Who’s (really) the best chess player?

You’ve just made it into the finals of a chess tournament. Although your opponent is “stronger” than you, you are trying to think of some strategy to beat him on the board.

There will be 2 (two) matches, and if they both result in a tie, a third match will be played to decide the winner, who will take home a $1,000 prize. If the play-off also ends in a draw, you’ll share the prize with your opponent.

The scoring system is the one used in chess tournaments since the middle of the 19th century: players who scored a win in a game are awarded 1 (one) point, while those scoring draws are given a 0.5 (half-point) each. Losing a game, as you might expect, is worth 0 (zero) point.

You know, from previous games against the same opponent, that you can choose between two different strategies: “play fearlessly”, in which you win 45% and lose 55% of the time, or “play defensively”, in which you draw 90% and lose 10% of the time (but selecting this alternative you’ll have no chance to win). Note that both strategies have the same expected value since 0.45 x 1.0 = 0.9 x 0.5.

If you play optimally, what are your chances of beating your opponent (and winning the $1,000 prize)?

The decision tree below shows all possible sequence of events that result in your winning the tournament. To keep the diagram simple, all remaining, non-winning sequences were omitted. Obviously this decision tree applies equally well to any other zero-sum game besides chess that can also result in a draw (e.g. checkers).

Tournament finals decision tree

It is really surprising that the probability of you, the “weaker” player, winning the finals is around 0.537 (or 0.536625 to be exact), which is greater than 0.5. Each strategy, analyzed separately, confirms that, on the average, your opponent will win more games than you. Yet the odds of grabbing the $1,000 are in your favor! How come?

N-ary m-sequence generator in Python

Maximum length sequences (or m-sequences) are bit sequences generated using maximal LFSRs (Linear Feedback Shift Registers) and are so called because they are periodic and reproduce every binary sequence that can be represented by the shift registers (i.e., for length-m registers they produce a sequence of length 2^{m}-1). Although originally defined for binary sequences, the concept of m-sequence can be extended to sequences over GF(n), n>2.

Despite being deterministically generated, m-sequences have nice statistical properties, which resemble those of white noise (they are spectrally flat, with the exception of a near-zero constant term). For this reason, m-sequences are often referred to as pseudo random noise code (PN code). Unfortunately though, unpredictability is not among those properties (see below).

The configuration of the feedback taps in a binary LFSR can be expressed in finite field arithmetic as a polynomial in GF(2^{n}). The LFSR is maximal-length if and only if the corresponding feedback polynomial is primitive. The linear complexity (LC) of a given periodic sequence is the number of cells in the shortest LFSR that can generate that sequence. The correlation between two sequences is the complex inner product of the first sequence with a shifted version of the second sequence. It is called an autocorrelation if the two sequences are the same or a cross-correlation if they are distinct. Particularly interesting for spread spectrum application are sequences (not necessarily m-sequences) that exhibit low autocorrelation and low cross-correlation (the ideal correlation function would then be the Kronecker’s delta).

Apart from spread spectrum communications, m-sequences are used in a variety of areas including digital watermarking and cryptography. As m-sequences are derived from linear recurrence relations — which lead to fairly easy cryptanalysis — they must be processed by nonlinear functions in order to be used in cryptographic applications. A good example is Trivium, a 80-bit stream cipher that consists of three shift registers of different lengths.

#!/usr/bin/python3
#
# Author: Joao H de A Franco (jhafranco@acm.org)
#
# Description: N-ary sequence generator in Python
#
# Version: 1
#
# Date: 2014-02-14
#
# License: Attribution-NonCommercial-ShareAlike 3.0 Unported
#          (CC BY-NC-SA 3.0)
#===========================================================

from functools import reduce
from fractions import Fraction
from cmath import rect,pi

def flatList(s,q):
    """Converts (list of) polynomials into (list of) elements
       of finite field mapped as integers.
         s: list of polynomials
         q: order of finite field"""
    if type(s[0]) is int:
        return s
    elif type(s[0]) is list:
        return [reduce(lambda i,j:(i*q)+j,e) for e in s]
    else:
        raise TypeError

def printListOfNumbers(S,L,*X):
    """Print list of numbers according to their particular
       format (integer, float or complex)"""
    print(S+" [",end="")
    for i,e in enumerate(L):
        if type(e) is complex:
            print("{:5.3f}{:+5.3f}i".format(e.real,e.imag),end="")
        elif type(e) is float:
            print("{:5.3f}".format(e),end="")
        elif type(e) is int:
            print("{:d}".format(e),end="")
        elif type(e) is Fraction:
            print("{:s}".format(e),end="")
        else:
            raise TypeError
        if i<len(L)-1:
            print(", ",end="")
    print("]",*X)

def xac(s,q=None,d=None,n=None):
    """Evaluates the complex auto-correlation of a periodic
       sequence with increasing delays within the period.
       Input length must correspond to a full period.
         s: q-ary periodic sequence
         q: order of finite field
         d: maximum denominator allowed in fractions
         n: number of decimal places in floating point numbers"""
    return xcc(s,s,q,d,n)
    
def xcc(s1,s2,q=None,d=None,n=None):
    """Evaluates the complex cross-correlation between two
       equally periodic q-ary sequences with increasing delays
       within the period. Input length must correspond to a full
       period.
         s: q-ary periodic sequence
         q: order of finite field
         d: maximum denominator allowed in fractions
         n: number of decimal places in floating point numbers"""
    
    def cc(s1,s2):
        """Evaluates the complex correlation between two equally
           periodic numerical sequences.        
             s1,s2: q-ary periodic sequences"""      
        assert type(s1[0]) == type(s2[0])
        if type(s1[0]) is list:
            s3 = [[(j-i)%q for i,j in zip(u,v)] for u,v in zip(s1,s2)]
            s4 = [reduce(lambda x,y:(x*q)+y,e) for e in s3]
            z = sum(rect(1,2*pi*i/q) for i in s4)/len(s1)
        elif type(s1[0]) is int:
            z = sum(rect(1,2*pi*(j-i)/q) for i,j in zip(s1,s2))/len(s1)
        else:
            raise TypeError
        zr,zi = round(z.real,n),round(z.imag,n)
        if abs(zi%1)<10**-n:
            if abs(zr-round(zr))<10**-n:
                return int(zr)
            elif Fraction(z.real).limit_denominator().denominator<=d:
                return Fraction(z.real).limit_denominator()
            else:
                return zr
        else:
            return complex(zr,zi)

    q = 2 if q is None else q
    d = 30 if d is None else d
    n = 3 if n is None else n
    assert len(s1) == len(s2)
    return [cc(s1,s2[i:]+s2[:i]) for i in range(len(s1))]

def LFSR(P,S,M,N,K):
    """Outputs K-ary sequence with N elements. Each element is
       derived from M successive values of the LFSR sequence
       generated by polynomial P and initial state S. Polynomial
       P is represented by a list of coefficients in decreasing
       power order."""
    def LFSR2():
        """Generates linear K-ary sequence according to polynomial
           P and initial state S. If P is primitive, sequence length
           is exactly one period."""
        seq,st = [S[-1]],S
        for j in range(K**len(S)-2):
            st0 = sum([i*j for i,j in zip(st,P[1:])])%K
            st = [st0]+st[:-1]
            seq += [st[-1]]
        return seq

    assert len(P) > 1 and len(P)-1 == len(S)
    s = LFSR2()
    L = len(s)
    assert M <= L
    return [s[i%L] if M == 1 else (s[i%L:]+s[:i%L])[:M] for i in range(N)] 

if __name__ == "__main__":
       
    # Binary sequence over GF(2) generated by the LFSR
    # defined by non primitive polynomial x⁴+x³+x+1.
    # This is not a m-sequence since its period is 6
    # (<15 = 2⁴-1)
    s01 = LFSR([1,1,0,1,1],[0,0,0,1],1,15,2)
    print("s01 =",s01,"\n")
s01 = [1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0]
    # Sequence over GF(2⁴) derived from sequence "s01".
    # The polynomial elements were also mapped to integers
    # in Z16, in order to show the period. 
    s04 = LFSR([1,1,0,1,1],[0,0,0,1],4,15,2)
    print("s04 =",s04,"\n")
    print("s04 =",flatList(s04,2),"\n")
s04 = [[1, 0, 0, 0], [0, 0, 0, 1], [0, 0, 1, 1], [0, 1, 1, 1],
       [1, 1, 1, 0], [1, 1, 0, 0], [1, 0, 0, 0], [0, 0, 0, 1],
       [0, 0, 1, 1], [0, 1, 1, 1], [1, 1, 1, 0], [1, 1, 0, 0],
       [1, 0, 0, 1], [0, 0, 1, 0], [0, 1, 0, 0]]

s04 = [8, 1, 3, 7, 14, 12, 8, 1, 3, 7, 14, 12, 9, 2, 4] 
    # m-sequence over GF(2) generated by the LFSR defined
    # by the primitive polynomial x⁴+x+1. Its period is
    # equal to 2⁴-1=15
    s11 = LFSR([1,1,0,0,1],[0,0,0,1],1,15,2)
    print("s11 =",s11,"\n")
s11 = [1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0] 
    # Autocorrelation of m-sequence "s11". This function
    # is two-valued and has the same period as sequence
    # "s11"
    printListOfNumbers("ac(s11) =",xac(s11),"\n")
ac(s11) = [1, -1/15, -1/15, -1/15, -1/15, -1/15, -1/15, -1/15,
           -1/15, -1/15, -1/15, -1/15, -1/15, -1/15, -1/15]
    # Sequence over GF(2²) derived from m-sequence "s11"
    s12 = LFSR([1,0,0,1,1],[0,0,0,1],2,15,2)
    print("s12 =",s12,"\n")
s12 = [[1, 0], [0, 0], [0, 0], [0, 1], [1, 0], [0, 0], [0, 1],
       [1, 1], [1, 0], [0, 1], [1, 0], [0, 1], [1, 1], [1, 1], [1, 1]] 
    
