Math poetry: prime numbers and Nemo

“How many computing operations have been performed by all machines across all of world history?” … To put it roughly, right around the turn of the century (2000 AD), about one mole — that is, the Avogadro number $6 \times 10^{23}$ of chemistry, call it $10^{24}$ — is the total operation count for all machines for all of history. In spite of the usual mystery and awe that surrounds the notion of industrial and government super-computing, it is the huge collection of personal computers that allows this $10^{24}$, this mole. The relevance is that a task such as trial dividing an integer $N \simeq 10^{50}$ directly for prime factors is hopeless in the sense that one would essentially have to replicate the machine effort of all time. To convey an idea of scale, a typical instance of the deepest factoring or primality-proving runs of the modern era involves perhaps $10^{16}$ to $10^{18}$ machine operations. Similarly, a full-length graphically rendered synthetic movie of today—for example, the 2003 Pixar/Disney movie Finding Nemo—involves operation counts in the $10^{18}$ range. It is amusing that for this kind of Herculean machine effort one may either obtain a single answer (a factor, maybe even a single “prime/composite” decision bit) or create a full-length animated feature whose character is as culturally separate from a one-bit answer as can be. It is interesting that a computational task of say $10^{18}$ operations is about one ten-millionth of the overall historical computing effort by all Earth-bound machinery.

[excerpt from Crandall & Pomerance, “Prime Numbers – A Computational Perspective”, 2nd edition, Springer (2005), p. 5]

RSA implementation in Python

This Python script below implements the basic RSA encryption and decryption operations without any concern about padding or character encoding. Nevertheless, it has all the primitive machinery needed to encrypt and decrypt messages using the RSA public-key algorithm. The getprime() function is in charge of generating primes of the required size and, for this purpose, uses a probabilistic algorithm, the Miller-Rabin primality test. The inv() function, which calculates modular multiplicative inverses, gets a helping hand from the Extended Euclidean algorithm xgcd() function. The genRSA() function is pretty straightforward: it generates the standard public-key small exponent, 3 or 65537 (hex 10001), depending on the size of modulus (a small public exponent speeds up both the encryption and signature verification operations).

#!/usr/bin/python3
#
# Author: Joao H de A Franco (jhafranco@acm.org)
#
# Description: RSA implementation in Python 3
#
# Date: 2012-01-30
#
#          (CC BY-NC-SA 3.0)
#===========================================================
from random import randrange, getrandbits
from itertools import repeat
from functools import reduce

def getPrime(n):
"""Get a n-bit pseudo-random prime"""
def isProbablePrime(n, t = 7):
"""Miller-Rabin primality test"""
def isComposite(a):
"""Check if n is composite"""
if pow(a, d, n) == 1:
return False
for i in range(s):
if pow(a, 2 ** i * d, n) == n - 1:
return False
return True

assert n > 0
if n < 3:
return [False, False, True][n]
elif not n & 1:
return False
else:
s, d = 0, n - 1
while not d & 1:
s += 1
d >>= 1
for _ in repeat(None, t):
if isComposite(randrange(2, n)):
return False
return True

p = getrandbits(n)
while not isProbablePrime(p):
p = getrandbits(n)
return p

def inv(p, q):
"""Multiplicative inverse"""
def xgcd(x, y):
"""Extended Euclidean Algorithm"""
s1, s0 = 0, 1
t1, t0 = 1, 0
while y:
q = x // y
x, y = y, x % y
s1, s0 = s0 - q * s1, s1
t1, t0 = t0 - q * t1, t1
return x, s0, t0

s, t = xgcd(p, q)[0:2]
assert s == 1
if t < 0:
t += q
return t

def genRSA(p, q):
"""Generate public and private keys"""
phi, mod = (p - 1) * (q - 1), p * q
if mod < 65537:
return (3, inv(3, phi), mod)
else:
return (65537, inv(65537, phi), mod)

def text2Int(text):
"""Convert a text string into an integer"""
return reduce(lambda x, y : (x << 8) + y, map(ord, text))

def int2Text(number, size):
"""Convert an integer into a text string"""
text = "".join([chr((number >> j) & 0xff)
for j in reversed(range(0, size << 3, 8))])
return text.lstrip("\x00")

def int2List(number, size):
"""Convert an integer into a list of small integers"""
return [(number >> j) & 0xff
for j in reversed(range(0, size << 3, 8))]

def list2Int(listInt):
"""Convert a list of small integers into an integer"""
return reduce(lambda x, y : (x << 8) + y, listInt)

def modSize(mod):
"""Return length (in bytes) of modulus"""
modSize = len("{:02x}".format(mod)) // 2
return modSize

def encrypt(ptext, pk, mod):
"""Encrypt message with public key"""
size = modSize(mod)
output = []
while ptext:
nbytes = min(len(ptext), size - 1)
aux1 = text2Int(ptext[:nbytes])
assert aux1 < mod
aux2 = pow(aux1, pk, mod)
output += int2List(aux2, size + 2)
ptext = ptext[size:]
return output

def decrypt(ctext, sk, p, q):
"""Decrypt message with private key
using the Chinese Remainder Theorem"""
mod = p * q
size = modSize(mod)
output = ""
while ctext:
aux3 = list2Int(ctext[:size + 2])
assert aux3 < mod
m1 = pow(aux3, sk % (p - 1), p)
m2 = pow(aux3, sk % (q - 1), q)
h = (inv(q, p) * (m1 - m2)) % p
aux4 = m2 + h * q
output += int2Text(aux4, size)
ctext = ctext[size + 2:]
return output

if __name__ == "__main__":

from math import log10
from time import time

def printHexList(intList):
"""Print ciphertext in hex"""
for index, elem in enumerate(intList):
if index % 32 == 0:
print()
print("{:02x}".format(elem), end = "")
print()

def printLargeInteger(number):
"""Print long primes in a formatted way"""
string = "{:02x}".format(number)
for j in range(len(string)):
if j % 64 == 0:
print()
print(string[j], end = "")
print()

def testCase(p, q, msg, nTimes = 1):
"""Execute test case: generate keys, encrypt message and
decrypt resulting ciphertext"""
print("Key size: {:0d} bits".format(round(log10(p * q) / log10(2))))
print("Prime #1:", end = "")
printLargeInteger(p)
print("Prime #2:", end = "")
printLargeInteger(q)
print("Plaintext:", msg)
pk, sk, mod = genRSA(p, q)
ctext = encrypt(msg, pk, mod)
print("Ciphertext:", end = "")
printHexList(ctext)
ptext = decrypt(ctext, sk, p, q)
print("Recovered plaintext:", ptext, "\n")

# First test: RSA-129 (see http://en.wikipedia.org/wiki/RSA_numbers#RSA-129)
p1 = 3490529510847650949147849619903898133417764638493387843990820577
p2 = 32769132993266709549961988190834461413177642967992942539798288533
testCase(p1, p2, "The Magic Words are Squeamish Ossifrage", 1000)

# Second test: random primes (key size: 512 to 4096 bits)
for n in [256, 512, 1024, 2048]:
t1 = time()
p5 = getPrime(n)
t2 = time()
print("Elapsed time for {:0d}-bit prime ".format(n), end = "")
print("generation: {:0.3f} s".format(round(t2 - t1, 3)))
t3 = time()
p6 = getPrime(n)
t4 = time()
print("Elapsed time for {:0d}-bit prime ".format(n), end = "")
print("generation: {:0.3f} s".format(round(t4 - t3, 3)))
testCase(p5, p6, "It's all greek to me")


The first test uses the famous 425-bit key pair belonging to RSA-129, a challenge published in Martin Gardner’s column in Scientific American back in 1977. RSA-129 owes its name to the 129-digit number that expresses the product of two (unknown at the time) primes.

