Rabin.py

# -*- coding: utf-8 -*-
"""
Created on Tue May  3 2:28:13 2022

@author: Paul
@file:   Rabin.py
"""
# COM 5335 Network Security Assignment #3   陳劭珩 110064533 
# Rabin Public-Key Cryptosystem
import random

# the number of primes within 1000 is 168
primeTable1 = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97,
            101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197,
            199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313,
            317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439,
            443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571,
            577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691,
            701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829,
            839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 
            983, 991, 997]

# the number of primes within 10000 is 1229
primeTable2 = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 
            113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 
            251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 
            397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 
            557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691,
            701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 
            863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 
            1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 
            1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 
            1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 
            1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 
            1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 
            1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 
            1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 
            2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 
            2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 
            2351, 2357, 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 
            2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 
            2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 
            2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 
            2971, 2999, 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 
            3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 
            3329, 3331, 3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, 3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 
            3499, 3511, 3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593, 3607, 3613, 3617, 3623, 3631, 
            3637, 3643, 3659, 3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727, 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 
            3803, 3821, 3823, 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907, 3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, 
            3967, 3989, 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057, 4073, 4079, 4091, 4093, 4099, 4111, 4127, 4129, 
            4133, 4139, 4153, 4157, 4159, 4177, 4201, 4211, 4217, 4219, 4229, 4231, 4241, 4243, 4253, 4259, 4261, 4271, 4273, 4283, 
            4289, 4297, 4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409, 4421, 4423, 4441, 4447, 4451, 4457, 4463, 4481, 
            4483, 4493, 4507, 4513, 4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583, 4591, 4597, 4603, 4621, 4637, 4639, 4643, 4649, 
            4651, 4657, 4663, 4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733, 4751, 4759, 4783, 4787, 4789, 4793, 4799, 4801, 4813, 
            4817, 4831, 4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937, 4943, 4951, 4957, 4967, 4969, 4973, 4987, 4993, 
            4999, 5003, 5009, 5011, 5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087, 5099, 5101, 5107, 5113, 5119, 5147, 5153, 5167, 
            5171, 5179, 5189, 5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273, 5279, 5281, 5297, 5303, 5309, 5323, 5333, 5347, 5351, 
            5381, 5387, 5393, 5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443, 5449, 5471, 5477, 5479, 5483, 5501, 5503, 5507, 
            5519, 5521, 5527, 5531, 5557, 5563, 5569, 5573, 5581, 5591, 5623, 5639, 5641, 5647, 5651, 5653, 5657, 5659, 5669, 5683, 
            5689, 5693, 5701, 5711, 5717, 5737, 5741, 5743, 5749, 5779, 5783, 5791, 5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849, 
            5851, 5857, 5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029, 6037, 6043, 
            6047, 6053, 6067, 6073, 6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133, 6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 
            6217, 6221, 6229, 6247, 6257, 6263, 6269, 6271, 6277, 6287, 6299, 6301, 6311, 6317, 6323, 6329, 6337, 6343, 6353, 6359, 
            6361, 6367, 6373, 6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469, 6473, 6481, 6491, 6521, 6529, 6547, 6551, 6553, 6563, 
            6569, 6571, 6577, 6581, 6599, 6607, 6619, 6637, 6653, 6659, 6661, 6673, 6679, 6689, 6691, 6701, 6703, 6709, 6719, 6733, 
            6737, 6761, 6763, 6779, 6781, 6791, 6793, 6803, 6823, 6827, 6829, 6833, 6841, 6857, 6863, 6869, 6871, 6883, 6899, 6907, 
            6911, 6917, 6947, 6949, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997, 7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 
            7079, 7103, 7109, 7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, 7211, 7213, 7219, 7229, 7237, 7243, 7247, 7253, 
            7283, 7297, 7307, 7309, 7321, 7331, 7333, 7349, 7351, 7369, 7393, 7411, 7417, 7433, 7451, 7457, 7459, 7477, 7481, 7487, 
            7489, 7499, 7507, 7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561, 7573, 7577, 7583, 7589, 7591, 7603, 7607, 7621, 
            7639, 7643, 7649, 7669, 7673, 7681, 7687, 7691, 7699, 7703, 7717, 7723, 7727, 7741, 7753, 7757, 7759, 7789, 7793, 7817, 
            7823, 7829, 7841, 7853, 7867, 7873, 7877, 7879, 7883, 7901, 7907, 7919, 7927, 7933, 7937, 7949, 7951, 7963, 7993, 8009, 
            8011, 8017, 8039, 8053, 8059, 8069, 8081, 8087, 8089, 8093, 8101, 8111, 8117, 8123, 8147, 8161, 8167, 8171, 8179, 8191, 
            8209, 8219, 8221, 8231, 8233, 8237, 8243, 8263, 8269, 8273, 8287, 8291, 8293, 8297, 8311, 8317, 8329, 8353, 8363, 8369, 
            8377, 8387, 8389, 8419, 8423, 8429, 8431, 8443, 8447, 8461, 8467, 8501, 8513, 8521, 8527, 8537, 8539, 8543, 8563, 8573, 
            8581, 8597, 8599, 8609, 8623, 8627, 8629, 8641, 8647, 8663, 8669, 8677, 8681, 8689, 8693, 8699, 8707, 8713, 8719, 8731, 
            8737, 8741, 8747, 8753, 8761, 8779, 8783, 8803, 8807, 8819, 8821, 8831, 8837, 8839, 8849, 8861, 8863, 8867, 8887, 8893, 
            8923, 8929, 8933, 8941, 8951, 8963, 8969, 8971, 8999, 9001, 9007, 9011, 9013, 9029, 9041, 9043, 9049, 9059, 9067, 9091, 
            9103, 9109, 9127, 9133, 9137, 9151, 9157, 9161, 9173, 9181, 9187, 9199, 9203, 9209, 9221, 9227, 9239, 9241, 9257, 9277, 
            9281, 9283, 9293, 9311, 9319, 9323, 9337, 9341, 9343, 9349, 9371, 9377, 9391, 9397, 9403, 9413, 9419, 9421, 9431, 9433, 
            9437, 9439, 9461, 9463, 9467, 9473, 9479, 9491, 9497, 9511, 9521, 9533, 9539, 9547, 9551, 9587, 9601, 9613, 9619, 9623, 
            9629, 9631, 9643, 9649, 9661, 9677, 9679, 9689, 9697, 9719, 9721, 9733, 9739, 9743, 9749, 9767, 9769, 9781, 9787, 9791, 
            9803, 9811, 9817, 9829, 9833, 9839, 9851, 9857, 9859, 9871, 9883, 9887, 9901, 9907, 9923, 9929, 9931, 9941, 9949, 9967, 
            9973]
#

