affine versioned pairing supporting for BitVM

ec/src/models/bn/g2.rs

@@ -40,6 +40,88 @@ pub struct G2HomProjective<P: BnConfig> {
     z: Fp2<P::Fp2Config>,
 }
 
+impl<P: BnConfig> G2Prepared<P> {
+    fn affine_double_in_place(t: &mut G2Affine<P>, three_div_two: &P::Fp) -> EllCoeff<P> {
+        //  for affine coordinates
+        //  slope: alpha = 3 * x^2 / 2 * y
+        let mut alpha = t.x.square();
+        alpha /= t.y;
+        alpha.mul_assign_by_fp(&three_div_two);
+        let bias = t.y - alpha * t.x;
+
+        // update T
+        // T.x = alpha^2 - 2 * t.x
+        // T.y = -bias - alpha * T.x
+        let tx = alpha.square() - t.x.double();
+        t.y = -bias - alpha * tx;
+        t.x = tx;
+
+        (Fp2::<P::Fp2Config>::ONE, alpha, -bias)
+    }
+
+    fn affine_add_in_place(t: &mut G2Affine<P>, q: &G2Affine<P>) -> EllCoeff<P> {
+        // alpha = (t.y - q.y) / (t.x - q.x)
+        // bias = t.y - alpha * t.x
+        let alpha = (t.y - q.y) / (t.x - q.x);
+        let bias = t.y - alpha * t.x;
+
+        // update T
+        // T.x = alpha^2 - t.x - q.x
+        // T.y = -bias - alpha * T.x
+        let tx = alpha.square() - t.x - q.x;
+        t.y = -bias - alpha * tx;
+        t.x = tx;
+
+        (Fp2::<P::Fp2Config>::ONE, alpha, -bias)
+    }
+
+    /// !!! this method cannot be used directly for users, so we need reuse the `from` trait already exists
+    fn from_affine(q: G2Affine<P>) -> Self {
+        if q.infinity {
+            G2Prepared {
+                ell_coeffs: vec![],
+                infinity: true,
+            }
+        } else {
+            // let two_inv = P::Fp::one().double().inverse().unwrap();
+            let two_inv = P::Fp::one().double().inverse().unwrap();
+            let three_div_two = (P::Fp::one().double() + P::Fp::one()) * two_inv;
+
+            let mut ell_coeffs = vec![];
+            let mut r = q.clone();
+
+            let neg_q = -q;
+
+            for bit in P::ATE_LOOP_COUNT.iter().rev().skip(1) {
+                ell_coeffs.push(Self::affine_double_in_place(&mut r, &three_div_two));
+
+                match bit {
+                    1 => ell_coeffs.push(Self::affine_add_in_place(&mut r, &q)),
+                    -1 => ell_coeffs.push(Self::affine_add_in_place(&mut r, &neg_q)),
+                    _ => continue,
+                }
+            }
+
+            let q1 = mul_by_char::<P>(q);
+            let mut q2 = mul_by_char::<P>(q1);
+
+            if P::X_IS_NEGATIVE {
+                r.y = -r.y;
+            }
+
+            q2.y = -q2.y;
+
+            ell_coeffs.push(Self::affine_add_in_place(&mut r, &q1));
+            ell_coeffs.push(Self::affine_add_in_place(&mut r, &q2));
+
+            Self {
+                ell_coeffs,
+                infinity: false,
+            }
+        }
+    }
+}
+
 impl<P: BnConfig> G2HomProjective<P> {
     pub fn double_in_place(&mut self, two_inv: &P::Fp) -> EllCoeff<P> {
         // Formula for line function when working with
@@ -96,8 +178,24 @@ impl<P: BnConfig> Default for G2Prepared<P> {
     }
 }
 
+/// !!! affine mode is for the purpose of verifying pairings
 impl<P: BnConfig> From<G2Affine<P>> for G2Prepared<P> {
     fn from(q: G2Affine<P>) -> Self {
+        if q.infinity {
+            G2Prepared {
+                ell_coeffs: vec![],
+                infinity: true,
+            }
+        } else {
+            Self::from_affine(q)
+        }
+    }
+}
+
+/// !!! projective mode is for the purpose of computing pairings
+impl<P: BnConfig> From<G2Projective<P>> for G2Prepared<P> {
+    fn from(q: G2Projective<P>) -> Self {
+        let q = q.into_affine();
         if q.infinity {
             G2Prepared {
                 ell_coeffs: vec![],
@@ -144,12 +242,6 @@ impl<P: BnConfig> From<G2Affine<P>> for G2Prepared<P> {
     }
 }
 
