​−zgt(x0​)−t(zg)​+γ4​x0​−zg2t(x0​)−t(zg2)​ The prover also needs to show that the trace evaluation \\(t(x_0)\\) belongs to the trace. To achieve this, it needs to commit the merkle roots of t and the merkle proof of \\(t(x_0)\\).","breadcrumbs":"STARKs » Recap » Consistency check","id":"59","title":"Consistency check"},"6":{"body":"For better clarity, we'll be using the following toy program throughout this recap. INPUT: x PRIVATE INPUT: e OUTPUT: e * x + x - 1 The observer would have noticed that this program could also be written as (e+1)∗x−1, which is more sensible. But the way it is written now serves us to better explain the arithmetization of PLONK. So we'll stick to it. The idea here is that the verifier holds some value x, say x=3. He gives it to the prover. She executes the program using her own chosen value e, and sends the output value, say 8, along with a proof π demonstrating correct execution of the program and obtaining the correct output. In the context of PLONK, both the inputs and outputs of the program are considered public inputs . This may sound odd, but it is because these are the inputs to the verification algorithm. This is the algorithm that takes, in this case, the tuple (3,8,π) and outputs Accept if the toy program was executed with input x=3, some private value e not revealed to the verifier, and out came 8. Otherwise it outputs Reject . PLONK can be used to delegate program executions to untrusted parties, but it can also be used as a proof of knowledge. Our program could be used by a prover to demostrate that she knows the multiplicative inverse of some value x in the finite field without revealing it. She would do it by sending the verifier the tuple (x,0,π), where π is the proof of the execution of our toy program. In our toy example this is pointless because inverting field elements is easily performed by any verifier. But change our program to the following and you get proofs of knowledge of the preimage of SHA256 digests. PRIVATE INPUT: e OUTPUT: SHA256(e) Here there's no input aside from the prover's private input. As we mentioned, the output h of the program is then part of the inputs to the verification algorithm. Which in this case just takes (h,π).","breadcrumbs":"Plonk » Recap » Programs. Our toy example","id":"6","title":"Programs. Our toy example"},"60":{"body":"We summarize below the steps required in a STARK proof for both prover and verifier.","breadcrumbs":"STARKs » Recap » Summary","id":"60","title":"Summary"},"61":{"body":"Compute the trace polynomial t by interpolating the trace column over a set of \\(2^n\\)-th roots of unity \\({g^i : 0 \\leq i < 2^n}\\). Compute the boundary polynomial B. Compute the transition constraint polynomial C. Construct the composition polynomial H from B and C. Sample an out of domain point z and provide the evaluations \\(H(z)\\), \\(t(z)\\), \\(t(zg)\\), and \\(t(zg^2)\\) to the verifier. Sample a domain point x_0 and provide the evaluations \\(H(x_0)\\) and \\(t(x_0)\\) to the verifier. Construct the deep composition polynomial Deep(x) from H, t, and the evaluations from the item above. Do FRI on Deep(x) and provide the resulting FRI commitment to the verifier. Provide the merkle root of t and the merkle proof of \\(t(x_0)\\).","breadcrumbs":"STARKs » Recap » Prover side","id":"61","title":"Prover side"},"62":{"body":"Take the evaluations \\(H(z)\\), \\(H(x_0)\\), \\(t(z)\\), \\(t(zg)\\), \\(t(zg^2)\\) and \\(t(x_0)\\) the prover provided. Reconstruct the evaluations \\(B(z)\\) and \\(C(z)\\) from the trace evaluations we were given. Check that the claimed evaluation \\(H(z)\\) the prover gave us actually satisfies H(z)=β1​B(z)+β2​C(z) Check that the claimed evaluation \\(Deep(x_0)\\) the prover gave us actually satisfies Deep(x0​)=γ1​x0​−zH(x0​)−H(z)​+γ2​x0​−zt(x0​)−t(z)​+γ3​x0​−zgt(x0​)−t(zg)​+γ4​x0​−zg2t(x0​)−t(zg2)​ Using the merkle root and the merkle proof the prover provided, check that \\(t(x_0)\\) belongs to the trace. Take the provided FRI commitment and check that it verifies.","breadcrumbs":"STARKs » Recap » Verifier side","id":"62","title":"Verifier side"},"63":{"body":"The walkthrough above was for the fibonacci example which, because of its simplicity, allowed us to sweep under the rug a few more complexities that we'll have to tackle on the implementation side. They are:","breadcrumbs":"STARKs » Recap » Simplifications and Omissions","id":"63","title":"Simplifications and Omissions"},"64":{"body":"Our trace contained only one column, but in the general setting there can be multiple (the Cairo AIR has around 30). This means there isn't just one trace polynomial, but several; one for each column. This also means there are multiple boundary constraint polynomials. The general idea, however, remains the same. The deep composition polynomial H is now the sum of several terms containing the boundary constraint polynomials \\(B_1(x), \\dots, B_k(x)\\) (one per column), and each \\(B_i\\) is in turn constructed from the \\(i\\)-th trace polynomial \\(t_i(x)\\).","breadcrumbs":"STARKs » Recap » Multiple trace columns","id":"64","title":"Multiple trace columns"},"65":{"body":"Much in the same way, our fibonacci AIR had only one transition constraint, but there could be several. We will therefore have multiple transition constraint polynomials \\(C_1(x), \\dots, C_n(x)\\), each of which encodes a different relationship between rows that must be satisfied. Also, because there are multiple trace columns, a transition constraint can mix different trace polynomials. One such constraint could be C1​(x)=t1​(gx)−t2​(x) which means \"The first column on the next row has to be equal to the second column in the current row\". Again, even though this seems way more complex, the ideas remain the same. The composition polynomial H will now include a term for every \\(C_i(x)\\), and for each one the prover will have to provide out of domain evaluations of the trace polynomials at the appropriate values. In our example above, to perform the consistency check on \\(C_1(x)\\) the prover will have to provide the evaluations \\(t_1(zg)\\) and \\(t_2(z)\\).","breadcrumbs":"STARKs » Recap » Multiple transition constraints","id":"65","title":"Multiple transition constraints"},"66":{"body":"In the actual implementation, we won't commit to \\(H\\), but rather to a decomposition of \\(H\\) into an even term \\(H_1(x)\\) and an odd term \\(H_2(x)\\), which satisfy H(x)=H1​(x2)+xH2​(x2) This way, we don't commit to \\(H\\) but to \\(H_1\\) and \\(H_2\\). This is just an optimization at the code level; once again, the ideas remain exactly the same.","breadcrumbs":"STARKs » Recap » Composition polynomial decomposition","id":"66","title":"Composition polynomial decomposition"},"67":{"body":"We treated FRI as a black box entirely. However, there is one thing we do need to understand about it: low degree extensions. When applying FRI to a polynomial of degree \\(n\\), we need to provide evaluations of it over a domain with more than \\(n\\) points. In our case, the DEEP composition polynomial's degree is around the same as the trace's, which is, at most, \\(2^n - 1\\) (because it interpolates the trace containing \\(2^n\\) points). The domain we are going to choose to evaluate our DEEP polynomial on will be a set of higher roots of unity. In our fibonacci example, we will take a primitive \\(16\\)-th root of unity \\(\\omega\\). As a reminder, this means: \\(\\omega\\) is an \\(16\\)-th root of unity, i.e., \\(\\omega^{16} = 1\\). Every \\(16\\)-th root of unity is of the form \\(\\omega^i\\) for some \\(0 \\leq i \\leq 15\\). Additionally, we also take it so that \\(\\omega\\) satisfies \\(\\omega^2 = g\\) (\\(g\\) being the \\(8\\)-th primitive root of unity we used to construct t). The evaluation of \\(t\\) on the set \\({\\omega^i : 0 \\leq i \\leq 15}\\) is called a low degree extension (LDE) of \\(t\\). Notice this is not a new polynomial, they're evaluations of \\(t\\) on some set of points. Also note that, because \\(\\omega^2 = g\\), the LDE contains all the evaluations of \\(t\\) on the set of powers of \\(g\\). In fact, {t(ω2i):0≤i≤15}={t(gi):0≤i≤7} This will be extremely important when we get to implementation. For our LDE, we chose \\(16\\)-th roots of unity, but we could have chosen any other power of two greater than \\(8\\). In general, this choice is called the blowup factor, so that if the trace has \\(2^n\\) elements, a blowup factor of \\(b\\) means our LDE evaluates over the \\(2^{n} * b\\) roots of unity (\\(b\\) needs to be a power of two). The blowup factor is a parameter of the protocol related to its security.","breadcrumbs":"STARKs » Recap » FRI, low degree extensions and roots of unity","id":"67","title":"FRI, low degree extensions and roots of unity"},"68":{"body":"In this section, we start diving deeper before showing the formal protocol. If you haven't done so, we recommend reading the \"Recap\" section first. At a high level, the protocol works as follows. The starting point is a matrix T that encodes the trace of a valid execution of the program. This matrix needs to be in a particular format so that its correctness is equivalent to checking a finite number of polynomial equations on its rows. Transforming the execution to this matrix is what's called the arithmetization process. Then a single polynomial F is constructed that encodes the set of all the polynomial constraints. The satisfiability of all these constraints is equivalent to F being divisible by some public polynomial G. So the prover constructs H as the quotient F/G called the composition polynomial. Then the verifier chooses a random point z and challenges the prover to reveal the values F(z) and H(z). Then the verifier checks that H(z)=F(z)/G(z), which convinces him that the same relation holds at a level of polynomials and, in consequence, convinces the verifier that the private trace T of the prover is valid. In summary, at a very high level, the STARK protocol can be organized into three major parts: Arithmetization and commitment of execution trace. Construction and commitment of composition polynomial H. Opening of polynomials at random z.","breadcrumbs":"STARKs » Protocol overview » Protocol Overview","id":"68","title":"Protocol Overview"},"69":{"body":"As the Recap mentions, the trace is a table containing the system's state at every step. In this section, we will denote the trace as T. A trace can have several columns to store different aspects or features of a particular state at a specific moment. We will refer to the j-th column as Tj​. You can think of a trace as a matrix T where the entry Tij​ is the j-th element of the i-th state. Most proving systems' primary tool is polynomials over a finite field F. Each column Tj​ of the trace T will be interpreted as evaluations of such a polynomial tj​. Consequently, any information about the states must be encoded somehow as an element in F. To ease notation, we will assume here and in the protocol that the constraints encoding transition rules depend only on a state and the previous one. Everything can be easily generalized to transitions that depend on many preceding states. Then, constraints can be expressed as multivariate polynomials in 2m variables PkT​(X1​,…,Xm​,Y1​,…,Ym​) A transition from state i to state i+1 will be valid if and only if when we plug row i of T in the first m variables and row i+1 in the second m variables of PkT​, we get 0 for all k. In mathematical notation, this is PkT​(Ti,0​,…,Ti,m​,Ti+1,0​,…,Ti+1,m​)=0 for all k These are called transition constraints and check the trace's local properties, where local means relative to specific rows. There is another type of constraint, called boundary constraint , and denoted PjB​. These enforce parts of the trace to take particular values. It is helpful, for example, to verify the initial states. So far, these constraints can only express the local properties of the trace. There are situations where the global properties of the trace need to be checked for consistency. For example, a column may need to take all values in a range but not in any predefined way. Several methods exist to express these global properties as local by adding redundant columns. Usually, they need to involve randomness from the verifier to make sense, and they turn into an interactive protocol called Randomized AIR with Preprocessing .","breadcrumbs":"STARKs » Protocol overview » Arithmetization","id":"69","title":"Arithmetization"},"7":{"body":"This is the process that takes the circuit of a particular program and produces a set of mathematical tools that can be used to generate and verify proofs of execution. The end result will be a set of eight polynomials. To compute them we need first to define two matrices. We call them the Q matrix and the V matrix. The polynomials and the matrices depend only on the program and not on any particular execution of it. So they can be computed once and used for every execution instance. To understand what they are useful for, we need to start from execution traces .","breadcrumbs":"Plonk » Recap » PLONK Arithmetization","id":"7","title":"PLONK Arithmetization"},"70":{"body":"To make interactions possible, a crucial cryptographic primitive is the Polynomial Commitment Scheme. This prevents the prover from changing the polynomials to adjust them to what the verifier expects. Such a scheme consists of the commit and the open protocols. STARK uses a univariate polynomial commitment scheme that internally combines a vector commitment scheme and a protocol called FRI. Let's begin with these two components and see how they build up the polynomial commitment scheme.","breadcrumbs":"STARKs » Protocol overview » Polynomial commitment scheme","id":"70","title":"Polynomial commitment scheme"},"71":{"body":"Given a vector Y=(y0​,…,yM​), commiting to Y means the following. The prover builds a Merkle tree out of it and sends its root to the verifier. The verifier can then ask the prover to reveal, or open , the value of the vector Y at some index i. The prover won't have any choice except to send the correct value. The verifier will expect the corresponding value yi​ and the authentication path to the tree's root to check its authenticity. The authentication path also encodes the vector's position i and its length M. The root of the Merkle tree is said to be the commitment of Y, and we denote it here by [Y].","breadcrumbs":"STARKs » Protocol overview » Vector commitments","id":"71","title":"Vector commitments"},"72":{"body":"In STARKs, all commited vectors are of the form Y=(p(d1​),…,p(dM​)) for some polynomial p and some fixed domain D=(d1​,…,dM​). The domain is always known to the prover and the verifier. It can be proved, as long as M is less than the total number of field elements, that every vector (y0​,…,yM​) is equal to (p(d1​),…,p(dM​)) for a unique polynomial p of degree at most M−1. This is called the Lagrange interpolation theorem. It means, there is a unique polynomial of degree at most M−1 such that p(di​)=yi​ for all i. And M−1 is an upper bound to the degree of p. It could be less. For example, the vector of all ones Y=(1,1,…,1) is the evaluation of the constant polynomial p=1, which has degree 0. Suppose the vector Y=(y1​,…,yM​) is the vector of evaluations of a polynomial p of degree strictly less than M−1. Suppose one party holds the vector Y and another party holds only the commitment [Y] of it. The FRI protocol is an efficient interactive protocol with which the former can convince the latter that the commitment they hold corresponds to the vector of evaluations of a polynomial p of degree strictly less than M. More precisely, the protocol depends on the following parameters Powers of two N=2n and M=2m with 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It is still a work in progress.","breadcrumbs":"Introduction » Introduction","id":"0","title":"Introduction"},"1":{"body":"","breadcrumbs":"FFT Library » Benchmarks » FFT Benchmarks","id":"1","title":"FFT Benchmarks"},"10":{"body":"The claim in the previous section is clearly not an \"if and only if\" statement because the following trace columns do satisfy the equations but do not correspond to a valid execution: A B C 2 3 6 0 0 0 20 - 19 The V matrix encodes the carry of the results from one gate to the right or left operand of a subsequent one. These are called wirings . Like the Q matrix, it's independent of the particular evaluation. It consists of indices for all input and intermediate variables. In this case that matrix is: L R O 0 1 2 2 1 3 3 - 4 Here 0 is the index of e, 1 is the index of x, 2 is the index of u, 3 is the index of v and 4 is the index of the output w. Now we can update the claim to have an \"if and only if\" statement. Claim: Let T be a matrix with columns A,B,C. It correspond to a valid evaluation of the circuit if and only if a) for all i the following equality holds Ai​(QL​)i​+Bi​(QR​)i​+Ai​Bi​QM​+Ci​(QO​)i​+(QC​)i​=0, b) for all i,j,k,l such that Vi,j​=Vk,l​ we have Ti,j​=Tk,l​. So now our malformed example does not pass the second check.","breadcrumbs":"Plonk » Recap » The V matrix","id":"10","title":"The V matrix"},"100":{"body":"The way we specify our fibonacci transition constraint looks like this: fn compute_transition( &self, frame: &air::frame::Frame, _rap_challenges: &Self::RAPChallenges,\n) -> Vec> { let first_row = frame.get_row(0); let second_row = frame.get_row(1); let third_row = frame.get_row(2); vec![third_row[0] - second_row[0] - first_row[0]]\n} It's not completely obvious why this is how we chose to express transition constraints, so let's talk a little about it. What we need to specify in this method is the relationship that has to hold between the current step of computation and the previous ones. For this, we get a Frame as an argument. This is a struct holding the current step (i.e. the current row of the trace) and all previous ones needed to encode our constraint. In our case, this is the current row and the two previous ones. To access rows we use the get_row method. The current step is always the last row (in our case 2), with the others coming before it. In our compute_transition method we get the three rows we need and return third_row[0] - second_row[0] - first_row[0] which is the value that needs to be zero for our constraint to hold. Because we support multiple transition constraints, we actually return a vector with one value per constraint, so the first element holds the first constraint value and so on.","breadcrumbs":"STARKs » Implementation » High Level API » Transition Constraints","id":"100","title":"Transition Constraints"},"101":{"body":"After defining our AIR, we create our specific trace to prove against it. let trace = fibonacci_trace([FE17::new(1), FE17::new(1)], 4); let trace_table = TraceTable { table: trace.clone(), num_cols: 1,\n}; TraceTable is the struct holding execution traces; the num_cols says how many columns the trace has, the table field is a vec holding the actual values of the trace in row-major form, meaning if the trace looks like this | 1 | 2 |\n| 3 | 4 |\n| 5 | 6 | then its corresponding TraceTable is let trace_table = TraceTable { table: vec![1, 2, 3, 4, 5, 6], num_cols: 2,\n}; In our example, fibonacci_trace is just a helper function we use to generate the fibonacci trace with 4 rows and [1, 1] as the first two values.","breadcrumbs":"STARKs » Implementation » High Level API » TraceTable","id":"101","title":"TraceTable"},"102":{"body":"After specifying our constraints and trace, the only thing left to do is provide a few parameters related to the STARK protocol and our AIR. These specify things such as the number of columns of the trace and proof configuration, among others. They are all encapsulated in the AirContext struct, which in our example we instantiate like this: let context = AirContext { options: ProofOptions { blowup_factor: 2, fri_number_of_queries: 1, coset_offset: 3, }, trace_columns: trace_table.n_cols, transition_degrees: vec![1], transition_exemptions: vec![2], transition_offsets: vec![0, 1, 2], num_transition_constraints: 1,\n}; Let's go over each of them: options requires a ProofOptions struct holding specific parameters related to the STARK protocol to be used when proving. They are: The blowup_factor used for the trace LDE extension, a parameter related to the security of the protocol. The number of queries performed by the verifier when doing FRI, also related to security. The offset used for the LDE coset. This depends on the field being used for the STARK proof. trace_columns are the number of columns of the trace, respectively. transition_degrees holds the degree of each transition constraint. transition_exemptions is a Vec which tells us, for each column, the number of rows the transition constraints should not apply, starting from the end of the trace. In the example, the transition constraints won't apply on the last two rows of the trace. transition_offsets holds the indexes that define a frame for our AIR. In our fibonacci case, these are [0, 1, 2] because we need the current row and the two previous one to define our transition constraint. num_transition_constraints simply says how many transition constraints our AIR has.","breadcrumbs":"STARKs » Implementation » High Level API » AIR Context","id":"102","title":"AIR Context"},"103":{"body":"Having defined all of the above, proving our fibonacci example amounts to instantiating the necessary structs and then calling prove passing the trace, public inputs and proof options. We use a simple implementation of a hasher called TestHasher to handle merkle proof building. let proof = prove(&trace_table, &pub_inputs, &proof_options); Verifying is then done by passing the proof of execution along with the same AIR to the verify function. assert!(verify(&proof, &pub_inputs, &proof_options));","breadcrumbs":"STARKs » Implementation » High Level API » Proving execution","id":"103","title":"Proving execution"},"104":{"body":"In this section we go over how a few things in the prove and verify functions are implemented. If you just need to use the prover, then you probably don't need to read this. If you're going through the code to try to understand it, read on. We will once again use the fibonacci example as an ilustration. Recall from the recap that the main steps for the prover and verifier are the following:","breadcrumbs":"STARKs » Implementation » Under the hood » How this works under the hood","id":"104","title":"How this works under the hood"},"105":{"body":"Compute the trace polynomial t by interpolating the trace column over a set of 2n-th roots of unity {gi:0≤i<2n}. Compute the boundary polynomial B. Compute the transition constraint polynomial C. Construct the composition polynomial H from B and C. Sample an out of domain point z and provide the evaluation H(z) and all the necessary trace evaluations to reconstruct it. In the fibonacci case, these are t(z), t(zg), and t(zg2). Sample a domain point x_0 and provide the evaluation H(x0​) and t(x0​). Construct the deep composition polynomial Deep(x) from H, t, and the evaluations from the item above. Do FRI on Deep(x) and provide the resulting FRI commitment to the verifier. Provide the merkle root of t and the merkle proof of t(x0​).","breadcrumbs":"STARKs » Implementation » Under the hood » Prover side","id":"105","title":"Prover side"},"106":{"body":"Take the evaluation H(z) along with the trace evaluations the prover provided. Reconstruct the evaluations B(z) and C(z) from the trace evaluations. Check that the claimed evaluation H(z) the prover gave us actually satisfies H(z)=β1​B(z)+β2​C(z) Take the evaluations H(x0​) and t(x0​). Check that the claimed evaluation Deep(x0​) the prover gave us actually satisfies Deep(x0​)=γ1​x0​−zH(x0​)−H(z)​+γ2​x0​−zt(x0​)−t(z)​+γ3​x0​−zgt(x0​)−t(zg)​+γ4​x0​−zg2t(x0​)−t(zg2)​ Take the provided FRI commitment and check that it verifies. Using the merkle root and the merkle proof the prover provided, check that t(x0​) belongs to the trace. Following along the code in the prove and verify functions, most of it maps pretty well to the steps above. The main things that are not immediately clear are: How we take the constraints defined in the AIR through the compute_transition method and map them to transition constraint polynomials. How we then construct H from them and the boundary constraint polynomials. What the composition polynomial even/odd decomposition is. What an ood frame is. What the transcript is.","breadcrumbs":"STARKs » Implementation » Under the hood » Verifier side","id":"106","title":"Verifier side"},"107":{"body":"This is possibly the most complex part of the code, so what follows is a long explanation for it. In our fibonacci example, after obtaining the trace polynomial t by interpolating, the transition constraint polynomial is C(x)=∏i=05​(x−gi)t(xg2)−t(xg)−t(x)​ On our prove code, if someone passes us a fibonacci AIR like the one we showed above used in one of our tests, we somehow need to construct C(x). However, what we are given is not a polynomial, but rather this method fn compute_transition( &self, frame: &air::frame::Frame, ) -> Vec> { let first_row = frame.get_row(0); let second_row = frame.get_row(1); let third_row = frame.get_row(2); vec![third_row[0] - second_row[0] - first_row[0]]\n} So how do we get to C(x) from this? The answer is interpolation. What the method above is doing is the following: if you pass it a frame that looks like this ​t(x0​)t(x0​g)t(x0​g2)​​ for any given point x0​, it will return the value t(x0​g2)−t(x0​g)−t(x0​) which is the numerator in C(x0​). Using the transition_exemptions field we defined in our AIR, we can also compute evaluations in the denominator, i.e. the zerofier evaluations. This is done under the hood by the transition_divisors() method. The above means that even though we don't explicitly have the polynomial C(x), we can evaluate it on points given an appropriate frame. If we can evaluate it on enough points, we can then interpolate them to recover C(x). This is exactly how we construct both transition constraint polynomials and subsequently the composition polynomial H. The job of evaluating H on enough points so we can then interpolate it is done by the ConstraintEvaluator struct. You'll notice prove does the following let constraint_evaluations = evaluator.evaluate( &lde_trace, &lde_roots_of_unity_coset, &alpha_and_beta_transition_coefficients, &alpha_and_beta_boundary_coefficients,\n); This function call will return the evaluations of the boundary terms βiB​Bi​(x) and constraint terms βiT​Ci​(x) for every i. The constraint_evaluations value returned is a ConstraintEvaluationTable struct, which is nothing more than a big list of evaluations of each polynomial required to construct H. With this in hand, we just call let composition_poly = constraint_evaluations.compute_composition_poly(& lde_roots_of_unity_coset); which simply interpolates the sum of all evaluations to obtain H. Let's go into more detail on how the evaluate method reconstructs C(x) in our fibonacci example. It receives the lde_trace as an argument, which is this: ​t(ω0)t(ω1)…t(ω15)​​ where ω is the primitive root of unity used for the LDE, that is, ω satisfies ω2=g. We need to recover C(x), a polynomial whose degree can't be more than t(x)'s. Because t was built by interpolating 8 points (the trace), we know we can recover C(x) by interpolating it on 16 points. We choose these points to be the LDE roots of unity {ω0,ω,ω2,…,ω15} Remember that to evaluate C(x) on these points, all we need are the evaluations of the polynomial t(xg2)−t(xg)−t(x) as the zerofier ones we can compute easily. These become: t(ω0g2)−t(ω0g)−t(ω0)t(ωg2)−t(ωg)−t(ω)t(ω2g2)−t(ω2g)−t(ω2)⋮t(ω15g2)−t(ω15g)−t(ω15) If we remember that ω2=g, this is t(ω4)−t(ω2)−t(ω0)t(ω5)−t(ω3)−t(ω)t(ω6)−t(ω4)−t(ω2)⋮t(ω3)−t(ω)−t(ω15) and we can compute each evaluation here by calling compute_transition on the appropriate frame built from the lde_trace. Specifically, for the first evaluation we can build the frame: ​t(ω0)t(ω2)t(ω4)​​ Calling compute_transition on this frame gives us the first evaluation. We can get the rest in a similar fashion, which is what this piece of code in the evaluate method does: for (i, d) in lde_domain.iter().enumerate() { let frame = Frame::read_from_trace( lde_trace, i, blowup_factor, &self.air.context().transition_offsets, ) let mut evaluations = self.air.compute_transition(&frame); ...\n} Each iteration builds a frame as above and computes one of the evaluations needed. The rest of the code just adds the zerofier evaluations, along with the alphas and betas. It then also computes boundary polynomial evaluations by explicitly constructing them.","breadcrumbs":"STARKs » Implementation » Under the hood » Reconstructing the transition constraint polynomials","id":"107","title":"Reconstructing the transition constraint polynomials"},"108":{"body":"The verifier employs the same trick to reconstruct the evaluations on the out of domain point Ci​(z) for the consistency check.","breadcrumbs":"STARKs » Implementation » Under the hood » Verifier","id":"108","title":"Verifier"},"109":{"body":"At the end of the recap we talked about how in our code we don't actually commit to H, but rather an even/odd decomposition for it. These are two polynomials H_1 and H_2 that satisfy H(x)=H1​(x2)+xH2​(x2) This all happens on this piece of code let composition_poly = constraint_evaluations.compute_composition_poly(&lde_roots_of_unity_coset); let (composition_poly_even, composition_poly_odd) = composition_poly.even_odd_decomposition(); // Evaluate H_1 and H_2 in z^2.