RSA-129 = 1143816257578888676692357799761466120102182967212423625625618429 35706935245733897830597123563958705058989075147599290026879543541


This challenge, posed by the inventors of the RSA algorithm (Rivest, Shamir & Adleman), was solved only in 1994. The decrypted plaintext was the phrase “The Magic Words are Squeamish Ossifrage”. The second test was done with pairs of randomly generated primes. Their key sizes (as a matter of fact, the size of the modulus) are 512, 1024 and 2048 bits (the key size most used in SSL certificates is 1024 bits). The output of this Python script is shown below.

Key size: 425 bits
Prime #1:
87c296ed480f9ab17885decd31197d617779c0dac70c3234996e1
Prime #2:
4fa84812157119acc8ecca98c404b2e5ee24ce18f60ea818091895
Plaintext: The Magic Words are Squeamish Ossifrage
Ciphertext:
1ae01ccda1945493cc02be7d739ded0c56d8cd9c996ee8
Recovered plaintext: The Magic Words are Squeamish Ossifrage

Elapsed time for 256-bit prime generation: 0.108 s
Elapsed time for 256-bit prime generation: 0.018 s
Key size: 511 bits
Prime #1:
7a29e838f6afe351c24b0dfee0b12d32e092c6ec129382b75b224f749e9785c9
Prime #2:
caffe728fa460b4f009a7647bdbb0763868206942a4ced79aefd36a6b05f539b
Plaintext: It's all greek to me
Ciphertext:
00005741fdf3f6f20fc16524c9a1c80f62e7a293aaff5238f319692e4c469443
10d2272ed76d6a9a884372c2eae03e2d3b8c10df04353d459659a3f8c3e63be3
2c6d
Recovered plaintext: It's all greek to me

Elapsed time for 512-bit prime generation: 0.162 s
Elapsed time for 512-bit prime generation: 0.576 s
Key size: 1015 bits
Prime #1:
5ef0fcba9c6044ecd3c98aa22981e1f75f8779ba5f72fa33ab4da89d4abfc357
Prime #2:
2aabce36a668c2f8e161521238d2be8541d286e612b6572e4c1531f8218dfe0e
08de2e177b02c2ab1a76056eca13920e2847bfdffffdb6b2b9fe874849c3637
Plaintext: It's all greek to me
Ciphertext:
00006d1c01974161e2c4710ec423076bbab8107cf31d2d2ecfd6127ca6dc142e
aa7f37d93281222684227892c82575ef93491ffa802003efac9085e3e8076000
a205c367ed6d7270e1ab02da80522264238e3cb93aff81e53eeec7910d5013c8
a4
Recovered plaintext: It's all greek to me

Elapsed time for 1024-bit prime generation: 7.189 s
Elapsed time for 1024-bit prime generation: 8.921 s
Key size: 2047 bits
Prime #1:
7843d991e4fa29466f4e0fa385f0b8612db5d51c0f09f0fca801339155cd1ca2
fde3b810569de371e99e8d1e3d127a9b4d7f07944d64c7d9da973252b73dd1ed
847183c1a855b65817411cfea22c05b58764d5bb016770feef93d6cb5e6274e3
fce2c5fb251bb41a8ba879cd5a2d6755c591e921c6aa08327ef6b6e8f1cf1a1b
Prime #2:
e751af1ed509106fdd814d52778da0cf6998b768bafd16a10ec91c4becf05856
9e96007661d4525cb0b40b20247aa879657c4550d3d58a74ab23666b504febaf
5de5b40f3cc7b8c766e5d426fa37d9832b91fef718e4f117984168a2eeb087bc
Plaintext: It's all greek to me
Ciphertext:
000056ce1bab323cc03876dc786cdc9f53e406a4e321a2e8233ed648f93d0026
ba3c5b899bd559222162d5580924ef073ba7118023710cebc901a20f51603acb
5a797e84ea7cf853d3b2031f3263879d3afb3d05bd74f15219b877753b3182ef
a93cdfc015db601b8fd93540e1e50d1b9676e152769268039484cdd821f9291b
93f408d1e8c851fb79b143a80d7ef03629f3c11c81f8188a35076df9a88df1ba
5da77de10857a9c7238122f3818c4011a6cbe4e702c8b481789edca59ab74606
3608ca5c98f6126ddec1a3eb972021b9297c7d4d2c38f50f8105745eb3110977
f76a
Recovered plaintext: It's all greek to me


I will comment on code details in another post.

Math curiosity: expansion of 1/998001

1/998001 = 0.0000010020030040050060070080090100110120130140150160170180190200210
22023024025026027028029030031032033034035036037038039040041042043044045046047048
04905005105205305405505605705805906006106206306406506606706806907007107207307407
50760770780790800810820830840850860870880890900910920930940950960970980991001011
02103104105106107108109110111112113114115116117118119120121122123124125126127128
12913013113213313413513613713813914014114214314414514614714814915015115215315415
51561571581591601611621631641651661671681691701711721731741751761771781791801811
82183184185186187188189190191192193194195196197198199200201202203204205206207208
20921021121221321421521621721821922022122222322422522622722822923023123223323423
52362372382392402412422432442452462472482492502512522532542552562572582592602612
62263264265266267268269270271272273274275276277278279280281282283284285286287288
28929029129229329429529629729829930030130230330430530630730830931031131231331431
53163173183193203213223233243253263273283293303313323333343353363373383393403413
42343344345346347348349350351352353354355356357358359360361362363364365366367368
36937037137237337437537637737837938038138238338438538638738838939039139239339439
53963973983994004014024034044054064074084094104114124134144154164174184194204214
22423424425426427428429430431432433434435436437438439440441442443444445446447448
44945045145245345445545645745845946046146246346446546646746846947047147247347447
54764774784794804814824834844854864874884894904914924934944954964974984995005015
02503504505506507508509510511512513514515516517518519520521522523524525526527528
52953053153253353453553653753853954054154254354454554654754854955055155255355455
55565575585595605615625635645655665675685695705715725735745755765775785795805815
82583584585586587588589590591592593594595596597598599600601602603604605606607608
60961061161261361461561661761861962062162262362462562662762862963063163263363463
56366376386396406416426436446456466476486496506516526536546556566576586596606616
62663664665666667668669670671672673674675676677678679680681682683684685686687688
68969069169269369469569669769869970070170270370470570670770870971071171271371471
57167177187197207217227237247257267277287297307317327337347357367377387397407417
42743744745746747748749750751752753754755756757758759760761762763764765766767768
76977077177277377477577677777877978078178278378478578678778878979079179279379479
57967977987998008018028038048058068078088098108118128138148158168178188198208218
22823824825826827828829830831832833834835836837838839840841842843844845846847848
84985085185285385485585685785885986086186286386486586686786886987087187287387487
58768778788798808818828838848858868878888898908918928938948958968978988999009019
02903904905906907908909910911912913914915916917918919920921922923924925926927928
92993093193293393493593693793893994094194294394494594694794894995095195295395495
595695795895996096196296396496596696796896997097197297397497

Note that the result contains the sequence “000”, “111”, “222” up to “999” (with exception of “998”), after which it starts again. This expansion was obtained evaluating the expression 1 + (1/999^2).n(digits = 3007) with the help of Sage Online (sagenb.org). A “1” was added to the expression to show the first zeroes, otherwise Sage uses exponential notation and omits them.