    # Autocorrelation of sequence "s12"
    printListOfNumbers("ac(s12) =",xac(s12,4),"\n")
ac(s12) = [1, 7/15, 7/15, 7/15, 7/15, 7/15, 7/15, 7/15, 7/15,
           7/15, 7/15, 7/15, 7/15, 7/15, 7/15]
    # Sequence over GF(2³) derived from m-sequence "s11"
    s13 = LFSR([1,0,0,1,1],[0,0,0,1],3,15,2)
    print("s13 =",s13,"\n")
s13 = [[1, 0, 0], [0, 0, 0], [0, 0, 1], [0, 1, 0], [1, 0, 0],
       [0, 0, 1], [0, 1, 1], [1, 1, 0], [1, 0, 1], [0, 1, 0],
       [1, 0, 1], [0, 1, 1], [1, 1, 1], [1, 1, 1], [1, 1, 0]] 
    # Autocorrelation of sequence "s13"
    printListOfNumbers("ac(s13) =",xac(s13,8,d=1000),"\n")
ac(s13) = [1, 0.844, 0.844, 0.844, 0.844, 0.844, 0.844, 0.844,
           0.844, 0.844, 0.844, 0.844, 0.844, 0.844, 0.844] 
    # Sequence over GF(2⁴) derived from the m-sequence "s11"
    s14 = LFSR([1,0,0,1,1],[0,0,0,1],4,15,2)
    print("s14 =",s14,"\n")
s14 = [[1, 0, 0, 0], [0, 0, 0, 1], [0, 0, 1, 0], [0, 1, 0, 0],
       [1, 0, 0, 1], [0, 0, 1, 1], [0, 1, 1, 0], [1, 1, 0, 1],
       [1, 0, 1, 0], [0, 1, 0, 1], [1, 0, 1, 1], [0, 1, 1, 1],
       [1, 1, 1, 1], [1, 1, 1, 0], [1, 1, 0, 0]] 
    # Autocorrelation of sequence "s14" 
    printListOfNumbers("ac(s14) =",xac(s14,16,d=1000),"\n")
ac(s14) = [1, 0.959, 0.959, 0.959, 0.959, 0.959, 0.959, 0.959,
           0.959, 0.959, 0.959, 0.959, 0.959, 0.959, 0.959] 
    # Binary sequence over GF(2) generated by the LFSR
    # defined by the primitive polynomial (x⁴+x³+1).
    # This is also a m-sequence with period equal to
    # 15 = 2⁴-1
    s21 = LFSR([1,0,0,1,1],[0,0,0,1],1,15,2)
    print("s21 =",s21,"\n")
s21 = [1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1]
    # Autocorrelation of sequence "s21"       
    printListOfNumbers("ac(s21) =",xac(s21),"\n")
ac(s21) = [1, -1/15, -1/15, -1/15, -1/15, -1/15, -1/15,
           -1/15, -1/15, -1/15, -1/15, -1/15, -1/15, -1/15, -1/15]
    # Cross-correlation between m-sequences "s11" and "s21".
    # This function is four-valued and has the same period
    # as sequences "s11" and "s21"
    printListOfNumbers("cc(s11,s21) =",xcc(s11,s21),"\n")
cc(s11,s21) = [-1/15, 1/5, -1/15, 7/15, 1/5, -1/3, -1/15,
               1/5, 7/15, -1/3, 1/5, -1/3, -1/3, -1/15, -1/15] 
    # m-sequence over GF(3) generated by the LFSR defined
    # by the primitive polynomial x²+2x+1. Its period is
    # equal to 3²-1=8 
    s31 = LFSR([1,2,1],[1,0],1,8,3)
    print("s31 =",s31,"\n")
s31 = [0, 1, 2, 2, 0, 2, 1, 1]
    # Autocorrelation of sequence "s31"
    printListOfNumbers("ac(s31) =",xac(s31,3),"\n")
ac(s31) = [1, -1/8, -1/8, -1/8, -1/8, -1/8, -1/8, -1/8]
    # Sequence over GF(3²) derived from m-sequence "s31"
    s32 = LFSR([1,2,1],[1,0],2,8,3)
    print("s32 =",s32,"\n")
s32 = [[0, 1], [1, 2], [2, 2], [2, 0], [0, 2], [2, 1], [1, 1], [1, 0]]
    # Autocorrelation of sequence "s32"
    printListOfNumbers("ac(s32) =",xac(s32,3),"\n")
ac(s32) = [1, -1/8, -1/8, -1/8, -1/8, -1/8, -1/8, -1/8]
    # Sequence over GF(3³) derived from m-sequence "s31"
    s33 = LFSR([1,2,1],[1,0],3,8,3)
    print("s33 =",s33,"\n")
s33 = [[0, 1, 2], [1, 2, 2], [2, 2, 0], [2, 0, 2],
       [0, 2, 1], [2, 1, 1], [1, 1, 0], [1, 0, 1]]
    # Autocorrelation of sequence "s33"
    printListOfNumbers("ac(s33) =",xac(s33,3),"\n")
ac(s33) = [1, -1/8, -1/8, -1/8, -1/8, -1/8, -1/8, -1/8]
    # m-sequence over GF(3) generated by the LFSR defined
    # by the primitive polynomial x³+x²+2. Its period is
    # equal to 3³-1=26 
    s41 = LFSR([1,1,0,2],[1,0,0],1,26,3)
    print("s41 =",s41,"\n")
s41 = [0, 0, 1, 1, 1, 0, 2, 1, 1, 2, 1, 0, 1, 0, 0, 2, 2, 2, 0, 1,
       2, 2, 1, 2, 0, 2] 
    # Autocorrelation of m-sequence "s41". This function is
    # two-valued and has the same period as sequence "s41"    
    printListOfNumbers("ac(s41) =",xac(s41,3),"\n")
ac(s41) = [1, -1/26, -1/26, -1/26, -1/26, -1/26, -1/26, -1/26, -1/26, 
           -1/26, -1/26, -1/26, -1/26, -1/26, -1/26, -1/26, -1/26, -1/26,
           -1/26, -1/26, -1/26, -1/26, -1/26, -1/26, -1/26, -1/26]
    # Sequence over GF(3²) derived from m-sequence "s41"
    s42 = LFSR([1,1,0,2],[1,0,0],2,26,3)
    print("s42 =",s42,"\n")
s42 = [[0, 0], [0, 1], [1, 1], [1, 1], [1, 0], [0, 2],
       [2, 1], [1, 1], [1, 2], [2, 1], [1, 0], [0, 1],
       [1, 0], [0, 0], [0, 2], [2, 2], [2, 2], [2, 0],
       [0, 1], [1, 2], [2, 2], [2, 1], [1, 2], [2, 0],
       [0, 2], [2, 0]]
    # Autocorrelation of sequence "s42"
    printListOfNumbers("ac(s42) =",xac(s42,9),"\n")
ac(s42) = [1, 0.701, 0.701, 0.701, 0.701, 0.701, 0.701,
           0.701, 0.701, 0.701, 0.701, 0.701, 0.701, 0.838,
           0.701, 0.701, 0.701, 0.701, 0.701, 0.701, 0.701,
           0.701, 0.701, 0.701, 0.701, 0.701]
    # Sequence over GF(3³) derived from m-sequence "s31"
    s43 = LFSR([1,1,0,2],[1,0,0],3,26,3)
    print("s43 =",s43,"\n")
s43 = [[0, 0, 1], [0, 1, 1], [1, 1, 1], [1, 1, 0],
       [1, 0, 2], [0, 2, 1], [2, 1, 1], [1, 1, 2],
       [1, 2, 1], [2, 1, 0], [1, 0, 1], [0, 1, 0],
       [1, 0, 0], [0, 0, 2], [0, 2, 2], [2, 2, 2],
       [2, 2, 0], [2, 0, 1], [0, 1, 2], [1, 2, 2],
       [2, 2, 1], [2, 1, 2], [1, 2, 0], [2, 0, 2],
       [0, 2, 0], [2, 0, 0]]
    # Autocorrelation of sequence "s43"
    printListOfNumbers("ac(s43) =",xac(s43,27),"\n")
ac(s43) = [1, 0.963, 0.963, 0.963, 0.963, 0.963, 0.963,
           0.963, 0.963, 0.963, 0.963, 0.963, 0.963, 0.981,
           0.963, 0.963, 0.963, 0.963, 0.963, 0.963, 0.963,
           0.963, 0.963, 0.963, 0.963, 0.963]
    # Sequence over Z20 with ideal autocorrelation function
    # (Kronecker's delta)
    s51 = [t**2%20 for t in range(10)]
    print("s51 =",s51,"\n")
s51 = [0, 1, 4, 9, 16, 5, 16, 9, 4, 1]
    # Autocorrelation of sequence "s51"  
    printListOfNumbers("xac(s51) =",xac(s51,20),"\n")
xac(s51) = [1, 0, 0, 0, 0, 0, 0, 0, 0, 0] 
    # Sequence over Z7 with ideal autocorrelation function
    # (Kronecker's delta)
    s61 = [(t*(t+1)//2)%7 for t in range(7)]
    print("s61 =",s61,"\n")
s61 = [0, 1, 3, 6, 3, 1, 0]
    # Autocorrelation of sequence "s61"  
    printListOfNumbers("xac(s61) =",xac(s61,7),"\n")
xac(s61) = [1, 0, 0, 0, 0, 0, 0]

HMAC-SHA-256 implementation in Python 3

A hash-based message authentication code (HMAC) is an algorithm for generating a message authentication code (MAC), which can be used to verify both the integrity and the authentication of a given message. Although both constructs, HMAC and MAC, are based on a cryptographic hash function (such as SHA-1, Whirlpool or RIPEMD-160), the former requires a key (shared between the sender and the receiver of the message) while the latter doesn’t. The HMAC concept was proposed by Bellare, Canetti, and Krawczyk in 1996 and is described in RFC 2104.

