Implement Karatsuba's Algorithm

[NMCarv]

As we now have a shift-and-add multiplication algorithm for unsigned integers, that is already faster than similar private compute libraries, we need to implement Karatsuba's algorithm for faster multiplication of larger integers.

This is a Python pseudocode for Karatsuba's:

def karatsuba_uint16(x, y):
    # Base case: if numbers are small, use simple multiplication
    if x <= 255 and y <= 255:
        return x * y

    # Split the numbers into high and low 8-bit parts
    x_high, x_low = x >> 8, x & 0xFF
    y_high, y_low = y >> 8, y & 0xFF

    # Compute the three products
    z0 = karatsuba_uint16(x_low, y_low)
    z2 = karatsuba_uint16(x_high, y_high)
    z1 = karatsuba_uint16((x_low + x_high), (y_low + y_high)) - z2 - z0

    # Combine the results
    return (z2 << 16) + (z1 << 8) + z0

Considerations (for 16-bit implementation):

1. Splitting numbers:
   x_high = x >> 8 (right shift by 8 bits)
   x_low = x & 0xFF (bitwise AND with 11111111)
2. Addition (for x_low + x_high and y_low + y_high):
   Implement an 8-bit adder using full adders made of XOR and AND gates.
3. Subtraction (for z1 - z2 - z0):
   Implement using two's complement addition:
   • Invert all bits (NOT operation)
   • Add 1
   • Then use the adder from step 2
4. Multiplication (for the base case and final combination):
   • For the base case (8-bit * 8-bit), use a series of AND gates and adders
   • For shifting (<<8 and <<16), simply connect wires to different positions
5. Recursive structure:
   • We'll need three separate 16-bit multipliers for z0, z1, and z2 (recursive calls)

Resources:

• https://en.wikipedia.org/wiki/Karatsuba_algorithm
• https://www.geeksforgeeks.org/karatsuba-algorithm-for-fast-multiplication-using-divide-and-conquer-algorithm/