AES implementation in Python

After implementing DES, the next obvious challenge was AES. I was expecting AES code to be simpler to write than DES’ because AES was designed to be implemented in hardware or software, while DES design was geared towards hardware. This time, however, I decided to write an object-oriented API supporting the three different key sizes AES inherited from Rijndael (128-, 192- and 256-bit). In addition, besides the ECB (Electronic Code Book) basic operation mode, this implementation also supports CBC (Cipher Block Chaining) mode.

#!/usr/bin/python3
#
# Author: Joao H de A Franco (jhafranco@acm.org)
#
# Description: AES implementation in Python 3
# (sundAES)
#
# Date: 2013-06-02 (version 1.1)
#       2012-01-16 (version 1.0)
#
# (CC BY-NC-SA 3.0)
#===========================================================
import sys
from itertools import repeat
from functools import reduce
from copy import copy

__all__ = ["setKey","encrypt","decrypt"]

def memoize(func):
"""Memoization function"""
memo = {}
def helper(x):
if x not in memo:
memo[x] = func(x)
return memo[x]
return helper

def mult(p1,p2):
"""Multiply two polynomials in GF(2^8)/x^8+x^4+x^3+x+1"""
p = 0
while p2:
if p2&0x01:
p ^= p1
p1 <<= 1
if p1&0x100:
p1 ^= 0x1b
p2 >>= 1
return p&0xff

# Auxiliary one-parameter functions defined for memoization
# (to speed up multiplication in GF(2^8))

@memoize
def x2(y):
"""Multiplication by 2"""
return mult(2,y)

@memoize
def x3(y):
"""Multiplication by 3"""
return mult(3,y)

@memoize
def x9(y):
"""Multiplication by 9"""
return mult(9,y)

@memoize
def x11(y):
"""Multiplication by 11"""
return mult(11,y)

@memoize
def x13(y):
"""Multiplication by 13"""
return mult(13,y)

@memoize
def x14(y):
"""Multiplication by 14"""
return mult(14,y)

class AES:
"""Class definition for AES objects"""
keySizeTable = {"SIZE_128":16,
"SIZE_192":24,
"SIZE_256":32}
wordSizeTable = {"SIZE_128":44,
"SIZE_192":52,
"SIZE_256":60}
numberOfRoundsTable = {"SIZE_128":10,
"SIZE_192":12,
"SIZE_256":14}
cipherModeTable = {"MODE_ECB":1,
"MODE_CBC":2}
# S-Box
sBox = (0x63,0x7c,0x77,0x7b,0xf2,0x6b,0x6f,0xc5,
0x30,0x01,0x67,0x2b,0xfe,0xd7,0xab,0x76,
0xca,0x82,0xc9,0x7d,0xfa,0x59,0x47,0xf0,
0xb7,0xfd,0x93,0x26,0x36,0x3f,0xf7,0xcc,
0x34,0xa5,0xe5,0xf1,0x71,0xd8,0x31,0x15,
0x04,0xc7,0x23,0xc3,0x18,0x96,0x05,0x9a,
0x07,0x12,0x80,0xe2,0xeb,0x27,0xb2,0x75,
0x09,0x83,0x2c,0x1a,0x1b,0x6e,0x5a,0xa0,
0x52,0x3b,0xd6,0xb3,0x29,0xe3,0x2f,0x84,
0x53,0xd1,0x00,0xed,0x20,0xfc,0xb1,0x5b,
0x6a,0xcb,0xbe,0x39,0x4a,0x4c,0x58,0xcf,
0xd0,0xef,0xaa,0xfb,0x43,0x4d,0x33,0x85,
0x45,0xf9,0x02,0x7f,0x50,0x3c,0x9f,0xa8,
0x51,0xa3,0x40,0x8f,0x92,0x9d,0x38,0xf5,
0xbc,0xb6,0xda,0x21,0x10,0xff,0xf3,0xd2,
0xcd,0x0c,0x13,0xec,0x5f,0x97,0x44,0x17,
0xc4,0xa7,0x7e,0x3d,0x64,0x5d,0x19,0x73,
0x60,0x81,0x4f,0xdc,0x22,0x2a,0x90,0x88,
0x46,0xee,0xb8,0x14,0xde,0x5e,0x0b,0xdb,
0xe0,0x32,0x3a,0x0a,0x49,0x06,0x24,0x5c,
0xc2,0xd3,0xac,0x62,0x91,0x95,0xe4,0x79,
0xe7,0xc8,0x37,0x6d,0x8d,0xd5,0x4e,0xa9,
0x6c,0x56,0xf4,0xea,0x65,0x7a,0xae,0x08,
0xba,0x78,0x25,0x2e,0x1c,0xa6,0xb4,0xc6,
0xe8,0xdd,0x74,0x1f,0x4b,0xbd,0x8b,0x8a,
0x70,0x3e,0xb5,0x66,0x48,0x03,0xf6,0x0e,
0x61,0x35,0x57,0xb9,0x86,0xc1,0x1d,0x9e,
0xe1,0xf8,0x98,0x11,0x69,0xd9,0x8e,0x94,
0x9b,0x1e,0x87,0xe9,0xce,0x55,0x28,0xdf,
0x8c,0xa1,0x89,0x0d,0xbf,0xe6,0x42,0x68,
0x41,0x99,0x2d,0x0f,0xb0,0x54,0xbb,0x16)
# Inverse S-Box
invSBox = (0x52,0x09,0x6a,0xd5,0x30,0x36,0xa5,0x38,
0xbf,0x40,0xa3,0x9e,0x81,0xf3,0xd7,0xfb,
0x7c,0xe3,0x39,0x82,0x9b,0x2f,0xff,0x87,
0x34,0x8e,0x43,0x44,0xc4,0xde,0xe9,0xcb,
0x54,0x7b,0x94,0x32,0xa6,0xc2,0x23,0x3d,
0xee,0x4c,0x95,0x0b,0x42,0xfa,0xc3,0x4e,
0x08,0x2e,0xa1,0x66,0x28,0xd9,0x24,0xb2,
0x76,0x5b,0xa2,0x49,0x6d,0x8b,0xd1,0x25,
0x72,0xf8,0xf6,0x64,0x86,0x68,0x98,0x16,
0xd4,0xa4,0x5c,0xcc,0x5d,0x65,0xb6,0x92,
0x6c,0x70,0x48,0x50,0xfd,0xed,0xb9,0xda,
0x5e,0x15,0x46,0x57,0xa7,0x8d,0x9d,0x84,
0x90,0xd8,0xab,0x00,0x8c,0xbc,0xd3,0x0a,
0xf7,0xe4,0x58,0x05,0xb8,0xb3,0x45,0x06,
0xd0,0x2c,0x1e,0x8f,0xca,0x3f,0x0f,0x02,
0xc1,0xaf,0xbd,0x03,0x01,0x13,0x8a,0x6b,
0x3a,0x91,0x11,0x41,0x4f,0x67,0xdc,0xea,
0x97,0xf2,0xcf,0xce,0xf0,0xb4,0xe6,0x73,
0xe2,0xf9,0x37,0xe8,0x1c,0x75,0xdf,0x6e,
0x47,0xf1,0x1a,0x71,0x1d,0x29,0xc5,0x89,
0x6f,0xb7,0x62,0x0e,0xaa,0x18,0xbe,0x1b,
0xfc,0x56,0x3e,0x4b,0xc6,0xd2,0x79,0x20,
0x9a,0xdb,0xc0,0xfe,0x78,0xcd,0x5a,0xf4,
0x1f,0xdd,0xa8,0x33,0x88,0x07,0xc7,0x31,
0xb1,0x12,0x10,0x59,0x27,0x80,0xec,0x5f,
0x60,0x51,0x7f,0xa9,0x19,0xb5,0x4a,0x0d,
0x2d,0xe5,0x7a,0x9f,0x93,0xc9,0x9c,0xef,
0xa0,0xe0,0x3b,0x4d,0xae,0x2a,0xf5,0xb0,
0xc8,0xeb,0xbb,0x3c,0x83,0x53,0x99,0x61,
0x17,0x2b,0x04,0x7e,0xba,0x77,0xd6,0x26,
0xe1,0x69,0x14,0x63,0x55,0x21,0x0c,0x7d)