\nlet composition_poly_evaluations = vec![ composition_poly_even.evaluate(&z_squared), composition_poly_odd.evaluate(&z_squared),\n]; After this, we don't really use H anymore, but rather H_1 and H_2. There's not that much to say other than that.","breadcrumbs":"STARKs » Implementation » Under the hood » Even/odd decomposition for H","id":"109","title":"Even/odd decomposition for H"},"11":{"body":"Our matrices are fine now. But they can be optimized. Let's do that to showcase this flexibility of PLONK and also reduce the size of our example. PLONK has the flexibility to construct more sophisticated gates as combinations of the five columns. And therefore the same program can be expressed in multiple ways. In our case all three gates can actually be merged into a single custom gate. The Q matrix ends up being a single row. QL​ QR​ QM​ QO​ QC​ 1 1 1 -1 1 and also the V matrix L R O 0 1 2 The trace matrix for this representation is just A B C 2 3 8 And we check that it satisfies the equation 2×1+3×1+2×3×1+8×(−1)+(−1)=0 Of course, we can't always squash an entire program into a single gate.","breadcrumbs":"Plonk » Recap » Custom gates","id":"11","title":"Custom gates"},"110":{"body":"As part of the consistency check, the prover needs to provide evaluations of the trace polynomials in all the points needed by the verifier to check that H was constructed correctly. In the fibonacci example, these are t(z), t(zg), and t(zg2). In code, the prover passes these evaluations as a Frame, which we call the out of domain (ood) frame. The reason we do this is simple: with the frame in hand, the verifier can reconstruct the evaluations of the constraint polynomials Ci​(z) by calling the compute_transition method on the ood frame and then adding the alphas, betas, and so on, just like we explained in the section above.","breadcrumbs":"STARKs » Implementation » Under the hood » Out of Domain Frame","id":"110","title":"Out of Domain Frame"},"111":{"body":"Throughout the protocol, there are a number of times where the verifier randomly samples some values that the prover needs to use (think of the alphas and betas used when constructing H). Because we don't actually have an interaction between prover and verifier, we emulate it by using a hash function, which we assume is a source of randomness the prover can't control. The job of providing these samples for both prover and verifier is done by the Transcript struct, which you can think of as a stateful rng; whenever you call challenge() on a transcript you get a random value and the internal state gets mutated, so the next time you call challenge() you get a different one. You can also call append on it to mutate its internal state yourself. This is done a number of times throughout the protocol to keep the prover honest so it can't predict or manipulate the outcome of challenge(). Notice that to sample the same values, both prover and verifier need to call challenge and append in the same order (and with the same values in the case of append) and the same number of times. The idea explained above is called the Fiat-Shamir heuristic or just Fiat-Shamir, and is more generally used throughout proving systems to remove interaction between prover and verifier. Though the concept is very simple, getting it right so the prover can't cheat is not, but we won't go into that here.","breadcrumbs":"STARKs » Implementation » Under the hood » Transcript","id":"111","title":"Transcript"},"112":{"body":"The generated proof has got all the information needed for the verifier to verify it: Trace length: The number of rows of the trace table, needed to know the max degree of the polynomials that appear in the system. LDE trace commitments. DEEP composition polynomial out of domain even and odd evaluations. DEEP composition polynomial root. FRI layers merkle roots. FRI last layer value. Query list. DEEP composition poly openings. Nonce: Proof of work setting used to generate the proof.","breadcrumbs":"STARKs » Implementation » Under the hood » Proof","id":"112","title":"Proof"},"113":{"body":"","breadcrumbs":"STARKs » Implementation » Under the hood » Special considerations","id":"113","title":"Special considerations"},"114":{"body":"When evaluating or interpolating a polynomial, if the input (be it coefficients or evaluations) size isn't a power of two then the FFT API will extend it with zero padding until this requirement is met. This is because the library currently only uses a radix-2 FFT algorithm. Also, right now FFT only supports inputs with a size up to 2264 elements.","breadcrumbs":"STARKs » Implementation » Under the hood » FFT evaluation and interpolation","id":"114","title":"FFT evaluation and interpolation"},"115":{"body":"","breadcrumbs":"STARKs » Implementation » Under the hood » Other","id":"115","title":"Other"},"116":{"body":"Whenever we interpolate or evaluate trace, boundary and constraint polynomials, we use some 2n-th roots of unity. There are a few reasons for this: Using roots of unity means we can use the Fast Fourier Transform and its inverse to evaluate and interpolate polynomials. This method is much faster than the naive Lagrange interpolation one. Since a huge part of the STARK protocol involves both evaluating and interpolating, this is a huge performance improvement. When computing boundary and constraint polynomials, we divide them by their zerofiers, polynomials that vanish on a few points (the trace elements where the constraints do not hold). These polynomials take the form Z(X)=∏(X−xi​) where the xi​ are the points where we want it to vanish. When implementing this, evaluating this polynomial can be very expensive as it involves a huge product. However, if we are using roots of unity, we can use the following trick. The vanishing polynomial for all the 2n roots of unity is X2n−1 Instead of expressing the zerofier as a product of the places where it should vanish, we express it as the vanishing polynomial above divided by the exemptions polynomial; the polynomial whose roots are the places where constraints don't need to hold. Z(X)=∏(X−ei​)X2n−1​ where the ei​ are now the points where we don't want it to vanish. This exemptions polynomial in the denominator is usually much smaller, and because the vanishing polynomial in the numerator is only two terms, evaluating it is really fast.","breadcrumbs":"STARKs » Implementation » Under the hood » Why use roots of unity?","id":"116","title":"Why use roots of unity?"},"117":{"body":"The n-th roots of unity are the numbers x that satisfy xn=1 There are n such numbers, because they are the roots of the polynomial Xn−1. The set of n-th roots of unity always has a generator, a root g that can be used to obtain every other root of unity by exponentiating. What this means is that the set of n-th roots of unity is {gi:0≤i1 and 0≤j≤J and I=∣Lj′′​∣. Extend M with additional L′′ rows to obtain a matrix M∈F(L+L′+L′′)×33 by appending copies of R′′ at the bottom. Pad M with copies of its last row until it has a power of two number of rows. As a result we obtain a matrix T∈F2n×33.","breadcrumbs":"Cairo » Trace Formal Description » Construction of execution trace T:","id":"121","title":"Construction of execution trace T:"},"122":{"body":"","breadcrumbs":"Cairo » Trace Detailed Description » Cairo execution trace","id":"122","title":"Cairo execution trace"},"123":{"body":"After the execution of a Cairo program in the Cairo VM, three files are generated that are the core components for the construction of the execution trace, needed for the proving system: trace file : Has the information on the state of the three Cairo VM registers ap, fp, and pc at every cycle of the execution of the program. To reduce ambiguity in terms, we should call these the register states of the Cairo VM, and leave the term trace to the final product that is passed to the prover to generate a proof. memory file : A file with the information of the VM's memory at the end of the program run, after the memory has been relocated. public inputs : A file with all the information that must be publicly available to the prover and verifier, such as the total number of execution steps, public memory, used builtins and their respective addresses range in memory. The next section will explain in detail how these elements are used to build the final execution trace.","breadcrumbs":"Cairo » Trace Detailed Description » Raw materials","id":"123","title":"Raw materials"},"124":{"body":"The execution trace is built in two stages. In the first one, the information on the files described in the previous section is aggregated to build a main trace table. In the second stage, there is an interaction with the verifier to add some extension columns to the main trace.","breadcrumbs":"Cairo » Trace Detailed Description » Construction details","id":"124","title":"Construction details"},"125":{"body":"The layout of the main execution trace is as follows: A. flags (16): Decoded instruction flags B. res (1): Res value C. pointers (2): Temporary memory pointers (ap and fp) D. mem_a (4): Memory addresses (pc, dst_addr, op0_addr, op1_addr) E. mem_v (4): Memory values (inst, dst, op0, op1) F. offsets (3) : (off_dst, off_op0, off_op1) G. derived (3) : (t0, t1, mul) A B C D E F G\n|xxxxxxxxxxxxxxxx|x|xx|xxxx|xxxx|xxx|xxx| Each letter from A to G represents some subsection of columns, and the number specifies how many columns correspond to that subsection. Cairo instructions It is important to have in mind the information that each executed Cairo instruction holds, since it is a key component of the construction of the execution trace. For a detailed explanation of how the building components of the instruction interact to change the VM state, refer to the Cairo whitepaper, sections 4.4 and 4.5. Structure of the 63-bit that forms the first word of each instruction: ┌─────────────────────────────────────────────────────────────────────────┐ │ off_dst (biased representation) │ ├─────────────────────────────────────────────────────────────────────────┤ │ off_op0 (biased representation) │ ├─────────────────────────────────────────────────────────────────────────┤ │ off_op1 (biased representation) │ ├─────┬─────┬───────┬───────┬───────────┬────────┬───────────────────┬────┤ │ dst │ op0 │ op1 │ res │ pc │ ap │ opcode │ 0 │ │ reg │ reg │ src │ logic │ update │ update │ │ │ ├─────┼─────┼───┬───┼───┬───┼───┬───┬───┼───┬────┼────┬────┬────┬────┼────┤ │ 0 │ 1 │ 2 │ 3 │ 4 │ 5 │ 6 │ 7 │ 8 │ 9 │ 10 │ 11 │ 12 │ 13 │ 14 │ 15 │ └─────┴─────┴───┴───┴───┴───┴───┴───┴───┴───┴────┴────┴────┴────┴────┴────┘ Columns The construction of the following columns corresponds to a colloquial explanation of what is done in the build_cairo_execution_trace function. Section A - Flags The flags section A corresponds to the 16 bits that represent the configuration of the dst_reg, op0_reg, op1_src, res_logic, pc_update, ap_update and opcode flags, as well as the zero flag. So there is one column for each bit of the flags decomposition. Section C - Temporary memory pointers The two columns in this section, as well as the pc column from section D , are the most trivial. For each step of the register states, the corresponding values are added to the columns, which are pointers to some memory cell in the VM's memory. Section D - Memory addresses As already mentioned, the first column of this section, pc, is trivially obtained from the register states for each cycle. Columns dst_addr, op0_addr, and op1_addr from section D are addresses constructed from pointers stored at ap or fp, and their respective offsets off_dst, off_op0 and off_op1. The exact way these are computed depends on the particular values of the flags for each instruction. Section E - Memory values The inst column, is obtained by fetching in memory the value stored at pointer pc, which corresponds to a 63-bit Cairo instruction. Columns dst, op0, and op1 are computed by fetching in memory by their respective addresses. Section F - Offsets/Range-checked values These columns represent integer values that are used to construct addresses dst_addr, op0_addr and op1_addr and are decoded directly from the instruction. These values have the property to be numbered in the range from 0 to 2^16. Section B - Res This column is computed depending on the decoded opcode and res_logic of every instruction. In some cases, res is unused in the instruction, and the value for (dst)^(-1) is used in that place as an optimization. Section G - Derived To have constraints of max degree two, some more columns are derived from the already calculated, t0, t1, and mul: t0 is the product of the values of DST and the PC_JNZ flag for each step. t1 is the product of t0 and res for each step. mul is the product of op0 and op1 for each step. Range check and Memory holes For the values constrained between ranges 0 and 216, the offsets, the prover uses a permutation argument to optimize enforcing this. In particular, it checks an ordered list with the offsets are the same as the original one, is continuous, the first value is rcmin​, and the last one is less than rcmax​. Since not all values are used, there may be unused values, and so the ordered offset may not be continuous. These unused values are called holes, and they need to be filled with the missing values, so the checks can be done. This is explained in section 9.9 of the Cairo Paper In the case of memory, something similar happens, where the values should be continuous, but if there are built-ins, this may not be the case. For example, the built-in may be using addresses in ranges higher than the ones used by the program. To fix this, holes in the memory cells are filled, just like the ones of the RC. It's something important to note that when filling the holes, we can't use dedicated columns like op0_addr, since this would break the constraints. For this to work, we either need new columns for holes, or make use of subcolumns, which are explained in their dedicated section. No matter which approach is used , either by subcolumns or columns, we will need cells where the constraints of the range check and memory are applied, but not the specific ones related to the instructions. Finally, using these columns, we can fill the holes without breaking the constraint system. Dummy memory accesses As part of proving the execution of a Cairo program, we need to prove that memory used by the program extends the public memory. This is important since public memory contains for example the bytecode that was executed and the outputs. The bytecode is something critical for the verifier to not only know that something was executed correctly but to know what was executed. To do this, we the permutation check, which that proves the memory is continuous and single-valued To do this, the permutation check of the memory is modified. Initially, we had M, V and M′, V′ pairs of memory addresses and values, where M and V are the values as used and without order, and M′, V′ the pairs ordered by address. For the public memory, we will add it directly to M′, V′. We also need to add dummy accesses to the pairs M V. These dummy accesses are just pairs of (M,V)=(0,0). This change makes the statement that (M′,V′) is a permutation of (M,V), which means that they are the same values in a different order, no longer true. This means that the two cumulative products used to check the statement are no longer equal, and their division Luckily, we can know the new expected value on the verifier, since we have access to the public memory. Even more, it's this fact that enables the verifier to check the memory is an extension of the public memory. The math is quite straightforward. When the memory was the same, we expected a final value pn−1​=1. This came from two cumulative products that should equal, one from the unordered pairs, and one from the ordered ones. Now, adding zeros to one side and the real values to the other unbalances the cumulative products, so the verifier will need to balance it by dividing pn−1​ by the extra factors that appeared with the addition of the public memory. Doing so will make the final value 1 again. Since they only depend on the public memory, the verifier has enough data to recalculate them and use them. Even more, if the prover lies, the equality that the verifier is expecting won't hold. In reality, instead of dividing and expecting the result to equal to 1, we can just check the equality against the new expected value, and avoid doing that inversion. All of this is explained in section 9.8 of the Cairo paper. Trace extension / Padding The last step is padding the trace to a power of two for efficiency. We may also need to pad the trace if for some reason some unbalance is given by the layout. For this, we will copy the last executed instruction until reaching the desired length. But there's a trick. If the last executed instruction is any instruction, and it's copied, the transition constraints won't be satisfied. To be able to do this, we need to use something called \"proof mode\". In proof mode, the main function of the program is wrapped in another one, which calls it and returns to an infinite loop. This loop is a jump relative 0. Since this loop can be executed many times without changing the validity of the trace, it can be copied as many times as needed, solving the issues mentioned before. Summary To construct the execution trace, we augment the RegisterStates with the information obtained from the Memory. This includes decoding instruction of each steps, and writing all the data needed to check the execution is valid. Additionally, memory holes have to be filled, public memory added, and a final pad using an infinite loop is needed for everything to work properly. Adding all of that, we create an execution trace that's ready for the prover to generate a proof.","breadcrumbs":"Cairo » Trace Detailed Description » Main trace construction","id":"125","title":"Main trace construction"},"126":{"body":"The verifier sends challenges α,z∈F (or the prover samples them from the transcript). Additional columns are added to incorporate the memory constraints. To define them the prover follows these steps: Stack the rows of the submatrix of T defined by the columns pc, dst_addr, op0_addr, op1_addr into a vector a of length 2n+2 (this means that the first entries of a are pc[0], dst_addr[0], op0_addr[0], op1_addr[0], pc[1], dst_addr[1],...). Stack the the rows of the submatrix defined by the columns inst, dst, op0, op1 into a vector v of length 2n+2. Define MMem​∈F2n+2×2 to be the matrix with columns a, v. Define MMemRepl​∈F2n+2×2 to be the matrix that's equal to MMem​ in the first 2n+2−Lpub​ rows, and its last Lpub​ entries are the addresses and values of the actual public memory (program code). Sort MMemRepl​ by the first column in increasing order. The result is a matrix MMemReplSorted​ of size 2n+2×2. Denote its columns by a′ and v′. Compute the vector p of size 2n+2 with entries pi​:=j=0∏i​z−(ai​+αvi​)z−(ai′​+αvi′​)​ Reshape the matrix MMemReplSorted​ into a 2n×8 in row-major. Reshape the vector p into a 2n×4 matrix in row-major. Concatenate these 12 rows. The result is a matrix MMemRAP2​ of size 2n×12 The verifier sends challenge z′∈F. Further columns are added to incorporate the range check constraints following these steps: Stack the rows of the submatrix of T defined by the columns in the group offsets into a vector b of length 3⋅2n. Sort the values of b in increasing order. Let b′ be the result. Compute the vector p′ of size 3⋅2n with entries pi′​:=j=0∏i​z′−bi​z′−bi′​​ Reshape b′ and p′ into matrices of size 2n×3 each and concatenate them into a matrix MRangeCheckRAP2​ of size 2n×6. Concatenate MMemRAP2​ and MRangeCheckRAP2​ into a matrix MRAP2​ of size 2n×18. Using the notation described at the beginning, m′=33, m′′=18 and m=52. They are respectively the columns of the first and second part of the rap, and the total number of columns. Putting all together, the final layout of the trace is the following A. flags (16) : Decoded instruction flags B. res (1) : Res value C. pointers (2) : Temporary memory pointers (ap and fp) D. mem_a (4) : Memory addresses (pc, dst_addr, op0_addr, op1_addr) E. mem_v (4) : Memory values (inst, dst, op0, op1) F. offsets (3) : (off_dst, off_op0, off_op1) G. derived (3) : (t0, t1, mul) H. mem_a' (4) : Sorted memory addresses I. mem_v' (4) : Sorted memory values J. mem_p (4) : Memory permutation argument columns K. offsets_b' (3) : Sorted offset columns L. offsets_p' (3) : Range check permutation argument columns A B C D E F G H I J K L\n|xxxxxxxxxxxxxxxx|x|xx|xxxx|xxxx|xxx|xxx|xxxx|xxxx|xxxx|xxx|xxx|","breadcrumbs":"Cairo » RAP » Extended columns","id":"126","title":"Extended columns"},"127":{"body":"","breadcrumbs":"Cairo » Virtual Columns » Virtual columns and Subcolumns","id":"127","title":"Virtual columns and Subcolumns"},"128":{"body":"In previous chapters, we have seen how the registers states and the memory are augmented to generate a provable trace. While we have shown a way of doing that, there isn't only one possible provable trace. In fact, there are multiple configurations possible. For example, in the Cairo VM, we have 15 flags. These flags include \"DstReg\", \"Op0Reg\", \"OpCode\" and others. For simplification, let's imagine we have 3 flags with letters from \"A\" to \"C\", where \"A\" is the first flag. Now, let's assume we have 4 steps in our trace. If we were to only use plain columns, the layout would look like this: FlagA FlagB FlagB A0 B0 C0 A1 B1 C1 A2 B2 C2 A3 B3 C3 But, we could also organize them like this Flags A0 B0 C0 A1 B1 C1 A2 B2 C2 A3 B3 C3 The only problem is that now the constraints for each transition of the rows are not the same. We will have to define then a concept called \"Virtual Column\". A Virtual Column is like a traditional column, which has its own set of constraints, but it exists interleaved with another one. In the previous example, each row is associated with a column, but in practice, we could have different ratios. We could have 3 rows corresponding to one Virtual Column, and the next one corresponding to another one. For the time being, let's focus on this simpler example. Each row corresponding to Flag A will have the constraints associated with its own Virtual Column, and the same will apply to Flag B and Flag C. Now, to do this, we will need to evaluate the multiple rows taking into account that they are part of the same step. For a real case, we will add a dummy flag D, whose purpose is to make the evaluation move in a number that is a power of 2. Let's see how it works. If we were evaluating the Frame where the constraints should give 0, the frame movement would look like this: + A0 | B0 | C0\n+ A1 | B1 | C1 A2 | B2 | C2 A3 | B3 | C3 A0 | B0 | C0\n+ A1 | B1 | C1\n+ A2 | B2 | C2 A3 | B3 | C3 A0 | B0 | C0 A1 | B1 | C1\n+ A2 | B2 | C2\n+ A3 | B3 | C3 In the second case, the evaluation would look like this: + A0 |\n+ B0 |\n+ C0 |\n+ D0 |\n+ A1 |\n+ B1 |\n+ C1 |\n+ D1 | A2 | B2 | C2 | D2 | A3 | B3 | C3 | D3 | A0 | B0 | C0 | D0 |\n+ A1 |\n+ B1 |\n+ C1 |\n+ D1 |\n+ A2 |\n+ B2 |\n+ C2 |\n+ D2 | A3 | B3 | C3 | D3 | A0 | B0 | C0 | D0 | A1 | B1 | C1 | D1 |\n+ A2 |\n+ B2 |\n+ C2 |\n+ D2 |\n+ A3 |\n+ B3 |\n+ C3 |\n+ D3 | When evaluating the composition polynomial, we will do it over the points on the LDE, where the constraints won't evaluate to 0, but we will use the same spacing. Assume we have three constraints for each flag, C0​, C1​, and C2​, and that they don't involve other trace cells. Let's call the index of the frame evaluation i, starting from 0. In the first case, the constraint C0​, C1​ and C2​ would be applied over the same rows, giving an equation that looks like this: ‘Ck​(wi,wi+1)‘ In the second case, the equations would look like: ‘C0​(wi∗4,wi∗4+4)‘ ‘C1​(wi∗4+1,wi∗4+5)‘ ‘C2​(wi∗4+2,wi∗4+6)‘","breadcrumbs":"Cairo » Virtual Columns » Virtual Columns","id":"128","title":"Virtual Columns"},"129":{"body":"Assume now we have 3 columns that share some constraints. For example, let's have three flags that can be either 0 or 1. Each flag will also have its own set of dedicated constraints. Let's denote the shared constraint B, and the independent constraints Ci​. What we can do is define a Column for the flags, where the binary constraint B is enforced. Additionally, we will define a subcolumn for each flag, which will enforce each Ci​. In summary, if we have a set of shared constraints to apply, we will be using a Column. If we want to mix or interleave Columns, we will define them as Virtual Columns. And if we want to apply more constraints to a subset of a Column of Virtual Columns, or share constraints between columns, we will define Virtual Subcolumns.","breadcrumbs":"Cairo » Virtual Columns » Virtual Subcolumns","id":"129","title":"Virtual Subcolumns"},"13":{"body":"In the previous section we showed how the arithmetization process works in PLONK. For a program with n public inputs and m gates, we constructed two matrices Q and V, of sizes (n+m+1)×5 and (n+m+1)×3 that satisfy the following. Let N=n+m+1. Claim: Let T be a N×3 matrix with columns A,B,C and PI a N×1 matrix. They correspond to a valid execution instance with public input given by PI if and only if a) for all i the following equality holds Ai​(QL​)i​+Bi​(QR​)i​+Ai​Bi​QM​+Ci​(QO​)i​+(QC​)i​+(PI)i​=0, b) for all i,j,k,l such that Vi,j​=Vk,l​ we have Ti,j​=Tk,l​, c) (PI)i​=0 for all i>n. Polynomials enter now to squash most of these equations. We will traduce the set of all equations in conditions (a) and (b) to just a few equations on polynomials. Let ω be a primitive N-th root of unity and let H=ωi:0≤in. Then we constructed polynomials qL​,qR​,qM​,qO​,qC​,Sσ1​,Sσ2​,Sσ3​, f, g off the matrices Q and V. They are the result of interpolating at a domain H={1,ω,ω2,…,ωN−1} for some N-th primitive root of unity and a few random values. And also constructed polynomials a,b,c,pi off the matrices T and PI. Loosely speaking, the above fact can be reformulated in terms of polynomial equations as follows. Fact: Let zH​=XN−1. Let T be a N×3 matrix with columns A,B,C and PI a N×1 matrix. They correspond to a valid execution instance with public input given by PI if and only if a) There is a polynomial t1​ such that the following equality holds aqL​+bqR​+abqM​+cqO​+qC​+pi=zH​t1​, b) There are polynomials t2​,t3​, z such that zf−gz′=zH​t2​ and (z−1)L1​=zH​t3​, where z′(X)=z(Xω) You might be wondering where the polynomials ti​ came from. Recall that for a polynomial F, we have F(h)=0 for all h∈H if and only if F=zH​t for some polynomial t. Finally both conditions (a) and (b) are equivalent to a single equation (c) if we let more randomness to come into play. This is: (c) Let α be a random field element. There is a polynomial t such that zH​t=​aqL​+bqR​+abqM​+cqO​+qC​+pi+α(gz′−fz)+α2(z−1)L1​​ This last step is not obvious. You can check the paper to see the proof. Anyway, this is the equation you'll recognize below in the description of the protocol. Randomness is a delicate matter and an important part of the protocol is where it comes from, who chooses it and when they choose it. Check out the protocol to see how it works.","breadcrumbs":"Plonk » Recap » Wrapping up the whole thing","id":"15","title":"Wrapping up the whole thing"},"16":{"body":"","breadcrumbs":"Plonk » Protocol » Protocol","id":"16","title":"Protocol"},"17":{"body":"","breadcrumbs":"Plonk » Protocol » Details and tricks","id":"17","title":"Details and tricks"},"18":{"body":"A polynomial commitment scheme (PCS) is a cryptographic tool that allows one party to commit to a polynomial, and later prove properties of that polynomial. This commitment polynomial hides the original polynomial's coefficients and can be publicly shared without revealing any information about the original polynomial. Later, the party can use the commitment to prove certain properties of the polynomial, such as that it satisfies certain constraints or that it evaluates to a certain value at a specific point. In the implementation section we'll explain the inner workings of the Kate-Zaverucha-Goldberg scheme, a popular PCS chosen in Lambdaworks for PLONK. For the moment we only need the following about it: It consists of a finite group G and the following algorithms: Commit(f) : This algorithm takes a polynomial f and produces an element of the group G. It is called the commitment of f and is denoted by [f]1​. It is homomorphic in the sense that [f+g]1​=[f]1​+[g]1​. The former sum being addition of polynomials. The latter is addition in the group G. Open(f,ζ) : It takes a polynomial f and a field element ζ and produces an element π of the group G. This element is called an opening proof for f(ζ). It is the proof that f evaluated at ζ gives f(ζ). Verify([f]1​, π, ζ, y) : It takes group elements [f]1​ and π, and also field elements ζ and y. With overwhelming probability it outputs Accept if f(z)=y and Reject otherwise.","breadcrumbs":"Plonk » Protocol » Polynomial commitment scheme","id":"18","title":"Polynomial commitment scheme"},"19":{"body":"As you will see in the protocol, the prover reveals the value taken by a bunch of the polynomials at a random ζ. In order for the protocol to be Honest Verifier Zero Knowledge , these polynomials need to be blinded . This is a process that makes the values of these polynomials at ζ seemingly random by forcing them to be of certain degree. Here's how it works. Let's take for example the polynomial a the prover constructs. This is the interpolation of the first column of the trace matrix T at the domain H. This matrix has all of the left operands of all the gates. The prover wishes to keep them secret. Say the trace matrix T has N rows. And so H is {1,ω,ω2,…,ωN−1}. The invariant that the prover cannot violate is that ablinded​(ωi) must take the value T0,i​, for all i. This is what the interpolation polynomial a satisfies. And is the unique such polynomial of degree at most N−1 with such property. But for higher degrees, there are many such polynomials. The blinding process takes a and a desired degree M≥N, and produces a new polynomial ablinded​ of degree exactly M. This new polynomial satisfies that ablinded​(ωi)=a(ωi) for all i. But outside H differs from a. This may seem hard but it's actually very simple. Let zH​ be the polynomial zH​=XN−1. If M=N+k, with k≥0, then sample random values b0​,…,bk​ and define ablinded​:=(b0​+b1​X+⋯+bk​Xk)zH​+a The reason why this does the job is that zH​(ωi)=0 for all i. Therefore the added term vanishes at H and leaves the values of a at H unchanged.","breadcrumbs":"Plonk » Protocol » Blindings","id":"19","title":"Blindings"},"2":{"body":"Three methods of polynomial interpolation were benchmarked, with different input sizes each time: CPU Lagrange: Finding the Lagrange polynomial of a set of random points via a naive algorithm (see math/src/polynomial.rs:interpolate()) CPU FFT: Finding the lowest degree polynomial that interpolates pairs of twiddle factors and Fourier coefficients (the results of applying the Fourier transform to the coefficients of a polynomial) (see math/src/polynomial.rs:interpolate_fft()). GPU FFT (Metal): Same as CPU FFT but the FFT algorithm is run on the GPU via the Metal API (Apple). these were run with criterion-rs in a MacBook Pro M1 (18.3), statistically measuring the total run time of one iteration. The field used was a 256 bit STARK-friendly prime field. All values of time are in milliseconds. Those cases which were greater than 30 seconds were marked respectively as they're too slow and weren't worth to be benchmarked. The input size refers to d + 1 where d is the polynomial's degree (so size is amount of coefficients). Input size CPU Lagrange CPU FFT GPU FFT (Metal) 2^4 2.2 ms 0.2 ms 2.5 ms 2^5 9.6 ms 0.4 ms 2.5 ms 2^6 42.6 ms 0.8 ms 2.5 ms 2^7 200.8 ms 1.7 ms 2.9 ms ... ... .. .. 