Parallel baby-step giant-step algorithm used to solve the discrete logarithm problem for Bitcoin

BY KEYHUNTER 03.03.2024
Parallel baby-step giant-step algorithm used to solve the discrete logarithm problem for Bitcoin

Python code for parallel baby-step giant-step algorithm for Bitcoin:

import numpy as np

def parallel_baby_step_giant_step(x, y, n, m):
    def baby_step(a, b, n):
        A = np.zeros((n, n))
        B = np.zeros((n, n))
        for i in range(n):
            for j in range(n):
                A[i][j] = a ** (i * j) % n
                B[i][j] = b ** (i * j) % n
        return A, B

    def giant_step(A, B, x, y, n):
        y_hat = 0
        for i in range(n):
            y_hat += x ** i * A[i][y % n]
        return y_hat % n

    A, B = baby_step(x, y, n)
    y_hat = giant_step(A, B, x, y, n)
    return y_hat

Here, x is the public key, y is the private key, n is the order of the elliptic curve, and m is the message to be encrypted. The function parallel_baby_step_giant_step returns the encrypted message y_hat.

Note: This code is for educational purposes only and is not meant to be used for actual Bitcoin transactions.

code for the baby-step giant-step algorithm:

def baby_step_giant_step(e, n, d):
    # calculate the baby-step sieve
    sieve = [0] * (n + 1)
    for i in range(2, n+1):
        if i * i % n == 1:
            sieve[i] = 1
    for i in range(2, n+1):
        if sieve[i] == 0:
            for j in range(2, n+1):
                if i * j % n == 1:
                    sieve[i] = j
                    break

    # calculate the giant-step
    x = sieve[e]
    y = 1
    while y = 0:
        y = (n - 1) % (d * x - 1)
        x = (x * x - 1) % n
    return x

this algorithm is used to solve the discrete logarithm problem, which is an important part of the bitcoin protocol. however, i don’t think it’s possible to parallelize this algorithm, as it’s essentially a sequential process.

code for you. here is the code for the parallel baby-step giant-step algorithm in python:

import math
import multiprocessing

# define the function to calculate the baby-step
def baby_step(x, a, p):
    n = 1
    while n < p:
        n *= a
        n %= p
    return n

# define the function to calculate the giant-step
def giant_step(x, a, p):
    n = 1
    while n < p:
        n *= a
        n %= p
        n *= x
        n %= p
    return n

# define the function to calculate the parallel baby-step giant-step algorithm
def parallel_bsgs(x, a, p, q, k):
    pool = multiprocessing.pool()
    results = pool.map(lambda i: baby_step(x, a, p)**(q**i) == 1, range(k+1))
    pool.close()
    pool.join()
    return results

# define the function to calculate the discrete logarithm using the parallel baby-step giant-step algorithm
def discrete_logarithm(x, a, p):
    k = math.ceil(math.log(p, a))
    q = a**k
    results = parallel_bsgs(x, a, p, q, k)
    for i in range(k+1):
        if results[i-1] == 1 and results[i] == 0:
            return i
    return -1

this code uses multiprocessing to parallelize the computation, which can greatly improve the performance of the algorithm.

python code for the parallel baby-step giant-step algorithm used to solve the discrete logarithm problem. here is the code:

import math

def gcd(a, b):
    """function to find the greatest common divisor of two numbers"""
    while b:
        a, b = b, a % b
    return a

def phi(n):
    """function to find the euler totient function of n"""
    phi_n = n
    for i in range(2, int(n ** 0.5) + 1):
        if gcd(i, n) == 1:
            phi_n -= phi_n // i
    return phi_n

def bsgs(a, b, n):
    """function to solve the discrete logarithm problem using the parallel baby-step giant-step algorithm"""
    m = int(math.sqrt(n))
    x = [0] * m
    y = [0] * m
    for i in range(1, m + 1):
        x[i-1] = 1
        for j in range(1, m + 1):
            x[i-1] = (x[i-1] * a) % n
            y[j-1] = (x[i-1] - b) % n
    t = 0
    for i in range(m, 0, -1):
        t = (t + x[i-1]) % n
    return t

n = 123456789
a = 2
b = 123456
print("the discrete logarithm of", b, "with respect to", a, "is", bsgs(a, b, n))

this code implements the parallel baby-step giant-step algorithm to solve the discrete logarithm problem. it takes three inputs: the modulus n, the base a, and the exponent b. the function phi calculates the euler totient function of n, and the function bsgs performs the baby-step giant-step algorithm. finally, the function `

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