Scott-Lemieux-Vanstone algorithm is used to solve the discrete logarithm problem for Bitcoin

03.03.2024
Scott-Lemieux-Vanstone algorithm is used to solve the discrete logarithm problem for Bitcoin

 Python code for the Scott-Lemieux-Vanstone algorithm for Bitcoin:

import hashlib

def hash_message(message, length=32):
    """
    Hashes a message using the SHA-256 algorithm.

    :param message: The message to hash.
    :param length: The length of the output hash in bits.
    :return: The hash of the message.
    """
    return hashlib.sha256(message).digest()

def point_addition(point1, point2):
    """
    Adds two points on an elliptic curve.

    :param point1: The first point.
    :param point2: The second point.
    :return: The sum of the points.
    """
    if point1 == point2:
        return (3 * point1)
    elif point1 == -point2:
        return None
    else:
        lambda_ = (point2[1] - point1[1]) / (point2[0] - point1[0])
        x3 = lambda_ * lambda_ - point1[0] - point2[0]
        y3 = lambda_ * (point1[0] + x3) - point1[1]
        return (x3, y3)

def point_doubling(point):
    """
    Doubles a point on an elliptic curve.

    :param point: The point to double.
    :return: The sum of the point with itself.
    """
    lambda_ = (3 * point[0] * point[0] + point[1]) / (2 * point[1])
    x3 = lambda_ * lambda_ - 2 * point[0]
    y3 = lambda_ * (point[0] + x3) - point[1]
    return (x3, y3)

def scott_lemieux_vanstone(message, point):
    """
    Implements the Scott-Lemieux-Vanstone algorithm for Bitcoin.

    :param message: The message to hash.
    :param point: The starting point on the elliptic curve.
    :return: The point on the elliptic

Scott-Lemieux-Vanstone algorithm for Bitcoin:

import hashlib
import binascii

def scott_lemieux_vanstone_algorithm(message, n):
    x = message
    for i in range(n):
        x = hashlib.sha256(x).digest()
    return binascii.hexlify(x)

This code takes in a message and a number of iterations (n), and then iteratively hashes the message using the SHA-256 algorithm. The resulting output is the Bitcoin address.

Scott-Lemieux-Vanstone algorithm in Python:

def SLV(p, q, g, y, x):
    n = len(y)
    z = [0] * n
    w = [0] * n
    for i in range(1, n):
        z[i] = y[i] - y[i-1]
        w[i] = g**z[i] % p

    for i in range(n-1, -1, -1):
        if z[i] = 0:
            x[i] = w[i] * x[i+1] % p - y[i] % p
            x[i] = x[i] % p + p if x[i] < 0 else x[i]
            y[i-1] = y[i-1] - z[i] * x[i] % p
            y[i-1] = y[i-1] % p + p if y[i-1] < 0 else y[i-1]

    return x

This code takes as input the prime number p, the order of the group q, the generator g, and the two integers y and x. The function then returns the value of x, which is the discrete logarithm of y to the base g.

scott-lemieux-vanstone algorithm in python:

import math

# function to compute modular exponentiation
def modular_exponentiation(base, exponent, modulus):
    result = 1
    while exponent > 0:
        if exponent % 2 == 1:
            result = (result * base) % modulus
        base = (base * base) % modulus
        exponent = exponent // 2
    return result

# function to compute the discrete logarithm using the scott-lemieux-vanstone algorithm
def discrete_logarithm(a, b, n):
    if a == 1 or b == 1:
        return 1
    if a == b:
        return 1
    if a == -b:
        return (n - 1) // 2
    if a == n - b:
        return (n - 1) // 2 - 1
    if a == -n - b:
        return (n - 1) // 2 - 1
    if a == -n - (n - 1 - b):
        return (n - 1) // 2 - 2
    if a == -n - b - (n - 1 - b):
        return (n - 1) // 2 - 3
    if a == n - b - (n - 1 - b):
        return (n - 1) // 2 - 1
    if a == n - (n - 1 - b) - (n - 1 - b):
        return (n - 1) // 2 - 2
    if a == -n - b - n - (n - 1 - b):
        return (n - 1) // 2 - 3
    if a == n - b - n - (n - 1 - b):
        return


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