# Instance variables
wordSize = None
w = [None]*60 # Round subkeys list
keyDefined = None # Key definition flag
numberOfRounds = None
cipherMode = None
ivEncrypt = None # Initialization
ivDecrypt = None # vectors

"""Create a new instance of an AES object"""
try:
assert mode in AES.cipherModeTable
except AssertionError:
print("Cipher mode not supported:",mode)
sys.exit("ValueError")
self.cipherMode = mode
try:
except AssertionError:
sys.exit(ValueError)
self.keyDefined = False

def intToList(self,number):
"""Convert an 16-byte number into a 16-element list"""
return [(number>>i)&0xff for i in reversed(range(0,128,8))]

def intToList2(self,number):
"""Converts an integer into one (or more) 16-element list"""
lst = []
while number:
lst.append(number&0xff)
number >>= 8
m = len(lst)%16
if m == 0 and len(lst) != 0:
return lst[::-1]
else:
return list(bytes(16-m)) + lst[::-1]

def listToInt(self,lst):
"""Convert a list into a number"""
return reduce(lambda x,y:(x<<8)+y,lst)

def wordToState(self,wordList):
"""Convert list of 4 words into a 16-element state list"""
return [(wordList[i]>>j)&0xff
for j in reversed(range(0,32,8)) for i in range(4)]

def listToState(self,list):
"""Convert a 16-element list into a 16-element state list"""
return [list[i+j] for j in range(4) for i in range(0,16,4)]

stateToList = listToState # this function is an involution

def subBytes(self,state):
"""SubBytes transformation"""
return [AES.sBox[e] for e in state]

def invSubBytes(self,state):
"""Inverse SubBytes transformation"""
return [AES.invSBox[e] for e in state]

def shiftRows(self,s):
"""ShiftRows transformation"""
return s[:4]+s[5:8]+s[4:5]+s[10:12]+s[8:10]+s[15:]+s[12:15]

def invShiftRows(self,s):
"""Inverse ShiftRows transformation"""
return s[:4]+s[7:8]+s[4:7]+s[10:12]+s[8:10]+s[13:]+s[12:13]

def mixColumns(self,s):
"""MixColumns transformation"""
return [x2(s[i])^x3(s[i+4])^ s[i+8] ^ s[i+12] for i in range(4)]+ \
[ s[i] ^x2(s[i+4])^x3(s[i+8])^ s[i+12] for i in range(4)]+ \
[ s[i] ^ s[i+4] ^x2(s[i+8])^x3(s[i+12]) for i in range(4)]+ \
[x3(s[i])^ s[i+4] ^ s[i+8] ^x2(s[i+12]) for i in range(4)]

def invMixColumns(self,s):
"""Inverse MixColumns transformation"""
return [x14(s[i])^x11(s[i+4])^x13(s[i+8])^ x9(s[i+12]) for i in range(4)]+ \
[ x9(s[i])^x14(s[i+4])^x11(s[i+8])^x13(s[i+12]) for i in range(4)]+ \
[x13(s[i])^ x9(s[i+4])^x14(s[i+8])^x11(s[i+12]) for i in range(4)]+ \
[x11(s[i])^x13(s[i+4])^ x9(s[i+8])^x14(s[i+12]) for i in range(4)]

return [i^j for i,j in zip(subkey,state)]

def rotWord(self,number):
"""Rotate subkey left"""
return (((number&0xff000000)>>24) +
((number&0xff0000)<<8) +
((number&0xff00)<<8) +
((number&0xff)<<8))

def subWord(self,key):
"""Substitute subkeys bytes using S-box"""
return ((AES.sBox[(key>>24)&0xff]<<24) +
(AES.sBox[(key>>16)&0xff]<<16) +
(AES.sBox[(key>>8)&0xff]<<8) +
AES.sBox[key&0xff])

def setKey(self,keySize,key,iv = None):
"""KeyExpansion transformation"""
rcon = (0x00,0x01,0x02,0x04,0x08,0x10,0x20,0x40,0x80,0x1B,0x36)
try:
assert keySize in AES.keySizeTable
except AssertionError:
print("Key size identifier not valid")
sys.exit("ValueError")
try:
assert isinstance(key,int)
except AssertionError:
print("Invalid key")
sys.exit("ValueError")
klen = len("{:02x}".format(key))//2
try:
assert klen <= AES.keySizeTable[keySize]
except AssertionError:
print("Key size mismatch")
sys.exit("ValueError")
try:
assert ((self.cipherMode == "MODE_CBC" and isinstance(iv,int)) or
self.cipherMode == "MODE_ECB")
except AssertionError:
print("IV is mandatory for CBC mode")
sys.exit(ValueError)

if self.cipherMode == "MODE_CBC":
temp = self.intToList(iv)
self.ivEncrypt = copy(temp)
self.ivDecrypt = copy(temp)
nr = AES.numberOfRoundsTable[keySize]
self.numberOfRounds = nr
self.wordSize = AES.wordSizeTable[keySize]
if nr == 10:
nk = 4
keyList = self.intToList(key)
elif nr == 12:
nk = 6
keyList = self.intToList(key>>64) + \
(self.intToList(key&int("ff"*32,16)))[8:]
else:
nk = 8
keyList = self.intToList(key>>128) + \
self.intToList(key&int("ff"*64,16))
for index in range(nk):
self.w[index] = (keyList[4*index]<<24) + \
(keyList[4*index+1]<<16) + \
(keyList[4*index+2]<<8) +\
keyList[4*index+3]
for index in range(nk,self.wordSize):
temp = self.w[index - 1]
if index % nk == 0:
temp = (self.subWord(self.rotWord(temp)) ^
rcon[index//nk]<<24)
elif self.numberOfRounds == 14 and index%nk == 4:
temp = self.subWord(temp)
self.w[index] = self.w[index-nk]^temp
self.keyDefined = True
return

def getKey(self,operation):
"""Return next round subkey for encryption or decryption"""
if operation == "encryption":
for i in range(0,self.wordSize,4):
yield self.wordToState(self.w[i:i+4])
else: # operation = "decryption":
for i in reversed(range(0,self.wordSize,4)):
yield self.wordToState(self.w[i:i+4])

def encryptBlock(self,plaintextBlock):
"""Encrypt a 16-byte block with key already defined"""
key = self.getKey("encryption")
state = self.listToState(plaintextBlock)
for _ in repeat(None,self.numberOfRounds - 1):
state = self.subBytes(state)
state = self.shiftRows(state)
state = self.mixColumns(state)
state = self.subBytes(state)
state = self.shiftRows(state)
return self.stateToList(state)

def decryptBlock(self,ciphertextBlock):
"""Decrypt a 16-byte block with key already defined"""
key = self.getKey("decryption")
state = self.listToState(ciphertextBlock)
for _ in repeat(None,self.numberOfRounds - 1):
state = self.invShiftRows(state)
state = self.invSubBytes(state)
state = self.invMixColumns(state)
state = self.invShiftRows(state)
state = self.invSubBytes(state)
return self.stateToList(state)

if type(data) is bytes:
else:

