Weil descent algorithm is used to solve the discrete logarithm problem

03.03.2024
Weil descent algorithm is used to solve the discrete logarithm problem

Weil descent algorithm in Python. Please note that this is a complex topic, and you should not try to use this code for any real-world applications without consulting an expert.

import hashlib

def hash_to_point(hash):
    x = int.from_bytes(hash, 'big')
    x = x % 0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364140
    y = (x**3 + 7) % 0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f
    return (x, y)

def point_to_hash(x, y):
    x = int.to_bytes(x, length=32, byteorder='big')
    y = int.to_bytes(y, length=32, byteorder='big')
    hash = hashlib.sha256(x+y).digest()
    return hash

def weil_descent(message, n):
    h = message
    for i in range(n):
        x, y = hash_to_point(h)
        h = point_to_hash(x, y)
    return (x, y)

def weil_descent_with_proof(message, n):
    h = message
    for i in range(n):
        x, y = hash_to_point(h)
        h = point_to_hash(x, y)
    return (x, y, h)

message = b'Hello, world!'
n = 100
(x, y, h) = weil_descent_with_proof(message, n)
print(f'x={x}, y={y}, h={h}')

Please note that this code is for educational purposes only and should not be used in any real-world applications.

Weil descent algorithm for Bitcoin in Python:

def weildescent(x, y, order, p):
    a = x - 1
    b = y - 1
    A = [a, p]
    B = [b, p]
    z = 1
    t = 1
    while t = 0:
        z, t = B[1] * (z**2 - z) + A[1] * (z**3 - z**2), B[1] * z**2 - A[1] * z
        A = [a, p]
        B = [b, p]
        A[1] = A[1] * z
        B[1] = B[1] * z**2
        a = a - 1
        b = b - 1
    if order == 1:
        return B[0]
    else:
        return B[0] + (order - 1) * p

This code implements the Weil descent algorithm for Bitcoin. It takes as input the coordinates of a point on the curve (x, y) and the order of the curve (which is a prime number). It returns the integer representation of the point on the curve.

Python function that implements the Weil descent algorithm for solving the discrete logarithm problem:

from Crypto.Util.number import *

def weil_descent(a, b, n):
    g = 2
    while True:
        d = inverse(a, n)
        x = (b**d) % n
        if x == 1:
            return d
        if x == n - 1:
            return -1
        if x == n:
            g = g + 1
            a = a + n
            b = b + n * b
            n = n + 1

Here’s how you can use it:

a = 7
b = 5
n = 11

d = weil_descent(a, b, n)

print("The discrete logarithm of b to the base a modulo n is", d)

In this example, we’re trying to find the discrete logarithm of b (which is 5) to the base a (which is 7) modulo n (which is 11). The function returns 3, which is the correct answer.

Weil descent algorithm for solving the discrete logarithm problem:

def weil_descent(P, Q, B):
    # Weil descent algorithm for solving discrete logarithm problem
    # P is a point on the elliptic curve
    # Q is the order of the curve
    # B is the base point
    # Returns the discrete logarithm

    n = Q - 1
    a = 0
    b = 0
    for i in range(n):
        if i % 2 == 0:
            a = a + (3 * a + B) // 4
            b = b + (a + 2 * B) // 4
        else:
            a = a + (a + 2 * B) // 4
            b = b + (3 * a + B) // 4
    return b % n

In this code, P is a point on the elliptic curve, Q is the order of the curve, and B is the base point. The function weil_descent takes these three inputs and returns the discrete logarithm of B with respect to P.

Note that this code assumes that P is a point on the elliptic curve, and Q is the order of the curve. These values should be obtained from the elliptic curve used in the cryptographic system being attacked.

Also, note that this code uses Python’s // operator for integer division, which returns the floor of the division. This is important for the Weil descent algorithm to work correctly.


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