Joux-Lercier algorithm for Bitcoin

03.03.2024
Joux-Lercier algorithm for Bitcoin

code for the joux-lercier algorithm in python:

def joux_lercier(seq, k):
    n = len(seq)
    for i in range(k, n):
        j = i - k + 1
        for j in range(i, -1, -1):
            if seq[j] < seq[j - 1]:
                seq[j], seq[j - 1] = seq[j - 1], seq[j]
    return seq

this algorithm sorts a sequence in descending order, using the following steps:

  1. start from the kth element of the sequence, where k is the number of elements in the sequence minus 1.
  2. for each element in the sequence starting from the kth element, compare it to the element before it.
  3. if the current element is less than the element before it, swap them.
  4. continue comparing and swapping elements until the beginning of the sequence is reached.
  5. return the sorted sequence.

Joux-Lercier algorithm is based on the Pollard’s rho algorithm, which is a well-known algorithm used for finding cycles in graphs. The Joux-Lercier algorithm uses the Pollard’s rho algorithm to find a cycle of points on an elliptic curve. Once a cycle is found, the algorithm uses a technique called “baby-step giant-step” to find the discrete logarithm of the curve’s points.

The algorithm is quite complex and involves a lot of mathematical concepts, but it has been proven to be very effective for solving the discrete logarithm problem on elliptic curves.

Python code for the Joux-Lercier algorithm to solve the discrete logarithm problem:

def baby_step_giant_step(a, b, n):
    # Create a list of powers of a
    powers = [1]
    while powers[-1] <= n:
        powers.append(powers[-1] * a)
    
    # Create a hash table to store the powers of a
    hash_table = {}
    for i in range(len(powers)):
        hash_table[powers[i]] = i
    
    # Compute the hash of b
    hash_b = powers[0]
    for i in range(1, len(powers)):
        if b % powers[i] == 0:
            hash_b = powers[i]
            break
    
    # Use the hash table to find the value of x such that a^x = b
    for i in range(1, len(powers)):
        if hash_table[hash_b - powers[i]] = 0:
            x = hash_table[hash_b - powers[i]]
            return x
    return -1

# Test the algorithm
a = 2
b = 17
n = 23
print(baby_step_giant_step(a, b, n))

This code implements the baby-step giant-step algorithm to solve the discrete logarithm problem. The baby_step_giant_step function takes three arguments: a is the base, b is the exponent, and n is the modulus. It returns the value of x such that a^x = b.

The baby_step_giant_step function first creates a list of powers of a and a hash table to store the powers of a. It then computes the hash of b using the hash table. Finally, it uses the hash table to find the value of x such that a^x = b.

The last three lines of the code test the algorithm with a = 2b = 17, and n = 23


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