-impl<P: BnConfig> From<G2Projective<P>> for G2Prepared<P> {
-    fn from(q: G2Projective<P>) -> Self {
-        q.into_affine().into()
-    }
-}
-
 impl<'a, P: BnConfig> From<&'a G2Affine<P>> for G2Prepared<P> {
     fn from(other: &'a G2Affine<P>) -> Self {
         (*other).into()
@@ -158,7 +250,7 @@ impl<'a, P: BnConfig> From<&'a G2Affine<P>> for G2Prepared<P> {
 
 impl<'a, P: BnConfig> From<&'a G2Projective<P>> for G2Prepared<P> {
     fn from(q: &'a G2Projective<P>) -> Self {
-        q.into_affine().into()
+        (*q).into()
     }
 }
 

ec/src/models/bn/mod.rs

@@ -102,6 +102,71 @@ pub trait BnConfig: 'static + Sized {
         MillerLoopOutput(f)
     }
 
+    fn multi_miller_loop_affine(
+        a: impl IntoIterator<Item = impl Into<G1Prepared<Self>>>,
+        b: impl IntoIterator<Item = impl Into<G2Prepared<Self>>>,
+    ) -> MillerLoopOutput<Bn<Self>> {
+        let mut pairs = a
+            .into_iter()
+            .zip_eq(b)
+            .filter_map(|(p, q)| {
+                // if input q is projective coordinates, then we will enter `into`` computing pairing mode
+                // otherwise if input q is affine coordinates, then we will enter `into` verifying pairing mode
+                let (p, q) = (p.into(), q.into());
+                match !p.is_zero() && !q.is_zero() {
+                    true => Some((
+                        -p.0.x / p.0.y,
+                        p.0.y.inverse().unwrap(),
+                        q.ell_coeffs.into_iter(),
+                    )),
+                    false => None,
+                }
+            })
+            .collect::<Vec<_>>();
+
+        let mut f = cfg_chunks_mut!(pairs, 4)
+            .map(|pairs| {
+                let mut f = <Bn<Self> as Pairing>::TargetField::one();
+                for i in (1..Self::ATE_LOOP_COUNT.len()).rev() {
+                    if i != Self::ATE_LOOP_COUNT.len() - 1 {
+                        f.square_in_place();
+                    }
+
+                    for (coeff_1, coeff_2, coeffs) in pairs.iter_mut() {
+                        Bn::<Self>::ell_affine(&mut f, &coeffs.next().unwrap(), &coeff_1, &coeff_2);
+                    }
+
+                    let bit = Self::ATE_LOOP_COUNT[i - 1];
+                    if bit == 1 || bit == -1 {
+                        for (coeff_1, coeff_2, coeffs) in pairs.iter_mut() {
+                            Bn::<Self>::ell_affine(
+                                &mut f,
+                                &coeffs.next().unwrap(),
+                                &coeff_1,
+                                &coeff_2,
+                            );
+                        }
+                    }
+                }
+                f
+            })
+            .product::<<Bn<Self> as Pairing>::TargetField>();
+
+        if Self::X_IS_NEGATIVE {
+            f.cyclotomic_inverse_in_place();
+        }
+
+        for (coeff_1, coeff_2, coeffs) in &mut pairs {
+            Bn::<Self>::ell_affine(&mut f, &coeffs.next().unwrap(), &coeff_1, &coeff_2);
+        }
+
+        for (coeff_1, coeff_2, coeffs) in &mut pairs {
+            Bn::<Self>::ell_affine(&mut f, &coeffs.next().unwrap(), &coeff_1, &coeff_2);
+        }
+
+        MillerLoopOutput(f)
+    }
+
     #[allow(clippy::let_and_return)]
     fn final_exponentiation(f: MillerLoopOutput<Bn<Self>>) -> Option<PairingOutput<Bn<Self>>> {
         // Easy part: result = elt^((q^6-1)*(q^2+1)).
@@ -180,7 +245,7 @@ pub use self::{
 pub struct Bn<P: BnConfig>(PhantomData<fn() -> P>);
 
 impl<P: BnConfig> Bn<P> {
-    /// Evaluates the line function at point p.
+    /// Evaluates the line function at point p, where the line function is in projective mode
     fn ell(f: &mut Fp12<P::Fp12Config>, coeffs: &g2::EllCoeff<P>, p: &G1Affine<P>) {
         let mut c0 = coeffs.0;
         let mut c1 = coeffs.1;
@@ -200,6 +265,35 @@ impl<P: BnConfig> Bn<P> {
         }
     }
 