As seen from its name, HMAC-SHA-256 uses as its engine the SHA-256 cryptographic hash function, which produces message digests of 256 bits in length. Like the other members of the SHA-2 family (and also MD-5 and SHA-1), SHA-256 is an iterative hash function (based on the Merkle–Damgård scheme) that works by breaking up the input message into blocks of a fixed size (512 bits for SHA-256) and iterating over them with a compression function.

#!/usr/bin/python3
#
# Author: Joao H de A Franco (jhafranco@acm.org)
#
# Description: HMAC-SHA256 implementation in Python 3
#
# Date: 2013-06-10
#
# License: Attribution-NonCommercial-ShareAlike 3.0 Unported
#          (CC BY-NC-SA 3.0)
#================================================================

from functools import reduce
from math import log,ceil

def intToList2(number,length):
    """Convert a number into a byte list
       with specified length"""
    return [(number >> i) & 0xff
            for i in reversed(range(0,length*8,8))]

def intToList(number):
    """Converts an integer of any length into an integer list"""
    L1 = log(number,256)
    L2 = ceil(L1)
    if L1 == L2:
        L2 += 1
    return [(number&(0xff<<8*i))>>8*i for i in reversed(range(L2))] 

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

def bitList32ToList4(lst):
    """Convert a 32-bit list into a 4-byte list"""
    def bitListToInt(lst):
        return reduce(lambda x,y:(x<<1)+y,lst)    
    
    lst2 = []
    for i in range(0,len(lst),8):
        lst2.append(bitListToInt(lst[i:i+8]))
    return list([0]*(4-len(lst2)))+lst2

def list4ToBitList32(lst):
    """Convert a 4-byte list into a 32-bit list"""
    def intToBitList2(number,length):
        """Convert an integer into a bit list
        with specified length"""
        return [(number>>n) & 1
                for n in reversed(range(length))]    
    
    lst2 = []
    for e in lst:
        lst2 += intToBitList2(e,8)
    return list([0]*(32-len(lst2)))+lst2

def add32(p,q,r=None,s=None,t=None):
    """Add up to five 32-bit numbers"""
    mask32 = (1<<32)-1
    p2,q2 = listToInt(p), listToInt(q)
    if t is None:
        if s is None:
            if r is None:
                return intToList2((p2+q2)&mask32,4)
            else:
                r2 = listToInt(r)
                return intToList2((p2+q2+r2)&mask32,4)
        else:
            r2,s2 = listToInt(r),listToInt(s)
            return intToList2((p2+q2+r2+s2)&mask32,4)
    else:
        r2,s2,t2 = listToInt(r),listToInt(s),listToInt(t)
        return intToList2((p2+q2+r2+s2+t2)&mask32,4)

def xor(x,y,z=None):
    """Evaluate the XOR on two or three operands"""
    if z is None:
        return list(i^j for i,j in zip(x,y))
    else:
        return list(i^j^k for i,j,k in zip(x,y,z))
   
def sha256(m):
    """Return the SHA-256 digest of input"""
    def padding(m):
        """Pad message according to SHA-256 rules"""
        def bitListToList(lst):
            """Convert a bit list into a byte list"""
            lst2 = [0]*((8-len(lst)%8)%8)+lst
            return [reduce(lambda x,y:(x<<1)+y,lst2[i*8:i*8+8])
                    for i in range(len(lst2)//8)]
    
        def intToBitList(number):
            """Convert an integer into a bit list"""
            return list(map(int,list(bin(number)[2:])))
    
        if type(m) is int:
            m1 = intToBitList(m)
            L = len(m1)
            k = (447-L)%512
            return bitListToList(m1+[1]+list([0]*k))+intToList2(L,8)
        else:
            m1 = m
            if type(m) is str:
                m1 = list(map(ord,m))
            if not(type(m) is list):
                raise TypeError
            L = len(m1)*8
            k = (447-L)%512
            return m1+bitListToList([1]+list([0]*k))+intToList2(L,8)    
    
    def compress(m):
        """Evaluates SHA-256 compression function to input"""
        def Ch(x,y,z):
            return list([(i&j)^((i^0xff)&k) for i,j,k in zip(x,y,z)])
        
        def Maj(x,y,z):
            return list([(i&j)^(i&k)^(j&k) for i,j,k in zip(x,y,z)])              
        
        def rotRight(p,n):
            """Rotate 32-bit word right by n bits"""
            p2 = list4ToBitList32(p)
            return bitList32ToList4(p2[-n:]+p2[:-n])
        
        def shiftRight(p,n):
            """Shift 32-bit right by n bits"""
            p2 = list4ToBitList32(p)
            return bitList32ToList4(list(bytes(n))+p2[:-n])
        
        def Sigma0(p):
            """SHA-256 function"""
            return xor(rotRight(p,2),rotRight(p,13),rotRight(p,22))
        
        def Sigma1(p):
            """SHA-256 function"""
            return xor(rotRight(p,6),rotRight(p,11),rotRight(p,25))
        
        def sigma0(p):
            """SHA-256 function"""
            return xor(rotRight(p,7),rotRight(p,18),shiftRight(p,3))
        
        def sigma1(p):
            """SHA-256 function"""
            return xor(rotRight(p,17),rotRight(p,19),shiftRight(p,10))            
        
        nonlocal H
        [a,b,c,d,e,f,g,h] = H
        K = [0x428a2f98, 0x71374491, 0xb5c0fbcf, 0xe9b5dba5,
             0x3956c25b, 0x59f111f1, 0x923f82a4, 0xab1c5ed5,
             0xd807aa98, 0x12835b01, 0x243185be, 0x550c7dc3,
             0x72be5d74, 0x80deb1fe, 0x9bdc06a7, 0xc19bf174,
             0xe49b69c1, 0xefbe4786, 0x0fc19dc6, 0x240ca1cc,
             0x2de92c6f, 0x4a7484aa, 0x5cb0a9dc, 0x76f988da,
             0x983e5152, 0xa831c66d, 0xb00327c8, 0xbf597fc7,
             0xc6e00bf3, 0xd5a79147, 0x06ca6351, 0x14292967,
             0x27b70a85, 0x2e1b2138, 0x4d2c6dfc, 0x53380d13,
             0x650a7354, 0x766a0abb, 0x81c2c92e, 0x92722c85,
             0xa2bfe8a1, 0xa81a664b, 0xc24b8b70, 0xc76c51a3,
             0xd192e819, 0xd6990624, 0xf40e3585, 0x106aa070,
             0x19a4c116, 0x1e376c08, 0x2748774c, 0x34b0bcb5,
             0x391c0cb3, 0x4ed8aa4a, 0x5b9cca4f, 0x682e6ff3,
             0x748f82ee, 0x78a5636f, 0x84c87814, 0x8cc70208,
             0x90befffa, 0xa4506ceb, 0xbef9a3f7, 0xc67178f2]    
        W = [None]*64
        for t in range(16):
            W[t] = m[t*4:t*4+4]
        for t in range(16,64):
            W[t] = add32(sigma1(W[t-2]),W[t-7],sigma0(W[t-15]),W[t-16])
        for t in range(64):
            T1 = add32(h,Sigma1(e),Ch(e,f,g),intToList2(K[t],4),W[t])
            T2 = add32(Sigma0(a),Maj(a,b,c))
            h = g; g = f; f = e; e = add32(d,T1)
            d = c; c = b; b = a; a = add32(T1,T2)
        H = [add32(x,y) for x,y in zip([a,b,c,d,e,f,g,h],H)]
        
    H0 = [0x6a09e667, 0xbb67ae85, 0x3c6ef372, 0xa54ff53a,
         0x510e527f, 0x9b05688c, 0x1f83d9ab, 0x5be0cd19]
    H = list(map(lambda x:intToList2(x,4),H0))
    mp = padding(m)
    for i in range(0,len(mp),64):
        compress(mp[i:i+64])
    return listToInt([s2 for s1 in H for s2 in s1])

def hmac_sha256(k,m):
    """Return the HMAC-SHA-256 of the input
       HMAC(k,m)=SHA-256((k⊕opad)∥SHA-256((k⊕ipad)∥m))"""
    opad = list([0x5c]*64); ipad = list([0x36]*64)
    if type(k) is int:
        k1 = intToList(k)
        L = len(k1)
        if L > 64:
            K = intToList2(sha256(k),32)+list([0]*32)
        else:
            K = k1+list([0]*(64-L))  
    else:
        k1 = list(map(ord,k))      
        L = len(k1)
        if L > 64:
            K = intToList(sha256(k1))
        else:
            K = k1+list([0]*(64-L))
    if type(m) is int:
        M = intToList(m)
    else:
        M = list(map(ord,m))
    arg1 = xor(K,opad)
    arg2 = xor(K,ipad)
    return sha256(arg1+intToList(sha256(arg2+M)))

if __name__ == '__main__':

    # Wikipedia's test case #1
    assert hmac_sha256("","") == 0xb613679a0814d9ec772f95d778c35fc5ff1697c493715653c6c712144292c5ad
    
    # Wikipedia's test case #2
    assert hmac_sha256("key", "The quick brown fox jumps over the lazy dog") == \
           0xf7bc83f430538424b13298e6aa6fb143ef4d59a14946175997479dbc2d1a3cd8
    