"""Remove PKCS7 padding (if present) from plaintext"""
return "".join(chr(e) for e in byteList[:-byteList[-1]])
else:
return "".join(chr(e) for e in byteList)

def encrypt(self,input):
"""Encrypt plaintext passed as a string or as an integer"""
try:
assert self.keyDefined
except AssertionError:
print("Key not defined")
sys.exit("ValueError")

if type(input) is int:
inList = self.intToList2(input)
else:
outList = []
if self.cipherMode == "MODE_CBC":
outBlock = self.ivEncrypt
for i in range(0,len(inList),16):
auxList = self.xorLists(outBlock,inList[i:i+16])
outBlock = self.encryptBlock(auxList)
outList += outBlock
self.ivEncrypt = outBlock
else:
for i in range(0,len(inList),16):
outList += self.encryptBlock(inList[i:i+16])
if type(input) is int:
return self.listToInt(outList)
else:
return outList

def decrypt(self,input):
"""Decrypt ciphertext passed as a string or as an integer"""
try:
assert self.keyDefined
except AssertionError:
print("Key not defined")
sys.exit("ValueError")
if type(input) is int:
inList = self.intToList2(input)
else:
inList = input
outList = []
if self.cipherMode == "MODE_CBC":
oldInBlock = self.ivDecrypt
for i in range(0,len(inList),16):
newInBlock = inList[i:i+16]
auxList = self.decryptBlock(newInBlock)
outList += self.xorLists(oldInBlock,auxList)
oldInBlock = newInBlock
self.ivDecrypt = oldInBlock
else:
for i in range(0,len(inList),16):
outList += self.decryptBlock(inList[i:i+16])
if type(input) is int:
return self.listToInt(outList)
else:


The code below informally verify the correctness of the implementation with the help of the test vectors described in NIST document SP800-38A, Recommendation for Block Cipher Modes of Operation – Methods and Techniques:

import pyAES

def intToList(number):
"""Convert a 16-byte number into a 16-element list"""
return [(number >> 120) & 0xff, (number >> 112) & 0xff,
(number >> 104) & 0xff, (number >> 96)  & 0xff,
(number >> 88)  & 0xff, (number >> 80)  & 0xff,
(number >> 72)  & 0xff, (number >> 64)  & 0xff,
(number >> 56)  & 0xff, (number >> 48)  & 0xff,
(number >> 40)  & 0xff, (number >> 32)  & 0xff,
(number >> 24)  & 0xff, (number >> 16)  & 0xff,
(number >> 8)   & 0xff,  number & 0xff]

def intToText(hexNumber):
"""Convert a 16-byte number into a 16 char text string"""
return "".join(chr(e) for e in intToList(hexNumber))

def checkTestVector1(mode, keySize, key, plaintext, ciphertext, iv = None):
"""Check test vectors for single block encryption and decryption"""
success = True
obj = pyAES.AES(mode)
obj.setKey(keySize, key, iv)
for i, (p, c) in enumerate(zip(plaintext, ciphertext)):
p_text = intToText(p)
c_text = intToList(c)
code = "{0:3s}-{1:3s}".format(mode[5:], keySize[5:])
try:
assert obj.encrypt(p_text) == c_text
except AssertionError:
print(code, "encryption #{:d} failed".format(i))
success = False
try:
assert obj.decrypt(c_text) == p_text
except AssertionError:
print(code, "decryption #{:d} failed".format(i))
success = False
if success:
print(code, "encryption/decryption ok")
return success

def checkTestVector2(mode, keySize, plaintext, key1, key2, iv1 = None, iv2 = None):
obj1.setKey(keySize, key1, iv1)
obj2.setKey(keySize, key2, iv2)
ciphertext1 = obj1.encrypt(plaintext)
ciphertext2 = obj2.encrypt(plaintext)
plaintext1 = obj1.decrypt(ciphertext1)
plaintext2 = obj2.decrypt(ciphertext2)
code = "{0:3s}-{1:3s}".format(mode[5:], keySize[5:])
try:
assert plaintext1 == plaintext and plaintext2 == plaintext
except AssertionError:
print("Multi-block", code, "encryption/decryption failed")
return False
print("Multi-block", code, "encryption/decryption ok")
return True

#===========================================================================

# Test vectors from NIST SP800-38A sections F.1.1 and F.1.2
key = 0x2b7e151628aed2a6abf7158809cf4f3c
plaintext =  [0x6bc1bee22e409f96e93d7e117393172a,
0xae2d8a571e03ac9c9eb76fac45af8e51,
0x30c81c46a35ce411e5fbc1191a0a52ef,
0xf5d3d58503b9699de785895a96fdbaaf,
0x43b1cd7f598ece23881b00e3ed030688,
return checkTestVector1("MODE_ECB", "SIZE_128", key, plaintext, ciphertext)

# Test vectors from NIST SP800-38A sections F.1.3 and F.1.4
plaintext =  [0x6bc1bee22e409f96e93d7e117393172a,
0xae2d8a571e03ac9c9eb76fac45af8e51,
0x30c81c46a35ce411e5fbc1191a0a52ef,
ciphertext = [0xbd334f1d6e45f25ff712a214571fa5cc,
0x9a4b41ba738d6c72fb16691603c18e0e]
return checkTestVector1("MODE_ECB", "SIZE_192", key, plaintext, ciphertext)

# Test vectors from NIST SP800-38A sections F.1.5 and F.1.6
key = 0x603deb1015ca71be2b73aef0857d77811f352c073b6108d72d9810a30914dff4
plaintext =  [0x6bc1bee22e409f96e93d7e117393172a,
0xae2d8a571e03ac9c9eb76fac45af8e51,
0x30c81c46a35ce411e5fbc1191a0a52ef,
ciphertext = [0xf3eed1bdb5d2a03c064b5a7e3db181f8,
0x591ccb10d410ed26dc5ba74a31362870,
0xb6ed21b99ca6f4f9f153e7b1beafed1d,
0x23304b7a39f9f3ff067d8d8f9e24ecc7]
return checkTestVector1("MODE_ECB", "SIZE_256", key, plaintext, ciphertext)

# Test vectors from NIST SP800-38A sections F.2.1 and F.2.2
key = 0x2b7e151628aed2a6abf7158809cf4f3c
iv = 0x000102030405060708090a0b0c0d0e0f
plaintext =  [0x6bc1bee22e409f96e93d7e117393172a,
0xae2d8a571e03ac9c9eb76fac45af8e51,
0x30c81c46a35ce411e5fbc1191a0a52ef,
ciphertext = [0x7649abac8119b246cee98e9b12e9197d,
0x5086cb9b507219ee95db113a917678b2,
0x73bed6b8e3c1743b7116e69e22229516,
0x3ff1caa1681fac09120eca307586e1a7]
return checkTestVector1("MODE_CBC", "SIZE_128", key, plaintext, ciphertext, iv)

