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README.md
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README.md
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RSA and Chinese Remainder Theorem (CRT)
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You can use as reference OpenSSL's RSA module that can produce following output, e.g.
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```bash
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$ openssl genrsa 32 |openssl rsa -noout -text
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Generating RSA private key, 32 bit long modulus
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.+++++++++++++++++++++++++++
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.+++++++++++++++++++++++++++
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e is 65537 (0x10001)
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Private-Key: (32 bit)
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modulus: 2719120549 (0xa2127ca5)
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publicExponent: 65537 (0x10001)
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privateExponent: 1790467441 (0x6ab85d71)
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prime1: 53267 (0xd013)
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prime2: 51047 (0xc767)
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exponent1: 37383 (0x9207)
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exponent2: 28991 (0x713f)
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coefficient: 16388 (0x4004)
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$ openssl genrsa 32 |openssl asn1parse
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```
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rsatest-ctr.py
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rsatest-ctr.py
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#!/usr/bin/env python
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# -*- coding: utf-8 -*-
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#
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# -------------------------------------
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# RSA and Chinese Remainder Theorem (CRT)
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# Copyright (C) 2014 Andrey Arapov
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#
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# This program is free software: you can redistribute it and/or modify
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# it under the terms of the GNU General Public License as published by
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# the Free Software Foundation, either version 3 of the License, or
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# (at your option) any later version.
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#
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# This program is distributed in the hope that it will be useful,
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# but WITHOUT ANY WARRANTY; without even the implied warranty of
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# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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# GNU General Public License for more details.
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#
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# You should have received a copy of the GNU General Public License
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# along with this program. If not, see <http://www.gnu.org/licenses/>.
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# -------------------------------------
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#
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#
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# Notes:
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# - Based on the https://en.wikipedia.org/wiki/RSA_%28cryptosystem%29#Key_generation
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# and http://www.di-mgt.com.au/crt_rsa.html
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# - The parameters used here are artificially small
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# - I've tried to apply KISS principle here
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#
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#
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import random
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from fractions import gcd
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class bcolors:
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RED = '\033[91m'
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DRED = '\033[31m'
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GREEN = '\033[92m'
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YELLOW = '\033[93m'
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BLUE = '\033[94m'
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PURPLE = '\033[95m'
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CYAN = '\033[96m'
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ENDC = '\033[0m'
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print "RSA and Chinese Remainder Theorem (CRT)\n"
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# -----------------------------------------------------------------------------
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# Generating the Public/Private keypair
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# -----------------------------------------------------------------------------
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# Primality test
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# https://en.wikipedia.org/wiki/Primality_test#Python_implementation
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def is_prime(num):
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if num <= 3:
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if num <= 1:
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return False
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return True
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if not num % 2 or not num % 3:
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return False
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for i in range(5, int(num ** 0.5) + 1, 6):
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if not num % i or not num % (i + 2):
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return False
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return True
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# 1. Choose two distinct prime numbers p and q.
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print "1. looking for two distinct prime numbers p and q in artificially small range..."
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i = 0
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while i < 2:
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rand = random.randint(0x8000, 0xffff)
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if is_prime(rand):
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if i == 1:
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q=rand
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break
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p=rand
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i += 1
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print "p =", bcolors.DRED, p, bcolors.ENDC, "\tprime?", is_prime(p), bcolors.YELLOW, "(prime1)", bcolors.ENDC
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print "q =", bcolors.DRED, q, bcolors.ENDC, "\tprime?", is_prime(q), bcolors.YELLOW, "(prime2)", bcolors.ENDC
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print
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# 2. Compute n = pq.
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print "2. computing the modulus n = pq ..."
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n = p * q
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print "n = p * q =", bcolors.DRED, p, bcolors.ENDC, "*", bcolors.DRED, q, bcolors.ENDC, "=", \
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bcolors.BLUE, n, bcolors.ENDC, bcolors.YELLOW, "(modulus)", bcolors.ENDC
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print
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print "Private-Key will be "+bcolors.YELLOW+str(n.bit_length())+bcolors.ENDC+" bit long\n"
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# 3. Compute φ(n) = φ(p)φ(q) = (p − 1)(q − 1) = n - (p + q - 1), where φ is Euler's totient function.