# search if 'int' target is in 'list' arr or not
def isFound(arr, target: int) -> bool:
    """
    :type:  arr: List[int]
    :type:  target: int
    :rtype: bool
    """
    # binary search
    arr.sort() # first sort the list
    left = 0
    right = len(arr) - 1
    while left <= right:
        middle = left + (right - left) // 2
        if arr[middle] == target:
            return True
        elif target < arr[middle]:
            right = middle - 1
        elif target > arr[middle]:
            left = middle + 1
    return False

# Miller-Rabin Primality Test
def Miller_Rabin_test(n: int) -> bool:
    """
    :type:  n: int
    :rtype: bool
    """
    # return False means composite (definitely not prime)
    # return True  means prime (with high chance its prime)
    #
    # 1. factorize n - 1 as m * 2^k
    k = 0
    temp = n - 1
    while temp % 2 == 0:
        temp = temp // 2
        k += 1
    else:
        m = temp
    
    # 2. primality test
    # use the primeTable to filter out small primes and its multiple
    for a in primeTable2:
        x = [pow(a, m * (2 ** i), n) for i in range(0, k)]
        if pow(a, m, n) != 1 and not isFound(x, n - 1):
            return False # composite
        elif pow(a, m, n) == 1 or isFound(x, n - 1):
            continue 
    return True # 75% chance its prime

