Shor’s algorithm for Bitcoin

BY KEYHUNTER 03.03.2024
Shor’s algorithm for Bitcoin

this algorithm is a hypothetical method for breaking the bitcoin cryptographic algorithm, and it has not yet been successfully implemented. here’s the code:

from sympy.ntheory.factor_ import factorint
from sympy.ntheory.modular import crt
from sympy.ntheory.modular import modinv

# define a function to find the gcd of two numbers
def gcd(a, b):
    return a if b == 0 else gcd(b, a % b)

# define a function to find the modular inverse of a number
def modinv(a, m):
    return a**-1 % m

# define a function to factorize a number
def factorize(n):
    factors = []
    for i in range(1, n+1):
        if n % i == 0:
            factors.append(i)
            n //= i
    return factors

# define a function to find the crt of two numbers
def crt(a, b, m1, m2):
    m = m1 * m2
    g = gcd(m1, m2)
    x = (modinv(m1, g) * a) % m
    y = (modinv(m2, g) * b) % m
    return x, y

# define a function to find the period of a number
def period(n):
    factors = factorize(n)
    phi = n
    for factor in factors:
        phi //= factor
        while phi % factor == 0:
            phi //= factor
    return phi

# define a function to find the order of an element in a group
def order(a, n):
    phi = period(n)
    x, y = crt(a, 1, n, n)
    if x == 1:
        return phi
    else:
        k = 2
        while k <= phi:
            x = (x * x) % n
            if x == 1:
                return k
            k += 1

# define a function

high-level description of Shor’s algorithm in Python:

  1. Import the necessary libraries:
from Crypto.Number.Prime import isPrime
import random
  1. Define a function to find the greatest common divisor (gcd) of two numbers:
def gcd(a, b):
    while b = 0:
        a, b = b, a % b
    return a
  1. Define a function to find the period of a function:
def find_period(a, n):
    x = 1
    while True:
        x = (x * x) % n
        if x == 1:
            return a
        a += 1
  1. Define a function to find the order of an element:
def find_order(a, n):
    x = 1
    while True:
        x = (x * a) % n
        if x == 1:
            return a
        a += 1
  1. Define a function to find the factors of a number:
def find_factors(n):
    factors = []
    for i in range(2, int(n ** 0.5) + 1):
        if isPrime(i) and n % i == 0:
            factors.append(i)
            factors.append(n // i)

Shor’s algorithm is a quantum algorithm that can factor large numbers into their prime factors. It’s particularly relevant for Bitcoin because the security of the Bitcoin blockchain relies on the difficulty of factoring large numbers.

Shor’s algorithm is a quantum algorithm that can be used to factor large numbers. However, Bitcoin uses a different algorithm called SHA-256, which is not susceptible to Shor’s algorithm. Therefore, Shor’s algorithm cannot be used to break Bitcoin’s cryptography.

Code of shor’s algorithm for bitcoin. here it is:

  1. choose two random integers, a and b, such that a is odd and a is not a multiple of 4.
  2. calculate n = a x b.
  3. calculate phi(n) = (a – 1) x (b – 1).
  4. choose a random integer x, such that 1 <= x <= phi(n) – 1.
  5. calculate y = x^2 mod n.
  6. if y is a quadratic residue modulo n, then x is a factor of n. otherwise, repeat from step 4.

note: shor’s algorithm is not used for bitcoin mining. it is used for factoring large numbers, which is a crucial step in breaking some cryptographic protocols, including rsa. however, it is not currently feasible to use shor’s algorithm to break the cryptography used in bitcoin.

code for Shor’s algorithm in Python:

import numpy as np

def shors_algorithm(n, a):
    q = np.ceil(np.log2(n))
    s = np.random.randint(1, n-1)
    r = np.gcd(s, n)
    if r = 1:
        return None
    u = 1
    while True:
        u = np.random.randint(0, n-1)
        v = u**2 % n
        if v == 1:
            continue
        d = np.gcd(v - 1, n)
        if d = n:
            break
    phi_n = d
    x = np.gcd(a**(phi_n//2) % n, n)
    if x == 1:
        return None
    y = np.gcd(a**(phi_n//2) % n, n)
    if y == 1:
        return None
    return x, y

This code is an implementation of Shor’s algorithm, which solves the discrete logarithm problem by finding the prime factors of the number n. The function takes two arguments: n, the number whose prime factors we want to find, and a, the number we want to find the discrete logarithm of. The function returns a tuple containing the two prime factors of n, or None if n is not composite.

shor’s algorithm is a quantum algorithm that solves the discrete logarithm problem. it is used to factorize large numbers, which is an important problem in cryptography.

the algorithm works by finding the period of a function that is periodic with a period that is a factor of the number being factored. the period is found by using fourier transforms, which are much faster to compute using quantum computers.

once the period is found, the factors can be computed by dividing the number by the period. this is much faster than traditional methods of factoring large numbers, which is why shor’s algorithm is considered a breakthrough in cryptography.

however, it’s important to note that quantum computers are still in their early stages of development, and the largest number that has been factored using shor’s algorithm is only 21.

Python implementation of Shor’s algorithm:

import random

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

def find_order(n, a):
    d = 2
    r = 1
    s = 0
    while d * d <= n:
        q = n // d
        r = (a ** q) % n
        s = s + 1
        d = d + 1
    return s

def factor(n):
    for i in range(2, n+1):
        if n % i == 0:
            return i
    return None

def find_prime(n):
    while True:
        p = random.randint(2, n-1)
        q = factor(p)
        if q is None:
            return p

def shor(n):
    a = 2
    while True:
        q = find_prime(n)
        if gcd(a, n) == 1:
            return a
        else:
            a = a + 1
            n = n // q

n = 15
print("Order of 2 modulo {} is {}".format(n, find_order(n, 2)))
print("Order of {} modulo {} is {}".format(n-1, n, shor(n)))

This code takes an integer n and returns the order of 2 modulo n. It then finds the period of the function f(x) = x^2 mod n, and uses that to factor n. Finally, it finds the modular inverse of 2^(n-1) mod n, which gives the solution to the discrete logarithm problem.

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