# Test vectors from NIST SP800-38A sections F.2.3 and F.2.4
iv = 0x000102030405060708090a0b0c0d0e0f
plaintext =  [0x6bc1bee22e409f96e93d7e117393172a,
0xae2d8a571e03ac9c9eb76fac45af8e51,
0x30c81c46a35ce411e5fbc1191a0a52ef,
ciphertext = [0x4f021db243bc633d7178183a9fa071e8,
0x571b242012fb7ae07fa9baac3df102e0,
0x08b0e27988598881d920a9e64f5615cd]
return checkTestVector1("MODE_CBC", "SIZE_192", key, plaintext, ciphertext, iv)

# Test vectors from NIST SP800-38A sections F.2.5 and F.2.6
key = 0x603deb1015ca71be2b73aef0857d77811f352c073b6108d72d9810a30914dff4
iv = 0x000102030405060708090a0b0c0d0e0f
plaintext=  [0x6bc1bee22e409f96e93d7e117393172a,
0xae2d8a571e03ac9c9eb76fac45af8e51,
0x30c81c46a35ce411e5fbc1191a0a52ef,
ciphertext = [0xf58c4c04d6e5f1ba779eabfb5f7bfbd6,
0x9cfc4e967edb808d679f777bc6702c7d,
0x39f23369a9d9bacfa530e26304231461,
0xb2eb05e2c39be9fcda6c19078c6a9d1b]
return checkTestVector1("MODE_CBC", "SIZE_256", key, plaintext, ciphertext, iv)

#===========================================================================

# Two multi-block encryptions in ECB mode with 128-bit key
def ECB_128_PKCS5():
plaintext = "The quick brown fox jumps over the lazy dog"
key1 = 0x000102030405060708090a0b0c0d0e0f
key2 = 0xffeeddccbbaa99887766554433221100
return checkTestVector2("MODE_ECB", "SIZE_128", plaintext, key1, key2)

# Two multi-block encryptions in ECB mode with 192-bit key
def ECB_192_PKCS5():
plaintext = "The quick brown fox jumps over the lazy dog"
key1 = 0x000102030405060708090a0b0c0d0e0f1011121314151617
key2 = 0x1716151413121110ffeeddccbbaa99887766554433221100
return checkTestVector2("MODE_ECB", "SIZE_192", plaintext, key1, key2)

# Two multi-block encryptions in ECB mode with 256-bit key
def ECB_256_PKCS5():
plaintext = "The quick brown fox jumps over the lazy dog"
key1 = 0x000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f
key2 = 0x1f1e1d1c1b1a19181716151413121110ffeeddccbbaa99887766554433221100
return checkTestVector2("MODE_ECB", "SIZE_256", plaintext, key1, key2)

# Two multi-block encryptions in CBC mode with 128-bit key
def CBC_128_PKCS5():
plaintext = "The quick brown fox jumps over the lazy dog"
key1 = 0x000102030405060708090a0b0c0d0e0f
key2 = 0xffeeddccbbaa99887766554433221100
iv1 = 0x0123cdef456789ab0123cdef456789ab
iv2 = 0xab0123cdef456789ab0123cdef456789
return checkTestVector2("MODE_CBC", "SIZE_128", plaintext, key1, key2, iv1, iv2)

# Two multi-block encryptions in CBC mode with 192-bit key
def CBC_192_PKCS5():
plaintext = "The quick brown fox jumps over the lazy dog"
key1 = 0x000102030405060708090a0b0c0d0e0f1011121314151617
key2 = 0x1716151413121110ffeeddccbbaa99887766554433221100
iv1 = 0x0123cdef456789ab0123cdef456789ab
iv2 = 0xab0123cdef456789ab0123cdef456789
return checkTestVector2("MODE_CBC", "SIZE_192", plaintext, key1, key2, iv1, iv2)

# Two multi-block encryptions in CBC mode with 256-bit key
def CBC_256_PKCS5():
plaintext = "The quick brown fox jumps over the lazy dog"
key1 = 0x000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f
key2 = 0x1f1e1d1c1b1a19181716151413121110ffeeddccbbaa99887766554433221100
iv1 = 0x0123cdef456789ab0123cdef456789ab
iv2 = 0xab0123cdef456789ab0123cdef456789
return checkTestVector2("MODE_CBC", "SIZE_256", plaintext, key1, key2, iv1, iv2)

def main():
# Perform vector tests
testSuccess = True
for mode in ["ECB", "CBC"]:
for size in ["128", "192", "256"]:
testVector = mode + "_" + size + "_" + pad + "()"
testSuccess = testSuccess & eval(testVector)
if testSuccess:
print("All tests passed!")

if __name__ == '__main__':
main()


DES implementation in Python

After coding RC4, I thought it was time to face a more difficult challenge: DES (Data Encryption Standard). Although it has been replaced by AES (Advanced Encryption Standard) more than a decade ago, it still has some appeal to me because it was the first symmetric algorithm I learned (and also because its creation, 40 years ago, was surrounded by mystery). Before fighting the monster, I faced a lighter opponent, S-DES (Simplified DES), to get used to the awkward bit manipulation DES take advantage of. Since S-DES is just a toy cryptographic algorithm, it isn’t worthwhile to spend much time and space writing about it. For this reason, without further ado, this is the DES Python code I wrote:

#!/usr/bin/python3
#
# Author: Joao H de A Franco (jhafranco@acm.org)
#
# Description: DES implementation in Python 3
#
# Note: only single DES in ECB mode (with PKCS5 padding)
#       is supported
#
#          (CC BY-NC-SA 3.0)
#===========================================================

subKeyList = 16*[[None]*8]

IPtable = (58, 50, 42, 34, 26, 18, 10, 2,
60, 52, 44, 36, 28, 20, 12, 4,
62, 54, 46, 38, 30, 22, 14, 6,
64, 56, 48, 40, 32, 24, 16, 8,
57, 49, 41, 33, 25, 17,  9, 1,
59, 51, 43, 35, 27, 19, 11, 3,
61, 53, 45, 37, 29, 21, 13, 5,
63, 55, 47, 39, 31, 23, 15, 7)

EPtable = (32,  1,  2,  3,  4,  5,
4,  5,  6,  7,  8,  9,
8,  9, 10, 11, 12, 13,
12, 13, 14, 15, 16, 17,
16, 17, 18, 19, 20, 21,
20, 21, 22, 23, 24, 25,
24, 25, 26, 27, 28, 29,
28, 29, 30, 31, 32,  1)

PFtable = (16,  7, 20, 21, 29, 12, 28, 17,
1, 15, 23, 26,  5, 18, 31, 10,
2,  8, 24, 14, 32, 27,  3,  9,
19, 13, 30,  6, 22, 11,  4, 25)