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print "3. computing φ(n) = φ(p)φ(q) = (p − 1)(q − 1) = n - (p + q -1), where φ is Euler's totient function ..."
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f_n = n - (p + q - 1)
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print "φ(n) =", bcolors.DRED, f_n, bcolors.ENDC
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print
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# 4. Choose an integer e such that 1 < e < φ(n) and gcd(e, φ(n)) = 1; i.e., e and φ(n) are coprime.
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print "4. looking for an integer e such that 1 < e < φ(n) and gcd(e, φ(n)) = 1; i.e., e and φ(n) are coprime ..."
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print "Setting e = 2^16+1 (65537) as per recommendation in\n"+ \
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"Dan Boneh's Twenty Years of Attacks on the RSA Cryptosystem - "+ \
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"http://crypto.stanford.edu/~dabo/pubs/papers/RSA-survey.pdf\n"
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#e_gcd = 2
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# TOFIX: add smth like --> while (e_gcd != 1) and (e > 3): as e should be > than 3
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#while e_gcd != 1:
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# e = random.randint(1, f_n)
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# e_gcd = gcd(e, f_n)
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e = 65537
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print "e =", bcolors.CYAN, e, bcolors.ENDC, bcolors.YELLOW, "(publicExponent)", bcolors.ENDC
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print
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# 5. Determine d as d ≡ e^−1 (mod φ(n)); i.e., d is the multiplicative inverse of e (modulo φ(n)).
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# http://en.wikibooks.org/wiki/Algorithm_Implementation/Mathematics/Extended_Euclidean_algorithm#Modular_inverse
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def egcd(a, b):
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if a == 0:
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return (b, 0, 1)
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else:
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g, y, x = egcd(b % a, a)
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return (g, x - (b // a) * y, y)
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def modinv(a, m):
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gcd, x, y = egcd(a, m)
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if gcd != 1:
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return None # modular inverse does not exist
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else:
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return x % m
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print "5. Determining d as d ≡ e^−1 (mod φ(n)); i.e., d is the multiplicative inverse of e (modulo φ(n)) ..."
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d = modinv(e, f_n)
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print "d =", bcolors.RED, d, bcolors.ENDC, bcolors.YELLOW, "(privateExponent)", bcolors.ENDC
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print
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#
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# Chinese Remainder Theorem (CRT)
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##
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print "6. Chinese Remainder Theorem (CRT) starts here"
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# exponent1
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# exponent2
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dP = modinv(e,p-1)
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dQ = modinv(e,q-1)
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# OR can be calculated this way
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# dP = d % (p-1)
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# dQ = d % (q-1)
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# coefficient
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coeff = modinv(q,p)
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print "dP = e^-1 mod (p-1) =", bcolors.CYAN, e, bcolors.ENDC, "^-1 mod", \
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bcolors.DRED, p-1, bcolors.ENDC, "=", bcolors.RED, dP, bcolors.ENDC, \
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bcolors.YELLOW, "(exponent1)", bcolors.ENDC
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print "dQ = e^-1 mod (q-1) =", bcolors.CYAN, e, bcolors.ENDC, "^-1 mod", \
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bcolors.DRED, q-1, bcolors.ENDC, "=", bcolors.RED, dQ, bcolors.ENDC, \
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bcolors.YELLOW, "(exponent2)", bcolors.ENDC
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print "coeff = q^-1 mod p = ", bcolors.DRED, q, bcolors.ENDC, "^-1 mod", \
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bcolors.DRED, p, bcolors.ENDC, "=", bcolors.RED, coeff, bcolors.ENDC, \
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bcolors.YELLOW, "(coefficient)", bcolors.ENDC
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print
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print "Public key is modulus n =", bcolors.BLUE, n, bcolors.ENDC, \
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"and the public (or encryption) exponent e =", bcolors.CYAN, e, \
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bcolors.ENDC
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print
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print "Private key is modulus n =", bcolors.BLUE, n, bcolors.ENDC, \
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"and the private (or decryption) exponent d =", bcolors.RED, d, \
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bcolors.ENDC, "and it must be kept secret"
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print bcolors.DRED+"p"+bcolors.ENDC+","+bcolors.DRED+" q"+bcolors.ENDC+","+ \
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bcolors.DRED+" φ(n) "+bcolors.ENDC+"and Chinese Remainder Theorem "+ \
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"exponents "+bcolors.DRED+"dP"+bcolors.ENDC+", "+bcolors.DRED+"dQ"+ \
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bcolors.ENDC+" with "+bcolors.DRED+"coefficient"+bcolors.ENDC+ \
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" must also be kept secret because they can be used to calculate "+ \
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bcolors.RED+"d"+bcolors.ENDC+"."