2^21 >30000 ms 28745 ms 574.2 ms 2^22 >30000 ms >30000 ms 1144.9 ms 2^23 >30000 ms >30000 ms 2340.1 ms 2^24 >30000 ms >30000 ms 4652.9 ms NOTE: Metal FFT execution includes the Metal state setup, twiddle factors generation and a bit-reverse permutation.","breadcrumbs":"FFT Library » Benchmarks » Polynomial interpolation methods comparison","id":"2","title":"Polynomial interpolation methods comparison"},"20":{"body":"This is an optimization in PLONK to reduce the number of checks of the verifier. One of the main checks in PLONK boils down to check that p(ζ)=zH​(ζ)t(ζ), with p some polynomial that looks like p=aqL​+bqR​+abqM​+⋯, and so on. In particular the verifier needs to get the value p(ζ) from somewhere. For the sake of simplicity, in this section assume p is exactly aqL​+bqR​. Secret to the prover here are only a,b. The polynomials qL​ and qR​ are known also to the verifier. The verifier will already have the commitments [a]1​,[b]1​,[qL​]1​ and [qR​]1​. So the prover could send just a(ζ), b(ζ) along with their opening proofs and let the verifier compute by himself qL​(ζ) and qR​(ζ). Then with all these values the verifier could compute p(ζ)=a(ζ)qL​(ζ)+b(ζ)qR​(ζ). And also use his commitments to validate the opening proofs of a(ζ) and b(ζ). This has the problem that computing qL​(ζ) and qR​(ζ) is expensive. The prover can instead save the verifier this by sending also qL​(ζ),qR​(ζ) along with opening proofs. Since the verifier will have the commitments [qL​]1​ and [qR​]1​ beforehand, he can check that the prover is not cheating and cheaply be convinced that the claimed values are actually qL​(ζ) and qR​(ζ). This is much better. It involves the check of four opening proofs and the computation of p(ζ) off the values received from the prover. But it can be further improved as follows. As before, the prover sends a(ζ),b(ζ) along with their opening proofs. She constructs the polynomial f=a(ζ)qL​+b(ζ)qR​. She sends the value f(ζ) along with an opening proof of it. Notice that the value of f(ζ) is exactly p(ζ). The verifier can compute by himself [f]1​ as a(ζ)[qL​]1​+b(ζ)[qR​]1​. The verifier has everything to check all three openings and get convinced that the claimed value f(ζ) is true. And this value is actually p(ζ). So this means no more work for the verifier. And the whole thing got reduced to three openings. This is called the linearization trick. The polynomial f is called the linearization of p.","breadcrumbs":"Plonk » Protocol » Linearization trick","id":"20","title":"Linearization trick"},"21":{"body":"There's a one time setup phase to compute some values common to any execution and proof of the particular circuit. Precisely, the following commitments are computed and published. [qL​]1​,[qR​]1​,[qM​]1​,[qO​]1​,[qC​]1​,[Sσ1​]1​,[Sσ2​]1​,[Sσ3​]1​","breadcrumbs":"Plonk » Protocol » Setup","id":"21","title":"Setup"},"22":{"body":"Next we describe the proving algorithm for a program of size N. That includes public inputs. Let ω be a primitive N-th root of unity. Let H={1,ω,ω2,…,ωN−1}. Define ZH​:=XN−1. Assume the eight polynomials of common preprocessed input are already given. The prover computes the trace matrix T as described in the first sections. That means, with the first rows corresponding to the public inputs. It should be a N×3 matrix.","breadcrumbs":"Plonk » Protocol » Proving algorithm","id":"22","title":"Proving algorithm"},"23":{"body":"Add to the transcript the following: [Sσ1​]1​,[Sσ2​]1​,[Sσ3​]1​,[qL​]1​,[qR​]1​,[qM​]1​,[qO​]1​,[qC​]1​ Compute polynomials a′,b′,c′ as the interpolation polynomials of the columns of T at the domain H. Sample random b1​,b2​,b3​,b4​,b5​,b6​ Let a:=(b1​X+b2​)ZH​+a′ b:=(b3​X+b4​)ZH​+b′ c:=(b5​X+b6​)ZH​+c′ Compute [a]1​,[b]1​,[c]1​ and add them to the transcript.","breadcrumbs":"Plonk » Protocol » Round 1","id":"23","title":"Round 1"},"24":{"body":"Sample β,γ from the transcript. Let z0​=1 and define recursively for 0≤k { pub n: usize, pub domain: Vec>, pub omega: FieldElement, pub k1: FieldElement, pub ql: Polynomial>, pub qr: Polynomial>, pub qo: Polynomial>, pub qm: Polynomial>, pub qc: Polynomial>, pub s1: Polynomial>, pub s2: Polynomial>, pub s3: Polynomial>, pub s1_lagrange: Vec>, pub s2_lagrange: Vec>, pub s3_lagrange: Vec>,\n} Apart from the eight polynomials in the canonical basis, we store also here the number of constraints n, the domain H, the primitive n-th of unity ω and the element k1​. The element k2​ will be k12​. For convenience, we also store the polynomials Sσi​ in Lagrange form. The following lines define the particular values of the program input x and the private input e. // Input\nlet x = FieldElement::from(2_u64); // Private input\nlet e = FieldElement::from(3_u64);\nlet y, witness = test_witness_2(x, e); The function test_witness_2(x, e) returns an instance of the following struct, that holds the polynomials that interpolate the columns A,B,C of the trace matrix T. pub struct Witness { pub a: Vec>, pub b: Vec>, pub c: Vec>,\n} Next the commitment scheme KZG (Kate-Zaverucha-Goldberg) is instantiated. let srs = test_srs(common_preprocessed_input.n);\nlet kzg = KZG::new(srs); The setup function performs the setup phase. It only needs the common preprocessed input and the commitment scheme. let verifying_key = setup(&common_preprocessed_input, &kzg); It outputs an instance of the struct VerificationKey. pub struct VerificationKey { pub qm_1: G1Point, pub ql_1: G1Point, pub qr_1: G1Point, pub qo_1: G1Point, pub qc_1: G1Point, pub s1_1: G1Point, pub s2_1: G1Point, pub s3_1: G1Point,\n} It stores the commitments of the eight polynomials of the common preprocessed input. The suffix _1 means it is a commitment. It comes from the notation [f]1​, where f is a polynomial. Then a prover is instantiated let random_generator = TestRandomFieldGenerator {};\nlet prover = Prover::new(kzg.clone(), random_generator); The prover is an instance of the struct Prover: pub struct Prover\nwhere F: IsField, CS: IsCommitmentScheme, R: IsRandomFieldElementGenerator { commitment_scheme: CS, random_generator: R, phantom: PhantomData,\n} It stores an instance of a commitment scheme and a random field element generator needed for blinding polynomials. Then the public input is defined. As we mentioned in the recap, the public input contains the output of the program. let public_input = vec![x.clone(), y]; We then generate a proof using the prover's method prove let proof = prover.prove( &witness, &public_input, &common_preprocessed_input, &verifying_key,\n); The output is an instance of the struct Proof. pub struct Proof> { // Round 1. /// Commitment to the wire polynomial `a(x)` pub a_1: CS::Commitment, /// Commitment to the wire polynomial `b(x)` pub b_1: CS::Commitment, /// Commitment to the wire polynomial `c(x)` pub c_1: CS::Commitment, // Round 2. /// Commitment to the copy constraints polynomial `z(x)` pub z_1: CS::Commitment, // Round 3. /// Commitment to the low part of the quotient polynomial t(X) pub t_lo_1: CS::Commitment, /// Commitment to the middle part of the quotient polynomial t(X) pub t_mid_1: CS::Commitment, /// Commitment to the high part of the quotient polynomial t(X) pub t_hi_1: CS::Commitment, // Round 4. /// Value of `a(ζ)`. pub a_zeta: FieldElement, /// Value of `b(ζ)`. pub b_zeta: FieldElement, /// Value of `c(ζ)`. pub c_zeta: FieldElement, /// Value of `S_σ1(ζ)`. pub s1_zeta: FieldElement, /// Value of `S_σ2(ζ)`. pub s2_zeta: FieldElement, /// Value of `z(ζω)`. pub z_zeta_omega: FieldElement, // Round 5 /// Value of `p_non_constant(ζ)`. pub p_non_constant_zeta: FieldElement, /// Value of `t(ζ)`. pub t_zeta: FieldElement, /// Batch opening proof for all the evaluations at ζ pub w_zeta_1: CS::Commitment, /// Single opening proof for `z(ζω)`. pub w_zeta_omega_1: CS::Commitment,\n} Finally, we instantiate a verifier. let verifier = Verifier::new(kzg); It's an instance of Verifier: struct Verifier> { commitment_scheme: CS, phantom: PhantomData,\n} Finally, we call the verifier's method verify that outputs a bool. assert!(verifier.verify( &proof, &public_input, &common_preprocessed_input, &verifying_key\n));","breadcrumbs":"Plonk » Implementation » Usage","id":"34","title":"Usage"},"35":{"body":"All the matrices Q,V,T,PI are padded with dummy rows so that their length is a power of two. To be able to interpolate their columns, we need a primitive root of unity ω of that order. Given the particular field used in our implementation, that means that the maximum possible size for a circuit is 232. The entries of the dummy rows are filled in with zeroes in the Q, V and PI matrices. The T matrix needs to be consistent with the V matrix. Therefore it is filled with the value of the variable with index 0. Some other rows in the V matrix have also dummy values. These are the rows corresponding to the B and C columns of the public input rows. In the recap we denoted them with the empty - symbol. They are filled in with the same logic as the padding rows, as well as the corresponding values in the T matrix.","breadcrumbs":"Plonk » Implementation » Padding","id":"35","title":"Padding"},"36":{"body":"The implementation pretty much follows the rounds as are described in the protocol section. There are a few details that are worth mentioning.","breadcrumbs":"Plonk » Implementation » Implementation details","id":"36","title":"Implementation details"},"37":{"body":"The commitment scheme we use is the Kate-Zaverucha-Goldberg scheme with the BLS 12 381 curve and the ate pairing. It can be found in the commitments module of the lambdaworks_crypto package. The order r of the cyclic subgroup is 0x73eda753299d7d483339d80809a1d80553bda402fffe5bfeffffffff00000001 The maximum power of two that divides r−1 is 232. Therefore, that is the maximum possible order for a primitive root of unity in Fr​ with order a power of two.","breadcrumbs":"Plonk » Implementation » Commitment Scheme","id":"37","title":"Commitment Scheme"},"38":{"body":"","breadcrumbs":"Plonk » Implementation » Fiat-Shamir","id":"38","title":"Fiat-Shamir"},"39":{"body":"Here we describe our implementation of the transcript used for the Fiat-Shamir heuristic. A Transcript exposes two methods: append and challenge. The method append adds a message to the transcript by updating the internal state of the hasher with the raw bytes of the message. The method challenge returns the result of the hasher using the current internal state of the hasher. It subsequently resets the hasher and updates the internal state with the last result. Here is an example of this process: Start a fresh transcript. Call append and pass message_1. Call append and pass message_2. The internal state of the hasher at this point is message_2 || message_1. Call challenge. The output is Hash(message_2 || message_1). Call append and pass message_3. Call challenge. The output is Hash(message_3 || Hash(message_2 || message_1)). Call append and pass message_4. The internal state of the hasher at the end of this exercise is message_4 || Hash(message_3 || Hash(message_2 || message_1)) The underlying hasher function we use is h=sha3.","breadcrumbs":"Plonk » Implementation » Transcript strategy","id":"39","title":"Transcript strategy"},"4":{"body":"We use the following notation. The symbol F denotes a finite field. It is fixed all along. The symbol ω denotes a primitive root of unity in F. All polynomials have coefficients in F and the variable is usually denoted by X. We denote polynomials by single letters like p,a,b,z. We only denote them as z(X) when we want to emphasize the fact that it is a polynomial in X, or we need that to explicitly define a polynomial from another one. For example when composing a polynomial z with the polynomial ωX, the result being denoted by z′:=z(ωX). The symbol ′ is not used to denote derivatives. When interpolating at a domain H={h0​,…,hn​}⊂F, the symbols Li​ denote the Lagrange basis. That is Li​ is the polynomial such that Li​(hj​)=0 for all j=i, and that Li​(hi​)=1. If M is a matrix, then Mi,j​ denotes the value at the row i and column j.","breadcrumbs":"Plonk » Recap » Notation","id":"4","title":"Notation"},"40":{"body":"The result of every challenge is a 256-bit string, which is interpreted as an integer in big-endian order. A field element is constructed out of it by taking modulo the field order. The prime field used in this implementation has a 255-bit order. Therefore some field elements are more probable to occur than others because they have more representatives as 256-bit integers.","breadcrumbs":"Plonk » Implementation » Field elements","id":"40","title":"Field elements"},"41":{"body":"The first messages added to the transcript are all commitments of the polynomials of the common preprocessed input and the values of the public inputs. This prevents a known vulnerability called \"weak Fiat-Shamir\". Check out the following resources to learn more about it. What can go wrong (zkdocs) How not to Prove Yourself: Pitfalls of the Fiat-Shamir Heuristic and Applications to Helios Weak Fiat-Shamir Attacks on Modern Proof Systems","breadcrumbs":"Plonk » Implementation » Strong Fiat-Shamir","id":"41","title":"Strong Fiat-Shamir"},"42":{"body":"In this section, we'll discuss how to build your own constraint system to prove the execution of a particular program.","breadcrumbs":"Plonk » Circuit API » Circuit API","id":"42","title":"Circuit API"},"43":{"body":"Let's take the following simple program as an example. We have two public inputs: x and y. We want to prove to a verifier that we know a private input e such that x * e = y. You can achieve this by building the following constraint system: use lambdaworks_plonk::constraint_system::ConstraintSystem;\nuse lambdaworks_math::elliptic_curve::short_weierstrass::curves::bls12_381::default_types::FrField; fn main() { let system = &mut ConstraintSystem::::new(); let x = system.new_public_input(); let y = system.new_public_input(); let e = system.new_variable(); let z = system.mul(&x, &e); // This constraint system asserts that x * e == y system.assert_eq(&y, &z);\n} This code creates a constraint system over the field of the BLS12381 curve. Then, it creates three variables: two public inputs x and y, and a private variable e. Note that every variable is private except for the public inputs. Finally, it adds the constraints that represent a multiplication and an assertion. Before generating proofs for this system, we need to run a setup and obtain a verifying key: let common = CommonPreprocessedInput::from_constraint_system(&system, &ORDER_R_MINUS_1_ROOT_UNITY);\nlet srs = test_srs(common.n);\nlet kzg = KZG::new(srs); // The commitment scheme for plonk.\nlet vk = setup(&common, &kzg); Now we can generate proofs for our system. We just need to specify the public inputs and obtain a witness that is a solution for our constraint system: let inputs = HashMap::from([(x, FieldElement::from(4)), (e, FieldElement::from(3))]);\nlet assignments = system.solve(inputs).unwrap();\nlet witness = Witness::new(assignments, &system); Once you have all these ingredients, you can call the prover: let public_inputs = system.public_input_values(&assignments);\nlet prover = Prover::new(kzg.clone(), TestRandomFieldGenerator {});\nlet proof = prover.prove(&witness, &public_inputs, &common, &vk); and verify: let verifier = Verifier::new(kzg);\nassert!(verifier.verify(&proof, &public_inputs, &common, &vk));","breadcrumbs":"Plonk » Circuit API » Simple Example","id":"43","title":"Simple Example"},"44":{"body":"Some operations are common, and it makes sense to wrap the set of constraints that do these operations in a function and use it several times. Lambdaworks comes with a collection of functions to help you build your own constraint systems, such as conditionals, inverses, and hash functions. However, if you have an operation that does not come with Lambdaworks, you can easily extend Lambdaworks functionality. Suppose that the exponentiation operation is something common in your program. You can write the square and multiply algorithm and put it inside a function: pub fn pow( system: &mut ConstraintSystem, base: Variable, exponent: Variable,\n) -> Variable { let exponent_bits = system.new_u32(&exponent); let mut result = system.new_constant(FieldElement::one()); for i in 0..32 { if i != 0 { result = system.mul(&result, &result); } let result_times_base = system.mul(&result, &base); result = system.if_else(&exponent_bits[i], &result_times_base, &result); } result\n} This function can then be used to modify our simple program from the previous section. The following circuit checks that the prover knows e such that pow(x, e) = y: use lambdaworks_plonk::constraint_system::ConstraintSystem;\nuse lambdaworks_math::elliptic_curve::short_weierstrass::curves::bls12_381::default_types::FrField; fn main() { let system = &mut ConstraintSystem::::new(); let x = system.new_public_input(); let y = system.new_public_input(); let e = system.new_variable(); let z = pow(system, &x, &e); system.assert_eq(&y, &z);\n} You can keep composing these functions in order to create more complex systems.","breadcrumbs":"Plonk » Circuit API » Building Complex Systems","id":"44","title":"Building Complex Systems"},"45":{"body":"The goal of this document is to give a good a understanding of our stark prover code. To this end, in the first section we go through a recap of how the proving system works at a high level mathematically; then we dive into how that's actually implemented in our code.","breadcrumbs":"STARKs » STARK Prover","id":"45","title":"STARK Prover"},"46":{"body":"","breadcrumbs":"STARKs » Recap » STARKs Recap","id":"46","title":"STARKs Recap"},"47":{"body":"In general, we express computation in our proving system by providing an execution trace satisfying certain constraints . The execution trace is a table containing the state of the system at every step of computation. This computation needs to follow certain rules to be valid; these rules are our constraints . The constraints for our computation are expressed using an Algebraic Intermediate Representation or AIR. This representation uses polynomials to encode constraints, which is why sometimes they are called polynomial constraints. To make all this less abstract, let's go through two examples.","breadcrumbs":"STARKs » Recap » Verifying Computation through Polynomials","id":"47","title":"Verifying Computation through Polynomials"},"48":{"body":"Throughout this section and the following we will use this example extensively to have a concrete example. Even though it's a bit contrived (no one cares about computing fibonacci numbers), it's simple enough to be useful. STARKs and proving systems in general are very abstract things; having an example in mind is essential to not get lost. Let's say our computation consists of calculating the k-th number in the fibonacci sequence. This is just the sequence of numbers \\(a_n\\) satisfying \\[ a_0 = 1 \\] \\[ a_1 = 1 \\] \\[ a_{n+2} = a_{n + 1} + a_n \\] An execution trace for this just consists of a table with one column, where each row is the i-th number in the sequence: a_i 1 1 2 3 5 8 13 21 A valid trace for this computation is a table satisfying two things: The first two rows are 1. The value on any other row is the sum of the two preceding ones. The first item is called a boundary constraint, it just enforces specific values on the trace at certain points. The second one is a transition constraint; it tells you how to go from one step of computation to the next.","breadcrumbs":"STARKs » Recap » Fibonacci numbers","id":"48","title":"Fibonacci numbers"},"49":{"body":"The example above is extremely useful to have a mental model, but it's not really useful for anything else. The problem is it just works for the very narrow example of computing fibonacci numbers. If we wanted to prove execution of something else, we would have to write an AIR for it. What we're actually aiming for is an AIR for an entire general purpose Virtual Machine. This way, we can provide proofs of execution for any computation using just one AIR. This is what cairo as a programming language does. Cairo code compiles to the bytecode of a virtual machine with an already defined AIR. The general flow when using cairo is the following: User writes a cairo program. The program is compiled into Cairo's VM bytecode. The VM executes said code and provides an execution trace for it. The trace is passed on to a STARK prover, which creates a proof of correct execution according to Cairo's AIR. The proof is passed to a verifier, who checks that the proof is valid. Ultimately, our goal is to give the tools to write a STARK prover for the cairo VM and do so. However, this is not a good example to start out as it's incredibly complex. The execution trace of a cairo program has around 30 columns, some for general purpose registers, some for other reasons. Cairo's AIR contains a lot of different transition constraints, encoding all the different possible instructions (arithmetic operations, jumps, etc). Use the fibonacci example as your go-to for understanding all the moving parts; keep the Cairo example in mind as the thing we are actually building towards.","breadcrumbs":"STARKs » Recap » Cairo","id":"49","title":"Cairo"},"5":{"body":"","breadcrumbs":"Plonk » Recap » The ideas and components","id":"5","title":"The ideas and components"},"50":{"body":"Below we go through a step by step explanation of a STARK prover. We will assume the trace of the fibonacci sequence mentioned above; it consists of only one column of length \\(2^n\\). In this case, we'll take n=3. The trace looks like this a_i a_0 a_1 a_2 a_3 a_4 a_5 a_6 a_7","breadcrumbs":"STARKs » Recap » Fibonacci step by step walkthrough","id":"50","title":"Fibonacci step by step walkthrough"},"51":{"body":"The first step is to interpolate these values to generate the trace polynomial. This will be a polynomial encoding all the information about the trace. The way we do it is the following: in the finite field we are working in, we take an 8-th primitive root of unity, let's call it g. It being a primitive root means two things: g is an 8-th root of unity, i.e., \\(g^8 = 1\\). Every 8-th root of unity is of the form \\(g^i\\) for some \\(0 \\leq i \\leq 7\\). With g in hand, we take the trace polynomial t to be the one satisfying t(gi)=ai​ From here onwards, we will talk about the validity of the trace in terms of properties that this polynomial must satisfy. We will also implicitly identify a certain power of \\(g\\) with its corresponding trace element, so for example we sometimes think of \\(g^5\\) as \\(a_5\\), the fifth row in the trace, even though technically it's \\(t\\) evaluated in \\(g^5\\) that equals \\(a_5\\). We talked about two different types of constraints the trace must satisfy to be valid. They were: The first two rows are 1. The value on any other row is the sum of the two preceding ones. In terms of t, this translates to \\(t(g^0) = 1\\) and \\(t(g) = 1\\). \\(t(x g^2) - t(xg) - t(x) = 0\\) for all \\(x \\in {g^0, g^1, g^2, g^3, g^4, g^5}\\). This is because multiplying by g is the same as advancing a row in the trace.","breadcrumbs":"STARKs » Recap » Trace polynomial","id":"51","title":"Trace polynomial"},"52":{"body":"To convince the verifier that the trace polynomial satisfies the relationships above, the prover will construct another polynomial that shows that both the boundary and transition constraints are satisfied and commit to it. We call this polynomial the composition polynomial, and usually denote it with \\(H\\). Constructing it involves a lot of different things, so we'll go step by step introducing all the moving parts required.","breadcrumbs":"STARKs » Recap » Composition Polynomial","id":"52","title":"Composition Polynomial"},"53":{"body":"To show that the boundary constraints are satisfied, we construct the boundary polynomial. Recall that our boundary constraints are \\(t(g^0) = t(g) = 1\\). Let's call \\(P\\) the polynomial that interpolates these constraints, that is, \\(P\\) satisfies: P(1)=1P(g)=1 The boundary polynomial \\(B\\) is defined as follows: B(x)=(x−1)(x−g)t(x)−P(x)​ The denominator here is called the boundary zerofier, and it's the polynomial whose roots are the elements of the trace where the boundary constraints must hold. How does \\(B\\) encode the boundary constraints? The idea is that, if the trace satisfies said constraints, then t(1)−P(1)=1−1=0 t(g)−P(g)=1−1=0 so \\(t(x) - P(x)\\) has \\(1\\) and \\(g\\) as roots. Showing these values are roots is the same as showing that \\(B(x)\\) is a polynomial instead of a rational function, and that's why we construct \\(B\\) this way.","breadcrumbs":"STARKs » Recap » Boundary polynomial","id":"53","title":"Boundary polynomial"},"54":{"body":"To convince the verifier that the transition constraints are satisfied, we construct the transition constraint polynomial and call it \\(C(x)\\). It's defined as follows: C(x)=∏i=05​(x−gi)t(xg2)−t(xg)−t(x)​ How does \\(C\\) encode the transition constraints? We mentioned above that these are satisfied if the polynomial in the numerator vanishes in the elements \\({g^0, g^1, g^2, g^3, g^4, g^5}\\). As with \\(B\\), this is the same as showing that \\(C(x)\\) is a polynomial instead of a rational function.","breadcrumbs":"STARKs » Recap » Transition constraint polynomial","id":"54","title":"Transition constraint polynomial"},"55":{"body":"With the boundary and transition constraint polynomials in hand, we build the composition polynomial \\(H\\) as follows: The verifier will sample four numbers \\(\\beta_1, \\beta_2\\) and \\(H\\) will be H(x)=β1​B(x)+β2​C(x) Why not just take \\(H(x) = B(x) + C(x)\\)? The reason for the betas is to make the resulting \\(H\\) be always different and unpredictable for the prover, so they can't precompute stuff beforehand. With what we discussed above, showing that the constraints are satisfied is equivalent to saying that H is a polynomial and not a rational function (we are simplifying things a bit here, but it works for our purposes).","breadcrumbs":"STARKs » Recap » Constructing \\(H\\)","id":"55","title":"Constructing \\(H\\)"},"56":{"body":"To show \\(H\\) is a polynomial we are going to use the FRI protocol, which we treat as a black box. For all we care, a FRI proof will verify if what we committed to is indeed a polynomial. Thus, the prover will provide a FRI commitment to H, and if it passes, the verifier will be convinced that the constraints are satisfied. There is one catch here though: how does the verifier know that FRI was applied to H and not any other polynomial? For this we need to add an additional step to the protocol.","breadcrumbs":"STARKs » Recap » Commiting to \\(H\\)","id":"56","title":"Commiting to \\(H\\)"},"57":{"body":"After commiting to H, the prover needs to show that H was constructed correctly according to the formula above. To do this, it will ask the prover to provide an evaluation of H on some random point z and evaluations of the trace at the points \\(t(z), t(zg)\\) and \\(t(zg^2)\\). Because the boundary and transition constraints are a public part of the protocol, the verifier knows them, and thus the only thing it needs to compute the evaluation \\((z)\\) by itself are the three trace evaluations mentioned above. Because it asked the prover for them, it can check both sides of the equation: H(z)=β1​B(z)+β2​C(z) and be convinced that \\(H\\) was constructed correctly. We are still not done, however, as the prover could have now cheated on the values of the trace or composition polynomial evaluations.","breadcrumbs":"STARKs » Recap » Consistency check","id":"57","title":"Consistency check"},"58":{"body":"There are two things left the prover needs to show to complete the proof: That \\(H\\) effectively is a polynomial, i.e., that the constraints are satisfied. That the evaluations the prover provided on the consistency check were indeed evaluations of the trace polynomial and composition polynomial on the out of domain point z. Earlier we said we would use the FRI protocol to commit to H and show the first item in the list. However, we can slightly modify the polynomial we do FRI on to show both the first and second items at the same time. This new modified polynomial is called the DEEP composition polynomial. We define it as follows: Deep(x)=γ1​x−zH(x)−H(z)​+γ2​x−zt(x)−t(z)​+γ3​x−zgt(x)−t(zg)​+γ4​x−zg2t(x)−t(zg2)​ where the numbers \\(\\gamma_i\\) are randomly sampled by the verifier. The high level idea is the following: If we apply FRI to this polynomial and it verifies, we are simultaneously showing that \\(H\\) is a polynomial and the prover indeed provided H(z) as one of the out of domain evaluations. This is the first summand in Deep(x). The trace evaluations provided by the prover were the correct ones, i.e., they were \\(t(z)\\), \\(t(zg)\\), and \\(t(zg^2)\\). These are the remaining summands of the Deep(x).","breadcrumbs":"STARKs » Recap » Deep Composition Polynomial","id":"58","title":"Deep Composition Polynomial"},"59":{"body":"The prover needs to show that Deep was constructed correctly according to the formula above. To do this, the verifier will ask the prover to provide: An evaluation of H on z and x_0 Evaluations of the trace at the points \\(t(z)\\), \\(t(zg)\\), \\(t(zg^2)\\) and \\(t(x_0)\\) Where z is the same random, out of domain point used in the consistency check of the composition polynomial, and x_0 is a random point that belongs to the trace domain. With the values provided by the prover, the verifier can check both sides of the equation: Deep(x0​)=γ1​x0​−zH(x0​)−H(z)​+γ2​x0​−zt(x0​)−t(z)​+γ3​x0​−zgt(x0​)−t(zg)​+γ4​x0​−zg2t(x0​)−t(zg2)​ The prover also needs to show that the trace evaluation \\(t(x_0)\\) belongs to the trace. To achieve this, it needs to commit the merkle roots of t and the merkle proof of \\(t(x_0)\\).","breadcrumbs":"STARKs » Recap » Consistency check","id":"59","title":"Consistency check"},"6":{"body":"For better clarity, we'll be using the following toy program throughout this recap. INPUT: x PRIVATE INPUT: e OUTPUT: e * x + x - 1 The observer would have noticed that this program could also be written as (e+1)∗x−1, which is more sensible. But the way it is written now serves us to better explain the arithmetization of PLONK. So we'll stick to it. The idea here is that the verifier holds some value x, say x=3. He gives it to the prover. She executes the program using her own chosen value e, and sends the output value, say 8, along with a proof π demonstrating correct execution of the program and obtaining the correct output. In the context of PLONK, both the inputs and outputs of the program are considered public inputs . This may sound odd, but it is because these are the inputs to the verification algorithm. This is the algorithm that takes, in this case, the tuple (3,8,π) and outputs Accept if the toy program was executed with input x=3, some private value e not revealed to the verifier, and out came 8. Otherwise it outputs Reject . PLONK can be used to delegate program executions to untrusted parties, but it can also be used as a proof of knowledge. Our program could be used by a prover to demostrate that she knows the multiplicative inverse of some value