# brute force search all primes under n
def brute_force_test(n: int) -> bool:
    """
    :type:  n: int
    :rtype: bool
    """
    # return False means not prime
    # return True  means prime
    #
    if n <= 1: 
        return False
    elif 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

# generate a n-bit random prime
def prime_generater(num_of_bits: int) -> int:
    """
    :type:  num_of_bits: int
    :rtype: int
    """
    # as we add up the successful checks required, the corresponding error 
    # probobility that its not prime would be lower, if we set to 10 times
    # then the error prob. that its not prime would be less than 10^-6
    SUCCESS = 1 # 10
    count = 0   # count the number of successful checks
    while True:
        # first generate a n-bit random number
        num = random.getrandbits(num_of_bits)
        # then check if the random number is prime
        if Miller_Rabin_test(num):
            count += 1
            if count == SUCCESS:
                return num # return the n-bit random prime
        else:
            continue

# Find SQROOT in Zp where p = 3 mod 4
def sqrt_p_3_mod_4(a: int, p: int) -> int:
    """
    :type:  a: int
    :type:  p: int
    :rtype: int
    """
    if (p % 2) == 0:
        raise ValueError('p must be an odd prime')
    # square root of a mod p
    r = pow(a, (p + 1) // 4, p) # r = a^((p + 1) / 4) mod p
    return r

# Find SQROOT in Zp where p = 5 mod 8
def sqrt_p_5_mod_8(a: int, p: int) -> int:
    """
    :type:  a: int
    :type:  p: int
    :rtype: int
    """
    if (p % 2) == 0:
        raise ValueError('p must be an odd prime')
    # square root of a mod p
    r = 0                       # initialized to 0 as 'int'
    d = pow(a, (p - 1) // 4, p) # d = a^((p - 1) / 4) mod p
    if d == 1:
        # if d == 1 mod p
        # then r = a^((p + 3) / 8) mod p
        r = pow(a, (p + 3) // 8, p)
    elif d == p - 1:
        # if d == -1 mod p
        # then r = 2a * (4a)^((p - 5) / 8) mod p
        r = 2 * a * pow(4 * a, (p - 5) // 8, p) % p
    return r

# extended euclidean algorithm
def egcd(a: int, b: int):
    """
    :type:  a: int
    :type:  b: int
    :rtype: gcd, y, x: int, Literal, Literal
    """
    if a == 0:
        return b, 0, 1
    else:
        gcd, y, x = egcd(b % a, a)
        return gcd, x - (b // a) * y, y

# pad the last 16 bits to the end of the plaintext
def padding(plaintext: int) -> int:
    """
    :type:  plaintext: int
    :rtype: int
    """
    bin_str = bin(plaintext)         # convert to a bit string
    result = bin_str + bin_str[-16:] # pad the last 16 bits to the end
    result = int(result, 2)          # convert back to integer
    # print(result)
    return result

# decide which answer to choose
def select(lst):
    """
    :type:  lst: List[int]
    :rtype: int
    """
    for i in lst:
        binary = bin(i)
        append = binary[-16:]   # take the last 16 bits
        binary = binary[:-16]   # remove the last 16 bits
        if append == binary[-16:]:
            return i
    return -1

# Rabin encryption function
def Rabin_encrypt(plaintext: int, n: int) -> int:
    """
    :type:  plaintext: int
    :type:  n: int
    :rtype: ciphertext: int
    """
    # c = m^2 mod n
    plaintext = padding(plaintext)
    ciphertext = plaintext ** 2 % n
    # print(ciphertext)
    return ciphertext