FPtable = (40, 8, 48, 16, 56, 24, 64, 32,
39, 7, 47, 15, 55, 23, 63, 31,
38, 6, 46, 14, 54, 22, 62, 30,
37, 5, 45, 13, 53, 21, 61, 29,
36, 4, 44, 12, 52, 20, 60, 28,
35, 3, 43, 11, 51, 19, 59, 27,
34, 2, 42, 10, 50, 18, 58, 26,
33, 1, 41,  9, 49, 17, 57, 25)

sBox = 8*[64*[0]]

sBox[0] = (14,  4, 13,  1,  2, 15, 11,  8,  3, 10,  6, 12,  5,  9,  0,  7,
0, 15,  7,  4, 14,  2, 13,  1, 10,  6, 12, 11,  9,  5,  3,  8,
4,  1, 14,  8, 13,  6,  2, 11, 15, 12,  9,  7,  3, 10,  5,  0,
15, 12,  8,  2,  4,  9,  1,  7,  5, 11,  3, 14, 10,  0,  6, 13)

sBox[1] = (15,  1,  8, 14,  6, 11,  3,  4,  9,  7,  2, 13, 12,  0,  5, 10,
3, 13,  4,  7, 15,  2,  8, 14, 12,  0,  1, 10,  6,  9, 11,  5,
0, 14,  7, 11, 10,  4, 13,  1,  5,  8, 12,  6,  9,  3,  2, 15,
13,  8, 10,  1,  3, 15,  4,  2, 11,  6,  7, 12,  0,  5, 14,  9)

sBox[2] = (10,  0,  9, 14,  6,  3, 15,  5,  1, 13, 12,  7, 11,  4,  2,  8,
13,  7,  0,  9,  3,  4,  6, 10,  2,  8,  5, 14, 12, 11, 15,  1,
13,  6,  4,  9,  8, 15,  3,  0, 11,  1,  2, 12,  5, 10, 14,  7,
1, 10, 13,  0,  6,  9,  8,  7,  4, 15, 14,  3, 11,  5,  2, 12)

sBox[3] = ( 7, 13, 14,  3,  0,  6,  9, 10,  1,  2,  8,  5, 11, 12,  4, 15,
13,  8, 11,  5,  6, 15,  0,  3,  4,  7,  2, 12,  1, 10, 14,  9,
10,  6,  9,  0, 12, 11,  7, 13, 15,  1,  3, 14,  5,  2,  8,  4,
3, 15,  0,  6, 10,  1, 13,  8,  9,  4,  5, 11, 12,  7,  2, 14)

sBox[4] = ( 2, 12,  4,  1,  7, 10, 11,  6,  8,  5,  3, 15, 13,  0, 14,  9,
14, 11,  2, 12,  4,  7, 13,  1,  5,  0, 15, 10,  3,  9,  8,  6,
4,  2,  1, 11, 10, 13,  7,  8, 15,  9, 12,  5,  6,  3,  0, 14,
11,  8, 12,  7,  1, 14,  2, 13,  6, 15,  0,  9, 10,  4,  5,  3)

sBox[5] = (12,  1, 10, 15,  9,  2,  6,  8,  0, 13,  3,  4, 14,  7,  5, 11,
10, 15,  4,  2,  7, 12,  9,  5,  6,  1, 13, 14,  0, 11,  3,  8,
9, 14, 15,  5,  2,  8, 12,  3,  7,  0,  4, 10,  1, 13, 11,  6,
4,  3,  2, 12,  9,  5, 15, 10, 11, 14,  1,  7,  6,  0,  8, 13)

sBox[6] = ( 4, 11,  2, 14, 15,  0,  8, 13,  3, 12,  9,  7,  5, 10,  6,  1,
13,  0, 11,  7,  4,  9,  1, 10, 14,  3,  5, 12,  2, 15,  8,  6,
1,  4, 11, 13, 12,  3,  7, 14, 10, 15,  6,  8,  0,  5,  9,  2,
6, 11, 13,  8,  1,  4, 10,  7,  9,  5,  0, 15, 14,  2,  3, 12)

sBox[7] = (13,  2,  8,  4,  6, 15, 11,  1, 10,  9,  3, 14,  5,  0, 12,  7,
1, 15, 13,  8, 10,  3,  7,  4, 12,  5,  6, 11,  0, 14,  9,  2,
7, 11,  4,  1,  9, 12, 14,  2,  0,  6, 10, 13, 15,  3,  5,  8,
2,  1, 14,  7,  4, 10,  8, 13, 15, 12,  9,  0,  3,  5,  6, 11)

def bit2Byte(bitList):
"""Convert bit list into a byte list"""
return [int("".join(map(str,bitList[i*8:i*8+8])),2) for i in range(len(bitList)//8)]

def byte2Bit(byteList):
"""Convert byte list into a bit list"""
return [(byteList[i//8]>>(7-(i%8)))&0x01 for i in range(8*len(byteList))]

def permBitList(inputBitList,permTable):
"""Permute input bit list according to input permutation table"""
return [inputBitList[e - 1] for e in permTable]

def permByteList(inByteList,permTable):
"""Permute input byte list according to input permutation table"""
outByteList = (len(permTable)>>3)*[0]
for index,elem in enumerate(permTable):
i = index%8
e = (elem-1)%8
if i>=e:
outByteList[index>>3] |= \
(inByteList[(elem-1)>>3]&(128>>e))>>(i-e)
else:
outByteList[index>>3] |= \
(inByteList[(elem-1)>>3]&(128>>e))<<(e-i)
return outByteList

def getIndex(inBitList):
"""Permute bits to properly index the S-boxes"""
return (inBitList[0]<<5)+(inBitList[1]<<3)+ \
(inBitList[2]<<2)+(inBitList[3]<<1)+ \
(inBitList[4]<<0)+(inBitList[5]<<4)

return "".join(chr(e) for e in byteList[:-byteList[-1]])

def setKey(keyByteList):
"""Generate all sixteen round subkeys"""
PC1table = (57, 49, 41, 33, 25, 17,  9,
1, 58, 50, 42, 34, 26, 18,
10,  2, 59, 51, 43, 35, 27,
19, 11,  3, 60, 52, 44, 36,
63, 55, 47, 39, 31, 23, 15,
7, 62, 54, 46, 38, 30, 22,
14,  6, 61, 53, 45, 37, 29,
21, 13,  5, 28, 20, 12,  4)

PC2table= (14, 17, 11, 24,  1,  5,  3, 28,
15,  6, 21, 10, 23, 19, 12,  4,
26,  8, 16,  7, 27, 20, 13,  2,
41, 52, 31, 37, 47, 55, 30, 40,
51, 45, 33, 48, 44, 49, 39, 56,
34, 53, 46, 42, 50, 36, 29, 32)

def leftShift(inKeyBitList,round):
"""Perform one (or two) circular left shift(s) on key"""
LStable = (1, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1)

outKeyBitList = 56*[0]
if LStable[round] == 2:
outKeyBitList[:26] = inKeyBitList[2:28]
outKeyBitList[26] = inKeyBitList[0]
outKeyBitList[27] = inKeyBitList[1]
outKeyBitList[28:54] = inKeyBitList[30:]
outKeyBitList[54] = inKeyBitList[28]
outKeyBitList[55] = inKeyBitList[29]
else:
outKeyBitList[:27] = inKeyBitList[1:28]
outKeyBitList[27] = inKeyBitList[0]
outKeyBitList[28:55] = inKeyBitList[29:]
outKeyBitList[55] = inKeyBitList[28]
return outKeyBitList

permKeyBitList = permBitList(byte2Bit(keyByteList),PC1table)
for round in range(16):
auxBitList = leftShift(permKeyBitList,round)
subKeyList[round] = bit2Byte(permBitList(auxBitList,PC2table))
permKeyBitList = auxBitList