x in the finite field without revealing it. She would do it by sending the verifier the tuple (x,0,π), where π is the proof of the execution of our toy program. In our toy example this is pointless because inverting field elements is easily performed by any verifier. But change our program to the following and you get proofs of knowledge of the preimage of SHA256 digests. PRIVATE INPUT: e OUTPUT: SHA256(e) Here there's no input aside from the prover's private input. As we mentioned, the output h of the program is then part of the inputs to the verification algorithm. Which in this case just takes (h,π).","breadcrumbs":"Plonk » Recap » Programs. Our toy example","id":"6","title":"Programs. Our toy example"},"60":{"body":"We summarize below the steps required in a STARK proof for both prover and verifier.","breadcrumbs":"STARKs » Recap » Summary","id":"60","title":"Summary"},"61":{"body":"Compute the trace polynomial t by interpolating the trace column over a set of \\(2^n\\)-th roots of unity \\({g^i : 0 \\leq i < 2^n}\\). Compute the boundary polynomial B. Compute the transition constraint polynomial C. Construct the composition polynomial H from B and C. Sample an out of domain point z and provide the evaluations \\(H(z)\\), \\(t(z)\\), \\(t(zg)\\), and \\(t(zg^2)\\) to the verifier. Sample a domain point x_0 and provide the evaluations \\(H(x_0)\\) and \\(t(x_0)\\) to the verifier. Construct the deep composition polynomial Deep(x) from H, t, and the evaluations from the item above. Do FRI on Deep(x) and provide the resulting FRI commitment to the verifier. Provide the merkle root of t and the merkle proof of \\(t(x_0)\\).","breadcrumbs":"STARKs » Recap » Prover side","id":"61","title":"Prover side"},"62":{"body":"Take the evaluations \\(H(z)\\), \\(H(x_0)\\), \\(t(z)\\), \\(t(zg)\\), \\(t(zg^2)\\) and \\(t(x_0)\\) the prover provided. Reconstruct the evaluations \\(B(z)\\) and \\(C(z)\\) from the trace evaluations we were given. Check that the claimed evaluation \\(H(z)\\) the prover gave us actually satisfies H(z)=β1​B(z)+β2​C(z) Check that the claimed evaluation \\(Deep(x_0)\\) the prover gave us actually satisfies Deep(x0​)=γ1​x0​−zH(x0​)−H(z)​+γ2​x0​−zt(x0​)−t(z)​+γ3​x0​−zgt(x0​)−t(zg)​+γ4​x0​−zg2t(x0​)−t(zg2)​ Using the merkle root and the merkle proof the prover provided, check that \\(t(x_0)\\) belongs to the trace. Take the provided FRI commitment and check that it verifies.","breadcrumbs":"STARKs » Recap » Verifier side","id":"62","title":"Verifier side"},"63":{"body":"The walkthrough above was for the fibonacci example which, because of its simplicity, allowed us to sweep under the rug a few more complexities that we'll have to tackle on the implementation side. They are:","breadcrumbs":"STARKs » Recap » Simplifications and Omissions","id":"63","title":"Simplifications and Omissions"},"64":{"body":"Our trace contained only one column, but in the general setting there can be multiple (the Cairo AIR has around 30). This means there isn't just one trace polynomial, but several; one for each column. This also means there are multiple boundary constraint polynomials. The general idea, however, remains the same. The deep composition polynomial H is now the sum of several terms containing the boundary constraint polynomials \\(B_1(x), \\dots, B_k(x)\\) (one per column), and each \\(B_i\\) is in turn constructed from the \\(i\\)-th trace polynomial \\(t_i(x)\\).","breadcrumbs":"STARKs » Recap » Multiple trace columns","id":"64","title":"Multiple trace columns"},"65":{"body":"Much in the same way, our fibonacci AIR had only one transition constraint, but there could be several. We will therefore have multiple transition constraint polynomials \\(C_1(x), \\dots, C_n(x)\\), each of which encodes a different relationship between rows that must be satisfied. Also, because there are multiple trace columns, a transition constraint can mix different trace polynomials. One such constraint could be C1​(x)=t1​(gx)−t2​(x) which means \"The first column on the next row has to be equal to the second column in the current row\". Again, even though this seems way more complex, the ideas remain the same. The composition polynomial H will now include a term for every \\(C_i(x)\\), and for each one the prover will have to provide out of domain evaluations of the trace polynomials at the appropriate values. In our example above, to perform the consistency check on \\(C_1(x)\\) the prover will have to provide the evaluations \\(t_1(zg)\\) and \\(t_2(z)\\).","breadcrumbs":"STARKs » Recap » Multiple transition constraints","id":"65","title":"Multiple transition constraints"},"66":{"body":"In the actual implementation, we won't commit to \\(H\\), but rather to a decomposition of \\(H\\) into an even term \\(H_1(x)\\) and an odd term \\(H_2(x)\\), which satisfy H(x)=H1​(x2)+xH2​(x2) This way, we don't commit to \\(H\\) but to \\(H_1\\) and \\(H_2\\). This is just an optimization at the code level; once again, the ideas remain exactly the same.","breadcrumbs":"STARKs » Recap » Composition polynomial decomposition","id":"66","title":"Composition polynomial decomposition"},"67":{"body":"We treated FRI as a black box entirely. However, there is one thing we do need to understand about it: low degree extensions. When applying FRI to a polynomial of degree \\(n\\), we need to provide evaluations of it over a domain with more than \\(n\\) points. In our case, the DEEP composition polynomial's degree is around the same as the trace's, which is, at most, \\(2^n - 1\\) (because it interpolates the trace containing \\(2^n\\) points). The domain we are going to choose to evaluate our DEEP polynomial on will be a set of higher roots of unity. In our fibonacci example, we will take a primitive \\(16\\)-th root of unity \\(\\omega\\). As a reminder, this means: \\(\\omega\\) is an \\(16\\)-th root of unity, i.e., \\(\\omega^{16} = 1\\). Every \\(16\\)-th root of unity is of the form \\(\\omega^i\\) for some \\(0 \\leq i \\leq 15\\). Additionally, we also take it so that \\(\\omega\\) satisfies \\(\\omega^2 = g\\) (\\(g\\) being the \\(8\\)-th primitive root of unity we used to construct t). The evaluation of \\(t\\) on the set \\({\\omega^i : 0 \\leq i \\leq 15}\\) is called a low degree extension (LDE) of \\(t\\). Notice this is not a new polynomial, they're evaluations of \\(t\\) on some set of points. Also note that, because \\(\\omega^2 = g\\), the LDE contains all the evaluations of \\(t\\) on the set of powers of \\(g\\). In fact, {t(ω2i):0≤i≤15}={t(gi):0≤i≤7} This will be extremely important when we get to implementation. For our LDE, we chose \\(16\\)-th roots of unity, but we could have chosen any other power of two greater than \\(8\\). In general, this choice is called the blowup factor, so that if the trace has \\(2^n\\) elements, a blowup factor of \\(b\\) means our LDE evaluates over the \\(2^{n} * b\\) roots of unity (\\(b\\) needs to be a power of two). The blowup factor is a parameter of the protocol related to its security.","breadcrumbs":"STARKs » Recap » FRI, low degree extensions and roots of unity","id":"67","title":"FRI, low degree extensions and roots of unity"},"68":{"body":"In this section, we start diving deeper before showing the formal protocol. If you haven't done so, we recommend reading the \"Recap\" section first. At a high level, the protocol works as follows. The starting point is a matrix T that encodes the trace of a valid execution of the program. This matrix needs to be in a particular format so that its correctness is equivalent to checking a finite number of polynomial equations on its rows. Transforming the execution to this matrix is what's called the arithmetization process. Then a single polynomial F is constructed that encodes the set of all the polynomial constraints. The satisfiability of all these constraints is equivalent to F being divisible by some public polynomial G. So the prover constructs H as the quotient F/G called the composition polynomial. Then the verifier chooses a random point z and challenges the prover to reveal the values F(z) and H(z). Then the verifier checks that H(z)=F(z)/G(z), which convinces him that the same relation holds at a level of polynomials and, in consequence, convinces the verifier that the private trace T of the prover is valid. In summary, at a very high level, the STARK protocol can be organized into three major parts: Arithmetization and commitment of execution trace. Construction of composition polynomial H. Opening of polynomials at random z.","breadcrumbs":"STARKs » Protocol overview » Protocol Overview","id":"68","title":"Protocol Overview"},"69":{"body":"As the Recap mentions, the trace is a table containing the system's state at every step. In this section, we will denote the trace as T. A trace can have several columns to store different aspects or features of a particular state at a specific moment. We will refer to the j-th column as Tj​. You can think of a trace as a matrix T where the entry Tij​ is the j-th element of the i-th state. Most proving systems' primary tool is polynomials over a finite field F. Each column Tj​ of the trace T will be interpreted as evaluations of such a polynomial tj​. Consequently, any information about the states must be encoded somehow as an element in F. To ease notation, we will assume here and in the protocol that the constraints encoding transition rules depend only on a state and the previous one. Everything can be easily generalized to transitions that depend on many preceding states. Then, constraints can be expressed as multivariate polynomials in 2m variables PkT​(X1​,…,Xm​,Y1​,…,Ym​) A transition from state i to state i+1 will be valid if and only if when we plug row i of T in the first m variables and row i+1 in the second m variables of PkT​, we get 0 for all k. In mathematical notation, this is PkT​(Ti,0​,…,Ti,m​,Ti+1,0​,…,Ti+1,m​)=0 for all k These are called transition constraints and check the trace's local properties, where local means relative to specific rows. There is another type of constraint, called boundary constraint , and denoted PjB​. These enforce parts of the trace to take particular values. It is helpful, for example, to verify the initial states. So far, these constraints can only express the local properties of the trace. There are situations where the global properties of the trace need to be checked for consistency. For example, a column may need to take all values in a range but not in any predefined way. Several methods exist to express these global properties as local by adding redundant columns. Usually, they need to involve randomness from the verifier to make sense, and they turn into an interactive protocol called Randomized AIR with Preprocessing .","breadcrumbs":"STARKs » Protocol overview » Arithmetization","id":"69","title":"Arithmetization"},"7":{"body":"This is the process that takes the circuit of a particular program and produces a set of mathematical tools that can be used to generate and verify proofs of execution. The end result will be a set of eight polynomials. To compute them we need first to define two matrices. We call them the Q matrix and the V matrix. The polynomials and the matrices depend only on the program and not on any particular execution of it. So they can be computed once and used for every execution instance. To understand what they are useful for, we need to start from execution traces .","breadcrumbs":"Plonk » Recap » PLONK Arithmetization","id":"7","title":"PLONK Arithmetization"},"70":{"body":"To make interactions possible, a crucial cryptographic primitive is the Polynomial Commitment Scheme. This prevents the prover from changing the polynomials to adjust them to what the verifier expects. Such a scheme consists of the commit and the open protocols. STARK uses a univariate polynomial commitment scheme that internally combines a vector commitment scheme and a protocol called FRI. Let's begin with these two components and see how they build up the polynomial commitment scheme.","breadcrumbs":"STARKs » Protocol overview » Polynomial commitment scheme","id":"70","title":"Polynomial commitment scheme"},"71":{"body":"Given a vector Y=(y0​,…,yM​), commiting to Y means the following. The prover builds a Merkle tree out of it and sends its root to the verifier. The verifier can then ask the prover to reveal, or open , the value of the vector Y at some index i. The prover won't have any choice except to send the correct value. The verifier will expect the corresponding value yi​ and the authentication path to the tree's root to check its authenticity. The authentication path also encodes the vector's position i and its length M. In STARKs, all commited vectors are of the form Y=(p(d1​),…,p(dM​)) for some polynomial p and some domain fixed domain D=(d1​,…,dM​). The domain is always known to the prover and the verifier.","breadcrumbs":"STARKs » Protocol overview » Vector commitments","id":"71","title":"Vector commitments"},"72":{"body":"The FRI protocol is a potent tool to prove that the commitment of a vector (p(d1​),…,p(dM​)) corresponds to the evaluations of a polynomial p of a certain degree. The degree is a configuration of the protocol.","breadcrumbs":"STARKs » Protocol overview » FRI","id":"72","title":"FRI"},"73":{"body":"STARK uses a univariate polynomial commitment scheme. The following is what is expected from the commit and open protocols: Commit : given a polynomial p, the prover produces a sort of hash of it. We denote it here by [p], called the commitment of p. This hash is unique to p. The prover usually sends [p] to the verifier. Open : this is an interactive protocol between the prover and the verifier. The prover holds the polynomial p. The verifier only has the commitment [p]. The verifier sends a value z to the prover at which he wants to know the value y=p(z). The prover sends a value y to the verifier, and then they engage in the Open protocol. As a result, the verifier gets convinced that the polynomial corresponding to the hash [p] evaluates to y at z. Let's see how both of these protocols work in detail. Some configuration parameters are needed: Powers of two N=2n and M=2m with n0. The commitment scheme will only work for polynomials of degree at most N. This means anyone can commit to any polynomial, but the Open protocol will pass only for polynomials satisfying that degree bound.","breadcrumbs":"STARKs » Protocol overview » Polynomial commitments","id":"73","title":"Polynomial commitments"},"74":{"body":"Given a polynomial p, the commitment [p] is just the commitment of the vector (p(d1​),…,p(dM​)). That is, [p] is the root of the Merkle tree of the vector of evaluations of p at D.","breadcrumbs":"STARKs » Protocol overview » Commit","id":"74","title":"Commit"},"75":{"body":"It is an interactive protocol. So assume there is a prover and a verifier. We describe the process considering an honest prover. In the next section, we analyze what happens for malicious provers. The prover holds the polynomial p, and the verifier only the commitment [p] of it. There is also an element z. The prover evaluates p(z) and sends the result to the verifier. As we mentioned, the goal is to generate proof of the validity of the evaluation. Let us denote y:=p(z). This is a value now both the prover and verifier have. The protocol has three steps: FRI on p : First, the prover and the verifier engage in a FRI protocol for polynomials of degree at most N to check that [p] is the commitment of such a polynomial. We will assume from now on that the degree of p is at most N. There is an optimization that completely removes this step—more on that later . FRI on (p−y)/(x−z) : Since p(z)=y, the polynomial p can be written as p=y+(x−z)q for some polynomial q. The prover computes the commitment [q] and sends it to the verifier. Now they engage in a FRI protocol for polynomials of degree at most N−1, which convinces the verifier that [q] is the commitment of a polynomial of degree at most N−1. Point checks : From the point of view of the verifier the commitments [p] and [q] are still potentially unrelated. Therefore, there is a check to ensure that q was computed properly from p following the formula q=(p−y)/(x−z) and that [q] is its commitment. To do this, the verifier challenges the prover to open [p] and [q] as vectors. They use the open protocol of the vector commitment scheme to reveal the values p(di​) and q(di​) for some random point di​∈D chosen by the verifier. Next he checks that p(di​)=y+(di​−z)q(di​). They repeat this last part k times and, as we'll analyze in the next section, this whole thing will convince the verifier that p=y+(x−z)q as polynomials with overwhelming probability (about (N/M)k). Finally, the verifier deduces that p(z)=y from this equality.","breadcrumbs":"STARKs » Protocol overview » Open","id":"75","title":"Open"},"76":{"body":"Let's analyze why the open protocol is sound. Keep in mind that the prover always has to provide a commitment [q] of a polynomial of degree at most N−1 that satisfies p(di​)=y+(di​−z)q(di​) for the chosen values of the verifier. That's the goal. An essential but subtle part of the soundness is that D is a vector of length M>N. To understand why let's see what would happen if N=M. Case M=N Suppose the prover tries to cheat the verifier by sending a value y different from p(z). This means that p−y is not divisible by X−z as polynomials. But as long as z is not in D the prover can interpolate the values d1​−zp(d1​)−y​,…,dN​−zp(dN​)−y​, at D and obtain some polynomial q of degree at most N−1. This polynomial satisfies p(di​)=y+(di​−z)q(di​) for all di​∈D, but the equality of polynomials p=y+(x−z)q does not hold. Mainly because that would imply that p(z)=y, and we are assuming the opposite case. Nevertheless, if they continue with the open protocol, FRI will pass since q is a polynomial of degree at most N−1. All subsequent point checks will pass since q was crafted especially for that. So how come p is different from y+(x−z)q but has the property that p(di​)=y+(di​−z)q(di​) for all di​∈D? FRI guarantees q is of degree at most N−1, which implies that y+(x−z)q is of degree at most N. Then, since p is of degree at most N, the difference f:=y+(X−z)q−p is again of degree at most N. A polynomial f of degree at most N is zero if and only if we can prove that f(d)=0 for N+1 distinct points. But the construction of q only shows that f(di​)=0 for all di​∈D, a set of N points. It's not enough. One extra point and the equality f=0 would hold. Of course, that point doesn't exist, again, because that would imply that p(z)=y, which we assume is not true. Case M=2N Let's see one more example where we double the de size of D but keep the same bound for the degree of the polynomials. So polynomials are of degree at most N, and the length of D is M=2N. Again let's assume the prover wants to cheat the verifier similarly and claims that y is p(z) but, in reality, y=p(z). Inevitably, for the Open protocol to succeed, the prover has to send a commitment [q] of some polynomial of degree at most N−1. The strategy of the prover is pretty much constrained to be the same. But now he can't interpolate q as before in every element of D because that would produce a polynomial of degree at most 2N−1. This most likely won't pass the FRI check. Alternatively, he can choose some subset of D′⊂D of size N and interpolate only there to get a polynomial q of degree at most N−1. That will succeed in the FRI phase, but what happens with the point checks? The check will pass if the verifier chooses a challenge di​ that belongs to D′. If di​∈/D′, then the check will inevitably fail. This is because, as we saw in the previous case, one extra successful point to the already N points where q was interpolated was enough to prove that p(z)=y, which we assume now not to hold. This means, if di​∈/D′, then p(di​) does not coincide with y+(di​−z)q(di​) and the point check fails. So if the verifier chooses the challenges following a uniform distribution, the chances of the prover being caught cheating are 1/2 for only one challenge. If the verifier performs k challenges, then the chance of the prover not getting caught is 1/2k. General case: the blowup factor If the ratio between M and N is 2, then k challenges give 1/2k of probability for a malicious prover to succeed. If the ratio is 4, that is, if M=4N, then that probability is 1/4k for the same number of point checks. This provides another way of improving the soundness error. The larger the ratio, the more complex it is to cheat in the open protocol. This ratio is what's called the blowup factor . It is a security parameter, and finding a balance between the number of challenges k and the size of D is part of the configuration of the protocol. Increasing the size of D makes committing an expensive operation since it involves building a Merkle tree with a vector of the size of D. But increasing the number of challenges makes the size of the final proof larger. We denote the blowup factor by b, and it's always assumed to be a power of 2.","breadcrumbs":"STARKs » Protocol overview » Soundness","id":"76","title":"Soundness"},"77":{"body":"During proof generation, polynomials are committed and opened several times. Computing these for each polynomial independently is costly. In this section, we'll see how batching polynomials can reduce the amount of computation. Let P={p1​,…,pL​} be a set of polynomials. We will commit and open P as a whole. We note this batch commitment as [P]. We need the same configuration parameters as before: N=2n, M=2m with N0. As described earlier, to commit to a single polynomial p, a Merkle tree is built over the vector (p(d1​),…,p(dm​)). When committing to a batch of polynomials P={p1​,…,pn​}, the leaves of the Merkle tree are instead the concatenation of the polynomial evaluations. That is, in the batch setting, the Merkle tree is built for the vector (p1​(d1​)∣∣…∣∣pL​(d1​),…,pL​(dm​)∣∣…∣∣pn​(dm​)). The commitment [P] is the root of this Merkle tree. This reduces the proof size: we only need one Merkle tree for L polynomials. The verifier can then only ask for values in batches. When the verifier chooses an index i, the prover sends p1​(di​),…,pL​(di​) along with one authentication path. The verifier on his side computes the concatenation p1​(di​)∣∣…∣∣pL​(di​) and validates it with the authentication path and [P]. This also reduces the computational time. By traversing the Merkle tree one time, it can reveal several components simultaneously. The batch open protocol proceeds similarly to the case of a single polynomial. The prover will try to convince the verifier that the committed polynomials P at z evaluate to some values yi​=pi​(z). In the batch case, the prover will construct the following polynomial: Q:=i=1∑L​γi​X−zpi​−yi​​ Where γi​ are challenges provided by the verifier. The prover commits to Q and sends [Q] to the verifier. Then the prover and verifier continue similarly to the normal open protocol for Q only. This means they engage in a FRI protocol for polynomials of degree at most N−1 for Q. Then they engage in the point checks for Q. Here, for each challenge di​, the prover uses one authentication path for [Q] to reveal Q(di​) and use one authentication path for [P] to batch reveal values p1​(di​),…,pL​(di​). Successful point checks here mean that Q(di​)=∑i​γi​(pi​(di​)−yi​)/(di​−z). This is equivalent to running the open protocol L times, one for each term pi​ and yi​. Note that this optimization makes a huge difference, as we only need to run the FRI protocol once instead of running it once for each polynomial.","breadcrumbs":"STARKs » Protocol overview » Batch","id":"77","title":"Batch"},"78":{"body":"There is an optimization for the open protocol to avoid running FRI to check that p is of degree at most N. The idea is as follows. Part of FRI protocol for [q], to check that it is of degree at most N−1, involves revealing values of q at other random points di​ also chosen by the verifier. These are part of the internal workings of FRI. These challenges are unrelated to what we mentioned before. So if one removes the FRI check for p, the point checks of the open protocol need to be performed on these challenges di​ of the FRI protocol for [q]. This optimization is included in the formal description of the protocol.","breadcrumbs":"STARKs » Protocol overview » Optimize the open protocol reusing FRI internal challenges","id":"78","title":"Optimize the open protocol reusing FRI internal challenges"},"79":{"body":"Transparent Polynomial Commitment Scheme with Polylogarithmic Communication Complexity Summary on FRI low degree test DEEP FRI Thank goodness it's FRIday Diving DEEP FRI","breadcrumbs":"STARKs » Protocol overview » References","id":"79","title":"References"},"8":{"body":"See the program as a sequence of gates that have left operand, a right operand and an output. The two most basic gates are multiplication and addition gates. In our example, one way of seeing our toy program is as a composition of three gates. Gate 1: left: e, right: x, output: u = e * x Gate 2: left: u, right: x, output: v = e + x Gate 3: left: v, right: 1, output: w = v - 1 On executing the circuit, all these variables will take a concrete value. All that information can be put in table form. It will be a matrix with all left, right and output values of all the gates. One row per gate. We call the columns of this matrix L,R,O. Let's build them for x=3 and e=2. We get u=6, v=9 and w=8. So the first matrix is: A B C 2 3 6 6 3 9 9 - 8 The last gate subtracts a constant value that is part of the program and is not a variable. So it actually has only one input instead of two. And the output is the result of subtracting 1 to it. That's why it is handled a bit different from the second gate. The symbol \"-\" in the R column is a consequence of that. With that we mean \"any value\" because it won't change the result. In the next section we'll see how we implement that. Here we'll use this notation when any value can be put there. In case we have to choose some, we'll default to 0. What we got is a valid execution trace. Not all matrices of that shape will be the trace of an execution. The matrices Q and V will be the tool we need to distinguish between valid and invalid execution traces.","breadcrumbs":"Plonk » Recap » Circuits and execution traces","id":"8","title":"Circuits and execution traces"},"80":{"body":"The protocol is split into rounds. Each round more or less represents an interaction with the verifier. Each round will generally start by getting a challenge from the verifier. The prover will need to interpolate polynomials, and he will always do it over the set DS​={gi}i=02n−1​⊆F, where g is a 2n root of unity in F. Also, the vector commitments will be performed over the set DLDE​=(h,hω,hω2,…,hω2n+l) where ω is a 2n+l root of unity and h is some field element. This is the set we denoted D in the commitment scheme section. The specific choices for the shapes of these sets are motivated by optimizations at a code level.","breadcrumbs":"STARKs » Protocol overview » High-level description of the protocol","id":"80","title":"High-level description of the protocol"},"81":{"body":"In round 1 , the prover commits to the columns of the trace T. He does so by interpolating each column j and obtaining univariate polynomials tj​. Then the prover commits to tj​ over DLDE​. In this way, we have Ti,j​=tj​(gi). From now on, the prover won't be able to change the trace values T. The verifier will leverage this and send challenges to the prover. The prover cannot know in advance what these challenges will be. Thus he cannot handcraft a trace to deceive the verifier. As mentioned before, if some constraints cannot be expressed locally, more columns can be added to make a constraint-friendly trace. This is done by committing to the first set of columns, then sampling challenges from the verifier and repeating round 1. The sampling of challenges serves to add new constraints. These constraints will ensure the new columns have some common structure with the original trace. In the protocol, extended columns are referred to as the RAP2 (Randomized AIR with Preprocessing). The matrix of the extended columns is denoted MRAP2​.","breadcrumbs":"STARKs » Protocol overview » Round 1: Arithmetization and commitment of the execution trace","id":"81","title":"Round 1: Arithmetization and commitment of the execution trace"},"82":{"body":"round 2 aims to build the composition polynomial H. This function will have the property that it is a polynomial if and only if the trace that the prover committed to at round 1 is valid and satisfies the agreed polynomial constraints. That is, H will be a polynomial if and only if T is a trace that satisfies all the transition and boundary constraints. Note that we can compose the polynomials tj​, the ones that interpolate the columns of the trace T, with the multivariate constraint polynomials as follows. QkT​(x)=PkT​(t1​(x),…,tm​(x),t1​(gx),…,tm​(ωx)) These result in univariate polynomials. And the same can be done for the boundary constraints. Since Ti,j​=tj​(gi), these univariate polynomials vanish at every element of D if and only if the trace T is valid. As we already mentioned, this is assuming that transitions only depend on the current and previous state. But it can be generalized to include frames with three or more rows or more context for each constraint. For example, in the Fibonacci case, the most natural way is to encode it as one transition constraint that depends on a row and the two preceding it, as we already did in the Recap section. The STARK protocol checks whether the function X2n−1QkT​​ is a polynomial instead of checking that the polynomial is zero over the domain D={gi​}i=02n−1​. The two statements are equivalent. The verifier could check that all X2n−1QkT​​ are polynomials one by one, and the same for the polynomials coming from the boundary constraints. But this is inefficient; the same can be obtained with a single polynomial. To do this, the prover samples challenges and obtains a random linear combination of these polynomials. The result of this is denoted by H and is called the composition polynomial. It integrates all the constraints by adding them up. So after computing H, the prover commits to it and sends the commitment to the verifier. The rest of the protocol aims to prove that H was constructed correctly and is a polynomial, which can only be true if the prover has a valid extension of the original trace.","breadcrumbs":"STARKs » Protocol overview » Round 2: Construction of composition polynomial H","id":"82","title":"Round 2: Construction of composition polynomial H"},"83":{"body":"The verifier must check that H was constructed according to the protocol rules. That is, H has to be a linear combination of all the functions X2n−1QkT​​ and similar terms for the boundary constraints. To do so, in round 3 the verifier chooses a random point z∈F and the prover computes H(z), tj​(z) and tj​(gz) for all j. With all these, the verifier can check that H and the expected linear combination coincide, at least when evaluated at z. Since z was chosen randomly, this proves with overwhelming probability that H was properly constructed.","breadcrumbs":"STARKs » Protocol overview » Round 3: Evaluation of polynomials at z","id":"83","title":"Round 3: Evaluation of polynomials at z"},"84":{"body":"In this round, the prover and verifier engage in the batch open protocol of the polynomial commitment scheme described above to validate all the evaluations at z from the previous round.","breadcrumbs":"STARKs » Protocol overview » Round 4: Run batch open protocol","id":"84","title":"Round 4: Run batch open protocol"},"85":{"body":"In this section we describe precisely the STARKs protocol used in Lambdaworks. We begin with some additional considerations and notation for most of the relevant objects and values to refer to them later on.","breadcrumbs":"STARKs » Protocol » STARKs protocol","id":"85","title":"STARKs protocol"},"86":{"body":"This is a technique to increase the soundness of the protocol by adding proof of work. It works as follows. At some fixed point in the protocol, the verifier sends a challenge x to the prover. The prover needs to find a string y such that H(x∣∣y) begins with a predefined number of zeroes. Here x∣∣y denotes the concatenation of x and y, seen as bit strings. The number of zeroes is called the grinding factor . The hash function H can be any hash function, independent of other hash functions used in the rest of the protocol. In Lambdaworks we use Keccak256.","breadcrumbs":"STARKs » Protocol » Grinding","id":"86","title":"Grinding"},"87":{"body":"The Fiat-Shamir heuristic is used to make the protocol noninteractive. We assume there is a transcript object to which values can be added and from which challenges can be sampled.","breadcrumbs":"STARKs » Protocol » Transcript","id":"87","title":"Transcript"},"88":{"body":"F denotes a finite field. Given a vector D=(y1​,…,yL​) and a function f:set(D)→F, denote by f(D) the vector (f(y1​),…,f(yL​)). Here set(D) denotes the underlying set of A. A polynomial p∈F[X] induces a function f:A→F for every subset A of F, where f(a):=p(a). Let p,q∈F[X] be two polynomials. A function f:A→F can be induced from them for every subset A disjoint from the set of roots of q, defined by f(a):=p(a)q(a)−1. We abuse notation and denote f by p/q.","breadcrumbs":"STARKs » Protocol » General notation","id":"88","title":"General notation"},"89":{"body":"We assume the prover has already obtained the trace of the execution of the program. This is a matrix T with entries in a finite field F. We assume the number of rows of T is 2n for some n in N. Values known by the prover and verifier prior to the interactions These values are determined the program, the specifications of the AIR being used and the security parameters chosen. m′ is the number of columns of the trace matrix T. r the number of RAP challenges. m′′ is the number of extended columns of the trace matrix in the (optional) second round of RAP. m is the total number of columns: m:=m′+m′′. PkT​ denote the transition constraint polynomials for k=1,…,nT​. We are assuming these are of degree at most 2. ZjT​ denote the transition constraint zerofiers for k=1,…,nT​. b=2l is the blowup factor . c is the grinding factor . Q is number of FRI queries. We assume there is a fixed hash function from F to binary strings. We also assume all Merkle trees are constructed using this hash function. Values computed by the prover These values are computed by the prover from the execution trace and are sent to the verifier along with the proof. 