Miller’s algorithm for Bitcoin

03.03.2024
Miller’s algorithm for Bitcoin

python implementation of miller’s algorithm for bitcoin. here is an example:

def miller_rabin(n, k):
    """
    miller-rabin primality test.
    n: integer to test for primality.
    k: number of iterations.
    """
    if n < 2:
        return false
    d = n - 1
    while d % 2 == 0:
        d //= 2
    r = 0
    x = pow(2, d, n)
    while k > 0:
        x = pow(x, 2, n)
        if x == 1:
            return true
        if x == n - 1:
            r += 1
        k -= 1
    return r == 0

this implementation takes two arguments: n and kn is the number to test for primality, and k is the number of iterations. it returns true if n is probably prime, and false otherwise.

to use this function to test whether a bitcoin private key is valid, you can generate a random 256-bit number and test it with miller’s algorithm. if the number passes the test, it is probably prime and can be used as a private key. here is an example:

import random

def miller_rabin(n, k):
    # implementation of miller-rabin primality test

while true:
    private_key = random.getrandbits(256)
    if miller_rabin(private_key, 10):
        break

print(f"valid private key: {private_key}")

this code generates a random 256-bit number, tests it with miller’s algorithm with 10 iterations, and prints it if it passes the test.

python code for miller’s algorithm to solve the discrete logarithm problem. however, i must remind you that miller’s algorithm is not very secure, and it’s not used in practice anymore.

here’s an example code:

def miller_rabin(n, k):
    # base case
    if n == 1:
        return true
    if n == 2:
        return false

    # find the factors of n
    factors = []
    for i in range(1, n):
        if n % i == 0:
            factors.append(i)

    # loop through the factors
    for factor in factors:
        if factor == n - 1:
            continue
        x = pow(factor, (n - 1) // factor, n)
        if x == 1 or x == n - 1:
            continue
        for i in range(k - 1):
            x = pow(x, 2, n)
            if x == n - 1:
                return false
        return false

    return true

in this code, n is the number whose discrete logarithm we want to find, and k is the number of iterations. the function returns true if n is a probable prime, and false otherwise.

note that this code is not optimized for speed or efficiency, and it may not work for very large values of n and k. also, as i mentioned earlier, miller’s algorithm is not very secure, and it’s not used in practice anymore.

Miller’s algorithm for solving the discrete logarithm problem in Python.

def miller_rabin(n, k):
    # Check if n is a prime number
    if n <= 1:
        return False
    if n <= 3:
        return True
    if n % 2 == 0 or n % 3 == 0:
        return False
    i = 5
    while i * i <= n:
        if n % i == 0 or n % (i + 2) == 0:
            return False
        i += 6
    return True

# Function to find the discrete logarithm of a number
def discrete_logarithm(n, k):
    # Check if n is a prime number
    if not miller_rabin(n, k):
        return -1
    
    # Calculate the value of b
    b = 2
    while b <= n:
        if n % b == 1:
            break
        b += 1
    
    # Calculate the value of x
    x = 2
    while x <= n:
        if n % x == 1:
            break
        x += 1
    
    # Calculate the value of a
    a = (b - 1) // x
    
    # Calculate the value of y
    y = 1
    while y <= n:
        if y == 1:
            break
        y = y * y % n
    
    # Calculate the value of r
    r = 1
    while r <= n:
        if r == n - 1:
            break
        r = r * r % n
    
    # Calculate the value of s
    s = 1
    while s <= n:
        if s == n - 1:
            break
        s = s * s % n
    
    # Calculate the value of t
    t = (s - 1) // r
    
    # Calculate the value of u
    u = t * r % (n - 1)
    
    # Calculate the value of v
    v = (u - 1) // x

Python implementation of Miller’s algorithm:

def miller_algorithm(base, exponent, n):
    if exponent == 0:
        return 1
    elif exponent % 2 == 0:
        x = miller_algorithm(base, exponent // 2, n)
        return (x * x) % n
    else:
        y = miller_algorithm(base, (n - 1) // exponent, n)
        return (base * y ** (n - 1)) % n

This function takes as input the base, exponent, and modulus n, and returns the solution to the discrete logarithm problem (i.e., the value of x such that base ** x = exponent mod n). The algorithm works by recursively computing the value of x using the formula:

x = (base ** (exponent // 2)) % n

If exponent is even, and (base ** (exponent // 2)) % n is not equal to 1, then we can continue recursing. Otherwise, we can compute the value of x as:

x = (base ** ((n - 1) // exponent)) % n

And then return x.


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