    # RFC 4231 - Identifiers and Test Vectors for HMAC-SHA-224, HMAC-SHA-256,
    # HMAC-SHA-384, and HMAC-SHA-512
    
    # RFC 4231 Test case 1
    Key1 = 0x0b0b0b0b0b0b0b0b0b0b0b0b0b0b0b0b0b0b0b0b
    Data1 = 0x4869205468657265
    HMAC1 = 0xb0344c61d8db38535ca8afceaf0bf12b881dc200c9833da726e9376c2e32cff7    
    assert hmac_sha256(Key1,Data1) == HMAC1      

    # RFC 4231 Test case 2
    Key2 = 0x4a656665
    Data2 = 0x7768617420646f2079612077616e7420666f72206e6f7468696e673f
    HMAC2 = 0x5bdcc146bf60754e6a042426089575c75a003f089d2739839dec58b964ec3843
    assert hmac_sha256(Key2,Data2) == HMAC2
    
    # RFC 4231 Test case 3    
    Key3 = 0xaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
    Data3 = 0xdddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd
    HMAC3 = 0x773ea91e36800e46854db8ebd09181a72959098b3ef8c122d9635514ced565fe
    assert hmac_sha256(Key3,Data3) == HMAC3

    # RFC 4231 Test case 4
    Key4 = 0x0102030405060708090a0b0c0d0e0f10111213141516171819
    Data4 = 0xcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcd
    HMAC4 = 0x82558a389a443c0ea4cc819899f2083a85f0faa3e578f8077a2e3ff46729665b
    assert hmac_sha256(Key4,Data4) == HMAC4

    # RFC 4231 Test case 5
    Key5 = 0x0c0c0c0c0c0c0c0c0c0c0c0c0c0c0c0c0c0c0c0c
    Data5 = 0x546573742057697468205472756e636174696f6e 
    HMAC5 = 0xa3b6167473100ee06e0c796c2955552b
    assert hmac_sha256(Key5,Data5)>>128 == HMAC5

    # RFC 4231 Test case 6
    Key6 = 0xaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
    Data6 = 0x54657374205573696e67204c6172676572205468616e20426c6f636b2d53697a65204b6579202d2048617368204b6579204669727374
    HMAC6 = 0x60e431591ee0b67f0d8a26aacbf5b77f8e0bc6213728c5140546040f0ee37f54
    assert hmac_sha256(Key6,Data6) == HMAC6

    # RFC 4231 Test case 7    
    Key7 = 0xaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
    Data7 = 0x5468697320697320612074657374207573696e672061206c6172676572207468616e20626c6f636b2d73697a65206b657920616e642061206c6172676572207468616e20626c6f636b2d73697a6520646174612e20546865206b6579206e6565647320746f20626520686173686564206265666f7265206265696e6720757365642062792074686520484d414320616c676f726974686d2e
    HMAC7 = 0x9b09ffa71b942fcb27635fbcd5b0e944bfdc63644f0713938a7f51535c3a35e2
    assert hmac_sha256(Key7,Data7) == HMAC7

    print("Ok!")

Validation of an AES-CFB implementation in Python 3

A symmetric block cipher such as AES (or Triple DES) operates on blocks of fixed size (128 bits for AES and 64 bits for TDES). It is possible, however, to convert a block cipher into a stream cipher using one of the three following modes: cipher feedback (CFB), output feedback (OFB), and counter (CTR). A stream cipher eliminates the need to pad a message to be an integral number of blocks and, for this reason, can operate in real time, making it the natural choice for encrypting streaming data (e.g. voice).

The Python code shown below implements the encryption and decryption operations for CFB-8 and CFB-128 modes. These functions rely on the “basic” AES mode (ECB) services provided by sundAES, an AES implementation in Python presented in a previous blog.

#!/usr/bin/python3

import sys
from functools import reduce
import sundAES

# Auxiliary functions

def xor(x,y):
    """Returns the xor between two lists"""
    return bytes(i^j for i,j in zip(x,y))

def bytesToInt(b):
    """Converts a bytes string into an integer"""
    return listToInt(list(b))

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

def intToList(number):
    """Converts an integer into an integer list"""
    if number == 0:
        return [0]
    lst = []
    while number:
        lst += [number&0xff]
        number >>= 8
    return lst[::-1]

def intToBytes(number):
    """Converts an integer into a bytes list"""
    return bytes(intToList(number))

def intToList2(number,length=None):
    """Converts an integer into an integer list with
       16, 24 or 32 elements"""
    lst = []
    while number:
        lst.append(number&0xff)
        number >>= 8
    L = len(lst)
    if length:
        pZero = length-L
        assert pZero >= 0
    else:
        if L <= 16:
            pZero = 16-L
        elif L <= 24:
            pZero = 24-L
        elif L <= 32:
            pZero = 32-L
        else:
            raise ValueError
    return list(bytes(pZero)) + lst[::-1]

def intToBytes2(number,length=None):
    """Converts an integer into a bytes list with
       16, 24 or 32 elements"""
    return bytes(intToList2(number,length))

# Real crypto stuff starts here...

def encryptCFB8(keysize,key,iv,input):
    """Encrypts single bytes of input block (CFB8 mode)"""
    inputBuffer = intToBytes2(iv)
    if type(input) is int:
        ptext = intToBytes(input)
    else:
        ptext = bytes(map(ord,input))
    obj = sundAES.AES("MODE_ECB")
    obj.setKey(keysize,key,iv)    
    ctext = bytes()
    for i in range(0,len(ptext),1):
        AESoutput = obj.encrypt(inputBuffer)
        cbyte = bytes([AESoutput[0] ^ ptext[i]])
        inputBuffer = inputBuffer[1:] + cbyte
        ctext += cbyte
    if type(input) is int:
        ctext = bytesToInt(ctext)
    else:
        ctext = list(ctext)
    return ctext

def decryptCFB8(keysize,key,iv,input):
    """Decrypts single bytes of input block (CFB8 mode)"""
    inputBuffer = intToBytes2(iv)
    if type(input) is int:
        ctext = intToBytes(input)
    else:
        ctext = bytes(map(ord,input))
    obj = sundAES.AES("MODE_ECB")
    obj.setKey(keysize,key,iv) 
    ptext = bytes()
    for i in range(0,len(ctext),1):
        AESoutput = obj.encrypt(inputBuffer)
        pbyte = bytes([AESoutput[0] ^ ctext[i]])
        inputBuffer = inputBuffer[1:] + bytes(ctext[i])
        ptext += pbyte
    if type(input) is int:
        ptext = bytesToInt(ptext)
    else:
        ptext = "".join(chr(e) for e in ptext)
    return ptext

# ... and goes on here

def encryptCFB128(keysize,key,iv,input):
    """Encrypts a 16-byte input block (CFB128 mode)"""
    inputBuffer = intToBytes2(iv)
    if type(input) is int:
        ptext = intToList2(input)
    else:
        ptext = list(map(ord,input))
        L = 16-len(ptext)
        ptext = list(bytes(L)) + ptext
    obj = sundAES.AES("MODE_ECB")
    obj.setKey(keysize,key,iv)    
    ctext = bytes()
    L3 = len(ptext)
    for i in range(0,L3,16):
        AESoutput = obj.encrypt(inputBuffer)
        inputBuffer = xor(AESoutput,ptext[i:min(L3,i+16)])
        ctext += bytes(inputBuffer)
    if type(input) is int:
        ctext = bytesToInt(ctext)
    else:
        ctext = list(ctext)
    return ctext

def decryptCFB128(keysize,key,iv,input):
    """Decrypts a 16-byte input block (CFB128 mode)"""
    inputBuffer = intToBytes2(iv)
    if type(input) is int:
        ctext = intToList2(input)
    else:
        ctext = list(map(ord,input))
        L = 16-len(ctext)
        ctext = list(bytes(L)) + ctext
    obj = sundAES.AES("MODE_ECB")
    obj.setKey(keysize,key,iv) 
    ptext = bytes()
    L3 = len(ctext)
    for i in range(0,L3,16):
        AESoutput = obj.encrypt(inputBuffer)
        inputBuffer = ctext[i:min(L3,i+16)]
        ptextBlock = xor(AESoutput,inputBuffer)
        ptext += ptextBlock
    if type(input) is int:
        ptext = bytesToInt(ptext)
    else:
        ptext = "".join(chr(e) for e in ptext)
    return ptext

CFB8 encrypts (or decrypts) a single byte while CFB128 operates on a 16-byte block. Like the other stream modes (OFB and CTR), and differently from Electronic codebook (ECB) and Cipher block chaining (CBC) “block modes”, CFB dispenses padding and uses only the ECB encryption operation. This latter feature is very convenient for AES, since AES encryption and decryption operations are somewhat different.

The US National Institute of Standards and Technology (NIST) defines four types of Known Answer Test (KAT): GFSbox, KeySbox, Variable Key and Variable Text. The contents of file CFB8GFSbox128.rsp (see below) describe the GFSbox encryption and decryption tests cases for CFB8 using 128-bits AES keys.

# CAVS 11.1
# Config info for aes_values
# AESVS GFSbox test data for CFB8
# State : Encrypt and Decrypt
# Key Length : 128
# Generated on Fri Apr 22 15:11:46 2011

[ENCRYPT]

COUNT = 0
KEY = 00000000000000000000000000000000
IV = f34481ec3cc627bacd5dc3fb08f273e6
PLAINTEXT = 00
CIPHERTEXT = 03

... 5 test cases omitted

[DECRYPT]

COUNT = 0
KEY = 00000000000000000000000000000000
IV = f34481ec3cc627bacd5dc3fb08f273e6
CIPHERTEXT = 03
PLAINTEXT = 00

... 5 test cases omitted

As a second example, the contents of file CFB128VarKey192.rsp file that follow describe the Variable Key encryption and decryption tests cases for the CFB128 mode using 192-bits AES keys.