2n is the number of rows of the trace matrix after RAP. ω a primitive 2n+l-th root of unity. g=ωb. An element h∈F∖{ωi}i≥0​. This is called the coset factor . Boundary constraints polynomials PjB​ for j=1,…,m. Boundary constraint zerofiers ZjB​ for j=1,…,m.. Derived values Both prover and verifier compute the following. The interpolation domain: the vector DS​=(1,g,…,g2n−1). The Low Degree Extension DLDE​=(h,hω,hω2,…,hω2n+l−1). Recall 2l is the blowup factor.","breadcrumbs":"STARKs » Protocol » Definitions","id":"89","title":"Definitions"},"9":{"body":"As we said, it only depends on the program itself and not on any particular evaluation of it. It has one row for each gate and its columns are called QL​,QR​,QO​,QM​,QC​. They encode the type of gate of the rows and are designed to satisfy the following. Claim: if columns L,R,O correspond to a valid evaluation of the circuit then for all i the following equality holds Ai​(QL​)i​+Bi​(QR​)i​+Ai​Bi​QM​+Ci​(QO​)i​+(QC​)i​=0 This is better seen with examples. A multiplication gate is represented by the row: QL​ QR​ QM​ QO​ QC​ 0 0 1 -1 0 And the row in the trace matrix that corresponds to the execution of that gate is A B C 2 3 6 The equation in the claim for that row is that 2×0+3×0+2×3×1+6×(−1)+0, which equals 0. The next is an addition gate. This is represented by the row QL​ QR​ QM​ QO​ QC​ 1 1 0 -1 0 The corresponding row in the trace matrix its A B C 6 3 9 And the equation of the claim is 6×1+3×1+2×3×0+9×(−1)+0, which adds up to 0. Our last row is the gate that adds a constant. Addition by constant C can be represented by the row QL​ QR​ QM​ QO​ QC​ 1 0 0 -1 C In our case C=−1. The corresponding row in the execution trace is A B C 9 - 8 And the equation of the claim is 9×1+0×0+9×0×0+8×(−1)+C. This is also zero. Putting it altogether, the full Q matrix is QL​ QR​ QM​ QO​ QC​ 0 0 1 -1 0 1 1 0 -1 0 1 0 0 -1 -1 And we saw that the claim is true for our particular execution: 2×0+3×0+2×3×1+6×(−1)+0=0 6×1+3×1+6×3×0+9×(−1)+0=0 9×1+0×0+9×0×0+8×(−1)+(−1)=0 Not important to our example, but multiplication by constant C can be represented by: QL​ QR​ QM​ QO​ QC​ C 0 0 -1 0 As you might have already noticed, there are several ways of representing the same gate in some cases. We'll exploit this in a moment.","breadcrumbs":"Plonk » Recap » The Q matrix","id":"9","title":"The Q matrix"},"90":{"body":"Vector commitment scheme Given a vector A=(y0​,…,yL​). The operation Commit(A) returns the root r of the Merkle tree that has the hash of the elements of A as leaves. For i∈[0,2n+k), the operation open(A,i) returns yi​ and the authentication path s to the Merkle tree root. The operation Verify(i,y,r,s) returns Accept or Reject depending on whether the i-th element of A is y. It checks whether the authentication path s is compatible with i, y and the Merkle tree root r. In our cases the sets A will be of the form A=(f(a),f(ab),f(ab2),…,f(abL)) for some elements a,b∈F. It will be convenient to use the following abuse of notation. We will write Open(A,abi) to mean Open(A,i). Similarly, we will write Verify(abi,y,r,s) instead of Verify(i,y,r,s). Note that this is only notation and Verify(abi,y,r,s) is only checking that the y is the i-th element of the commited vector.","breadcrumbs":"STARKs » Protocol » Notation of important operations","id":"90","title":"Notation of important operations"},"91":{"body":"","breadcrumbs":"STARKs » Protocol » Protocol","id":"91","title":"Protocol"},"92":{"body":"Round 0: Transcript initialization Start a new transcript. (Strong Fiat Shamir) Add to it all the public values. Round 1: Arithmetization and commitment of the execution trace Round 1.1: Commit main trace For each column Mj​ of the execution trace matrix T, interpolate its values at the domain DS​ and obtain polynomials tj​ such that tj​(gi)=Ti,j​. Compute [tj​]:=Commit(tj​(DLED​)) for all j=1,…,m′ ( Batch commitment optimization applies here ). Add [tj​] to the transcript in increasing order. Round 1.2: Commit extended trace Sample random values a1​,…,al​ in F from the transcript. Use a1​,…,al​ to build MRAP2​∈F2n×m′′ following the specifications of the RAP process. For each column M^j​ of the matrix MRAP2​, interpolate its values at the domain DS​ and obtain polynomials tm′+1​,…,tm′+m′′​ such that tj​(gi)=M^i,j​. Compute [tj​]:=Commit(tj​(DLED​)) for all j=m′+1,…,m′+m′′ ( Batch commitment optimization applies here ). Add [tj​] to the transcript in increasing order for all j=m′+1,…,m′+m′′. Round 2: Construction of composition polynomial H Sample α1B​,…,αmB​ and β1B​,…,βmB​ in F from the transcript. Sample α1T​,…,αnT​T​ and β1T​,…,βnT​T​ in F from the transcript. Compute Bj​:=ZjB​tj​−PjB​​. Compute Ck​:=ZkT​PkT​(t1​,…,tm​,t1​(gX),…,tm​(gX))​. Compute the composition polynomial H:=k∑​βkT​Ck​+j∑​βjB​Bj​ Decompose H as H=H1​(X2)+XH2​(X2) Compute commitments [H1​] and [H2​]. Add [H1​] and [H2​] to the transcript. Round 3: Evaluation of polynomials at z Sample from the transcript until obtaining z∈F∖DLDE​. Compute H1​(z2), H2​(z2), and tj​(z) and tj​(gz) for all j. Add H1​(z2), H2​(z2), and tj​(z) and tj​(gz) for all j to the transcript. Round 4: Run batch open protocol Sample γ, γ′, and γ1​,…,γm​, γ1′​,…,γm′​ in F from the transcript. Compute p0​ as γX−z2H1​−H1​(z2)​+γ′X−z2H2​−H2​(z2)​+j∑​γj​X−ztj​−tj​(z)​+γj′​X−gztj​−tj​(gz)​ Round 4.1.k: FRI commit phase Let D0​:=DLDE​, and [p0​]:=Commit(p0​(D0​)). Add [p0​] to the transcript. For k=1,…,n do the following: Sample ζk−1​ from the transcript. Decompose pk−1​ into even and odd parts, that is, pk−1​=pk−1odd​(X2)+Xpk−1even​(X2). Define pk​:=pk−1odd​(X)+ζk−1​pk−1even​(X). If k>(&trace, &pub_inputs, &proof_options).unwrap(); assert!(verify::>(&proof, &pub_inputs, &proof_options));\n} The proving system revolves around the prove function, that takes a trace, public inputs and proof options as inputs to generate a proof, and a verify function that takes the generated proof, the public inputs and the proof options as inputs, outputting true when the proof is verified correctly and false otherwise. Note that the public inputs and proof options should be the same for both. Public inputs should be shared by the Cairo runner to prover and verifier, and the proof options should have been agreed on beforehand by the two entities beforehand. Below we go over the main things involved in this code.","breadcrumbs":"STARKs » Implementation » High Level API » High level API: Fibonacci example","id":"97","title":"High level API: Fibonacci example"},"98":{"body":"To prove the integrity of a fibonacci trace, we first need to define what it means for a trace to be valid. As we've talked about in the recap, this involves defining an AIR for our computation where we specify both the boundary and transition constraints for a fibonacci sequence. In code, this is done through the AIR trait. Implementing AIR requires defining a couple methods, but the two most important ones are boundary_constraints and compute_transition, which encode the boundary and transition constraints of our computation.","breadcrumbs":"STARKs » Implementation » High Level API » AIR","id":"98","title":"AIR"},"99":{"body":"For our Fibonacci AIR, boundary constraints look like this: fn boundary_constraints( &self, _rap_challenges: &Self::RAPChallenges,\n) -> BoundaryConstraints { let a0 = BoundaryConstraint::new_simple(0, self.pub_inputs.a0.clone()); let a1 = BoundaryConstraint::new_simple(1, self.pub_inputs.a1.clone()); BoundaryConstraints::from_constraints(vec![a0, a1])\n} The BoundaryConstraint struct represents a specific boundary constraint, meaning \"column i at row j should be equal to x\". In this case, because we have only one column, we are using the new_simple method to simply say Row 0 should equal the public input a0, which in the typical fibonacci is set to 1. Row 1 should equal the public input a1, which in the typical fibonacci is set to 1. In the case of multiple columns, the new method exists so you can also specify column number. After instantiating each of these constraints, we return all of them through the struct BoundaryConstraints.","breadcrumbs":"STARKs » Implementation » High Level API » Boundary Constraints","id":"99","title":"Boundary 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Which means that a single authentication path serves to validate both points simultaneously.","breadcrumbs":"STARKs » Protocol » Bit-reversal ordering of Merkle tree leaves","id":"100","title":"Bit-reversal ordering of Merkle tree leaves"},"101":{"body":"The prover opens the polynomials pk​ of the FRI layers at υs2k​ and −υs2k​ for all k>1. Later on, the verifier uses each of those pairs to reconstruct one of the values of the next layer, namely pk+1​(υ2k+1). So there's no need to add the value pk​(υ2k+1) to the proof, as the verifier reconstructs them. The prover only needs to send the authentication paths Pk​ for them. The protocol is only modified at Step 3 of the verifier as follows. Checking that Verify((υs2k​,πkυs2k​​),Pk​,Pk​) is skipped. After computing x:=G+ζk​H, the verifier uses it to check that Verify((υs2k​,x),Pk​,Pk​) is Accept , which proves that x is actually πkυs2k​​, and continues to the next iteration of the loop.","breadcrumbs":"STARKs » Protocol » Redundant values in the proof","id":"101","title":"Redundant values in the proof"},"102":{"body":"The goal of this section will be to go over the details of the implementation of the proving system. To this end, we will follow the flow the example in the recap chapter, diving deeper into the code when necessary and explaining how it fits into a more general case. This implementation couldn't be done without checking Facebook's Winterfell and Max Gillett's Giza . We want to thank everyone involved in them, along with Shahar Papini and Lior Goldberg from Starkware who also provided us valuable insight.","breadcrumbs":"STARKs » Implementation » STARKs Prover Lambdaworks Implementation","id":"102","title":"STARKs Prover Lambdaworks Implementation"},"103":{"body":"Let's go over the main test we use for our prover, where we compute a STARK proof for a fibonacci trace with 4 rows and then verify it. fn test_prove_fib() { let trace = simple_fibonacci::fibonacci_trace([FE::from(1), FE::from(1)], 8); let proof_options = ProofOptions::default_test_options(); let pub_inputs = FibonacciPublicInputs { a0: FE::one(), a1: FE::one(), }; let proof = prove::>(&trace, &pub_inputs, &proof_options).unwrap(); assert!(verify::>(&proof, &pub_inputs, &proof_options));\n} The proving system revolves around the prove function, that takes a trace, public inputs and proof options as inputs to generate a proof, and a verify function that takes the generated proof, the public inputs and the proof options as inputs, outputting true when the proof is verified correctly and false otherwise. Note that the public inputs and proof options should be the same for both. Public inputs should be shared by the Cairo runner to prover and verifier, and the proof options should have been agreed on beforehand by the two entities beforehand. Below we go over the main things involved in this code.","breadcrumbs":"STARKs » Implementation » High Level API » High level API: Fibonacci example","id":"103","title":"High level API: Fibonacci example"},"104":{"body":"To prove the integrity of a fibonacci trace, we first need to define what it means for a trace to be valid. As we've talked about in the recap, this involves defining an AIR for our computation where we specify both the boundary and transition constraints for a fibonacci sequence. In code, this is done through the AIR trait. Implementing AIR requires defining a couple methods, but the two most important ones are boundary_constraints and compute_transition, which encode the boundary and transition constraints of our computation.","breadcrumbs":"STARKs » Implementation » High Level API » AIR","id":"104","title":"AIR"},"105":{"body":"For our Fibonacci AIR, boundary constraints look like this: fn boundary_constraints( &self, _rap_challenges: &Self::RAPChallenges,\n) -> BoundaryConstraints { let a0 = BoundaryConstraint::new_simple(0, self.pub_inputs.a0.clone()); let a1 = BoundaryConstraint::new_simple(1, self.pub_inputs.a1.clone()); BoundaryConstraints::from_constraints(vec![a0, a1])\n} The BoundaryConstraint struct represents a specific boundary constraint, meaning \"column i at row j should be equal to x\". In this case, because we have only one column, we are using the new_simple method to simply say Row 0 should equal the public input a0, which in the typical fibonacci is set to 1. Row 1 should equal the public input a1, which in the typical fibonacci is set to 1. In the case of multiple columns, the new method exists so you can also specify column number. After instantiating each of these constraints, we return all of them through the struct BoundaryConstraints.","breadcrumbs":"STARKs » Implementation » High Level API » Boundary Constraints","id":"105","title":"Boundary Constraints"},"106":{"body":"The way we specify our fibonacci transition constraint looks like this: fn compute_transition( &self, frame: &air::frame::Frame, _rap_challenges: &Self::RAPChallenges,\n) -> Vec> { let first_row = frame.get_row(0); let second_row = frame.get_row(1); let third_row = frame.get_row(2); vec![third_row[0] - second_row[0] - first_row[0]]\n} It's not completely obvious why this is how we chose to express transition constraints, so let's talk a little about it. What we need to specify in this method is the relationship that has to hold between the current step of computation and the previous ones. For this, we get a Frame as an argument. This is a struct holding the current step (i.e. the current row of the trace) and all previous ones needed to encode our constraint. In our case, this is the current row and the two previous ones. To access rows we use the get_row method. The current step is always the last row (in our case 2), with the others coming before it. In our compute_transition method we get the three rows we need and return third_row[0] - second_row[0] - first_row[0] which is the value that needs to be zero for our constraint to hold. Because we support multiple transition constraints, we actually return a vector with one value per constraint, so the first element holds the first constraint value and so on.","breadcrumbs":"STARKs » Implementation » High Level API » Transition Constraints","id":"106","title":"Transition Constraints"},"107":{"body":"After defining our AIR, we create our specific trace to prove against it. let trace = fibonacci_trace([FE17::new(1), FE17::new(1)], 4); let trace_table = TraceTable { table: trace.clone(), num_cols: 1,\n}; TraceTable is the struct holding execution traces; the num_cols says how many columns the trace has, the table field is a vec holding the actual values of the trace in row-major form, meaning if the trace looks like this | 1 | 2 |\n| 3 | 4 |\n| 5 | 6 | then its corresponding TraceTable is let trace_table = TraceTable { table: vec![1, 2, 3, 4, 5, 6], num_cols: 2,\n}; In our example, fibonacci_trace is just a helper function we use to generate the fibonacci trace with 4 rows and [1, 1] as the first two values.","breadcrumbs":"STARKs » Implementation » High Level API » TraceTable","id":"107","title":"TraceTable"},"108":{"body":"After specifying our constraints and trace, the only thing left to do is provide a few parameters related to the STARK protocol and our AIR. These specify things such as the number of columns of the trace and proof configuration, among others. They are all encapsulated in the AirContext struct, which in our example we instantiate like this: let context = AirContext { options: ProofOptions { blowup_factor: 2, fri_number_of_queries: 1, coset_offset: 3, }, trace_columns: trace_table.n_cols, transition_degrees: vec![1], transition_exemptions: vec![2], transition_offsets: vec![0, 1, 2], num_transition_constraints: 1,\n}; Let's go over each of them: options requires a ProofOptions struct holding specific parameters related to the STARK protocol to be used when proving. They are: The blowup_factor used for the trace LDE extension, a parameter related to the security of the protocol. The number of queries performed by the verifier when doing FRI, also related to security. The offset used for the LDE coset. This depends on the field being used for the STARK proof. trace_columns are the number of columns of the trace, respectively. transition_degrees holds the degree of each transition constraint. transition_exemptions is a Vec which tells us, for each column, the number of rows the transition constraints should not apply, starting from the end of the trace. In the example, the transition constraints won't apply on the last two rows of the trace. transition_offsets holds the indexes that define a frame for our AIR. In our fibonacci case, these are [0, 1, 2] because we need the current row and the two previous one to define our transition constraint. num_transition_constraints simply says how many transition constraints our AIR has.","breadcrumbs":"STARKs » Implementation » High Level API » AIR Context","id":"108","title":"AIR Context"},"109":{"body":"Having defined all of the above, proving our fibonacci example amounts to instantiating the necessary structs and then calling prove passing the trace, public inputs and proof options. We use a simple implementation of a hasher called TestHasher to handle merkle proof building. let proof = prove(&trace_table, &pub_inputs, &proof_options); Verifying is then done by passing the proof of execution along with the same AIR to the verify function. assert!(verify(&proof, &pub_inputs, &proof_options));","breadcrumbs":"STARKs » Implementation » High Level API » Proving execution","id":"109","title":"Proving execution"},"11":{"body":"Our matrices are fine now. But they can be optimized. Let's do that to showcase this flexibility of PLONK and also reduce the size of our example. PLONK has the flexibility to construct more sophisticated gates as combinations of the five columns. And therefore the same program can be expressed in multiple ways. In our case all three gates can actually be merged into a single custom gate. The Q matrix ends up being a single row. QL​ QR​ QM​ QO​ QC​ 1 1 1 -1 1 and also the V matrix L R O 0 1 2 The trace matrix for this representation is just A B C 2 3 8 And we check that it satisfies the equation 2×1+3×1+2×3×1+8×(−1)+(−1)=0 Of course, we can't always squash an entire program into a single gate.","breadcrumbs":"Plonk » Recap » Custom gates","id":"11","title":"Custom gates"},"110":{"body":"In this section we go over how a few things in the prove and verify functions are implemented. If you just need to use the prover, then you probably don't need to read this. If you're going through the code to try to understand it, read on. We will once again use the fibonacci example as an ilustration. Recall from the recap that the main steps for the prover and verifier are the following:","breadcrumbs":"STARKs » Implementation » Under the hood » How this works under the hood","id":"110","title":"How this works under the hood"},"111":{"body":"Compute the trace polynomial t by interpolating the trace column over a set of 2n-th roots of unity {gi:0≤i<2n}. Compute the boundary polynomial B. Compute the transition constraint polynomial C. Construct the composition polynomial H from B and C. Sample an out of domain point z and provide the evaluation H(z) and all the necessary trace evaluations to reconstruct it. In the fibonacci case, these are t(z), t(zg), and t(zg2). Sample a domain point x_0 and provide the evaluation H(x0​) and t(x0​). Construct the deep composition polynomial Deep(x) from H, t, and the evaluations from the item above. Do FRI on Deep(x) and provide the resulting FRI commitment to the verifier. Provide the merkle root of t and the merkle proof of t(x0​).","breadcrumbs":"STARKs » Implementation » Under the hood » Prover side","id":"111","title":"Prover side"},"112":{"body":"Take the evaluation H(z) along with the trace evaluations the prover provided. Reconstruct the evaluations B(z) and C(z) from the trace evaluations. Check that the claimed evaluation H(z) the prover gave us actually satisfies H(z)=β1​B(z)+β2​C(z) Take the evaluations H(x0​) and t(x0​). Check that the claimed evaluation Deep(x0​) the prover gave us actually satisfies Deep(x0​)=γ1​x0​−zH(x0​)−H(z)​+γ2​x0​−zt(x0​)−t(z)​+γ3​x0​−zgt(x0​)−t(zg)​+γ4​x0​−zg2t(x0​)−t(zg2)​ Take the provided FRI commitment and check that it verifies. Using the merkle root and the merkle proof the prover provided, check that t(x0​) belongs to the trace. Following along the code in the prove and verify functions, most of it maps pretty well to the steps above. The main things that are not immediately clear are: How we take the constraints defined in the AIR through the compute_transition method and map them to transition constraint polynomials. How we then construct H from them and the boundary constraint polynomials. What the composition polynomial even/odd decomposition is. What an ood frame is. What the transcript is.","breadcrumbs":"STARKs » Implementation » Under the hood » Verifier side","id":"112","title":"Verifier side"},"113":{"body":"This is possibly the most complex part of the code, so what follows is a long explanation for it. In our fibonacci example, after obtaining the trace polynomial t by interpolating, the transition constraint polynomial is C(x)=∏i=05​(x−gi)t(xg2)−t(xg)−t(x)​ On our prove code, if someone passes us a fibonacci AIR like the one we showed above used in one of our tests, we somehow need to construct C(x). However, what we are given is not a polynomial, but rather this method fn compute_transition( &self, frame: &air::frame::Frame, ) -> Vec> { let first_row = frame.get_row(0); let second_row = frame.get_row(1); let third_row = frame.get_row(2); vec![third_row[0] - second_row[0] - first_row[0]]\n} So how do we get to C(x) from this? The answer is interpolation. What the method above is doing is the following: if you pass it a frame that looks like this ​t(x0​)t(x0​g)t(x0​g2)​​ for any given point x0​, it will return the value t(x0​g2)−t(x0​g)−t(x0​) which is the numerator in C(x0​). Using the transition_exemptions field we defined in our AIR, we can also compute evaluations in the denominator, i.e. the zerofier evaluations. This is done under the hood by the transition_divisors() method. The above means that even though we don't explicitly have the polynomial C(x), we can evaluate it on points given an appropriate frame. If we can evaluate it on enough points, we can then interpolate them to recover C(x). This is exactly how we construct both transition constraint polynomials and subsequently the composition polynomial H. The job of evaluating H on enough points so we can then interpolate it is done by the ConstraintEvaluator struct. You'll notice prove does the following let constraint_evaluations = evaluator.evaluate( &lde_trace, &lde_roots_of_unity_coset, &alpha_and_beta_transition_coefficients, &alpha_and_beta_boundary_coefficients,\n); This function call will return the evaluations of the boundary terms βiB​Bi​(x) and constraint terms βiT​Ci​(x) for every i. The constraint_evaluations value returned is a ConstraintEvaluationTable struct, which is nothing more than a big list of evaluations of each polynomial required to construct H. With this in hand, we just call let composition_poly = constraint_evaluations.compute_composition_poly(& lde_roots_of_unity_coset); which simply interpolates the sum of all evaluations to obtain H. Let's go into more detail on how the evaluate method reconstructs C(x) in our fibonacci example. It receives the lde_trace as an argument, which is this: ​t(ω0)t(ω1)…t(ω15)​​ where ω is the primitive root of unity used for the LDE, that is, ω satisfies ω2=g. We need to recover C(x), a polynomial whose degree can't be more than t(x)'s. Because t was built by interpolating 8 points (the trace), we know we can recover C(x) by interpolating it on 16 points. We choose these points to be the LDE roots of unity {ω0,ω,ω2,…,ω15} Remember that to evaluate C(x) on these points, all we need are the evaluations of the polynomial t(xg2)−t(xg)−t(x) as the zerofier ones we can compute easily. These become: t(ω0g2)−t(ω0g)−t(ω0)t(ωg2)−t(ωg)−t(ω)t(ω2g2)−t(ω2g)−t(ω2)⋮t(ω15g2)−t(ω15g)−t(ω15) If we remember that ω2=g, this is t(ω4)−t(ω2)−t(ω0)t(ω5)−t(ω3)−t(ω)t(ω6)−t(ω4)−t(ω2)⋮t(ω3)−t(ω)−t(ω15) and we can compute each evaluation here by calling compute_transition on the appropriate frame built from the lde_trace. Specifically, for the first evaluation we can build the frame: ​t(ω0)t(ω2)t(ω4)​​ Calling compute_transition on this frame gives us the first evaluation. We can get the rest in a similar fashion, which is what this piece of code in the evaluate method does: for (i, d) in lde_domain.iter().enumerate() { let frame = Frame::read_from_trace( lde_trace, i, blowup_factor, &self.air.context().transition_offsets, ) let mut evaluations = self.air.compute_transition(&frame); ...\n} Each iteration builds a frame as above and computes one of the evaluations needed. The rest of the code just adds the zerofier evaluations, along with the alphas and betas. It then also computes boundary polynomial evaluations by explicitly constructing them.","breadcrumbs":"STARKs » Implementation » Under the hood » Reconstructing the transition constraint polynomials","id":"113","title":"Reconstructing the transition constraint polynomials"},"114":{"body":"The verifier employs the same trick to reconstruct the evaluations on the out of domain point Ci​(z) for the consistency check.","breadcrumbs":"STARKs » Implementation » Under the hood » Verifier","id":"114","title":"Verifier"},"115":{"body":"At the end of the recap we talked about how in our code we don't actually commit to H, but rather an even/odd decomposition for it. These are two polynomials H_1 and H_2 that satisfy H(x)=H1​(x2)+xH2​(x2) This all happens on this piece of code let composition_poly = constraint_evaluations.compute_composition_poly(&lde_roots_of_unity_coset); let (composition_poly_even, composition_poly_odd) = composition_poly.even_odd_decomposition(); // Evaluate H_1 and H_2 in z^2.