+    /// Evaluates the line function at point p, where the line function is in affine mode
+    /// input:
+    ///     f, Fq12
+    ///     coeffs, (1, alpha, bias)
+    ///     x' = -p.x / p.y
+    ///     y' = 1 / p.y
+    /// output:
+    ///     f = f * f_Q(P)', where f_Q(P)' is a vairant of f_Q(P), f_Q(P) = y' * f_Q(P)
+    fn ell_affine(f: &mut Fp12<P::Fp12Config>, coeffs: &g2::EllCoeff<P>, xx: &P::Fp, yy: &P::Fp) {
+        // c0 is a trival value 1
+        let c0 = coeffs.0;
+        let mut c1 = coeffs.1;
+        let mut c2 = coeffs.2;
+
+        match P::TWIST_TYPE {
+            TwistType::M => {
+                c1.mul_assign_by_fp(&xx);
+                c2.mul_assign_by_fp(&yy);
+                f.mul_by_014(&c0, &c1, &c2);
+            },
+            // line evaluation is y' * f_Q(P), coefficients are (1, x' * lambda, -y' * bias)
+            TwistType::D => {
+                c1.mul_assign_by_fp(&xx);
+                c2.mul_assign_by_fp(&yy);
+                f.mul_by_034(&c0, &c1, &(c2));
+            },
+        }
+    }
+
     fn exp_by_neg_x(mut f: Fp12<P::Fp12Config>) -> Fp12<P::Fp12Config> {
         f = f.cyclotomic_exp(P::X);
         if !P::X_IS_NEGATIVE {
@@ -227,6 +321,13 @@ impl<P: BnConfig> Pairing for Bn<P> {
         P::multi_miller_loop(a, b)
     }
 
+    fn multi_miller_loop_affine(
+        a: impl IntoIterator<Item = impl Into<Self::G1Prepared>>,
+        b: impl IntoIterator<Item = impl Into<Self::G2Prepared>>,
+    ) -> MillerLoopOutput<Self> {
+        P::multi_miller_loop_affine(a, b)
+    }
+
     fn final_exponentiation(f: MillerLoopOutput<Self>) -> Option<PairingOutput<Self>> {
         P::final_exponentiation(f)
     }

ec/src/pairing.rs

@@ -88,6 +88,14 @@
         b: impl IntoIterator<Item = impl Into<Self::G2Prepared>>,
     ) -> MillerLoopOutput<Self>;
 
+    /// Computes the product of Miller loops for some number of (G1, G2) pairs, where the line functions are in affine mode
+    fn multi_miller_loop_affine(
+        a: impl IntoIterator<Item = impl Into<Self::G1Prepared>>,
+        b: impl IntoIterator<Item = impl Into<Self::G2Prepared>>,
+    ) -> MillerLoopOutput<Self> {
+        unimplemented!()
+    }
+
     /// Computes the Miller loop over `a` and `b`.
     fn miller_loop(
         a: impl Into<Self::G1Prepared>,
@@ -108,13 +116,29 @@
         Self::final_exponentiation(Self::multi_miller_loop(a, b)).unwrap()
     }
 
+    /// Computes a "product" of pairings, where the line functions are in affine mode
+    fn multi_pairing_affine(
+        a: impl IntoIterator<Item = impl Into<Self::G1Prepared>>,
+        b: impl IntoIterator<Item = impl Into<Self::G2Prepared>>,
+    ) -> PairingOutput<Self> {
+        Self::final_exponentiation(Self::multi_miller_loop_affine(a, b)).unwrap()
+    }
+
     /// Performs multiple pairing operations
     fn pairing(
         p: impl Into<Self::G1Prepared>,
         q: impl Into<Self::G2Prepared>,
     ) -> PairingOutput<Self> {
         Self::multi_pairing([p], [q])
     }
+
+    /// Performs multiple pairing operations, where the line functions are in affine mode
+    fn pairing_affine(
+        p: impl Into<Self::G1Prepared>,
+        q: impl Into<Self::G2Prepared>,
+    ) -> PairingOutput<Self> {
+        Self::multi_pairing_affine([p], [q])
+    }
 }
 
 /// Represents the target group of a pairing. This struct is a