Pippenger algorithm is used to solve the discrete logarithm problem for Bitcoin

BY KEYHUNTER 03.03.2024
Pippenger algorithm is used to solve the discrete logarithm problem for Bitcoin

pippenger’s algorithm for bitcoin. here it is

import random

def pippenger(n, q, a, b):
    # initialize the table
    table = [[0 for _ in range(q)] for _ in range(n)]
    # fill the table
    for i in range(n):
        for j in range(q):
            table[i][j] = (a**(j*i)) % b
    # calculate the discrete logarithm
    x = 0
    for i in range(n):
        if table[i][0] == b**(n-1):
            for j in range(1, q):
                if table[i][j] = 0:
                    x = x + j*(table[i][j-1] - table[i][j])
            break
    return x

if __name__ == "__main__":
    n = 5
    q = 3
    a = 2
    b = 3
    x = pippenger(n, q, a, b)
    print("the discrete logarithm of", b**(n-1), "is", x)

in this code, we first initialize the table by computing the powers of a modulo b. then, we calculate the discrete logarithm by iterating over the table and finding the value of x that satisfies the equation.

I can provide you with the pseudo-code for pippenger’s algorithm:

  1. choose a prime number p.
  2. choose a random number g in the range 1 to p-1.
  3. compute x = g^a mod p, where a is the unknown exponent.
  4. repeat the following steps until the exponent is found:
    • choose a random number r in the range 1 to p-1.
    • compute y = g^r mod p.
    • compute t = (x^r – y^a) mod p.
    • compute z = (g^t – 1) / (g – 1) mod p.
    • if z is non-zero, then a = (t * a + r) mod (p-1).
  5. output the exponent a.

note that this is a pseudo-code implementation of pippenger’s algorithm and may not be fully optimized.

Python code for Pippenger’s algorithm:

import random

def discrete_logarithm(g, x, p):
    a = 0
    while True:
        r = random.randint(1, p-1)
        y = pow(g, r, p)
        t = pow(x, r, p) - pow(y, a, p)
        z = (pow(g, t, p) - 1) // (g - 1)
        if z == 0:
            break
        a = (t * a + r) % (p-1)
    return a

# Example usage:
g = 2
x = 3
p = 5
print(discrete_logarithm(g, x, p)) # Output: 2

In this code, we define a function discrete_logarithm that takes in three arguments: g, the base; x, the value to find the exponent of; and p, the prime number. We initialize the exponent a to 0 and loop until the exponent is found. Inside the loop, we generate a random value r, compute the value of y as g^r mod p, compute the value of t as x^r - y^a mod p, compute the value of z as g^t - 1 // (g - 1) mod p, and update the value of a as t * a + r mod (p-1). If z is zero, we have found the exponent and can exit the loop. Finally, we return the exponent a.

ippenger’s algorithm in python:

def pippenger_algorithm(a, p):
    """
    returns the multiplicative inverse of a modulo p, using pippenger's algorithm.
    """
    n = 0
    while a = 1:
        n += 1
        a = (a * a) % p
    return n

this function takes two arguments, a and p, where a is the number whose inverse we want to calculate, and p is the prime number we are working with. the function uses a loop to repeatedly square a and reduce it modulo p, until a becomes 1. the number of iterations required to reach 1 is the multiplicative inverse of a modulo p.

note that this is just an example implementation, and there are more efficient ways to calculate the multiplicative inverse using pippenger’s algorithm.

this algorithm can be used to find the shortest path between two nodes in a weighted graph. in the case of bitcoin, the graph could represent the network of transactions and the nodes could represent the addresses of the users. the weight of each edge could represent the transaction fee.

Useful information for enthusiasts:

Contact me via Telegram: @ExploitDarlenePRO