\nlet composition_poly_evaluations = vec![ composition_poly_even.evaluate(&z_squared), composition_poly_odd.evaluate(&z_squared),\n]; After this, we don't really use H anymore, but rather H_1 and H_2. There's not that much to say other than that.","breadcrumbs":"STARKs » Implementation » Under the hood » Even/odd decomposition for H","id":"115","title":"Even/odd decomposition for H"},"116":{"body":"As part of the consistency check, the prover needs to provide evaluations of the trace polynomials in all the points needed by the verifier to check that H was constructed correctly. In the fibonacci example, these are t(z), t(zg), and t(zg2). In code, the prover passes these evaluations as a Frame, which we call the out of domain (ood) frame. The reason we do this is simple: with the frame in hand, the verifier can reconstruct the evaluations of the constraint polynomials Ci​(z) by calling the compute_transition method on the ood frame and then adding the alphas, betas, and so on, just like we explained in the section above.","breadcrumbs":"STARKs » Implementation » Under the hood » Out of Domain Frame","id":"116","title":"Out of Domain Frame"},"117":{"body":"Throughout the protocol, there are a number of times where the verifier randomly samples some values that the prover needs to use (think of the alphas and betas used when constructing H). Because we don't actually have an interaction between prover and verifier, we emulate it by using a hash function, which we assume is a source of randomness the prover can't control. The job of providing these samples for both prover and verifier is done by the Transcript struct, which you can think of as a stateful rng; whenever you call challenge() on a transcript you get a random value and the internal state gets mutated, so the next time you call challenge() you get a different one. You can also call append on it to mutate its internal state yourself. This is done a number of times throughout the protocol to keep the prover honest so it can't predict or manipulate the outcome of challenge(). Notice that to sample the same values, both prover and verifier need to call challenge and append in the same order (and with the same values in the case of append) and the same number of times. The idea explained above is called the Fiat-Shamir heuristic or just Fiat-Shamir, and is more generally used throughout proving systems to remove interaction between prover and verifier. Though the concept is very simple, getting it right so the prover can't cheat is not, but we won't go into that here.","breadcrumbs":"STARKs » Implementation » Under the hood » Transcript","id":"117","title":"Transcript"},"118":{"body":"The generated proof has got all the information needed for the verifier to verify it: Trace length: The number of rows of the trace table, needed to know the max degree of the polynomials that appear in the system. LDE trace commitments. DEEP composition polynomial out of domain even and odd evaluations. DEEP composition polynomial root. FRI layers merkle roots. FRI last layer value. Query list. DEEP composition poly openings. Nonce: Proof of work setting used to generate the proof.","breadcrumbs":"STARKs » Implementation » Under the hood » Proof","id":"118","title":"Proof"},"119":{"body":"","breadcrumbs":"STARKs » Implementation » Under the hood » Special considerations","id":"119","title":"Special considerations"},"12":{"body":"Aside from the gates that execute the program operations, additional rows must be incorporated into these matrices. This is due to the fact that the prover must demonstrate not only that she executed the program, but also that she used the appropriate inputs. Furthermore, the proof must include an assertion of the output value. As a result, a few extra rows are necessary. In our case these are the first two and the last one. The original one sits now in the third row. QL​ QR​ QM​ QO​ QC​ -1 0 0 0 3 -1 0 0 0 8 1 1 1 -1 1 1 -1 0 0 0 And this is the updated V matrix L R O 0 - - 1 - - 2 0 3 1 3 - The first row is there to force the variable with index 0 to take the value 3. Similarly the second row forces variable with index 1 to take the value 8. These two will be the public inputs of the verifier. The last row checks that the output of the program is the claimed one. And the trace matrix is now A B C 3 - - 8 - - 2 3 8 8 8 - With these extra rows, equations add up to zero only for valid executions of the program with input 3 and output 8. An astute observer would notice that by incorporating these new rows, the matrix Q is no longer independent of the specific evaluation. This is because the first two rows of the QC​ column contain concrete values that are specific to a particular execution instance. To maintain independence, we can remove these values and consider them as part of an extra one-column matrix called PI (stands for Public Input). This column has zeros in all rows not related to public inputs. We put zeros in the QC​ columns. The responsibility of filling in the PI matrix is of the prover and verifier. In our example it is PI 3 8 0 0 And the final Q matrix is QL​ QR​ QM​ QO​ QC​ -1 0 0 0 0 -1 0 0 0 0 1 1 1 -1 1 1 -1 0 0 0 We ended up with two matrices that depend only on the program, Q and V. And two matrices that depend on a particular evaluation, namely the ABC and PI matrices. The updated version of the claim is the following: Claim: Let T be a matrix with columns A,B,C. It corresponds to a evaluation of the circuit if and only if a) for all i the following equality holds Ai​(QL​)i​+Bi​(QR​)i​+Ai​Bi​QM​+Ci​(QO​)i​+(QC​)i​+(PI)i​=0, b) for all i,j,k,l such that Vi,j​=Vk,l​ we have Ti,j​=Tk,l​.","breadcrumbs":"Plonk » Recap » Public inputs","id":"12","title":"Public inputs"},"120":{"body":"When evaluating or interpolating a polynomial, if the input (be it coefficients or evaluations) size isn't a power of two then the FFT API will extend it with zero padding until this requirement is met. This is because the library currently only uses a radix-2 FFT algorithm. Also, right now FFT only supports inputs with a size up to 2264 elements.","breadcrumbs":"STARKs » Implementation » Under the hood » FFT evaluation and interpolation","id":"120","title":"FFT evaluation and interpolation"},"121":{"body":"","breadcrumbs":"STARKs » Implementation » Under the hood » Other","id":"121","title":"Other"},"122":{"body":"Whenever we interpolate or evaluate trace, boundary and constraint polynomials, we use some 2n-th roots of unity. There are a few reasons for this: Using roots of unity means we can use the Fast Fourier Transform and its inverse to evaluate and interpolate polynomials. This method is much faster than the naive Lagrange interpolation one. Since a huge part of the STARK protocol involves both evaluating and interpolating, this is a huge performance improvement. When computing boundary and constraint polynomials, we divide them by their zerofiers, polynomials that vanish on a few points (the trace elements where the constraints do not hold). These polynomials take the form Z(X)=∏(X−xi​) where the xi​ are the points where we want it to vanish. When implementing this, evaluating this polynomial can be very expensive as it involves a huge product. However, if we are using roots of unity, we can use the following trick. The vanishing polynomial for all the 2n roots of unity is X2n−1 Instead of expressing the zerofier as a product of the places where it should vanish, we express it as the vanishing polynomial above divided by the exemptions polynomial; the polynomial whose roots are the places where constraints don't need to hold. Z(X)=∏(X−ei​)X2n−1​ where the ei​ are now the points where we don't want it to vanish. This exemptions polynomial in the denominator is usually much smaller, and because the vanishing polynomial in the numerator is only two terms, evaluating it is really fast.","breadcrumbs":"STARKs » Implementation » Under the hood » Why use roots of unity?","id":"122","title":"Why use roots of unity?"},"123":{"body":"The n-th roots of unity are the numbers x that satisfy xn=1 There are n such numbers, because they are the roots of the polynomial Xn−1. The set of n-th roots of unity always has a generator, a root g that can be used to obtain every other root of unity by exponentiating. What this means is that the set of n-th roots of unity is {gi:0≤i1 and 0≤j≤J and I=∣Lj′′​∣. Extend M with additional L′′ rows to obtain a matrix M∈F(L+L′+L′′)×33 by appending copies of R′′ at the bottom. Pad M with copies of its last row until it has a power of two number of rows. As a result we obtain a matrix T∈F2n×33.","breadcrumbs":"Cairo » Trace Formal Description » Construction of execution trace T:","id":"127","title":"Construction of execution trace T:"},"128":{"body":"","breadcrumbs":"Cairo » Trace Detailed Description » Cairo execution trace","id":"128","title":"Cairo execution trace"},"129":{"body":"After the execution of a Cairo program in the Cairo VM, three files are generated that are the core components for the construction of the execution trace, needed for the proving system: trace file : Has the information on the state of the three Cairo VM registers ap, fp, and pc at every cycle of the execution of the program. To reduce ambiguity in terms, we should call these the register states of the Cairo VM, and leave the term trace to the final product that is passed to the prover to generate a proof. memory file : A file with the information of the VM's memory at the end of the program run, after the memory has been relocated. public inputs : A file with all the information that must be publicly available to the prover and verifier, such as the total number of execution steps, public memory, used builtins and their respective addresses range in memory. The next section will explain in detail how these elements are used to build the final execution trace.","breadcrumbs":"Cairo » Trace Detailed Description » Raw materials","id":"129","title":"Raw materials"},"13":{"body":"In the previous section we showed how the arithmetization process works in PLONK. For a program with n public inputs and m gates, we constructed two matrices Q and V, of sizes (n+m+1)×5 and (n+m+1)×3 that satisfy the following. Let N=n+m+1. Claim: Let T be a N×3 matrix with columns A,B,C and PI a N×1 matrix. They correspond to a valid execution instance with public input given by PI if and only if a) for all i the following equality holds Ai​(QL​)i​+Bi​(QR​)i​+Ai​Bi​QM​+Ci​(QO​)i​+(QC​)i​+(PI)i​=0, b) for all i,j,k,l such that Vi,j​=Vk,l​ we have Ti,j​=Tk,l​, c) (PI)i​=0 for all i>n. Polynomials enter now to squash most of these equations. We will traduce the set of all equations in conditions (a) and (b) to just a few equations on polynomials. Let ω be a primitive N-th root of unity and let H=ωi:0≤in. Then we constructed polynomials qL​,qR​,qM​,qO​,qC​,Sσ1​,Sσ2​,Sσ3​, f, g off the matrices Q and V. They are the result of interpolating at a domain H={1,ω,ω2,…,ωN−1} for some N-th primitive root of unity and a few random values. And also constructed polynomials a,b,c,pi off the matrices T and PI. Loosely speaking, the above fact can be reformulated in terms of polynomial equations as follows. Fact: Let zH​=XN−1. Let T be a N×3 matrix with columns A,B,C and PI a N×1 matrix. They correspond to a valid execution instance with public input given by PI if and only if a) There is a polynomial t1​ such that the following equality holds aqL​+bqR​+abqM​+cqO​+qC​+pi=zH​t1​, b) There are polynomials t2​,t3​, z such that zf−gz′=zH​t2​ and (z−1)L1​=zH​t3​, where z′(X)=z(Xω) You might be wondering where the polynomials ti​ came from. Recall that for a polynomial F, we have F(h)=0 for all h∈H if and only if F=zH​t for some polynomial t. Finally both conditions (a) and (b) are equivalent to a single equation (c) if we let more randomness to come into play. This is: (c) Let α be a random field element. There is a polynomial t such that zH​t=​aqL​+bqR​+abqM​+cqO​+qC​+pi+α(gz′−fz)+α2(z−1)L1​​ This last step is not obvious. You can check the paper to see the proof. Anyway, this is the equation you'll recognize below in the description of the protocol. Randomness is a delicate matter and an important part of the protocol is where it comes from, who chooses it and when they choose it. Check out the protocol to see how it works.","breadcrumbs":"Plonk » Recap » Wrapping up the whole thing","id":"15","title":"Wrapping up the whole thing"},"16":{"body":"","breadcrumbs":"Plonk » Protocol » Protocol","id":"16","title":"Protocol"},"17":{"body":"","breadcrumbs":"Plonk » Protocol » Details and tricks","id":"17","title":"Details and tricks"},"18":{"body":"A polynomial commitment scheme (PCS) is a cryptographic tool that allows one party to commit to a polynomial, and later prove properties of that polynomial. This commitment polynomial hides the original polynomial's coefficients and can be publicly shared without revealing any information about the original polynomial. Later, the party can use the commitment to prove certain properties of the polynomial, such as that it satisfies certain constraints or that it evaluates to a certain value at a specific point. In the implementation section we'll explain the inner workings of the Kate-Zaverucha-Goldberg scheme, a popular PCS chosen in Lambdaworks for PLONK. For the moment we only need the following about it: It consists of a finite group G and the following algorithms: Commit(f) : This algorithm takes a polynomial f and produces an element of the group G. It is called the commitment of f and is denoted by [f]1​. It is homomorphic in the sense that [f+g]1​=[f]1​+[g]1​. The former sum being addition of polynomials. The latter is addition in the group G. Open(f,ζ) : It takes a polynomial f and a field element ζ and produces an element π of the group G. This element is called an opening proof for f(ζ). It is the proof that f evaluated at ζ gives f(ζ). Verify([f]1​, π, ζ, y) : It takes group elements [f]1​ and π, and also field elements ζ and y. With overwhelming probability it outputs Accept if f(z)=y and Reject otherwise.","breadcrumbs":"Plonk » Protocol » Polynomial commitment scheme","id":"18","title":"Polynomial commitment scheme"},"19":{"body":"As you will see in the protocol, the prover reveals the value taken by a bunch of the polynomials at a random ζ. In order for the protocol to be Honest Verifier Zero Knowledge , these polynomials need to be blinded . This is a process that makes the values of these polynomials at ζ seemingly random by forcing them to be of certain degree. Here's how it works. Let's take for example the polynomial a the prover constructs. This is the interpolation of the first column of the trace matrix T at the domain H. This matrix has all of the left operands of all the gates. The prover wishes to keep them secret. Say the trace matrix T has N rows. And so H is {1,ω,ω2,…,ωN−1}. The invariant that the prover cannot violate is that ablinded​(ωi) must take the value T0,i​, for all i. This is what the interpolation polynomial a satisfies. And is the unique such polynomial of degree at most N−1 with such property. But for higher degrees, there are many such polynomials. The blinding process takes a and a desired degree M≥N, and produces a new polynomial ablinded​ of degree exactly M. This new polynomial satisfies that ablinded​(ωi)=a(ωi) for all i. But outside H differs from a. This may seem hard but it's actually very simple. Let zH​ be the polynomial zH​=XN−1. If M=N+k, with k≥0, then sample random values b0​,…,bk​ and define ablinded​:=(b0​+b1​X+⋯+bk​Xk)zH​+a The reason why this does the job is that zH​(ωi)=0 for all i. Therefore the added term vanishes at H and leaves the values of a at H unchanged.","breadcrumbs":"Plonk » Protocol » Blindings","id":"19","title":"Blindings"},"2":{"body":"Three methods of polynomial interpolation were benchmarked, with different input sizes each time: CPU Lagrange: Finding the Lagrange polynomial of a set of random points via a naive algorithm (see math/src/polynomial.rs:interpolate()) CPU FFT: Finding the lowest degree polynomial that interpolates pairs of twiddle factors and Fourier coefficients (the results of applying the Fourier transform to the coefficients of a polynomial) (see math/src/polynomial.rs:interpolate_fft()). GPU FFT (Metal): Same as CPU FFT but the FFT algorithm is run on the GPU via the Metal API (Apple). these were run with criterion-rs in a MacBook Pro M1 (18.3), statistically measuring the total run time of one iteration. The field used was a 256 bit STARK-friendly prime field. All values of time are in milliseconds. Those cases which were greater than 30 seconds were marked respectively as they're too slow and weren't worth to be benchmarked. The input size refers to d + 1 where d is the polynomial's degree (so size is amount of coefficients). Input size CPU Lagrange CPU FFT GPU FFT (Metal) 2^4 2.2 ms 0.2 ms 2.5 ms 2^5 9.6 ms 0.4 ms 2.5 ms 2^6 42.6 ms 0.8 ms 2.5 ms 2^7 200.8 ms 1.7 ms 2.9 ms ... ... .. .. 2^21 >30000 ms 28745 ms 574.2 ms 2^22 >30000 ms >30000 ms 1144.9 ms 2^23 >30000 ms >30000 ms 2340.1 ms 2^24 >30000 ms >30000 ms 4652.9 ms NOTE: Metal FFT execution includes the Metal state setup, twiddle factors generation and a bit-reverse permutation.","breadcrumbs":"FFT Library » Benchmarks » Polynomial interpolation methods comparison","id":"2","title":"Polynomial interpolation methods comparison"},"20":{"body":"This is an optimization in PLONK to reduce the number of checks of the verifier. One of the main checks in PLONK boils down to check that p(ζ)=zH​(ζ)t(ζ), with p some polynomial that looks like p=aqL​+bqR​+abqM​+⋯, and so on. In particular the verifier needs to get the value p(ζ) from somewhere. For the sake of simplicity, in this section assume p is exactly aqL​+bqR​. Secret to the prover here are only a,b. The polynomials qL​ and qR​ are known also to the verifier. The verifier will already have the commitments [a]1​,[b]1​,[qL​]1​ and [qR​]1​. So the prover could send just a(ζ), b(ζ) along with their opening proofs and let the verifier compute by himself qL​(ζ) and qR​(ζ). Then with all these values the verifier could compute p(ζ)=a(ζ)qL​(ζ)+b(ζ)qR​(ζ). And also use his commitments to validate the opening proofs of a(ζ) and b(ζ). This has the problem that computing qL​(ζ) and qR​(ζ) is expensive. The prover can instead save the verifier this by sending also qL​(ζ),qR​(ζ) along with opening proofs. Since the verifier will have the commitments [qL​]1​ and [qR​]1​ beforehand, he can check that the prover is not cheating and cheaply be convinced that the claimed values are actually qL​(ζ) and qR​(ζ). This is much better. It involves the check of four opening proofs and the computation of p(ζ) off the values received from the prover. But it can be further improved as follows. As before, the prover sends a(ζ),b(ζ) along with their opening proofs. She constructs the polynomial f=a(ζ)qL​+b(ζ)qR​. She sends the value f(ζ) along with an opening proof of it. Notice that the value of f(ζ) is exactly p(ζ). The verifier can compute by himself [f]1​ as a(ζ)[qL​]1​+b(ζ)[qR​]1​. The verifier has everything to check all three openings and get convinced that the claimed value f(ζ) is true. And this value is actually p(ζ). So this means no more work for the verifier. And the whole thing got reduced to three openings. This is called the linearization trick. The polynomial f is called the linearization of p.","breadcrumbs":"Plonk » Protocol » Linearization trick","id":"20","title":"Linearization trick"},"21":{"body":"There's a one time setup phase to compute some values common to any execution and proof of the particular circuit. Precisely, the following commitments are computed and published. [qL​]1​,[qR​]1​,[qM​]1​,[qO​]1​,[qC​]1​,[Sσ1​]1​,[Sσ2​]1​,[Sσ3​]1​","breadcrumbs":"Plonk » Protocol » Setup","id":"21","title":"Setup"},"22":{"body":"Next we describe the proving algorithm for a program of size N. That includes public inputs. Let ω be a primitive N-th root of unity. Let H={1,ω,ω2,…,ωN−1}. Define ZH​:=XN−1. Assume the eight polynomials of common preprocessed input are already given. The prover computes the trace matrix T as described in the first sections. That means, with the first rows corresponding to the public inputs. It should be a N×3 matrix.","breadcrumbs":"Plonk » Protocol » Proving algorithm","id":"22","title":"Proving algorithm"},"23":{"body":"Add to the transcript the following: [Sσ1​]1​,[Sσ2​]1​,[Sσ3​]1​,[qL​]1​,[qR​]1​,[qM​]1​,[qO​]1​,[qC​]1​ Compute polynomials a′,b′,c′ as the interpolation polynomials of the columns of T at the domain H. Sample random b1​,b2​,b3​,b4​,b5​,b6​ Let a:=(b1​X+b2​)ZH​+a′ b:=(b3​X+b4​)ZH​+b′ c:=(b5​X+b6​)ZH​+c′ Compute [a]1​,[b]1​,[c]1​ and add them to the transcript.","breadcrumbs":"Plonk » Protocol » Round 1","id":"23","title":"Round 1"},"24":{"body":"Sample β,γ from the transcript. Let z0​=1 and define recursively for 0≤k { pub n: usize, pub domain: Vec>, pub omega: FieldElement, pub k1: FieldElement, pub ql: Polynomial>, pub qr: Polynomial>, pub qo: Polynomial>, pub qm: Polynomial>, pub qc: Polynomial>, pub s1: Polynomial>, pub s2: Polynomial>, pub s3: Polynomial>, pub s1_lagrange: Vec>, pub s2_lagrange: Vec>, pub s3_lagrange: Vec>,\n} Apart from the eight polynomials in the canonical basis, we store also here the number of constraints n, the domain H, the primitive n-th of unity ω and the element k1​. The element k2​ will be k12​. For convenience, we also store the polynomials Sσi​ in Lagrange form. The following lines define the particular values of the program input x and the private input e. // Input\nlet x = FieldElement::from(2_u64); // Private input\nlet e = FieldElement::from(3_u64);\nlet y, witness = test_witness_2(x, e); The function test_witness_2(x, e) returns an instance of the following struct, that holds the polynomials that interpolate the columns A,B,C of the trace matrix T. pub struct Witness { pub a: Vec>, pub b: Vec>, pub c: Vec>,\n} Next the commitment scheme KZG (Kate-Zaverucha-Goldberg) is instantiated. let srs = test_srs(common_preprocessed_input.n);\nlet kzg = KZG::new(srs); The setup function performs the setup phase. It only needs the common preprocessed input and the commitment scheme. let verifying_key = setup(&common_preprocessed_input, &kzg); It outputs an instance of the struct VerificationKey. pub struct VerificationKey { pub qm_1: G1Point, pub ql_1: G1Point, pub qr_1: G1Point, pub qo_1: G1Point, pub qc_1: G1Point, pub s1_1: G1Point, pub s2_1: G1Point, pub s3_1: G1Point,\n} It stores the commitments of the eight polynomials of the common preprocessed input. The suffix _1 means it is a commitment. It comes from the notation [f]1​, where f is a polynomial. Then a prover is instantiated let random_generator = TestRandomFieldGenerator {};\nlet prover = Prover::new(kzg.clone(), random_generator); The prover is an instance of the struct Prover: pub struct Prover\nwhere F: IsField, CS: IsCommitmentScheme, R: IsRandomFieldElementGenerator { commitment_scheme: CS, random_generator: R, phantom: PhantomData,\n} It stores an instance of a commitment scheme and a random field element generator needed for blinding polynomials. Then the public input is defined. As we mentioned in the recap, the public input contains the output of the program. let public_input = vec![x.clone(), y]; We then generate a proof using the prover's method prove let proof = prover.prove( &witness, &public_input, &common_preprocessed_input, &verifying_key,\n); The output is an instance of the struct Proof. pub struct Proof> { // Round 1. /// Commitment to the wire polynomial `a(x)` pub a_1: CS::Commitment, /// Commitment to the wire polynomial `b(x)` pub b_1: CS::Commitment, /// Commitment to the wire polynomial `c(x)` pub c_1: CS::Commitment, // Round 2. /// Commitment to the copy constraints polynomial `z(x)` pub z_1: CS::Commitment, // Round 3. /// Commitment to the low part of the quotient polynomial t(X) pub t_lo_1: CS::Commitment, /// Commitment to the middle part of the quotient polynomial t(X) pub t_mid_1: CS::Commitment, /// Commitment to the high part of the quotient polynomial t(X) pub t_hi_1: CS::Commitment, // Round 4. /// Value of `a(ζ)`. pub a_zeta: FieldElement, /// Value of `b(ζ)`. pub b_zeta: FieldElement, /// Value of `c(ζ)`. pub c_zeta: FieldElement, /// Value of `S_σ1(ζ)`. pub s1_zeta: FieldElement, /// Value of `S_σ2(ζ)`. pub s2_zeta: FieldElement, /// Value of `z(ζω)`. pub z_zeta_omega: FieldElement, // Round 5 /// Value of `p_non_constant(ζ)`. pub p_non_constant_zeta: FieldElement, /// Value of `t(ζ)`. pub t_zeta: FieldElement, /// Batch opening proof for all the evaluations at ζ pub w_zeta_1: CS::Commitment, /// Single opening proof for `z(ζω)`. pub w_zeta_omega_1: CS::Commitment,\n} Finally, we instantiate a verifier. let verifier = Verifier::new(kzg); It's an instance of Verifier: struct Verifier> { commitment_scheme: CS, phantom: PhantomData,\n} Finally, we call the verifier's method verify that outputs a bool. assert!(verifier.verify( &proof, &public_input, &common_preprocessed_input, &verifying_key\n));","breadcrumbs":"Plonk » Implementation » Usage","id":"34","title":"Usage"},"35":{"body":"All the matrices Q,V,T,PI are padded with dummy rows so that their length is a power of two. To be able to interpolate their columns, we need a primitive root of unity ω of that order. Given the particular field used in our implementation, that means that the maximum possible size for a circuit is 232. The entries of the dummy rows are filled in with zeroes in the Q, V and PI matrices. The T matrix needs to be consistent with the V matrix. Therefore it is filled with the value of the variable with index 0. Some other rows in the V matrix have also dummy values. These are the rows corresponding to the B and C columns of the public input rows. In the recap we denoted them with the empty - symbol. They are filled in with the same logic as the padding rows, as well as the corresponding values in the T matrix.","breadcrumbs":"Plonk » Implementation » Padding","id":"35","title":"Padding"},"36":{"body":"The implementation pretty much follows the rounds as are described in the protocol section. There are a few details that are worth mentioning.","breadcrumbs":"Plonk » Implementation » Implementation details","id":"36","title":"Implementation details"},"37":{"body":"The commitment scheme we use is the Kate-Zaverucha-Goldberg scheme with the BLS 12 381 curve and the ate pairing. It can be found in the commitments module of the lambdaworks_crypto package. The order r of the cyclic subgroup is 0x73eda753299d7d483339d80809a1d80553bda402fffe5bfeffffffff00000001 The maximum power of two that divides r−1 is 232. Therefore, that is the maximum possible order for a primitive root of unity in Fr​ with order a power of two.","breadcrumbs":"Plonk » Implementation » Commitment Scheme","id":"37","title":"Commitment Scheme"},"38":{"body":"","breadcrumbs":"Plonk » Implementation » Fiat-Shamir","id":"38","title":"Fiat-Shamir"},"39":{"body":"Here we describe our implementation of the transcript used for the Fiat-Shamir heuristic. A Transcript exposes two methods: append and challenge. The method append adds a message to the transcript by updating the internal state of the hasher with the raw bytes of the message. The method challenge returns the result of the hasher using the current internal state of the hasher. It subsequently resets the hasher and updates the internal state with the last result. Here is an example of this process: Start a fresh transcript. Call append and pass message_1. Call append and pass message_2. The internal state of the hasher at this point is message_2 || message_1. Call challenge. The output is Hash(message_2 || message_1). Call append and pass message_3. Call challenge. The output is Hash(message_3 || Hash(message_2 || message_1)). Call append and pass message_4. The internal state of the hasher at the end of this exercise is message_4 || Hash(message_3 || Hash(message_2 || message_1)) The underlying hasher function we use is h=sha3.","breadcrumbs":"Plonk » Implementation » Transcript strategy","id":"39","title":"Transcript strategy"},"4":{"body":"We use the following notation. The symbol F denotes a finite field. It is fixed all along. The symbol ω denotes a primitive root of unity in F. All polynomials have coefficients in F and the variable is usually denoted by X. We denote polynomials by single letters like p,a,b,z. We only denote them as z(X) when we want to emphasize the fact that it is a polynomial in X, or we need that to explicitly define a polynomial from another one. For example when composing a polynomial z with the polynomial ωX, the result being denoted by z′:=z(ωX). The symbol ′ is not used to denote derivatives. When interpolating at a domain H={h0​,…,hn​}⊂F, the symbols Li​ denote the Lagrange basis. That is Li​ is the polynomial such that Li​(hj​)=0 for all j=i, and that Li​(hi​)=1. If M is a matrix, then Mi,j​ denotes the value at the row i and column j.","breadcrumbs":"Plonk » Recap » Notation","id":"4","title":"Notation"},"40":{"body":"The result of every challenge is a 256-bit string, which is interpreted as an integer in big-endian order. A field element is constructed out of it by taking modulo the field order. The prime field used in this implementation has a 255-bit order. Therefore some field elements are more probable to occur than others because they have more representatives as 256-bit integers.","breadcrumbs":"Plonk » Implementation » Field elements","id":"40","title":"Field elements"},"41":{"body":"The first messages added to the transcript are all commitments of the polynomials of the common preprocessed input and the values of the public inputs. This prevents a known vulnerability called \"weak Fiat-Shamir\". Check out the following resources to learn more about it. What can go wrong (zkdocs) How not to Prove Yourself: Pitfalls of the Fiat-Shamir Heuristic and Applications to Helios Weak Fiat-Shamir Attacks on Modern Proof Systems","breadcrumbs":"Plonk » Implementation » Strong Fiat-Shamir","id":"41","title":"Strong Fiat-Shamir"},"42":{"body":"In this section, we'll discuss how to build your own constraint system to prove the execution of a particular program.","breadcrumbs":"Plonk » Circuit API » Circuit API","id":"42","title":"Circuit API"},"43":{"body":"Let's take the following simple program as an example. We have two public inputs: x and y. We want to prove to a verifier that we know a private input e such that x * e = y. You can achieve this by building the following constraint system: use lambdaworks_plonk::constraint_system::ConstraintSystem;\nuse lambdaworks_math::elliptic_curve::short_weierstrass::curves::bls12_381::default_types::FrField; fn main() { let system = &mut ConstraintSystem::::new(); let x = system.new_public_input(); let y = system.new_public_input(); let e = system.new_variable(); let z = system.mul(&x, &e); // This constraint system asserts that x * e == y system.assert_eq(&y, &z);\n} This code creates a constraint system over the field of the BLS12381 curve. Then, it creates three variables: two public inputs x and y, and a private variable e. Note that every variable is private except for the public inputs. Finally, it adds the constraints that represent a multiplication and an assertion. Before generating proofs for this system, we need to run a setup and obtain a verifying key: let common = CommonPreprocessedInput::from_constraint_system(&system, &ORDER_R_MINUS_1_ROOT_UNITY);\nlet srs = test_srs(common.n);\nlet kzg = KZG::new(srs); // The commitment scheme for plonk.