# CAVS 11.1
# Config info for aes_values
# AESVS VarKey test data for CFB128
# State : Encrypt and Decrypt
# Key Length : 192
# Generated on Fri Apr 22 15:11:55 2011

[ENCRYPT]

COUNT = 0
KEY = 800000000000000000000000000000000000000000000000
IV = 00000000000000000000000000000000
PLAINTEXT = 00000000000000000000000000000000
CIPHERTEXT = de885dc87f5a92594082d02cc1e1b42c

... 190 test cases omitted

[DECRYPT]

COUNT = 0
KEY = 800000000000000000000000000000000000000000000000
IV = 00000000000000000000000000000000
CIPHERTEXT = de885dc87f5a92594082d02cc1e1b42c
PLAINTEXT = 00000000000000000000000000000000

... 190 test cases omitted

Next is presented the Python program that extracts data from the files contained in the KAT_AES directory which names start with “CFB8″ or “CFB128″, then builds the test cases and finally executes them.

#!/usr/bin/python3
#
# Author: Joao H de A Franco (jhafranco@acm.org)
#
# Description: Validation of AES-CFB8 and AES-CFB128
#              implementations in Python
#
# Date: 2013-06-05
#
# License: Attribution-NonCommercial-ShareAlike 3.0 Unported
#          (CC BY-NC-SA 3.0)
#===========================================================

import os,sys,re
from functools import reduce
from glob import glob
import AES_CFB

# Global counters
noFilesTested = noFilesSkipped = 0
counterOK = counterNOK = 0

class AEStester:
    def buildTestCases(self,filename):
        """Build test cases described in a given file"""
        global noFilesTested,noFilesSkipped
        
        self.basename = os.path.basename(filename)
        if self.basename.startswith('CFB'):
            if self.basename.startswith('CFB8'):
                self.mode = "MODE_CFB8"
                result = re.search("CFB8(\D{6,})\d{3}",self.basename)
                self.typeTest = result.group(1)                
            elif self.basename.startswith('CFB128'):
                self.mode = "MODE_CFB128"
                result = re.search("CFB128(\D{6,})\d{3}",self.basename)
                self.typeTest = result.group(1)                
            else: # CFB1 files not considered
                noFilesSkipped += 1
                return
        else: # not CFB files
            noFilesSkipped += 1
            return
    
        noFilesTested += 1
        digits = re.search("(\d{3})\.",self.basename)
        self.keysize = 'SIZE_' + digits.group(1)
        self.iv = None
        for line in open(filename):
            line = line.strip()
            if (line == "") or line.startswith('#'):
                continue
            elif line == '[ENCRYPT]':
                self.operation = 'encrypt'
                continue
            elif line == '[DECRYPT]':
                self.operation = 'decrypt'
                continue
            param,_,value = line.split(' ',2)
            if param == "COUNT":
                self.count = int(value)
                continue
            else:           
                self.__setattr__(param.lower(),int(value,16))
                if (self.operation == 'encrypt') and (param == "CIPHERTEXT") or \
                   (self.operation == 'decrypt' and param == "PLAINTEXT"):
                    self.runTestCase()
    
    def runTestCase(self):
        """Execute test case and report result"""
        global counterOK,counterNOK      
        
        def printTestCase(result):
            print("Type={0:s} Mode={1:s} Keysize={2:s} Function={3:s} Count={4:03d} {5:s}"\
                  .format(self.typeTest,self.mode[5:],\
                          self.keysize[5:],self.operation.upper(),\
                          self.count,result))              
        if self.operation == 'encrypt':
            if self.mode == "MODE_CFB8":
                CIPHERTEXT = AES_CFB5.encryptCFB8(self.keysize,self.key,self.iv,self.plaintext)
            else:
                CIPHERTEXT = AES_CFB5.encryptCFB128(self.keysize,self.key,self.iv,self.plaintext)
            try:
                assert self.ciphertext == CIPHERTEXT
                counterOK += 1
                printTestCase("OK")   
            except AssertionError:
                counterNOK +=1
                print(self.basename,end=" ")
                printTestCase("failed")
                print("Expected ciphertext={0:0x}".format(self.ciphertext))
                print("Returned ciphertext={0:0x}".format(CIPHERTEXT))
        else:
            if self.mode == "MODE_CFB8":
                PLAINTEXT = AES_CFB5.decryptCFB8(self.keysize,self.key,self.iv,self.ciphertext)
            else:
                PLAINTEXT = AES_CFB5.decryptCFB128(self.keysize,self.key,self.iv,self.ciphertext)
            try:
                assert self.plaintext == PLAINTEXT
                counterOK += 1
                printTestCase("OK")
            except AssertionError:
                counterNOK +=1
                print(self.basename,end=" ")
                printTestCase("failed")
                print("Expected plaintext={0:0x}".format(self.plaintext))
                print("Returned plaintext={0:0x}".format(PLAINTEXT))

if __name__ == '__main__': 
    path = os.path.dirname(__file__)
    files = sys.argv[1:]
    if not files:
        files = glob(os.path.join(path,'KAT_AES','*.rsp'))
        files.sort()
    for file in files:
        AEStester().buildTestCases(file)
    print("Files tested={0:d}".format(noFilesTested))
    print("Files skipped={0:d}".format(noFilesSkipped))
    print("Test cases OK={0:d}".format(counterOK))
    print("Test cases NOK={0:d}".format(counterNOK))

This program, when executed without arguments, will generate the following report:

Type=GFSbox Mode=CFB128 Keysize=128 Function=ENCRYPT Count=000 OK
Type=GFSbox Mode=CFB128 Keysize=128 Function=ENCRYPT Count=001 OK
Type=GFSbox Mode=CFB128 Keysize=128 Function=ENCRYPT Count=002 OK
Type=GFSbox Mode=CFB128 Keysize=128 Function=ENCRYPT Count=003 OK
Type=GFSbox Mode=CFB128 Keysize=128 Function=ENCRYPT Count=004 OK

... 4,146 test cases omitted

Type=VarTxt Mode=CFB8 Keysize=256 Function=DECRYPT Count=123 OK
Type=VarTxt Mode=CFB8 Keysize=256 Function=DECRYPT Count=124 OK
Type=VarTxt Mode=CFB8 Keysize=256 Function=DECRYPT Count=125 OK
Type=VarTxt Mode=CFB8 Keysize=256 Function=DECRYPT Count=126 OK
Type=VarTxt Mode=CFB8 Keysize=256 Function=DECRYPT Count=127 OK
Files tested=24
Files skipped=48
Test cases OK=4156
Test cases NOK=0

Validation of an AES implementation in Python 3

The Cryptographic Algorithm Validation Program (CAVP) defines validation testing for cryptographic algorithms approved by the US National Institute of Standards and Technology (NIST). All of the tests under CAVP, established by NIST and its Canadian counterpart (CSEC) in 1995, are handled by accredited third-party laboratories.

To assist prospective vendors in checking their implementations, NIST provides electronic versions of the vectors for the Known Answer Test (KAT) for the three NIST-approved symmetric cryptographic algorithms: AES, Triple-DES, and Skipjack. Also available are sample values for the Monte Carlo (MCT) test and the Multiblock Message (MMT) test for the same algorithms, thus completing the set of tests a cryptographic implementation (dubbed “Implementation Under Test”) will face during a formal validation. A detailed account of the procedures involved in validating AES implementations can be found in the NIST document The Advanced Encryption Standard Algorithm Validation Suite (AESAVS).

The NIST KAT validation suite for AES contains 72 files describing test vectors for different AES modes of operation: ECB (Electronic Codebook), CBC (Cipher Block Chaining), CFB (Cipher Feedback) and OFB (Output Feedback). Besides this partition, there are also separated tests for AES encryption and decryption, exemplified by the abridged contents of the file ECBKeySbox128.rsp (see below).

# CAVS 11.1
# Config info for aes_values
# AESVS KeySbox test data for ECB
# State : Encrypt and Decrypt
# Key Length : 128
# Generated on Fri Apr 22 15:11:26 2011

[ENCRYPT]

COUNT = 0
KEY = 10a58869d74be5a374cf867cfb473859
PLAINTEXT = 00000000000000000000000000000000
CIPHERTEXT = 6d251e6944b051e04eaa6fb4dbf78465

COUNT = 1
KEY = caea65cdbb75e9169ecd22ebe6e54675
PLAINTEXT = 00000000000000000000000000000000
CIPHERTEXT = 6e29201190152df4ee058139def610bb

... (18 test cases omitted)

[DECRYPT]

COUNT = 0
KEY = 10a58869d74be5a374cf867cfb473859
CIPHERTEXT = 6d251e6944b051e04eaa6fb4dbf78465
PLAINTEXT = 00000000000000000000000000000000

COUNT = 1
KEY = caea65cdbb75e9169ecd22ebe6e54675
CIPHERTEXT = 6e29201190152df4ee058139def610bb
PLAINTEXT = 00000000000000000000000000000000

... (18 test cases omitted)

These vectors can be used to informally verify the correctness of an AES implementation, such as the one presented below, which successfully passed all 4,156 KAT tests involving ECB and CBC modes. This Python code differs from the one already presented in a previous blog only with respect to the input/output types accepted: if the input plaintext (ciphertext) is of integer type, so will be the correspondent ciphertext (plaintext) output. This feature simplifies validation testing, since the integer plaintexts and ciphertexts usually employed in those scenarios can be deal with directly without any type conversion.

