Weil pairing algorithm for Bitcoin

03.03.2024
Weil pairing algorithm for Bitcoin

 Python code for weil pairing algorithm for Bitcoin:

import ecc

# Generate public and private keys
private_key = ecc.generate_private_key(curve='secp256k1')
public_key = private_key.pubkey

# Generate a random point on the curve
point = ecc.generate_random_point(curve='secp256k1')

# Perform Weil pairing
result = ecc.weil_pairing(public_key, point)

print("Weil pairing result:", result)

This code uses the ecc library to generate a private and public key, a random point on the curve, and performs the Weil pairing algorithm for Bitcoin.

The Weil pairing algorithm is a mathematical algorithm used in cryptography to calculate a shared secret between two parties. It’s used in Bitcoin to calculate the public and private keys for a Bitcoin address.

The Weil pairing algorithm is based on elliptic curve cryptography (ECC), which is a type of public-key cryptography. ECC is used in Bitcoin to create a secure, decentralized network.

The Weil pairing algorithm is a complex algorithm, and implementing it correctly requires a deep understanding of cryptography and mathematics. It’s not recommended to try to implement it yourself unless you’re an experienced cryptographer or mathematician.

Weil pairing algorithm in Python:

import hashlib

# Define the parameters
p = 21888242871839275222246405745257275088548364400416034343698204186575808495617
a = -3
b = 41018428876551020177132617442589261862607196850031167551342877870051005937681
G1 = (2, 41018428876551020177132617442589261862607196850031167551342877870051005937681)
G2 = (p - 1, 2)

# Define the functions
def add(x, y):
    return (x[0] + y[0], x[1] + y[1])

def double(x):
    return (x[0] * 2, x[1] * 2)

def subtract(x, y):
    return (x[0] - y[0], x[1] - y[1])

def multiply(x, y):
    return ((y[0] + x[0]) * (y[1] + x[1]), (y[0] + x[0]) * (y[1] - x[1]) - (y[0] - x[0]) * (y[1] + x[1]))

def negate(x):
    return (x[0], -x[1])

def miller_loop(a, b, P):
    Q = (0, 1)
    R = P
    for i in range(a - 1):
        Q = double(Q)
        if a % 2 == 1:
            R = add(R, Q)
    return R

# Define the Weil pairing function
def weil_pairing(P, Q):
    R = miller_loop(a, b, P)
    T = miller_loop(a, b, Q)
    return (T[0] - R[0]) / (T[1] - R

Python implementation of the Weil Pairing algorithm:

import hashlib
import binascii

def miller_loop(a, P, r, s):
    n = len(P) - 1
    d = P[n] - (a ** (r - 1)) % P[n]
    for i in range(r - 2, -1, -1):
        d = (d ** 2) % P[i] - (a ** (2 ** (r - 1 - i))) % P[i] + P[i]
    return d == 0

def weil_pairing(a, b, P):
    n = len(P) - 1
    r = 2 ** n
    s = (r - 1) // 2
    if miller_loop(a, P, r, s):
        return pow(b, s, P)
    else:
        return pow(b, -s, P)

def main():
    P = [2, 5, 17, 257, 65537, 4294967297]
    a = 5
    b = 17
    print("Weil Pairing (a, b, P):", weil_pairing(a, b, P))

if __name__ == "__main__":
    main()

In this code, we first define the miller_loop function, which is used to check if a given point is on the elliptic curve. The weil_pairing function is the main function that performs the Weil Pairing algorithm. It takes in three parameters: ab, and P, where P is the elliptic curve. Finally, we define the main function, which sets the values of ab, and P and prints the result of the Weil Pairing algorithm.

Python function that implements the Weil pairing algorithm:

import math

def weil_pairing(p, q, g, g_sq, a, b):
    k = math.floor((p + 1) // 4)
    x = g_sq ** ((p - 1) // 2)
    y = a ** (p - 1) % p
    z = b ** (q - 1) % q
    r = g ** x % p
    s = g ** y % p
    t = g ** z % q
    u = s ** (q - 1) % q
    v = t ** (p - 1) % p
    w = (u - 1) // q
    x = (v - 1) // p
    y = math.gcd(w, q)
    z = math.gcd(x, p)
    if y = 1 or z = 1:
        return None
    return (r ** (k * w) % p, s ** (k * x) % p, t ** (k * z) % q)

This function takes in the prime numbers p and q, the generator g of the order q, the square of g raised to the power of q - 1 (i.e. g_sq), and the two elements a and b of the groups Z_p and Z_q respectively. It returns the tuple (r, s, t) where r is the pairing value of a and gs is the pairing value of b and g, and t is the pairing value of a and b.


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