\nlet vk = setup(&common, &kzg); Now we can generate proofs for our system. We just need to specify the public inputs and obtain a witness that is a solution for our constraint system: let inputs = HashMap::from([(x, FieldElement::from(4)), (e, FieldElement::from(3))]);\nlet assignments = system.solve(inputs).unwrap();\nlet witness = Witness::new(assignments, &system); Once you have all these ingredients, you can call the prover: let public_inputs = system.public_input_values(&assignments);\nlet prover = Prover::new(kzg.clone(), TestRandomFieldGenerator {});\nlet proof = prover.prove(&witness, &public_inputs, &common, &vk); and verify: let verifier = Verifier::new(kzg);\nassert!(verifier.verify(&proof, &public_inputs, &common, &vk));","breadcrumbs":"Plonk » Circuit API » Simple Example","id":"43","title":"Simple Example"},"44":{"body":"Some operations are common, and it makes sense to wrap the set of constraints that do these operations in a function and use it several times. Lambdaworks comes with a collection of functions to help you build your own constraint systems, such as conditionals, inverses, and hash functions. However, if you have an operation that does not come with Lambdaworks, you can easily extend Lambdaworks functionality. Suppose that the exponentiation operation is something common in your program. You can write the square and multiply algorithm and put it inside a function: pub fn pow( system: &mut ConstraintSystem, base: Variable, exponent: Variable,\n) -> Variable { let exponent_bits = system.new_u32(&exponent); let mut result = system.new_constant(FieldElement::one()); for i in 0..32 { if i != 0 { result = system.mul(&result, &result); } let result_times_base = system.mul(&result, &base); result = system.if_else(&exponent_bits[i], &result_times_base, &result); } result\n} This function can then be used to modify our simple program from the previous section. The following circuit checks that the prover knows e such that pow(x, e) = y: use lambdaworks_plonk::constraint_system::ConstraintSystem;\nuse lambdaworks_math::elliptic_curve::short_weierstrass::curves::bls12_381::default_types::FrField; fn main() { let system = &mut ConstraintSystem::::new(); let x = system.new_public_input(); let y = system.new_public_input(); let e = system.new_variable(); let z = pow(system, &x, &e); system.assert_eq(&y, &z);\n} You can keep composing these functions in order to create more complex systems.","breadcrumbs":"Plonk » Circuit API » Building Complex Systems","id":"44","title":"Building Complex Systems"},"45":{"body":"The goal of this document is to give a good a understanding of our stark prover code. To this end, in the first section we go through a recap of how the proving system works at a high level mathematically; then we dive into how that's actually implemented in our code.","breadcrumbs":"STARKs » STARK Prover","id":"45","title":"STARK Prover"},"46":{"body":"","breadcrumbs":"STARKs » Recap » STARKs Recap","id":"46","title":"STARKs Recap"},"47":{"body":"In general, we express computation in our proving system by providing an execution trace satisfying certain constraints . The execution trace is a table containing the state of the system at every step of computation. This computation needs to follow certain rules to be valid; these rules are our constraints . The constraints for our computation are expressed using an Algebraic Intermediate Representation or AIR. This representation uses polynomials to encode constraints, which is why sometimes they are called polynomial constraints. To make all this less abstract, let's go through two examples.","breadcrumbs":"STARKs » Recap » Verifying Computation through Polynomials","id":"47","title":"Verifying Computation through Polynomials"},"48":{"body":"Throughout this section and the following we will use this example extensively to have a concrete example. Even though it's a bit contrived (no one cares about computing fibonacci numbers), it's simple enough to be useful. STARKs and proving systems in general are very abstract things; having an example in mind is essential to not get lost. Let's say our computation consists of calculating the k-th number in the fibonacci sequence. This is just the sequence of numbers \\(a_n\\) satisfying \\[ a_0 = 1 \\] \\[ a_1 = 1 \\] \\[ a_{n+2} = a_{n + 1} + a_n \\] An execution trace for this just consists of a table with one column, where each row is the i-th number in the sequence: a_i 1 1 2 3 5 8 13 21 A valid trace for this computation is a table satisfying two things: The first two rows are 1. The value on any other row is the sum of the two preceding ones. The first item is called a boundary constraint, it just enforces specific values on the trace at certain points. The second one is a transition constraint; it tells you how to go from one step of computation to the next.","breadcrumbs":"STARKs » Recap » Fibonacci numbers","id":"48","title":"Fibonacci numbers"},"49":{"body":"The example above is extremely useful to have a mental model, but it's not really useful for anything else. The problem is it just works for the very narrow example of computing fibonacci numbers. If we wanted to prove execution of something else, we would have to write an AIR for it. What we're actually aiming for is an AIR for an entire general purpose Virtual Machine. This way, we can provide proofs of execution for any computation using just one AIR. This is what cairo as a programming language does. Cairo code compiles to the bytecode of a virtual machine with an already defined AIR. The general flow when using cairo is the following: User writes a cairo program. The program is compiled into Cairo's VM bytecode. The VM executes said code and provides an execution trace for it. The trace is passed on to a STARK prover, which creates a proof of correct execution according to Cairo's AIR. The proof is passed to a verifier, who checks that the proof is valid. Ultimately, our goal is to give the tools to write a STARK prover for the cairo VM and do so. However, this is not a good example to start out as it's incredibly complex. The execution trace of a cairo program has around 30 columns, some for general purpose registers, some for other reasons. Cairo's AIR contains a lot of different transition constraints, encoding all the different possible instructions (arithmetic operations, jumps, etc). Use the fibonacci example as your go-to for understanding all the moving parts; keep the Cairo example in mind as the thing we are actually building towards.","breadcrumbs":"STARKs » Recap » Cairo","id":"49","title":"Cairo"},"5":{"body":"","breadcrumbs":"Plonk » Recap » The ideas and components","id":"5","title":"The ideas and components"},"50":{"body":"Below we go through a step by step explanation of a STARK prover. We will assume the trace of the fibonacci sequence mentioned above; it consists of only one column of length \\(2^n\\). In this case, we'll take n=3. The trace looks like this a_i a_0 a_1 a_2 a_3 a_4 a_5 a_6 a_7","breadcrumbs":"STARKs » Recap » Fibonacci step by step walkthrough","id":"50","title":"Fibonacci step by step walkthrough"},"51":{"body":"The first step is to interpolate these values to generate the trace polynomial. This will be a polynomial encoding all the information about the trace. The way we do it is the following: in the finite field we are working in, we take an 8-th primitive root of unity, let's call it g. It being a primitive root means two things: g is an 8-th root of unity, i.e., \\(g^8 = 1\\). Every 8-th root of unity is of the form \\(g^i\\) for some \\(0 \\leq i \\leq 7\\). With g in hand, we take the trace polynomial t to be the one satisfying t(gi)=ai​ From here onwards, we will talk about the validity of the trace in terms of properties that this polynomial must satisfy. We will also implicitly identify a certain power of \\(g\\) with its corresponding trace element, so for example we sometimes think of \\(g^5\\) as \\(a_5\\), the fifth row in the trace, even though technically it's \\(t\\) evaluated in \\(g^5\\) that equals \\(a_5\\). We talked about two different types of constraints the trace must satisfy to be valid. They were: The first two rows are 1. The value on any other row is the sum of the two preceding ones. In terms of t, this translates to \\(t(g^0) = 1\\) and \\(t(g) = 1\\). \\(t(x g^2) - t(xg) - t(x) = 0\\) for all \\(x \\in {g^0, g^1, g^2, g^3, g^4, g^5}\\). This is because multiplying by g is the same as advancing a row in the trace.","breadcrumbs":"STARKs » Recap » Trace polynomial","id":"51","title":"Trace polynomial"},"52":{"body":"To convince the verifier that the trace polynomial satisfies the relationships above, the prover will construct another polynomial that shows that both the boundary and transition constraints are satisfied and commit to it. We call this polynomial the composition polynomial, and usually denote it with \\(H\\). Constructing it involves a lot of different things, so we'll go step by step introducing all the moving parts required.","breadcrumbs":"STARKs » Recap » Composition Polynomial","id":"52","title":"Composition Polynomial"},"53":{"body":"To show that the boundary constraints are satisfied, we construct the boundary polynomial. Recall that our boundary constraints are \\(t(g^0) = t(g) = 1\\). Let's call \\(P\\) the polynomial that interpolates these constraints, that is, \\(P\\) satisfies: P(1)=1P(g)=1 The boundary polynomial \\(B\\) is defined as follows: B(x)=(x−1)(x−g)t(x)−P(x)​ The denominator here is called the boundary zerofier, and it's the polynomial whose roots are the elements of the trace where the boundary constraints must hold. How does \\(B\\) encode the boundary constraints? The idea is that, if the trace satisfies said constraints, then t(1)−P(1)=1−1=0 t(g)−P(g)=1−1=0 so \\(t(x) - P(x)\\) has \\(1\\) and \\(g\\) as roots. Showing these values are roots is the same as showing that \\(B(x)\\) is a polynomial instead of a rational function, and that's why we construct \\(B\\) this way.","breadcrumbs":"STARKs » Recap » Boundary polynomial","id":"53","title":"Boundary polynomial"},"54":{"body":"To convince the verifier that the transition constraints are satisfied, we construct the transition constraint polynomial and call it \\(C(x)\\). It's defined as follows: C(x)=∏i=05​(x−gi)t(xg2)−t(xg)−t(x)​ How does \\(C\\) encode the transition constraints? We mentioned above that these are satisfied if the polynomial in the numerator vanishes in the elements \\({g^0, g^1, g^2, g^3, g^4, g^5}\\). As with \\(B\\), this is the same as showing that \\(C(x)\\) is a polynomial instead of a rational function.","breadcrumbs":"STARKs » Recap » Transition constraint polynomial","id":"54","title":"Transition constraint polynomial"},"55":{"body":"With the boundary and transition constraint polynomials in hand, we build the composition polynomial \\(H\\) as follows: The verifier will sample four numbers \\(\\beta_1, \\beta_2\\) and \\(H\\) will be H(x)=β1​B(x)+β2​C(x) Why not just take \\(H(x) = B(x) + C(x)\\)? The reason for the betas is to make the resulting \\(H\\) be always different and unpredictable for the prover, so they can't precompute stuff beforehand. With what we discussed above, showing that the constraints are satisfied is equivalent to saying that H is a polynomial and not a rational function (we are simplifying things a bit here, but it works for our purposes).","breadcrumbs":"STARKs » Recap » Constructing \\(H\\)","id":"55","title":"Constructing \\(H\\)"},"56":{"body":"To show \\(H\\) is a polynomial we are going to use the FRI protocol, which we treat as a black box. For all we care, a FRI proof will verify if what we committed to is indeed a polynomial. Thus, the prover will provide a FRI commitment to H, and if it passes, the verifier will be convinced that the constraints are satisfied. There is one catch here though: how does the verifier know that FRI was applied to H and not any other polynomial? For this we need to add an additional step to the protocol.","breadcrumbs":"STARKs » Recap » Commiting to \\(H\\)","id":"56","title":"Commiting to \\(H\\)"},"57":{"body":"After commiting to H, the prover needs to show that H was constructed correctly according to the formula above. To do this, it will ask the prover to provide an evaluation of H on some random point z and evaluations of the trace at the points \\(t(z), t(zg)\\) and \\(t(zg^2)\\). Because the boundary and transition constraints are a public part of the protocol, the verifier knows them, and thus the only thing it needs to compute the evaluation \\((z)\\) by itself are the three trace evaluations mentioned above. Because it asked the prover for them, it can check both sides of the equation: H(z)=β1​B(z)+β2​C(z) and be convinced that \\(H\\) was constructed correctly. We are still not done, however, as the prover could have now cheated on the values of the trace or composition polynomial evaluations.","breadcrumbs":"STARKs » Recap » Consistency check","id":"57","title":"Consistency check"},"58":{"body":"There are two things left the prover needs to show to complete the proof: That \\(H\\) effectively is a polynomial, i.e., that the constraints are satisfied. That the evaluations the prover provided on the consistency check were indeed evaluations of the trace polynomial and composition polynomial on the out of domain point z. Earlier we said we would use the FRI protocol to commit to H and show the first item in the list. However, we can slightly modify the polynomial we do FRI on to show both the first and second items at the same time. This new modified polynomial is called the DEEP composition polynomial. We define it as follows: Deep(x)=γ1​x−zH(x)−H(z)​+γ2​x−zt(x)−t(z)​+γ3​x−zgt(x)−t(zg)​+γ4​x−zg2t(x)−t(zg2)​ where the numbers \\(\\gamma_i\\) are randomly sampled by the verifier. The high level idea is the following: If we apply FRI to this polynomial and it verifies, we are simultaneously showing that \\(H\\) is a polynomial and the prover indeed provided H(z) as one of the out of domain evaluations. This is the first summand in Deep(x). The trace evaluations provided by the prover were the correct ones, i.e., they were \\(t(z)\\), \\(t(zg)\\), and \\(t(zg^2)\\). These are the remaining summands of the Deep(x).","breadcrumbs":"STARKs » Recap » Deep Composition Polynomial","id":"58","title":"Deep Composition Polynomial"},"59":{"body":"The prover needs to show that Deep was constructed correctly according to the formula above. To do this, the verifier will ask the prover to provide: An evaluation of H on z and x_0 Evaluations of the trace at the points \\(t(z)\\), \\(t(zg)\\), \\(t(zg^2)\\) and \\(t(x_0)\\) Where z is the same random, out of domain point used in the consistency check of the composition polynomial, and x_0 is a random point that belongs to the trace domain. With the values provided by the prover, the verifier can check both sides of the equation: Deep(x0​)=γ1​x0​−zH(x0​)−H(z)​+γ2​x0​−zt(x0​)−t(z)​+γ3​x0​−zgt(x0​)−t(zg)​+γ4​x0​−zg2t(x0​)−t(zg2)​ The prover also needs to show that the trace evaluation \\(t(x_0)\\) belongs to the trace. To achieve this, it needs to commit the merkle roots of t and the merkle proof of \\(t(x_0)\\).","breadcrumbs":"STARKs » Recap » Consistency check","id":"59","title":"Consistency check"},"6":{"body":"For better clarity, we'll be using the following toy program throughout this recap. INPUT: x PRIVATE INPUT: e OUTPUT: e * x + x - 1 The observer would have noticed that this program could also be written as (e+1)∗x−1, which is more sensible. But the way it is written now serves us to better explain the arithmetization of PLONK. So we'll stick to it. The idea here is that the verifier holds some value x, say x=3. He gives it to the prover. She executes the program using her own chosen value e, and sends the output value, say 8, along with a proof π demonstrating correct execution of the program and obtaining the correct output. In the context of PLONK, both the inputs and outputs of the program are considered public inputs . This may sound odd, but it is because these are the inputs to the verification algorithm. This is the algorithm that takes, in this case, the tuple (3,8,π) and outputs Accept if the toy program was executed with input x=3, some private value e not revealed to the verifier, and out came 8. Otherwise it outputs Reject . PLONK can be used to delegate program executions to untrusted parties, but it can also be used as a proof of knowledge. Our program could be used by a prover to demostrate that she knows the multiplicative inverse of some value x in the finite field without revealing it. She would do it by sending the verifier the tuple (x,0,π), where π is the proof of the execution of our toy program. In our toy example this is pointless because inverting field elements is easily performed by any verifier. But change our program to the following and you get proofs of knowledge of the preimage of SHA256 digests. PRIVATE INPUT: e OUTPUT: SHA256(e) Here there's no input aside from the prover's private input. As we mentioned, the output h of the program is then part of the inputs to the verification algorithm. Which in this case just takes (h,π).","breadcrumbs":"Plonk » Recap » Programs. Our toy example","id":"6","title":"Programs. Our toy example"},"60":{"body":"We summarize below the steps required in a STARK proof for both prover and verifier.","breadcrumbs":"STARKs » Recap » Summary","id":"60","title":"Summary"},"61":{"body":"Compute the trace polynomial t by interpolating the trace column over a set of \\(2^n\\)-th roots of unity \\({g^i : 0 \\leq i < 2^n}\\). Compute the boundary polynomial B. Compute the transition constraint polynomial C. Construct the composition polynomial H from B and C. Sample an out of domain point z and provide the evaluations \\(H(z)\\), \\(t(z)\\), \\(t(zg)\\), and \\(t(zg^2)\\) to the verifier. Sample a domain point x_0 and provide the evaluations \\(H(x_0)\\) and \\(t(x_0)\\) to the verifier. Construct the deep composition polynomial Deep(x) from H, t, and the evaluations from the item above. Do FRI on Deep(x) and provide the resulting FRI commitment to the verifier. Provide the merkle root of t and the merkle proof of \\(t(x_0)\\).","breadcrumbs":"STARKs » Recap » Prover side","id":"61","title":"Prover side"},"62":{"body":"Take the evaluations \\(H(z)\\), \\(H(x_0)\\), \\(t(z)\\), \\(t(zg)\\), \\(t(zg^2)\\) and \\(t(x_0)\\) the prover provided. Reconstruct the evaluations \\(B(z)\\) and \\(C(z)\\) from the trace evaluations we were given. Check that the claimed evaluation \\(H(z)\\) the prover gave us actually satisfies H(z)=β1​B(z)+β2​C(z) Check that the claimed evaluation \\(Deep(x_0)\\) the prover gave us actually satisfies Deep(x0​)=γ1​x0​−zH(x0​)−H(z)​+γ2​x0​−zt(x0​)−t(z)​+γ3​x0​−zgt(x0​)−t(zg)​+γ4​x0​−zg2t(x0​)−t(zg2)​ Using the merkle root and the merkle proof the prover provided, check that \\(t(x_0)\\) belongs to the trace. Take the provided FRI commitment and check that it verifies.","breadcrumbs":"STARKs » Recap » Verifier side","id":"62","title":"Verifier side"},"63":{"body":"The walkthrough above was for the fibonacci example which, because of its simplicity, allowed us to sweep under the rug a few more complexities that we'll have to tackle on the implementation side. They are:","breadcrumbs":"STARKs » Recap » Simplifications and Omissions","id":"63","title":"Simplifications and Omissions"},"64":{"body":"Our trace contained only one column, but in the general setting there can be multiple (the Cairo AIR has around 30). This means there isn't just one trace polynomial, but several; one for each column. This also means there are multiple boundary constraint polynomials. The general idea, however, remains the same. The deep composition polynomial H is now the sum of several terms containing the boundary constraint polynomials \\(B_1(x), \\dots, B_k(x)\\) (one per column), and each \\(B_i\\) is in turn constructed from the \\(i\\)-th trace polynomial \\(t_i(x)\\).","breadcrumbs":"STARKs » Recap » Multiple trace columns","id":"64","title":"Multiple trace columns"},"65":{"body":"Much in the same way, our fibonacci AIR had only one transition constraint, but there could be several. We will therefore have multiple transition constraint polynomials \\(C_1(x), \\dots, C_n(x)\\), each of which encodes a different relationship between rows that must be satisfied. Also, because there are multiple trace columns, a transition constraint can mix different trace polynomials. One such constraint could be C1​(x)=t1​(gx)−t2​(x) which means \"The first column on the next row has to be equal to the second column in the current row\". Again, even though this seems way more complex, the ideas remain the same. The composition polynomial H will now include a term for every \\(C_i(x)\\), and for each one the prover will have to provide out of domain evaluations of the trace polynomials at the appropriate values. In our example above, to perform the consistency check on \\(C_1(x)\\) the prover will have to provide the evaluations \\(t_1(zg)\\) and \\(t_2(z)\\).","breadcrumbs":"STARKs » Recap » Multiple transition constraints","id":"65","title":"Multiple transition constraints"},"66":{"body":"In the actual implementation, we won't commit to \\(H\\), but rather to a decomposition of \\(H\\) into an even term \\(H_1(x)\\) and an odd term \\(H_2(x)\\), which satisfy H(x)=H1​(x2)+xH2​(x2) This way, we don't commit to \\(H\\) but to \\(H_1\\) and \\(H_2\\). This is just an optimization at the code level; once again, the ideas remain exactly the same.","breadcrumbs":"STARKs » Recap » Composition polynomial decomposition","id":"66","title":"Composition polynomial decomposition"},"67":{"body":"We treated FRI as a black box entirely. However, there is one thing we do need to understand about it: low degree extensions. When applying FRI to a polynomial of degree \\(n\\), we need to provide evaluations of it over a domain with more than \\(n\\) points. In our case, the DEEP composition polynomial's degree is around the same as the trace's, which is, at most, \\(2^n - 1\\) (because it interpolates the trace containing \\(2^n\\) points). The domain we are going to choose to evaluate our DEEP polynomial on will be a set of higher roots of unity. In our fibonacci example, we will take a primitive \\(16\\)-th root of unity \\(\\omega\\). As a reminder, this means: \\(\\omega\\) is an \\(16\\)-th root of unity, i.e., \\(\\omega^{16} = 1\\). Every \\(16\\)-th root of unity is of the form \\(\\omega^i\\) for some \\(0 \\leq i \\leq 15\\). Additionally, we also take it so that \\(\\omega\\) satisfies \\(\\omega^2 = g\\) (\\(g\\) being the \\(8\\)-th primitive root of unity we used to construct t). The evaluation of \\(t\\) on the set \\({\\omega^i : 0 \\leq i \\leq 15}\\) is called a low degree extension (LDE) of \\(t\\). Notice this is not a new polynomial, they're evaluations of \\(t\\) on some set of points. Also note that, because \\(\\omega^2 = g\\), the LDE contains all the evaluations of \\(t\\) on the set of powers of \\(g\\). In fact, {t(ω2i):0≤i≤15}={t(gi):0≤i≤7} This will be extremely important when we get to implementation. For our LDE, we chose \\(16\\)-th roots of unity, but we could have chosen any other power of two greater than \\(8\\). In general, this choice is called the blowup factor, so that if the trace has \\(2^n\\) elements, a blowup factor of \\(b\\) means our LDE evaluates over the \\(2^{n} * b\\) roots of unity (\\(b\\) needs to be a power of two). The blowup factor is a parameter of the protocol related to its security.","breadcrumbs":"STARKs » Recap » FRI, low degree extensions and roots of unity","id":"67","title":"FRI, low degree extensions and roots of unity"},"68":{"body":"In this section, we start diving deeper before showing the formal protocol. If you haven't done so, we recommend reading the \"Recap\" section first. At a high level, the protocol works as follows. The starting point is a matrix T that encodes the trace of a valid execution of the program. This matrix needs to be in a particular format so that its correctness is equivalent to checking a finite number of polynomial equations on its rows. Transforming the execution to this matrix is what's called the arithmetization process. Then a single polynomial F is constructed that encodes the set of all the polynomial constraints. The satisfiability of all these constraints is equivalent to F being divisible by some public polynomial G. So the prover constructs H as the quotient F/G called the composition polynomial. Then the verifier chooses a random point z and challenges the prover to reveal the values F(z) and H(z). Then the verifier checks that H(z)=F(z)/G(z), which convinces him that the same relation holds at a level of polynomials and, in consequence, convinces the verifier that the private trace T of the prover is valid. In summary, at a very high level, the STARK protocol can be organized into three major parts: Arithmetization and commitment of execution trace. Construction and commitment of composition polynomial H. Opening of polynomials at random z.","breadcrumbs":"STARKs » Protocol overview » Protocol Overview","id":"68","title":"Protocol Overview"},"69":{"body":"As the Recap mentions, the trace is a table containing the system's state at every step. In this section, we will denote the trace as T. A trace can have several columns to store different aspects or features of a particular state at a specific moment. We will refer to the j-th column as Tj​. You can think of a trace as a matrix T where the entry Tij​ is the j-th element of the i-th state. Most proving systems' primary tool is polynomials over a finite field F. Each column Tj​ of the trace T will be interpreted as evaluations of such a polynomial tj​. Consequently, any information about the states must be encoded somehow as an element in F. To ease notation, we will assume here and in the protocol that the constraints encoding transition rules depend only on a state and the previous one. Everything can be easily generalized to transitions that depend on many preceding states. Then, constraints can be expressed as multivariate polynomials in 2m variables PkT​(X1​,…,Xm​,Y1​,…,Ym​) A transition from state i to state i+1 will be valid if and only if when we plug row i of T in the first m variables and row i+1 in the second m variables of PkT​, we get 0 for all k. In mathematical notation, this is PkT​(Ti,0​,…,Ti,m​,Ti+1,0​,…,Ti+1,m​)=0 for all k These are called transition constraints and check the trace's local properties, where local means relative to specific rows. There is another type of constraint, called boundary constraint , and denoted PjB​. These enforce parts of the trace to take particular values. It is helpful, for example, to verify the initial states. So far, these constraints can only express the local properties of the trace. There are situations where the global properties of the trace need to be checked for consistency. For example, a column may need to take all values in a range but not in any predefined way. Several methods exist to express these global properties as local by adding redundant columns. Usually, they need to involve randomness from the verifier to make sense, and they turn into an interactive protocol called Randomized AIR with Preprocessing .","breadcrumbs":"STARKs » Protocol overview » Arithmetization","id":"69","title":"Arithmetization"},"7":{"body":"This is the process that takes the circuit of a particular program and produces a set of mathematical tools that can be used to generate and verify proofs of execution. The end result will be a set of eight polynomials. To compute them we need first to define two matrices. We call them the Q matrix and the V matrix. The polynomials and the matrices depend only on the program and not on any particular execution of it. So they can be computed once and used for every execution instance. To understand what they are useful for, we need to start from execution traces .","breadcrumbs":"Plonk » Recap » PLONK Arithmetization","id":"7","title":"PLONK Arithmetization"},"70":{"body":"To make interactions possible, a crucial cryptographic primitive is the Polynomial Commitment Scheme. This prevents the prover from changing the polynomials to adjust them to what the verifier expects. Such a scheme consists of the commit and the open protocols. STARK uses a univariate polynomial commitment scheme that internally combines a vector commitment scheme and a protocol called FRI. Let's begin with these two components and see how they build up the polynomial commitment scheme.","breadcrumbs":"STARKs » Protocol overview » Polynomial commitment scheme","id":"70","title":"Polynomial commitment scheme"},"71":{"body":"Given a vector Y=(y0​,…,yM​), commiting to Y means the following. The prover builds a Merkle tree out of it and sends its root to the verifier. The verifier can then ask the prover to reveal, or open , the value of the vector Y at some index i. The prover won't have any choice except to send the correct value. The verifier will expect the corresponding value yi​ and the authentication path to the tree's root to check its authenticity. The authentication path also encodes the vector's position i and its length M. The root of the Merkle tree is said to be the commitment of Y, and we denote it here by [Y].","breadcrumbs":"STARKs » Protocol overview » Vector commitments","id":"71","title":"Vector commitments"},"72":{"body":"In STARKs, all commited vectors are of the form Y=(p(d1​),…,p(dM​)) for some polynomial p and some fixed domain D=(d1​,…,dM​). The domain is always known to the prover and the verifier. It can be proved, as long as M is less than the total number of field elements, that every vector (y0​,…,yM​) is equal to (p(d1​),…,p(dM​)) for a unique polynomial p of degree at most M−1. This is called the Lagrange interpolation theorem. It means, there is a unique polynomial of degree at most M−1 such that p(di​)=yi​ for all i. And M−1 is an upper bound to the degree of p. It could be less. For example, the vector of all ones Y=(1,1,…,1) is the evaluation of the constant polynomial p=1, which has degree 0. Suppose the vector Y=(y1​,…,yM​) is the vector of evaluations of a polynomial p of degree strictly less than M−1. Suppose one party holds the vector Y and another party holds only the commitment [Y] of it. The FRI protocol is an efficient interactive protocol with which the former can convince the latter that the commitment they hold corresponds to the vector of evaluations of a polynomial p of degree strictly less than M. More precisely, the protocol depends on the following parameters Powers of two N=2n and M=2m with 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\ No newline at end of file diff --git a/starks/protocol.html b/starks/protocol.html index c029f87c5..77fb10be2 100644 --- a/starks/protocol.html +++ b/starks/protocol.html @@ -147,11 +147,11 @@