#!/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)
#
# License: Attribution-NonCommercial-ShareAlike 3.0 Unported
#          (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}
    paddingTable = {"NoPadding":0,
                    "PKCS5Padding":1}
    # 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,
            0xad,0xd4,0xa2,0xaf,0x9c,0xa4,0x72,0xc0,
            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,
               0x96,0xac,0x74,0x22,0xe7,0xad,0x35,0x85,
               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
    padding = None # Padding scheme
    ivEncrypt = None # Initialization
    ivDecrypt = None #  vectors

    def __init__(self,mode,padding = "NoPadding"):
        """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:
            assert padding in AES.paddingTable
        except AssertionError:
            print("Padding scheme not supported:",padding)
            sys.exit(ValueError)
        self.padding = padding
        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)]

    def addRoundKey (self,subkey,state):
        """AddRoundKey transformation"""
        return [i^j for i,j in zip(subkey,state)]

    xorLists = addRoundKey

    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)
        state = self.addRoundKey(next(key),state)
        for _ in repeat(None,self.numberOfRounds - 1):
            state = self.subBytes(state)
            state = self.shiftRows(state)
            state = self.mixColumns(state)
            state = self.addRoundKey(next(key),state)
        state = self.subBytes(state)
        state = self.shiftRows(state)
        state = self.addRoundKey(next(key),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)
        state = self.addRoundKey(next(key),state)
        for _ in repeat(None,self.numberOfRounds - 1):
            state = self.invShiftRows(state)
            state = self.invSubBytes(state)
            state = self.addRoundKey(next(key),state)
            state = self.invMixColumns(state)
        state = self.invShiftRows(state)
        state = self.invSubBytes(state)
        state = self.addRoundKey(next(key),state)
        return self.stateToList(state)

    def padData(self,data):
        """Add PKCS5 padding to plaintext (or just add bytes to fill a block)"""
        paddingLength = 16-(len(data)%16)
        if self.padding == "NoPadding":
            paddingLength %= 16
        if type(data) is bytes:
            return data+bytes(list([paddingLength]*paddingLength))
        else:
            return [ord(s) for s in data]+[paddingLength]*paddingLength

    def unpadData(self,byteList):
        """Remove PKCS5 padding (if present) from plaintext"""
        if self.padding == "PKCS5Padding":
            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:
            inList = self.padData(input)
        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:
            return self.unpadData(outList)

The Python program below extracts data from the files contained in the KAT_AES directory, then builds the test cases and finally executes them. Only the files applicable to ECB and CBC operation modes (the modes supported by this AES implementation) are considered.

#!/usr/bin/python3
#
# Author: Joao H de A Franco (jhafranco@acm.org)
#
# Description: Validation of an AES implementation in Python
#
# Date: 2013-06-02
#
# License: Attribution-NonCommercial-ShareAlike 3.0 Unported
#          (CC BY-NC-SA 3.0)
#===========================================================

import os,sys,re
from functools import reduce
from glob import glob
import sundAES

# Global counters
noFilesTested = noFilesSkipped = 0
counterOK = counterNOK = 0

class AEStester:
    """"""
    def buildTestCases(self,filename):
        """Build test cases described in a given file"""
        global noFilesTested,noFilesSkipped

        self.basename = os.path.basename(filename)
        if self.basename.startswith('ECB'):
            self.mode = "MODE_ECB"
            noFilesTested += 1
        elif self.basename.startswith('CBC'):
            self.mode = "MODE_CBC"
            noFilesTested += 1
        else:
            noFilesSkipped += 1
            return

        digits = re.search("\d{3}",self.basename)
        self.keysize = 'SIZE_' + digits.group()
        result = re.search("CFB\d*(\D{6,})\d{3}",self.basename)
        if result != None:
            self.typeTest = result.group(1)
        self.typeTest = re.search("\w{3}(\D{6,})\d{3}",self.basename).group(1)
        self.iv = None
        for line in open(filename):
            line = line.strip()
            if (line == "") or line.startswith('#'):
                continue
            elif line == '[ENCRYPT]':
                self.operation = 'encrypt'
                continue
            elif line == '[DECRYPT]':
                self.operation = 'decrypt'
                continue
            param,_,value = line.split(' ',2)
            if param == "COUNT":
                self.count = int(value)
                continue
            else:
                self.__setattr__(param.lower(),int(value,16))
                if (self.operation == 'encrypt') and (param == "CIPHERTEXT") or \
                   (self.operation == 'decrypt' and param == "PLAINTEXT"):
                    self.runTestCase()

    def runTestCase(self):
        """Execute test case and report result"""
        global counterOK,counterNOK

        def printTestCase(result):
            print("Type={0:s} Mode={1:s} Keysize={2:s} Function={3:s} Count={4:03d} {5:s}"\
                  .format(self.typeTest,self.mode[5:],\
                          self.keysize[5:],self.operation.upper(),\
                          self.count,result))

        obj = sundAES3.AES(self.mode)
        obj.setKey(self.keysize,self.key,self.iv)
        if self.operation == 'encrypt':
            CIPHERTEXT = obj.encrypt(self.plaintext)
            try:
                assert self.ciphertext == CIPHERTEXT
                counterOK += 1
                printTestCase("OK")
            except AssertionError:
                counterNOK +=1
                print(self.basename)
                printTestCase("failed")
                print("Expected ciphertext={0:0x}".format(self.ciphertext))
                print("Returned ciphertext={0:0x}".format(CIPHERTEXT))
        else:
            PLAINTEXT = obj.decrypt(self.ciphertext)
            try:
                assert self.plaintext == PLAINTEXT
                counterOK += 1
                printTestCase("OK")
            except AssertionError:
                counterNOK +=1
                print(self.basename)
                printTestCase("failed")
                print("Expected plaintext={0:0x}".format(self.plaintext))
                print("Returned plaintext={0:0x}".format(PLAINTEXT))

# Main program
path = os.path.dirname(__file__)
files = sys.argv[1:]
if not files:
    files = glob(os.path.join(path,'KAT_AES','*.rsp'))
    files.sort()
for file in files:
    AEStester().buildTestCases(file)
print("Files tested={0:d}".format(noFilesTested))
print("Files skipped={0:d}".format(noFilesSkipped))
print("Test cases OK={0:d}".format(counterOK))
print("Test cases NOK={0:d}".format(counterNOK))

Executed without arguments, this program will apply all ECB and CBC tests described in the files against the AES implementation, producing the following output:

Type=GFSbox Mode=CBC Keysize=128 Function=ENCRYPT Count=000 OK
Type=GFSbox Mode=CBC Keysize=128 Function=ENCRYPT Count=001 OK
Type=GFSbox Mode=CBC Keysize=128 Function=ENCRYPT Count=002 OK
Type=GFSbox Mode=CBC Keysize=128 Function=ENCRYPT Count=003 OK
Type=GFSbox Mode=CBC Keysize=128 Function=ENCRYPT Count=004 OK

... (4,147 lines omitted)

Type=VarTxt Mode=ECB Keysize=256 Function=DECRYPT Count=124 OK
Type=VarTxt Mode=ECB Keysize=256 Function=DECRYPT Count=125 OK
Type=VarTxt Mode=ECB Keysize=256 Function=DECRYPT Count=126 OK
Type=VarTxt Mode=ECB Keysize=256 Function=DECRYPT Count=127 OK
Files tested=24
Files skipped=48
Test cases OK=4156
Test cases NOK=0

If, however, this program is run with one or more files in the KAT_AES directory as argument(s), it will instead build and execute the tests contained in the indicated file(s).

AES-GCM implementation in Python 3

Galois/Counter Mode (GCM) is a mode of operation for symmetric key cryptographic block ciphers that provides authenticated encryption.

Proposed by David McGrew and John Viega in 2005, GCM is suited for high-speed secure computing and communication. Acknowledging this fact, the US National Institute of Standards and Technology (NIST) standardized GCM and its companion algorithm, Galois Message Authentication Code (GMAC), in 2007. The latter is an authentication-only variant of GCM which can be used as an incremental message authentication code (MAC). AES-GCM (the Advanced Encryption Algorithm operating in Galois/Counter Mode) has also been included in NSA Suite B Cryptography.

GCM’s confidentiality service is based on a variation of the Counter mode (CTR) while its authenticity assurance relies on a universal hash function defined over the binary Galois field GF(2^{128})/x^{128}+x^7+x^2+x+1.

GCM is defined for block ciphers with block sizes of 128, 192, and 256 bits (AES uses 128-bit blocks). Both GCM and GMAC can accept initialization vectors (IVs) of arbitrary length (AES and other symmetric ciphers, on the other hand, require IVs to be of the same size as the cipher’s block size).

Its interesting to note that Intel high-end CPUs support a special instruction, PCLMULQDQ, that computes the 128-bit product (the carry-less multiplication) of two 64-bit operands. This instruction can be used as a building block to perform the carry-less multiplication of two 128-bit operands required by GCM. The other step in GCM is reduction modulo the irreducible polynomial x^{128}+x^7+x^2+x+1, an operation that takes advantage of the PSRLD, PSLLD and PSHUFD instructions. Together with Intel’s Advanced Encryption Standard Instructions (AES-NI), these instructions allow GCM to offer authenticated encryption services at very high rates without using hardware-based solutions (e.g. FPGAs or ASICs). According to Intel, “an AES-GCM implementation based on the AES-NI and PCLMULQDQ instructions delivered a 400% throughput performance gain when compared to a non-AES-NI enabled software solution on the same platform.”

The Python code below implements AES-GCM using the AES implementation already presented and supports the three key sizes used by AES (128, 192 and 256 bits). All eighteen test cases proposed by McGrew & Viega were used to validate this implementation. Please note that this code is not of production quality.