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print
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# -----------------------------------------------------------------------------
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# Encrypting: c = m^e (mod n)
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# -----------------------------------------------------------------------------
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print "To encrypt message m: c = m^e (mod n)"
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#mstr = "hi"
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cstr = ""
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mstr = raw_input("Enter your message m: ")
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for m in [elem.encode("hex") for elem in mstr]:
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print ":: encrypting ", bcolors.YELLOW, '"'+chr(int(m, 16))+'"', \
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int(m, 16), bcolors.ENDC, " >>>", bcolors.YELLOW, int(m, 16), \
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bcolors.ENDC, "^", bcolors.CYAN, e, bcolors.ENDC, \
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"( mod", bcolors.BLUE, n, bcolors.ENDC, ")", ">>> ",
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c = ( int(m, 16) ** e ) % n
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print bcolors.PURPLE, c, bcolors.ENDC
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cstr += str(c)
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cstr += ","
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cstr = cstr[:-1]
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print "Your encrypted message m is now a ciphertext c =", bcolors.YELLOW, cstr, bcolors.ENDC
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print
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# -----------------------------------------------------------------------------
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# Decrypting: m = c^d (mod n)
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# -----------------------------------------------------------------------------
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#print "To decrypt ciphertext c: m = c^d (mod n)"
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#mstr = ""
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#for c in cstr.split(","):
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# print ":: decrypting ", bcolors.PURPLE, c, bcolors.ENDC, " >>>", \
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# bcolors.PURPLE, int(c), "^", bcolors.ENDC, bcolors.RED, \
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# d, bcolors.ENDC, "( mod", bcolors.BLUE, n, bcolors.ENDC, ")", ">>>",
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# m = ( int(c) ** d ) % n
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# print bcolors.YELLOW, m, '"'+chr(m)+'"', bcolors.ENDC
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# mstr += chr(m)
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print "To decrypt ciphertext "+bcolors.PURPLE+"c"+bcolors.ENDC+" we apply Chinese Remainder Theorem:"
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print " m1 = (c ** dP) % p"
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print " m2 = (c ** dQ) % q"
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print " h = coeff * (m1 - m2) % p"
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print " m = m2 + h*q"
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print
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mstr = ""
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for c in cstr.split(","):
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print ":: decrypting ", bcolors.PURPLE, c, bcolors.ENDC, " >>> "
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m1 = ( int(c) ** dP ) % p
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m2 = ( int(c) ** dQ ) % q
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h = coeff * (m1 - m2) % p
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m = m2 + h*q
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print "\t\t\t\tm1 = (", c, " ^ ", dP, ") mod ", p, "=", m1
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print "\t\t\t\tm2 = (", c, " ^ ", dQ, ") mod ", q, "=", m2
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print "\t\t\t\th =", coeff, "* (", m1, "-", m2, ") mod", p, "=", h
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print "\t\t\t\tm =", m2, "+", h, "*", q, "=", m
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print "\t\t\t\t"+bcolors.YELLOW, m, '"'+chr(m)+'"', bcolors.ENDC
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mstr += chr(m)
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print
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print "Decrypted ciphertext is now m =", bcolors.YELLOW, mstr, bcolors.ENDC
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print
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