          docs

          STARKs protocol

          In this section we describe precisely the STARKs protocol used in Lambdaworks.

          We begin with some additional considerations and notation for most of the relevant objects and values to refer to them later on.

          -

          Grinding

          -

          This is a technique to increase the soundness of the protocol by adding proof of work. It works as follows. At some fixed point in the protocol, the verifier sends a challenge to the prover. The prover needs to find a string such that begins with a predefined number of zeroes. Here denotes the concatenation of and , seen as bit strings. -The number of zeroes is called the grinding factor. The hash function can be any hash function, independent of other hash functions used in the rest of the protocol. In Lambdaworks we use Keccak256.

          Transcript

          The Fiat-Shamir heuristic is used to make the protocol noninteractive. We assume there is a transcript object to which values can be added and from which challenges can be sampled.

          +

          Grinding

          +

          This is a technique to increase the soundness of the protocol by adding proof of work. It works as follows. At some fixed point in the protocol, a value is derived in a deterministic way from all the interactions between the prover and the verifier up to that point (the state of the transcript). The prover needs to find a string such that begins with a predefined number of zeroes. Here denotes the concatenation of and , seen as bit strings. +The number of zeroes is called the grinding factor. The hash function can be any hash function, independent of other hash functions used in the rest of the protocol. In Lambdaworks we use Keccak256.

          General notation

          • denotes a finite field.
            • @@ -170,7 +170,7 @@

            is the total number of columns: .
          • denote the transition constraint polynomials for . We are assuming these are of degree at most 2.
          • denote the transition constraint zerofiers for .
            • -
          • is the blowup factor.
            • +
          • is the blowup factor.
          • is the grinding factor.
          • is number of FRI queries.
          • We assume there is a fixed hash function from to binary strings. We also assume all Merkle trees are constructed using this hash function.
            • @@ -194,9 +194,11 @@

            Derived values<

            Notation of important operations

            Vector commitment scheme

            Given a vector . The operation returns the root of the Merkle tree that has the hash of the elements of as leaves.

            -

            For , the operation returns and the authentication path to the Merkle tree root.

            +

            For , the operation returns the pair , where is the authentication path to the Merkle tree root.

            The operation returns Accept or Reject depending on whether the -th element of is . It checks whether the authentication path is compatible with , and the Merkle tree root .

            In our cases the sets will be of the form for some elements . It will be convenient to use the following abuse of notation. We will write to mean . Similarly, we will write instead of . Note that this is only notation and is only checking that the is the -th element of the commited vector.

            +
            Batch
            +

            As we mentioned in the protocol overview. When committing to multiple vectors , where one can build a single Merkle tree. Its -th leaf is the concatenation of all the -th coordinates of all vectors, that is, . The commitment to this batch of vectors is the root of this Merkle tree.

            Protocol

            Prover

            Round 0: Transcript initialization

            @@ -221,15 +223,15 @@
            Round 2: Construction of composition polynomial
              -
            • Sample and in from the transcript.
              • -
            • Sample and in from the transcript.
              • +
            • Sample in from the transcript.
              • +
            • Sample in from the transcript.
            • Compute .
            • Compute .
            • Compute the composition polynomial
            • Decompose as
              • -
            • Compute commitments and .
              • +
            • Compute commitments and (Batch commitment optimization applies here).
            • Add and to the transcript.

            Round 3: Evaluation of polynomials at

            @@ -245,8 +247,7 @@

            Round 4.1.k: FRI commit phase

              -
            • Let , and .
              • -
            • Add to the transcript.
              • +
            • Let .
            • For do the following: -
              Round 4.4: Open deep composition polynomial components
              -
                -
              • For do the following: -
                  -
                • Compute , .
                  • -
                • Compute for all .
                  • +
                • Sample random index from the transcript and let .
                  • +
                • Compute and for all .
                  • +
                • Compute and .
                  • +
                • Compute and .
                  • +
                • Compute and for all .

              Build proof

              • Send the proof to the verifier: -
                • +

              Verifier

              -

              Notation

              +

              From the point of view of the verifier, the proof they receive is a bunch of values that may or may not be what they claim to be. To make this explicit, we avoid denoting values like as such, because that implicitly assumes that the value was obtained after evaluating a polynomial at . And that's something the verifier can't assume. We use the following convention.

                -
              • Bold capital letters refer to commitments. For example is the claimed commitment .
                • -
              • Greek letters with superscripts refer to claimed function evaluations. For example is the claimed evaluation .
                • -
              • Gothic letters refer to authentication paths (e.g. is the authentication path of the opening of ).
                • +
              • Bold capital letters refer to commitments. For example is the claimed commitment .
                • +
              • Greek letters with superscripts refer to claimed function evaluations. For example is the claimed evaluation and is the claimed evaluation of . Note that field elements in superscripts never indicate powers. They are just notation.
                • +
              • Gothic letters refer to authentication paths. For example is the authentication path of a opening of .
                • +
              • Recall that every opening is a pair , where is the claimed value at index and is the authentication path. So for example, is denoted as from the verifier's end.

              Input

              -

              +

              This is the proof using the notation described above. The elements appear in the same exact order as they are in the Prover section, serving also as a complete reference of the meaning of each value.

              +

              Step 1: Replay interactions and recover challenges

              • Start a transcript
                • -
              • (Strong Fiat Shamir) Commit to the set of coefficients of the transition and boundary polynomials, and add the commitments to the transcript.
                • +
              • (Strong Fiat Shamir) Add all public values to the transcript.
              • Add to the transcript for all .
              • Sample random values from the transcript.
              • Add to the transcript for .
                • @@ -315,7 +311,6 @@

                Sample from the transcript.
              • Add , , and to the transcript.
              • Sample , , and from the transcript.
                • -
              • Add to the transcript
              • For do the following: @@ -343,40 +337,56 @@

                Step 3: Verify FRI

                  -
                • Check that the following are all Accept: +
                • Reconstruct the deep composition polynomial values at and . That is, define +
                  • +
                • For all :
                    -
                  • for all .
                    • -
                  • for all , .
                    • +
                  • For all : +
                      +
                    • Check that and are Accept.
                      • +
                    • Solve the following system of equations on the variables +
                      • +
                    • If , check that equals
                      • +
                    • If , check that equals .
                      • +
                    +
                  • -
                • For all : - For all : - Solve the following system of equations on the variables - -- Define -- Check that is equal to .
                -

                Step 4: Verify deep composition polynomial is FRI first layer

                +

                Step 4: Verify trace and composition polynomials openings

                • For do the following:
                  • Check that the following are all Accept:
                      +
                    • for all .
                    • .
                    • .
                      • -
                    • for all .
                      • +
                    • for all .
                      • +
                    • .
                      • +
                    • .
                    • -
                  • Check that is equal to the following: -
                -

                Other

                -

                Notes on Optimizations

                -
                  -
                • Inversions of finite field elements are slow. There is a very well known trick to batch invert many elements at once replacing inversions by multiplications. See here for the algorithm.
                  • -
                • One of the most computationally intensive operations performed is polynomial division. These can be optimized by utilizing Fast Fourier Transform (FFT) to divide each field element in Lagrange form.
                  • -
                • In specific scenarios, such as dividing by a polynomial of the form (x-a), Ruffini's rule can be employed to further enhance performance.
                  • -
                +

                Notes on Optimizations and variants

                +

                Sampling of challenges variant

                +

                To build the composition the prover samples challenges and for and . A variant of this is sampling a single challenge and defining and as powers of . That is, define for and for .

                +

                The same variant applies for the challenges for used to build the deep composition polynomial. In this case the variant samples a single challenge and defines , for all , and .

                +

                Batch inversion

                +

                Inversions of finite field elements are slow. There is a very well known trick to batch invert many elements at once replacing inversions by multiplications. See here for the algorithm.

                +

                FFT

                +

                One of the most computationally intensive operations performed is polynomial division. These can be optimized by utilizing Fast Fourier Transform (FFT) to divide each field element in Lagrange form.

                +

                Ruffini's rule

                +

                In specific scenarios, such as dividing by a polynomial of the form , for example when building the deep composition polynomial, Ruffini's rule can be employed to further enhance performance.

                +

                Bit-reversal ordering of Merkle tree leaves

                +

                As one can see from inspecting the protocol, there are multiple times where, for a polynomial , the prover sends both openings and . This implies, a priori, sending two authentication paths. Domains can be indexed using bit-reverse ordering to reduce this to a single authentication path for both openings, as follows.

                +

                The natural way of building a Merkle tree to commit to a vector , is assigning the value to leaf . If this is the case, the value is at position and the value is at position . This is because equals for the value used in the protocol.

                +

                Instead of this naive approach, a better solution is to assign the value to leaf , where is the bit-reversal permutation. This is the permutation that maps to the index whose binary representation (padded to bits), is the binary representation of but in reverse order. For example, if and , then its binary representation is , which reversed is . Therefore . In the same way and . Check out the wikipedia article. With this ordering of the leaves, if is even, element is at index and is at index . Which means that a single authentication path serves to validate both points simultaneously.

                +

                Redundant values in the proof

                +

                The prover opens the polynomials of the FRI layers at and for all . Later on, the verifier uses each of those pairs to reconstruct one of the values of the next layer, namely . So there's no need to add the value to the proof, as the verifier reconstructs them. The prover only needs to send the authentication paths for them.

                +

                The protocol is only modified at Step 3 of the verifier as follows. Checking that is skipped. After computing , the verifier uses it to check that is Accept, which proves that is actually , and continues to the next iteration of the loop.

                diff --git a/starks/protocol_overview.html b/starks/protocol_overview.html index a604b8d09..04989f5c2 100644 --- a/starks/protocol_overview.html +++ b/starks/protocol_overview.html @@ -152,12 +152,12 @@

                Protocol

                In summary, at a very high level, the STARK protocol can be organized into three major parts:

                • Arithmetization and commitment of execution trace.
                  • -
                • Construction of composition polynomial .
                  • +
                • Construction and commitment of composition polynomial .
                • Opening of polynomials at random .

                Arithmetization

                As the Recap mentions, the trace is a table containing the system's state at every step. In this section, we will denote the trace as . A trace can have several columns to store different aspects or features of a particular state at a specific moment. We will refer to the -th column as . You can think of a trace as a matrix where the entry is the -th element of the -th state.

                -

                Most proving systems' primary tool is polynomials over a finite field . Each column of the trace will be interpreted as evaluations of such a polynomial . Consequently, any information about the states must be encoded somehow as an element in .

                +

                Most proving systems' primary tool is polynomials over a finite field . Each column of the trace will be interpreted as evaluations of such a polynomial . Consequently, any information about the states must be encoded somehow as an element in .

                To ease notation, we will assume here and in the protocol that the constraints encoding transition rules depend only on a state and the previous one. Everything can be easily generalized to transitions that depend on many preceding states. Then, constraints can be expressed as multivariate polynomials in variables A transition from state to state will be valid if and only if when we plug row of in the first variables and row in the second variables of , we get for all . In mathematical notation, this is @@ -169,73 +169,68 @@

                Vector commitments

                Given a vector , commiting to means the following. The prover builds a Merkle tree out of it and sends its root to the verifier. The verifier can then ask the prover to reveal, or open, the value of the vector at some index . The prover won't have any choice except to send the correct value. The verifier will expect the corresponding value and the authentication path to the tree's root to check its authenticity. The authentication path also encodes the vector's position and its length .

                -

                In STARKs, all commited vectors are of the form for some polynomial and some domain fixed domain . The domain is always known to the prover and the verifier.

                +

                The root of the Merkle tree is said to be the commitment of , and we denote it here by .

                FRI

                -

                The FRI protocol is a potent tool to prove that the commitment of a vector corresponds to the evaluations of a polynomial of a certain degree. The degree is a configuration of the protocol.

                +

                In STARKs, all commited vectors are of the form for some polynomial and some fixed domain . The domain is always known to the prover and the verifier. It can be proved, as long as is less than the total number of field elements, that every vector is equal to for a unique polynomial of degree at most . This is called the Lagrange interpolation theorem. It means, there is a unique polynomial of degree at most such that for all . And is an upper bound to the degree of . It could be less. For example, the vector of all ones is the evaluation of the constant polynomial , which has degree .

                +

                Suppose the vector is the vector of evaluations of a polynomial of degree strictly less than . Suppose one party holds the vector and another party holds only the commitment of it. The FRI protocol is an efficient interactive protocol with which the former can convince the latter that the commitment they hold corresponds to the vector of evaluations of a polynomial of degree strictly less than .

                +

                More precisely, the protocol depends on the following parameters

                +
                  +
                • Powers of two and with .
                  • +
                • A vector , with , with a nonzero value in and a primitive -root of unity
                  • +
                +

                A prover holds a vector , and the verifier holds the commitment of it. The result of the FRI protocol will be Accept if the unique polynomial of degree less than such that has degree less than . Even more precisely, the protocol proves that is very close to a vector with of degree less than , but it may differ in negligible proportion of the coordinates.

                +

                The number is called the blowup factor and the security of the protocol depends in part on this parameter. The specific shape of the domain set has some symmetric properties important for the inner workings of FRI, such as for all .

                +

                Variant useful for STARKs

                +

                FRI is usually described as above. In STARK, FRI is used as a building block for the polynomial commitment scheme of the next section. For that, a small variant of FRI is needed.

                +

                Suppose the prover holds a vector and the verifier holds its commitment as before. Suppose further that both parties know a function that takes two field elements and outputs another field element. For example could be the function . More precisely, the kind of functions we need are .

                +

                The protocol can be used to prove that the transformed vector is the vector of evaluations of a polynomial of degree at most . Note that in this variant, the verifier holds originally the commitment of the vector and not the commitment of the transformed vector. In the example, the verifier holds the commitment and FRI will return Accept if is the vector of evaluations of a polynomial of degree at most .

                Polynomial commitments

                STARK uses a univariate polynomial commitment scheme. The following is what is expected from the commit and open protocols:

                • Commit: given a polynomial , the prover produces a sort of hash of it. We denote it here by , called the commitment of . This hash is unique to . The prover usually sends to the verifier.
                • Open: this is an interactive protocol between the prover and the verifier. The prover holds the polynomial . The verifier only has the commitment . The verifier sends a value to the prover at which he wants to know the value . The prover sends a value to the verifier, and then they engage in the Open protocol. As a result, the verifier gets convinced that the polynomial corresponding to the hash evaluates to at .
                -

                Let's see how both of these protocols work in detail. Some configuration parameters are needed:

                +

                Let's see how both of these protocols work in detail. The same configuration parameters of FRI are needed:

                • Powers of two and with .
                  • -
                • A vector , with in for all and for all .
                  • -
                • An integer .
                  • +
                • A vector , with , with a nonzero value in and a primitive -root of unity
                -

                The commitment scheme will only work for polynomials of degree at most . This means anyone can commit to any polynomial, but the Open protocol will pass only for polynomials satisfying that degree bound.

                +

                The commitment scheme will only work for polynomials of degree at most (polynomials of degree are allowed). This means: anyone can commit to any polynomial, but the Open protocol will pass only for polynomials satisfying that degree bound.

                Commit

                Given a polynomial , the commitment is just the commitment of the vector . That is, is the root of the Merkle tree of the vector of evaluations of at .

                Open

                It is an interactive protocol. So assume there is a prover and a verifier. We describe the process considering an honest prover. In the next section, we analyze what happens for malicious provers.

                -

                The prover holds the polynomial , and the verifier only the commitment of it. There is also an element . The prover evaluates and sends the result to the verifier. As we mentioned, the goal is to generate proof of the validity of the evaluation. Let us denote . This is a value now both the prover and verifier have.

                -

                The protocol has three steps:

                -
                  -
                • -

                  FRI on : First, the prover and the verifier engage in a FRI protocol for polynomials of degree at most to check that is the commitment of such a polynomial. We will assume from now on that the degree of is at most . There is an optimization that completely removes this step—more on that later.

                  -
                  • -
                • -

                  FRI on : Since , the polynomial can be written as for some polynomial . The prover computes the commitment and sends it to the verifier. Now they engage in a FRI protocol for polynomials of degree at most , which convinces the verifier that is the commitment of a polynomial of degree at most .

                  -
                  • -
                • -

                  Point checks: From the point of view of the verifier the commitments and are still potentially unrelated. Therefore, there is a check to ensure that was computed properly from following the formula and that is its commitment. To do this, the verifier challenges the prover to open and as vectors. They use the open protocol of the vector commitment scheme to reveal the values and for some random point chosen by the verifier. Next he checks that . They repeat this last part times and, as we'll analyze in the next section, this whole thing will convince the verifier that as polynomials with overwhelming probability (about ). Finally, the verifier deduces that from this equality.

                  -
                  • -
                +

                The prover holds the polynomial , and the verifier only the commitment of it. There is also an element chosen by the verifier. The prover evaluates and sends the result back. As we mentioned, the goal is to generate proof of the validity of the evaluation. Let us denote the value received by the verifier.

                +

                Now they engage in the variant of the FRI protocol for the function . The verifier accepts the value if and only if the result of FRI is Accept.

                +

                Let's see why this makes sense.

                +

                Completeness

                +

                If the prover is honest, is of degree at most and equals . That means that + +for some polynomial . Since is of degree at most , then is of degree at most . The vector is then a vector of evaluations of a polynomial of degree at most . And it is equal to . So the FRI protocol will succeed.

                Soundness

                -

                Let's analyze why the open protocol is sound. Keep in mind that the prover always has to provide a commitment of a polynomial of degree at most that satisfies for the chosen values of the verifier. That's the goal. An essential but subtle part of the soundness is that is a vector of length . To understand why let's see what would happen if .

                -

                Case

                -

                Suppose the prover tries to cheat the verifier by sending a value different from . This means that is not divisible by as polynomials. But as long as is not in the prover can interpolate the values -at and obtain some polynomial of degree at most . This polynomial satisfies for all , but the equality of polynomials does not hold. Mainly because that would imply that , and we are assuming the opposite case. Nevertheless, if they continue with the open protocol, FRI will pass since is a polynomial of degree at most . All subsequent point checks will pass since was crafted especially for that.

                -

                So how come is different from but has the property that for all ? FRI guarantees is of degree at most , which implies that is of degree at most . Then, since is of degree at most , the difference is again of degree at most . A polynomial of degree at most is zero if and only if we can prove that for distinct points. But the construction of only shows that for all , a set of points. It's not enough. One extra point and the equality would hold. Of course, that point doesn't exist, again, because that would imply that , which we assume is not true.

                -

                Case

                -

                Let's see one more example where we double the de size of but keep the same bound for the degree of the polynomials. So polynomials are of degree at most , and the length of is .

                -

                Again let's assume the prover wants to cheat the verifier similarly and claims that is but, in reality, . Inevitably, for the Open protocol to succeed, the prover has to send a commitment of some polynomial of degree at most . The strategy of the prover is pretty much constrained to be the same. But now he can't interpolate as before in every element of because that would produce a polynomial of degree at most . This most likely won't pass the FRI check.

                -

                Alternatively, he can choose some subset of of size and interpolate only there to get a polynomial of degree at most . That will succeed in the FRI phase, but what happens with the point checks? The check will pass if the verifier chooses a challenge that belongs to . If , then the check will inevitably fail. This is because, as we saw in the previous case, one extra successful point to the already points where was interpolated was enough to prove that , which we assume now not to hold. This means, if , then does not coincide with and the point check fails. So if the verifier chooses the challenges following a uniform distribution, the chances of the prover being caught cheating are for only one challenge. If the verifier performs challenges, then the chance of the prover not getting caught is .

                -

                General case: the blowup factor

                -

                If the ratio between and is , then challenges give of probability for a malicious prover to succeed. If the ratio is , that is, if , then that probability is for the same number of point checks. This provides another way of improving the soundness error. The larger the ratio, the more complex it is to cheat in the open protocol. This ratio is what's called the blowup factor. It is a security parameter, and finding a balance between the number of challenges and the size of is part of the configuration of the protocol. Increasing the size of makes committing an expensive operation since it involves building a Merkle tree with a vector of the size of . But increasing the number of challenges makes the size of the final proof larger.

                -

                We denote the blowup factor by , and it's always assumed to be a power of .

                +

                Let's sketch an idea of the soundness. Note that the value is chosen by the verifier after receiving the commitment of . So the prover does not know in advance, at the moment of sending , what will be.

                +

                Suppose the prover is trying to cheat and sends the commitment of a vector that's not the vector of evaluations of a polynomial of degree at most . Then the coordinates of the transformed vector are . Since was chosen by the verifier, dividing by shuffles all the elements in a very unpredictable way for the prover. So it is extremely unlikely that the cheating prover can craft an invalid vector such that the transformed vector turns out to be of degree at most . The expected degree of the polynomial associated with a random vector is .

                Batch

                During proof generation, polynomials are committed and opened several times. Computing these for each polynomial independently is costly. In this section, we'll see how batching polynomials can reduce the amount of computation. Let be a set of polynomials. We will commit and open as a whole. We note this batch commitment as .

                -

                We need the same configuration parameters as before: , with , a vector and .

                -

                As described earlier, to commit to a single polynomial , a Merkle tree is built over the vector . When committing to a batch of polynomials , the leaves of the Merkle tree are instead the concatenation of the polynomial evaluations. That is, in the batch setting, the Merkle tree is built for the vector . The commitment is the root of this Merkle tree. This reduces the proof size: we only need one Merkle tree for polynomials. The verifier can then only ask for values in batches. When the verifier chooses an index , the prover sends along with one authentication path. The verifier on his side computes the concatenation and validates it with the authentication path and . This also reduces the computational time. By traversing the Merkle tree one time, it can reveal several components simultaneously.

                -

                The batch open protocol proceeds similarly to the case of a single polynomial. The prover will try to convince the verifier that the committed polynomials at evaluate to some values . In the batch case, the prover will construct the following polynomial:

                -

                -

                Where are challenges provided by the verifier. The prover commits to and sends to the verifier. Then the prover and verifier continue similarly to the normal open protocol for only. This means they engage in a FRI protocol for polynomials of degree at most for . Then they engage in the point checks for . Here, for each challenge , the prover uses one authentication path for to reveal and use one authentication path for to batch reveal values . Successful point checks here mean that .

                +

                We need the same configuration parameters as before: , with , a vector .

                +

                As described earlier, to commit to a single polynomial , a Merkle tree is built over the vector . When committing to a batch of polynomials , the leaves of the Merkle tree are instead the concatenation of the polynomial evaluations. That is, in the batch setting, the Merkle tree is built for the vector + +The commitment is the root of this Merkle tree. This reduces the proof size: we only need one Merkle tree for polynomials. The verifier can then only ask for values in batches. When the verifier chooses an index , the prover sends along with one authentication path. The verifier on his side computes the concatenation and validates it with the authentication path and . This also reduces the computational time. By traversing the Merkle tree one time, it can reveal several components simultaneously.

                +

                The batch open protocol proceeds similarly to the case of a single polynomial. The verifier sends evaluations points to the prover at which they wish to know the value of . The prover will try to convince the verifier that the committed polynomials , evaluate to some values . There is a generalization of the variant of FRI where the function takes more parameters, and in this case is +Where are challenges provided by the verifier. Then FRI return Accept if and only if the vector +is close to the vector of evaluations of a polynomial of degree at most . If this is the case, the verifier accepts the openings. In the context of STARKs, the polynomial is called the DEEP composition polynomial.

                This is equivalent to running the open protocol times, one for each term and . Note that this optimization makes a huge difference, as we only need to run the FRI protocol once instead of running it once for each polynomial.

                -

                Optimize the open protocol reusing FRI internal challenges

                -

                There is an optimization for the open protocol to avoid running FRI to check that is of degree at most . The idea is as follows. Part of FRI protocol for , to check that it is of degree at most , involves revealing values of at other random points also chosen by the verifier. These are part of the internal workings of FRI. These challenges are unrelated to what we mentioned before. So if one removes the FRI check for , the point checks of the open protocol need to be performed on these challenges of the FRI protocol for . This optimization is included in the formal description of the protocol.

                References

                High-level description of the protocol

                -

                The protocol is split into rounds. Each round more or less represents an interaction with the verifier. Each round will generally start by getting a challenge from the verifier.

                -

                The prover will need to interpolate polynomials, and he will always do it over the set , where is a root of unity in . Also, the vector commitments will be performed over the set where is a root of unity and is some field element. This is the set we denoted in the commitment scheme section. The specific choices for the shapes of these sets are motivated by optimizations at a code level.

                +

                The protocol is split into rounds. Each round more or less represents an interaction with the verifier. Each round will generally start by getting a challenge from the verifier.

                +

                The prover will need to interpolate polynomials, and he will always do it over the set , where is a root of unity in . Also, the vector commitments will be performed over the set where is a root of unity and is some field element. This is the set we denoted in the commitment scheme section.

                Round 1: Arithmetization and commitment of the execution trace

                In round 1, the prover commits to the columns of the trace . He does so by interpolating each column and obtaining univariate polynomials . Then the prover commits to over . In this way, we have . @@ -245,9 +240,9 @@

                . This function will have the property that it is a polynomial if and only if the trace that the prover committed to at round 1 is valid and satisfies the agreed polynomial constraints. That is, will be a polynomial if and only if is a trace that satisfies all the transition and boundary constraints.

                Note that we can compose the polynomials , the ones that interpolate the columns of the trace , with the multivariate constraint polynomials as follows. -These result in univariate polynomials. And the same can be done for the boundary constraints. Since , these univariate polynomials vanish at every element of if and only if the trace is valid.

                +These result in univariate polynomials. The same can be done for the boundary constraints. Since , these univariate polynomials vanish at every element of if and only if the trace is valid.

                As we already mentioned, this is assuming that transitions only depend on the current and previous state. But it can be generalized to include frames with three or more rows or more context for each constraint. For example, in the Fibonacci case, the most natural way is to encode it as one transition constraint that depends on a row and the two preceding it, as we already did in the Recap section. The STARK protocol checks whether the function is a polynomial instead of checking that the polynomial is zero over the domain . The two statements are equivalent.

                -

                The verifier could check that all are polynomials one by one, and the same for the polynomials coming from the boundary constraints. But this is inefficient; the same can be obtained with a single polynomial. To do this, the prover samples challenges and obtains a random linear combination of these polynomials. The result of this is denoted by and is called the composition polynomial. It integrates all the constraints by adding them up. So after computing , the prover commits to it and sends the commitment to the verifier. The rest of the protocol aims to prove that was constructed correctly and is a polynomial, which can only be true if the prover has a valid extension of the original trace.

                +

                The verifier could check that all are polynomials one by one, and the same for the polynomials coming from the boundary constraints. However, this is inefficient; the same can be obtained with a single polynomial. To do this, the prover samples challenges and obtains a random linear combination of these polynomials. The result of this is denoted by and is called the composition polynomial. It integrates all the constraints by adding them up. So after computing , the prover commits to it and sends the commitment to the verifier. The rest of the protocol aims to prove that was constructed correctly and is a polynomial, which can only be true if the prover has a valid extension of the original trace.

                Round 3: Evaluation of polynomials at

                The verifier must check that was constructed according to the protocol rules. That is, has to be a linear combination of all the functions and similar terms for the boundary constraints. To do so, in round 3 the verifier chooses a random point and the prover computes , and for all . With all these, the verifier can check that and the expected linear combination coincide, at least when evaluated at . Since was chosen randomly, this proves with overwhelming probability that was properly constructed.

                Round 4: Run batch open protocol