如何使用python查找数字的素因

Question

I'm writing a program that will calulate the private key for a weak RSA public key. I am wondering how I would go about determining the values for p and q from the value n . Here is the Python code so far:

from Crypto.PublicKey import RSA #PyCryptoDome
import .math as cm # My own module

with open(public_keyfile, 'rb') as key: # Public Keyfile Is in PEM format
    public_key = RSA.import_key(key)

n = public_key.n # N value of the public_key

e = public_key.e # E value of the public_key

p, q = get_factors_of(n) # This I don't know how to do, though there is a question that might help [see bottom]

t = cm.lcm(p-1, q-1) # Get the lowest common multiple of q and q

d = cm.mod_inverse(e, t) # Get d, the modular inverse of e % t

private_key = RSA.construct((n, e, d, p, q) # Construct the RSA private_key

The .math module referenced above:

from math import gcd


def mod_inverse(a, b):
    a = a % b
    for x in range(1, b):
        if (a * x) % b == 1:
            return x
    return 1


def lcm(x, y):
    return x * y // gcd(x, y)

What I need to do appears to be referenced here but this code is in Java.

If anyone knows how to get p and q from n with python, help would be appreciated.

Many thanks, Legorooj.

Answer 1

Mandatory warning: if you are after performance, you will need to investigate the details of the algorithms yourself. Even "weak" public keys will take forever to crack with a simplistic algorithm (eg Erathostene's sieve).

That being said, sympy.ntheory.factorint() might be what you need:

from sympy.ntheory import factorint

print(factorint(54))  # {2: 1, 3: 3} i.e. 54 == 2**1 * 3**3

Answer 2

After lots of googling, and pdf reading, I found an algorithm that works. Here is a python implementation:

import math
def get_factors_of(num):
    poss_p = math.floor(math.sqrt(num)) 

    if poss_p % 2 == 0: # Only checks odd numbers, it reduces time by orders of magnitude
        poss_p += 1
    while poss_p < num:
        if num % poss_p == 0:
            return poss_p
        poss_p += 2

This algorithm effectively finds the P/Q factors of a small RSA key. (I have tested it against a 64-bit PEM public key)