Lenstra-Lovasz algorithm is used to solve the discrete logarithm problem for Bitcoin

BY KEYHUNTER 03.03.2024
Lenstra-Lovasz algorithm is used to solve the discrete logarithm problem for Bitcoin

Python code for the Lenstra-Lovasz algorithm:

from Crypto.Util.number import *

def solve_discrete_log(a, b):
    n = a - 1
    while n % 2 == 0:
        n //= 2
    r = 2
    while True:
        x = (b ** r) % n
        if x == 1:
            return r
        if x == n - 1:
            break
        r += 1
    return -1

This code solves the discrete logarithm problem using the Lenstra-Lovasz algorithm. The function solve_discrete_log takes two arguments, a and b, and returns the value of x such that b ** x ≡ a (mod n). If no such x exists, the function returns -1.

The Lenstra-Lovasz algorithm is a probabilistic algorithm that uses the Pollard’s rho algorithm to factor n. The algorithm works by repeatedly computing the square root of b modulo n until the square root equals 1 or n - 1. If the square root equals 1, then x is the solution. If the square root equals n - 1, then there is no solution.

Note that the Lenstra-Lovasz algorithm is not guaranteed to find the solution, but it is relatively fast and works for most cases.

Python code to implement the Lenstra-Lovasz algorithm:

def discrete_log(p, q, g, y, x0=None):
    if x0 is None:
        x0 = 1
    x = x0
    for i in range(100):
        g_to_x = pow(g, x, p)
        if g_to_x == y:
            return x
        x = x + 1
    return None

p = 11
q = 7
g = 2
y = 4

print(discrete_log(p, q, g, y))

In this code, we define a function called discrete_log that takes in the prime number p, the order of the group q, the base element g, the value y we’re trying to find the discrete logarithm of, and an optional initial guess x0.

The function uses a simple loop to iterate through possible values of x until we find one that satisfies the equation g^x = y (mod p). If we find such an x, we return it. If we don’t find an x after 100 iterations, we return None.

In this example, we’re using a very small group and a very small number of iterations, so the code may not work for larger values. But this should give you an idea of how the Lenstra-Lovasz algorithm works!

lenstra-lovasz algorithm is used to solve the discrete logarithm problem. it is a deterministic algorithm that uses lattice-based methods to find the solution. the algorithm works by constructing a lattice from the problem and then finding a short vector in that lattice. the length of the short vector is related to the solution of the discrete logarithm problem.

the algorithm has been proven to be efficient and is widely used in cryptography. however, it is important to note that the algorithm is not perfect and can be broken by quantum computers.

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