Pohlig-Hellman algorithm for Bitcoin

BY KEYHUNTER 03.03.2024
Pohlig-Hellman algorithm for Bitcoin

Python code for the Pohlig-Hellman algorithm for Bitcoin:

import hashlib
import math

# Function to calculate the discrete logarithm of x and y modulo p
def discrete_log(x, y, p):
    a = 0
    b = 1
    z = 1
    while z = 0:
        a = a + b
        b = b * x
        z = hashlib.sha256(str(b).encode('utf-8')).hexdigest()
        z = int(z, 16) % p
    return a

# Function to calculate the modular exponentiation of x to the power of y modulo p
def modular_exponentiation(x, y, p):
    z = 1
    while y > 0:
        if y % 2 == 1:
            z = z * x % p
        x = x * x % p
        y = y // 2
    return z

# Function to calculate the Pohlig-Hellman algorithm
def pohlig_hellman(x, y, p, q, a, b):
    z = 1
    for i in range(1, q):
        z = z * modular_exponentiation(x, i, p) % p
    return discrete_log(y, z, p)

# Example usage:
p = 2**256 - 2**32 + 2**9 - 2**8 + 2**7 - 2**6 + 2**4 - 1
q = 2**256
a = 7
b = 5
x = 2
y = 3
print(pohlig_hellman(x, y, p, q, a, b))

In this example, we set p to the Bitcoin prime number, q to 2^256, a to 7, b to 5, x to 2, and y to 3. We then use the pohlig_hellman function to calculate the discrete logarithm of x and y modulo p.

Python implementation of the Pohlig-Hellman algorithm:

import gmpy2

# Function to compute the discrete logarithm using the Pohlig-Hellman algorithm
def discrete_log(a, p, q):
    g = 2
    m = q

    # Compute the order of 2 modulo q
    while (m % 2 == 0):
        m //= 2

    # Compute the order of 2 modulo p
    k = 1
    while (gmpy2.is_perfect_power(g, k) == False):
        k += 1

    # Compute the discrete logarithm
    b = 1
    for i in range(k):
        g = gmpy2.powmod(g, gmpy2.gcd(m, k), p)
        x = gmpy2.powmod(g, (m // k) - 1, p)
        b = (b * x) % p
        m //= k

    return b

# Example usage
a = 7
p = 13
q = 11

print("Discrete logarithm of", a, "modulo", p, "is", discrete_log(a, p, q))

In this implementation, we use the GMPY2 library to perform modular arithmetic and perfect power computations. The discrete_log function takes three arguments: a is the number whose discrete logarithm we want to compute, p is the modulus, and q is a small prime factor of p-1.

The function first computes the order of 2 modulo q, which is the smallest positive integer k such that 2^k ≡ 1 (mod q). It then computes the order of 2 modulo p, which is the smallest positive integer k such that 2^k ≡ 1 (mod p).

The function then uses these orders to compute the discrete logarithm of a modulo p using a loop that iterates k times. At each iteration

Python code for the Pohlig-Hellman algorithm:

import math

# Function to find the discrete logarithm of a number using the Pohlig-Hellman algorithm
def discrete_logarithm(a, p, q):
    # Compute the order of the number a modulo p
    order = math.ceil(math.log(p-1, 2))
    
    # Compute the prime factorization of p-1
    factors = []
    i = 2
    while i <= p-1:
        if p % i == 0:
            factors.append(i)
            p //= i
        else:
            i += 1
            
    # Compute the discrete logarithm of a modulo p using the Pohlig-Hellman algorithm
    for factor in factors:
        k = 1
        while (k * factor) % order = 1:
            k += 1
            
        b = math.ceil(math.log(k, 2))
        m = math.ceil(b / math.log(2, factor))
        
        for i in range(m):
            x = math.pow(a, (k**(i+1) - 1) // (k - 1), p)
            y = math.pow(q, i, p)
            if x == y:
                return i * factor
            
    return -1

In this code, a is the number for which we want to find the discrete logarithm modulo p, and q is a random number between 1 and p-1. The function uses the Pohlig-Hellman algorithm to find the discrete logarithm of a modulo p.

The function first computes the order of a modulo p, which is the smallest positive integer k such that a^k mod p = 1. It then computes the prime factorization of p-1, and iterates over each factor. For each factor, it computes the discrete logarithm of a modulo p using the Pohlig-Hellman algorithm. If a discrete log

Python code for the Pohlig-Hellman algorithm:

import math

def discrete_logarithm(a, b, p):
    n = int(math.log(p, 2))
    phi = p - 1
    g = 2
    m = 1
    while m < n:
        g = gcd(g, phi)
        if g > 1:
            m = m + 1
            phi = phi // g
        else:
            break
    g = 2
    m = 1
    while m < n:
        g = gcd(g, phi)
        if g > 1:
            m = m + 1
            phi = phi // g
        else:
            break
    x = 0
    k = 0
    while k < n:
        if gcd(a, p) == 1:
            x = x + (b**(2**k) * ((g**(2**k)) % p)) % phi
        g = g * g
        k = k + 1
    x = x % phi
    return x

def gcd(a, b):
    while b = 0:
        a, b = b, a % b
    return a

This code implements the Pohlig-Hellman algorithm to solve the discrete logarithm problem. It takes three arguments: a, b, and p. a is the number for which we want to find the logarithm, b is a number whose logarithm is known, and p is a prime number.

The algorithm works by first finding the smallest value of n such that 2^n is greater than or equal to p. It then uses the Pohlig-Hellman algorithm to find the discrete logarithm of a with respect to b.

I hope this helps! Let me know if you have any questions.

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