simplersa/main.py

#!/usr/bin/env python3
import math
import random
from itertools import combinations


def euclid(a, b):
    """returns the Greatest Common Divisor of a and b"""
    a = abs(a)
    b = abs(b)
    if a < b:
        a, b = b, a
    while b != 0:
        a, b = b, a % b
    return a


def coPrime(l):
    """returns 'True' if the values in the list L are all co-prime
       otherwise, it returns 'False'. """
    for i, j in combinations(l, 2):
        if euclid(i, j) != 1:
            return False
    return True


def extendedEuclid(a, b):
    """return a tuple of three values: x, y and z, such that x is
    the GCD of a and b, and x = y * a + z * b"""
    if a == 0:
        return b, 0, 1
    else:
        g, y, x = extendedEuclid(b % a, a)
        return g, x - (b // a) * y, y


def modInv(a, m):
    """returns the multiplicative inverse of a in modulo m as a
       positive value between zero and m-1"""
    # notice that a and m need to be co-prime to each other.
    if coPrime([a, m]):
        linearCombination = extendedEuclid(a, m)
        return linearCombination[1] % m
    else:
        return 0


def extractTwos(m):
    """m is a positive integer. A tuple (s, d) of integers is returned
    such that m = (2 ** s) * d."""
    # the problem can be reduced to counting how many '0's there are in
    # the end of bin(m). This can be done this way: m & a stretch of '1's
    # which can be represent as (2 ** n) - 1.
    assert m >= 0
    i = 0
    while m & (2 ** i) == 0:
        i += 1
    return i, m >> i


def millerRabin(n, k):
    """
    Miller Rabin pseudo-prime test
    return True means likely a prime, (how sure about that, depending on k)
    return False means definitely a composite.
    Raise assertion error when n, k are not positive integers
    and n is not 1
    """
    assert n >= 1
    # ensure n is bigger than 1
    assert k > 0
    # ensure k is a positive integer so everything down here makes sense

    if n == 2:
        return True
    # make sure to return True if n == 2

    if n % 2 == 0:
        return False
    # immediately return False for all the even numbers bigger than 2

    extract2 = extractTwos(n - 1)
    s = extract2[0]
    d = extract2[1]
    assert 2 ** s * d == n - 1

    def tryComposite(a):
        """Inner function which will inspect whether a given witness
        will reveal the true identity of n. Will only be called within
        millerRabin"""
        x = pow(a,d,n)
        if x == 1 or x == n - 1:
            return None
        else:
            for j in range(1, s):
                x = pow(x,2,n)
                if x == 1:
                    return False
                elif x == n - 1:
                    return None
            return False

    for i in range(0, k):
        a = random.randint(2, n - 2)
        if tryComposite(a) == False:
            return False
    return True  # actually, we should return probably true.


def findAPrime(a, b, k):
    """Return a pseudo prime number roughly between a and b,
    (could be larger than b). Raise ValueError if cannot find a
    pseudo prime after 10 * ln(x) + 3 tries. """
    x = random.randint(a, b)
    for i in range(0, int(10 * math.log(x) + 3)):
        if millerRabin(x, k):
            return x
        else:
            x += 1
    raise ValueError


def newKey(a, b, k):
    """ Try to find two large pseudo primes roughly between a and b.
    Generate public and private keys for RSA encryption.
    Raises ValueError if it fails to find one"""
    try:
        p = findAPrime(a, b, k)
        while True:
            q = findAPrime(a, b, k)
            if q != p:
                break
    except:
        raise ValueError

    n = p * q
    m = (p - 1) * (q - 1)

    while True:
        e = random.randint(1, m)
        if coPrime([e, m]):
            break

    d = modInv(e, m)
    return (n, e, d)


def encrypt(message, modN, e, blockSize):
    """given a string message, public keys and blockSize, encrypt using
    RSA algorithms."""
    return pow(message, e, modN)


def decrypt(secret, modN, d, blockSize):
    """reverse function of encrypt"""
    return pow(secret, d, modN)

if __name__ == '__main__':

    n, e, d = newKey(2**40, 2 ** 41, 20)
    message = 35274764

    print("original message is {}".format(message))
    print("-"*80)
    cipher = encrypt(message, n, e, 15)
    print("cipher text is {}".format(cipher))
    print("-"*80)
    deciphered = decrypt(cipher, n, d, 15)
    print("decrypted message is {}".format(deciphered))