def encryptBlock(inputBlock):
"""Encrypt an 8-byte block with already defined key"""
inputData = permByteList(inputBlock,IPtable)
leftPart,rightPart = inputData[:4],inputData[4:]
for round in range(16):
expRightPart = permByteList(rightPart,EPtable)
key = subKeyList[round]
indexList = byte2Bit([i^j for i,j in zip(key,expRightPart)])
sBoxOutput = 4*[0]
for nBox in range(4):
nBox12 = 12*nBox
leftIndex = getIndex(indexList[nBox12:nBox12+6])
rightIndex = getIndex(indexList[nBox12+6:nBox12+12])
sBoxOutput[nBox] = (sBox[nBox<<1][leftIndex]<<4)+ \
sBox[(nBox<<1)+1][rightIndex]
aux = permByteList(sBoxOutput,PFtable)
newRightPart = [i^j for i,j in zip(aux,leftPart)]
leftPart = rightPart
rightPart = newRightPart
return permByteList(rightPart+leftPart,FPtable)

def decryptBlock(inputBlock):
"""Decrypt an 8-byte block with already defined key"""
inputData = permByteList(inputBlock,IPtable)
leftPart,rightPart = inputData[:4],inputData[4:]
for round in range(16):
expRightPart = permByteList(rightPart,EPtable)
key = subKeyList[15-round]
indexList = byte2Bit([i^j for i,j in zip(key,expRightPart)])
sBoxOutput = 4*[0]
for nBox in range(4):
nBox12 = 12*nBox
leftIndex = getIndex(indexList[nBox12:nBox12+6])
rightIndex = getIndex(indexList[nBox12+6:nBox12+12])
sBoxOutput[nBox] = (sBox[nBox*2][leftIndex]<<4)+ \
sBox[nBox*2+1][rightIndex]
aux = permByteList(sBoxOutput,PFtable)
newRightPart = [i^j for i,j in zip(aux,leftPart)]
leftPart = rightPart
rightPart = newRightPart
return permByteList(rightPart+leftPart,FPtable)

def encrypt(key, inString):
"""Encrypt plaintext with given key"""
setKey(key)
for i in range(0,len(inByteList),8):
outByteList += encryptBlock(inByteList[i:i+8])
return outByteList

def decrypt(key, inByteList):
"""Decrypt ciphertext with given key"""
setKey(key)
outByteList = []
for i in range(0,len(inByteList),8):
outByteList += decryptBlock(inByteList[i:i+8])


To verify this DES implementation, I also wrote a separate Python module (shown below) containing an interesting algorithm proposed by Ron Rivest a long time ago (1985) in the paper Testing Implementation of DES. I also added a simple multi-block test.

import sys
from pyDES import setKey, encryptBlock, decryptBlock, encrypt, decrypt

def sanityCheck1():
"""Tests single-block DES encryption & decryption
using algorithm proposed by Ronald Rivest
(http://people.csail.mit.edu/rivest/Destest.txt)"""
x0 =  [0x94, 0x74, 0xb8, 0xe8, 0xc7, 0x3b, 0xca, 0x7d]
x16 = [0x1b, 0x1a, 0x2d, 0xdb, 0x4c, 0x64, 0x24, 0x38]
x = x0
for i in range(16):
setKey(x)
if i % 2 == 0:
x = encryptBlock(x) # if i is even, x[i+1] = E(x[i], x[i)
else:
x = decryptBlock(x) # if i is odd, x[i+1] = D(x[i], x[i)
try:
assert x == x16
except AssertionError:
return False
return True

def sanityCheck2():
"""Tests multi-block DES encryption and decryption"""
try:
key = [0x0f, 0x15, 0x71, 0xc9, 0x47, 0xd9, 0xe8, 0x59]
plaintext = "The quick brown fox jumps over the lazy dog"
ciphertext = encrypt(key, plaintext)
assert decrypt(key, ciphertext) == plaintext
except AssertionError:
return False
return True

def main():
if sanityCheck1() and sanityCheck2():
print("All DES tests ok!")
else:
sys.exit(1)
sys.exit()

if __name__ == '__main__':
main()


RC4 implementation in Python

I started learning Python two months ago. To get the most out of the process, I decided to combine it with another interest of mine, cryptography, by trying to implement a very simple symmetric algorithm, RC4. Here is the code:

#!/usr/bin/python3
#
# Author: Joao H de A Franco (jhafranco@acm.org)
#
# Description: RC4 implementation in Python 3
#
#          (CC BY-NC-SA 3.0)
#===========================================================

# Global variables
state = [None] * 256
p = q = None

def setKey(key):
"""RC4 Key Scheduling Algorithm (KSA)"""
global p, q, state
state = [n for n in range(256)]
p = q = j = 0
for i in range(256):
if len(key) > 0:
j = (j + state[i] + key[i % len(key)]) % 256
else:
j = (j + state[i]) % 256
state[i], state[j] = state[j], state[i]

def byteGenerator():
"""RC4 Pseudo-Random Generation Algorithm (PRGA)"""
global p, q, state
p = (p + 1) % 256
q = (q + state[p]) % 256
state[p], state[q] = state[q], state[p]
return state[(state[p] + state[q]) % 256]

def encrypt(inputString):
"""Encrypt input string returning a byte list"""
return [ord(p) ^ byteGenerator() for p in inputString]

def decrypt(inputByteList):
"""Decrypt input byte list returning a string"""
return "".join([chr(c ^ byteGenerator()) for c in inputByteList])

To informally verify the correctness of this implementation, I wrote a separate Python module that uses Wikipedia’s RC4 test vectors for this purpose:

import sys
import pyRC4

def main():
"""Verify the correctness of RC4 implementation using Wikipedia
test vectors"""

def intToList(inputNumber):
"""Convert a number into a byte list"""
inputString = "{:02x}".format(inputNumber)
return [int(inputString[i:i + 2], 16) for i in range(0, len(inputString), 2)]

def string_to_list(inputString):
"""Convert a string into a byte list"""
return [ord(c) for c in inputString]

def test(key, plaintext, ciphertext, testNumber):
success = True
pyRC4.setKey(string_to_list(key))
try:
assert pyRC4.encrypt(plaintext) == intToList(ciphertext)
print("RC4 encryption test #{:d} ok!".format(testNumber))
except AssertionError:
print("RC4 encryption test #{:d} failed".format(testNumber))
success = False
pyRC4.setKey(string_to_list(key))
try:
assert pyRC4.decrypt(intToList(ciphertext)) == plaintext
print("RC4 decryption test #{:d} ok!".format(testNumber))
except AssertionError:
print("RC4 decryption test #{:d} failed".format(testNumber))
success = False
return success

# Test vectors definition section
numberOfTests = 3
testVectorList = [{}] * numberOfTests

# Wikipedia test vector #1
testVectorList[0] = dict(key = "Key",
plaintext = "Plaintext",
testNumber = 1)

# Wikipedia test vector #2
testVectorList[1] = dict(key = "Wiki",
plaintext = "pedia",
ciphertext = 0x1021BF0420,
testNumber = 2)

# Wikipedia test vector #3
testVectorList[2] = dict(key = "Secret",
plaintext = "Attack at dawn",
ciphertext = 0x45A01F645FC35B383552544B9BF5,
testNumber = 3)

# Testing section
testSuccess = True
for p in range(numberOfTests):
testSuccess &= test(**testVectorList[p])
if testSuccess:
print("All RC4 tests succeeded!")
else:
print("At least one RC4 test failed")
sys.exit(1)
sys.exit()

if __name__ == '__main__':
main()