#!/usr/bin/python3
#
# Author: Joao H de A Franco (jhafranco@acm.org)
#
# Description: AES-GCM (Galois Counter/Mode) implementation
#              in Python 3
#
# Date: 2013-05-30
#
# License: Attribution-NonCommercial-ShareAlike 3.0 Unported
#          (CC BY-NC-SA 3.0)
#================================================================

import pyAES
from functools import reduce

def xor(x,y):
    """Returns the exclusive or (xor) between two vectors"""
    return bytes(i^j for i,j in zip(x,y))

def intToList(number,listSize):
    """Convert a number into a byte list""" 
    return [(number >> i) & 0xff
            for i in reversed(range(0,listSize*8,8))]

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

def GHASH (hkey,aad,ctext):
    """GCM's GHASH function"""
    def xorMultH (p,q):
        """Multiply (p^q) by hash key"""
        def multGF2(x,y):
            """Multiply two polynomials in GF(2^m)/g(w)
               g(w) = w^128 + w^7 + w^2 + w + 1
               (operands and result bits reflected)"""      
            (x,y) = map(lambda z:listToInt(list(z)),(x,y))
            z = 0
            while y & ((1<<128)-1):
                if y & (1<<127):
                    z ^= x
                y <<= 1
                if x & 1:
                    x = (x>>1)^(0xe1<<120)
                else:
                    x >>= 1
            return bytes(intToList(z,16))

        return bytes(multGF2(hkey,xor(p,q)))
    
    def gLen(s):
        """Evaluate length of input in bits and returns
           it in the LSB bytes of a 64-bit string"""
        return bytes(intToList(len(s)*8,8))  
  
    x = bytes(16)    
    aadP = aad + bytes((16-len(aad)%16)%16)
    ctextP = ctext + bytes((16-len(ctext)%16)%16)
    for i in range(0,len(aadP),16):
        x = xorMultH(x,aadP[i:i+16])
    for i in range(0,len(ctextP),16):
        x = xorMultH(x,ctextP[i:i+16])
    return xorMultH(x,gLen(aad) + gLen(ctext))

def GCM_crypt(keysize,key,iv,input,aad):
    """GCM's Authenticated Encryption/Decryption Operations"""
    def incr(m):
        """Increment the LSB 32 bits of input counter"""
        n = list(m)
        n12 = bytes(n[:12])
        ctr = listToInt(n[12:])
        if ctr == (1<<32)-1:
            return n12 + bytes(4)
        else:
            return n12 + bytes(intToList(ctr+1,4))
       
    obj = pyAES.AES('MODE_ECB')
    obj.setKey(keysize,key)    
    h = bytes(obj.encrypt(bytes(16)))
    output = bytes()
    L = len(input)
    if len(iv) == 12:
        y0 = bytes(iv) + bytes(b'\x00\x00\x00\x01')
    else:
        y0 = bytes(GHASH(h,bytes(),iv))
    y = y0
    for i in range(0,len(input),16):
        y = incr(y)
        ctextBlock = xor(bytes(obj.encrypt(y)),
                         input[i:min(i+16,L)])
        output += bytes(ctextBlock)
    g = obj.encrypt(y0)
    tag = xor(GHASH(h,aad,output),g)
    return output,tag,g,h
 
def GCM_encrypt(keysize,key,iv,ptext,aad):
    """GCM's Authenticated Encryption Operation"""
    (ctext,tag,g,h) = GCM_crypt(keysize,key,iv,ptext,aad)
    return ctext,xor(GHASH(h,aad,ctext),g)

def GCM_decrypt(keysize,key,iv,ctext,aad,tag):
    """GCM's Authenticated Decryption Operation"""
    (ptext,_,g,h) = GCM_crypt(keysize,key,iv,ctext,aad)
    if tag == xor(GHASH(h,aad,ctext),g):
        return True,ptext
    else:
        return False,None

######################## Testing section ########################

if __name__ == '__main__':

    def printHex(s):
        """Prints a bytes string in hex format"""
        print('{0:#0x}'.format(bytesToInt(s)))
    
    def bytesToInt(b):
        """Converts a bytes string into an integer"""
        return listToInt(list(b))
    
    def listToInt(list):
        """Convert a byte list into a number""" 
        return reduce(lambda x,y:(x<<8)+y,list)
    
    def checkTestVector(id,keysize,key,ptext,aad,iv,ctext,tag):
        def intToBytes(n):
            """Converts an integer into a bytes string"""
            lst = []
            while n:
                lst.append(n&0xff)
                n >>= 8
            return bytes(reversed(lst))

        def convertType(v):
            """Convert input variable type to bytes"""
            if type(v) is int:
                return intToBytes(v)
            elif type(v) is bytes:
                return v
            else:
                return bytes(v,'ISO-8859-1')

        def printOutputs():
            """Prints expected/evaluated tags and
                      expected/evaluated ciphertexts"""
            print("Tag expected:  ", end="")
            printHex(tag)
            print("Tag evaluated: ", end="")
            printHex(TAG)
            print("Ciphertext expected:  ", end="")
            printHex(ctext)
            print("Ciphertext evaluated: ", end="")
            printHex(CTEXT)            
          
        (ptext,aad,iv,ctext,tag) = map(convertType,(ptext,aad,iv,ctext,tag))
        (CTEXT,TAG) = GCM_encrypt(keysize,key,iv,ptext,aad)
        (SUCCESS,PTEXT) = GCM_decrypt(keysize,key,iv,CTEXT,aad,TAG)
    
        try:
            assert SUCCESS & (CTEXT == ctext) & (TAG == tag)
            print("Test case {0:0d} succeeded".format(id))
        except AssertionError:
            print("Test case {0:0d} failed".format(id))
            printOutputs()           
    
    # Test cases extracted from
    # "The Galois/Counter Mode of Operation(GCM)", McGrew & Viega, 2005 (http://goo.gl/DWJPK)
    
    # Some useful constants
    emptyString = bytes() # zero length bit string
    nullBitString128 = bytes(16) # 128-bit null string 
    nullBitString96 = bytes(12) # 96-bit null IV
    
    # Test Case #1
    testcase1 = {'id':1,'keysize':"SIZE_128",'key':0x0,'ptext':emptyString,
                 'aad':emptyString,'iv':nullBitString96,'ctext':emptyString,
                 'tag':0x58e2fccefa7e3061367f1d57a4e7455a}
    
    # Test Case #2
    testcase2 = {'id':2,'keysize':"SIZE_128",'key':0x0,'ptext':nullBitString128,
                 'aad':emptyString,'iv':nullBitString96,'ctext':0x0388dace60b6a392f328c2b971b2fe78,
                 'tag':0xab6e47d42cec13bdf53a67b21257bddf}
    
    # Test Case #3
    testcase3 = {'id':3,'keysize':"SIZE_128",'key':0xfeffe9928665731c6d6a8f9467308308,
                 'ptext':0xd9313225f88406e5a55909c5aff5269a86a7a9531534f7da2e4c303d8a318a721c3c0c95956809532fcf0e2449a6b525b16aedf5aa0de657ba637b391aafd255,
                 'aad':emptyString,'iv':0xcafebabefacedbaddecaf888,
                 'ctext':0x42831ec2217774244b7221b784d0d49ce3aa212f2c02a4e035c17e2329aca12e21d514b25466931c7d8f6a5aac84aa051ba30b396a0aac973d58e091473f5985,
                 'tag':0x4d5c2af327cd64a62cf35abd2ba6fab4}
    
    ## Test Case #4
    testcase4 = {'id':4,'keysize':"SIZE_128",'key':0xfeffe9928665731c6d6a8f9467308308,
                 'ptext':0xd9313225f88406e5a55909c5aff5269a86a7a9531534f7da2e4c303d8a318a721c3c0c95956809532fcf0e2449a6b525b16aedf5aa0de657ba637b39,
                 'aad':0xfeedfacedeadbeeffeedfacedeadbeefabaddad2,'iv':0xcafebabefacedbaddecaf888,
                 'ctext':0x42831ec2217774244b7221b784d0d49ce3aa212f2c02a4e035c17e2329aca12e21d514b25466931c7d8f6a5aac84aa051ba30b396a0aac973d58e091,
                 'tag':0x5bc94fbc3221a5db94fae95ae7121a47}

    ## Test Case #5
    testcase5 = {'id':5,'keysize':"SIZE_128",'key':0xfeffe9928665731c6d6a8f9467308308,
                 'ptext':0xd9313225f88406e5a55909c5aff5269a86a7a9531534f7da2e4c303d8a318a721c3c0c95956809532fcf0e2449a6b525b16aedf5aa0de657ba637b39,
                 'aad':0xfeedfacedeadbeeffeedfacedeadbeefabaddad2,'iv':0xcafebabefacedbad,
                 'ctext':0x61353b4c2806934a777ff51fa22a4755699b2a714fcdc6f83766e5f97b6c742373806900e49f24b22b097544d4896b424989b5e1ebac0f07c23f4598,
                 'tag':0x3612d2e79e3b0785561be14aaca2fccb}

    ## Test Case #6
    testcase6 = {'id':6,'keysize':"SIZE_128",'key':0xfeffe9928665731c6d6a8f9467308308,
                 'ptext':0xd9313225f88406e5a55909c5aff5269a86a7a9531534f7da2e4c303d8a318a721c3c0c95956809532fcf0e2449a6b525b16aedf5aa0de657ba637b39,
                 'aad':0xfeedfacedeadbeeffeedfacedeadbeefabaddad2,
                 'iv':0x09313225df88406e555909c5aff5269aa6a7a9538534f7da1e4c303d2a318a728c3c0c95156809539fcf0e2429a6b525416aedbf5a0de6a57a637b39b,
                 'ctext':0x8ce24998625615b603a033aca13fb894be9112a5c3a211a8ba262a3cca7e2ca701e4a9a4fba43c90ccdcb281d48c7c6fd62875d2aca417034c34aee5,
                 'tag':0x619cc5aefffe0bfa462af43c1699d050}
        