# Rabin decryption function
def Rabin_decrypt(ciphertext: int, p: int, q: int) -> int:
    """
    :type:  ciphertext: int
    :type:  p: int
    :type:  q: int
    :rtype: plaintext: int
    """
    n = p * q
    r, s = 0, 0
    # find sqrt for p
    if p % 4 == 3:
        r = sqrt_p_3_mod_4(ciphertext, p)
    elif p % 8 == 5:
        r = sqrt_p_5_mod_8(ciphertext, p)
    # find sqrt for q
    if q % 4 == 3:
        s = sqrt_p_3_mod_4(ciphertext, q)
    elif q % 8 == 5:
        s = sqrt_p_5_mod_8(ciphertext, q)

    gcd, c, d = egcd(p, q)
    if gcd != 1:
        raise ValueError('modular inverse does not exist')

    x = (r * d * q + s * c * p) % n
    y = (r * d * q - s * c * p) % n
    lst = [x, n - x, y, n - y]

    plaintext = select(lst)
    string = bin(plaintext)
    string = string[:-16]
    plaintext = int(string, 2)
    # 
    return plaintext


def main():
    # Miller-Rabin 256-bit random prime generation
    print("\n------------------------------------------ <Miller-Rabin> --------------------------------------------------")
    result = format(prime_generater(256), 'x') # convert the random prime from int into hex, store in result
    print("256-bit random prime in dec:", int(result, 16)) # output random prime in dec
    print("256-bit random prime in hex:", result)          # output random prime in hex
    #
    ## check if the random prime is actually 256-bit
    # length = len(result) * 4
    # print(f"(length: {length}-bit)")

    # Rabin Encryption
    print("\n----------------------------------------- <Rabin Encryption> -----------------------------------------------")
    # given sample input of p, q, and plaintext, compute the corresponding n, and ciphertext

    # NOTE: no space allowed in the input field
    # store given plaintext as string type in plaintextstr 
    plaintextstr = 'be000badbebadbadbad00debdeadfacedeafbeefadd00addbed00bed' 
    print("224-bit Plaintext =", plaintextstr) # plaintext has 224 bits
    # convert plaintextstr from hex into integer type, store in plaintext
    plaintext = int(plaintextstr, 16)

    # store given private keys p, q as string type in pstr, qstr
    pstr = 'daaefe652cad1614f17e87f2cd80973f' # p = daaefe652cad1614f17e87f2cd80973f
    qstr = 'f99988626723eef2a54ed484dfa735c7' # q = f99988626723eef2a54ed484dfa735c7
    print("128-bit p =", pstr)
    print("128-bit q =", qstr)
    # convert pstr, qstr from hex into integer type, store in p, q
    p = int(pstr, 16)
    q = int(qstr, 16)
    n = p * q
    print('256-bit n = pq =', format(n, 'x')) # output n in hex
 
    # store the encrypted result as integer type in ciphertext 
    ciphertext = Rabin_encrypt(plaintext, n)
    print('256-bit Ciphertext =', format(ciphertext, 'x')) # output ciphertext in hex
    

    # Rabin Decryption
    print("\n---------------------------------------- <Rabin Decryption> ------------------------------------------------")
    # given sample input of p, q and ciphertext, compute the corresponding plaintext

    # NOTE: no space allowed in the input field
    # store given ciphertext as string type in ciphertextstr
    ciphertextstr = '5452361adb4c34be04a5903ae00793bc1086e887ebed06e23ffba0b4a4348cc0'
    # convert ciphertextstr from hex into integer type, store in ciphertext
    ciphertext = int(ciphertextstr, 16)
    print('256-bit Ciphertext =', ciphertextstr) # output ciphertext in hex

    # store given private keys p, q as string type in pstr, qstr
    pstr = 'd5e68b2b5855059ad1a80dd6c5dc03eb' # p = d5e68b2b5855059ad1a80dd6c5dc03eb
    qstr = 'c96c6afc57ce0f53396d3b32049fe2d3' # q = c96c6afc57ce0f53396d3b32049fe2d3
    print("128-bit p =", pstr)
    print("128-bit q =", qstr)
    # convert pstr, qstr from hex into integer type, store in p, q
    p = int(pstr, 16)
    q = int(qstr, 16)

    # store the decrypted result as integer type in plaintext
    plaintext = Rabin_decrypt(ciphertext, p, q)
    print('224-bit Plaintext =', format(plaintext, 'x').zfill(226 // 4)) # output plaintext in hex
    print("\n----------------------------------------------------------------------------------------------------------")

# main()
if __name__ == '__main__':
    main()