    ## Test Case #7
    testcase7 = {'id':7,'keysize':"SIZE_192",'key':0x0,'ptext':emptyString,
                 'aad':emptyString,'iv':nullBitString96,'ctext':emptyString,
                 'tag':0xcd33b28ac773f74ba00ed1f312572435}
   
    ## Test Case #8
    testcase8 = {'id':8,'keysize':"SIZE_192",'key':0x0,'ptext':nullBitString128,
                 'aad':emptyString,'iv':nullBitString96,
                 'ctext':0x98e7247c07f0fe411c267e4384b0f600,
                 'tag':0x2ff58d80033927ab8ef4d4587514f0fb}

    ## Test Case #9
    testcase9 = {'id':9,'keysize':"SIZE_192",'key':0xfeffe9928665731c6d6a8f9467308308feffe9928665731c,
                 'ptext':0xd9313225f88406e5a55909c5aff5269a86a7a9531534f7da2e4c303d8a318a721c3c0c95956809532fcf0e2449a6b525b16aedf5aa0de657ba637b391aafd255,
                 'aad':emptyString,'iv':0xcafebabefacedbaddecaf888,
                 'ctext':0x3980ca0b3c00e841eb06fac4872a2757859e1ceaa6efd984628593b40ca1e19c7d773d00c144c525ac619d18c84a3f4718e2448b2fe324d9ccda2710acade256,
                 'tag':0x9924a7c8587336bfb118024db8674a14}
                   
    ## Test Case #10
    testcase10 = {'id':10,'keysize':"SIZE_192",'key':0xfeffe9928665731c6d6a8f9467308308feffe9928665731c,
                  'ptext':0xd9313225f88406e5a55909c5aff5269a86a7a9531534f7da2e4c303d8a318a721c3c0c95956809532fcf0e2449a6b525b16aedf5aa0de657ba637b39,
                  'aad':0xfeedfacedeadbeeffeedfacedeadbeefabaddad2,'iv':0xcafebabefacedbaddecaf888,
                  'ctext':0x3980ca0b3c00e841eb06fac4872a2757859e1ceaa6efd984628593b40ca1e19c7d773d00c144c525ac619d18c84a3f4718e2448b2fe324d9ccda2710,
                  'tag':0x2519498e80f1478f37ba55bd6d27618c}
                                    
    ## Test Case #11
    testcase11 = {'id':11,'keysize':"SIZE_192",'key':0xfeffe9928665731c6d6a8f9467308308feffe9928665731c,
                  'ptext':0xd9313225f88406e5a55909c5aff5269a86a7a9531534f7da2e4c303d8a318a721c3c0c95956809532fcf0e2449a6b525b16aedf5aa0de657ba637b39,
                  'aad':0xfeedfacedeadbeeffeedfacedeadbeefabaddad2,'iv':0xcafebabefacedbad,
                  'ctext':0x0f10f599ae14a154ed24b36e25324db8c566632ef2bbb34f8347280fc4507057fddc29df9a471f75c66541d4d4dad1c9e93a19a58e8b473fa0f062f7,
                  'tag':0x65dcc57fcf623a24094fcca40d3533f8}
                                                    
    ## Test Case #12
    testcase12 = {'id':12,'keysize':"SIZE_192",'key':0xfeffe9928665731c6d6a8f9467308308feffe9928665731c,
                  'ptext':0xd9313225f88406e5a55909c5aff5269a86a7a9531534f7da2e4c303d8a318a721c3c0c95956809532fcf0e2449a6b525b16aedf5aa0de657ba637b39,
                  'aad':0xfeedfacedeadbeeffeedfacedeadbeefabaddad2,
                  'iv':0x9313225df88406e555909c5aff5269aa6a7a9538534f7da1e4c303d2a318a728c3c0c95156809539fcf0e2429a6b525416aedbf5a0de6a57a637b39b,
                  'ctext':0xd27e88681ce3243c4830165a8fdcf9ff1de9a1d8e6b447ef6ef7b79828666e4581e79012af34ddd9e2f037589b292db3e67c036745fa22e7e9b7373b,
                  'tag':0xdcf566ff291c25bbb8568fc3d376a6d9}

    ## Test Case #13
    testcase13 = {'id':13,'keysize':"SIZE_256",'key':0x0,'ptext':emptyString,
                  'aad':emptyString,'iv':nullBitString96,'ctext':emptyString,
                  'tag':0x530f8afbc74536b9a963b4f1c4cb738b}

    ## Test Case #14
    testcase14 = {'id':14,'keysize':"SIZE_256",'key':0x0,'ptext':nullBitString128,
                   'aad':emptyString,'iv':nullBitString96,'ctext':0xcea7403d4d606b6e074ec5d3baf39d18,
                   'tag':0xd0d1c8a799996bf0265b98b5d48ab919}             

    ## Test Case #15
    testcase15 = {'id':15,'keysize':"SIZE_256",'key':0xfeffe9928665731c6d6a8f9467308308feffe9928665731c6d6a8f9467308308,
                  'ptext':0xd9313225f88406e5a55909c5aff5269a86a7a9531534f7da2e4c303d8a318a721c3c0c95956809532fcf0e2449a6b525b16aedf5aa0de657ba637b391aafd255,
                  'aad':emptyString,'iv':0xcafebabefacedbaddecaf888,
                  'ctext':0x522dc1f099567d07f47f37a32a84427d643a8cdcbfe5c0c97598a2bd2555d1aa8cb08e48590dbb3da7b08b1056828838c5f61e6393ba7a0abcc9f662898015ad,
                  'tag':0xb094dac5d93471bdec1a502270e3cc6c}

    ## Test Case #16
    testcase16 = {'id':16,'keysize':"SIZE_256",'key':0xfeffe9928665731c6d6a8f9467308308feffe9928665731c6d6a8f9467308308,
                  'ptext':0xd9313225f88406e5a55909c5aff5269a86a7a9531534f7da2e4c303d8a318a721c3c0c95956809532fcf0e2449a6b525b16aedf5aa0de657ba637b39,
                  'aad':0xfeedfacedeadbeeffeedfacedeadbeefabaddad2,'iv':0xcafebabefacedbaddecaf888,
                  'ctext':0x522dc1f099567d07f47f37a32a84427d643a8cdcbfe5c0c97598a2bd2555d1aa8cb08e48590dbb3da7b08b1056828838c5f61e6393ba7a0abcc9f662,
                  'tag':0x76fc6ece0f4e1768cddf8853bb2d551b}

    ## Test Case #17
    testcase17 = {'id':17,'keysize':"SIZE_256",'key':0xfeffe9928665731c6d6a8f9467308308feffe9928665731c6d6a8f9467308308,
                  'ptext':0xd9313225f88406e5a55909c5aff5269a86a7a9531534f7da2e4c303d8a318a721c3c0c95956809532fcf0e2449a6b525b16aedf5aa0de657ba637b39,
                  'aad':0xfeedfacedeadbeeffeedfacedeadbeefabaddad2,'iv':0xcafebabefacedbad,
                  'ctext':0xc3762df1ca787d32ae47c13bf19844cbaf1ae14d0b976afac52ff7d79bba9de0feb582d33934a4f0954cc2363bc73f7862ac430e64abe499f47c9b1f,
                  'tag':0x3a337dbf46a792c45e454913fe2ea8f2} 

    ## Test Case #18
    testcase18 = {'id':18,'keysize':"SIZE_256",'key':0xfeffe9928665731c6d6a8f9467308308feffe9928665731c6d6a8f9467308308,
                  'ptext':0xd9313225f88406e5a55909c5aff5269a86a7a9531534f7da2e4c303d8a318a721c3c0c95956809532fcf0e2449a6b525b16aedf5aa0de657ba637b39,
                  'aad':0xfeedfacedeadbeeffeedfacedeadbeefabaddad2,
                  'iv':0x9313225df88406e555909c5aff5269aa6a7a9538534f7da1e4c303d2a318a728c3c0c95156809539fcf0e2429a6b525416aedbf5a0de6a57a637b39b,
                  'ctext':0x5a8def2f0c9e53f1f75d7853659e2a20eeb2b22aafde6419a058ab4f6f746bf40fc0c3b780f244452da3ebf1c5d82cdea2418997200ef82e44ae7e3f,
                  'tag':0xa44a8266ee1c8eb0c8b5d4cf5ae9f19a}
                  
    for i in range(1,19):    
        checkTestVector(**(eval("testcase" + str(i))))

Fast Fourier Transform in Haskell

The Discrete Fourier Transform (DFT) converts a finite list of equally-spaced samples of a function into the list of coefficients of a finite combination of complex sinusoids, ordered by their frequencies, that has those same sample values.

The DFT is used to perform Fourier analysis in different applications like digital signal processing, polynomial multiplication, partial differential equations solving, large integers multiplication etc. It has also been used in statistical tests of random and pseudorandom number generators for cryptographic applications (see NIST Special Publication 800-22).

The Haskell function dft (shown below) evaluates the DFT for any number of points. Since the vectors [e^{2{\pi}kn/N} | n = 0,1..N-1] constitute an orthogonal basis over the set of N-dimensional complex vectors, the inverse DFT can be calculated by the same algorithm using the complex conjugate vectors (see function idft). In order to make the forward and the inverse transforms unitary, a normalization factor equal to \sqrt{1/N} is used in both functions.

-------------------------------------------------------------
--
-- Author: Joao H de A Franco (jhafranco@acm.org)
--
-- Description: DFT implementation in Haskell
--
-- Date: 2013-Mar-28
--
-- License: Attribution-NonCommercial-ShareAlike 3.0 Unported
--          (CC BY-NC-SA 3.0)
--
-------------------------------------------------------------
module Aux.DFT (dft,idft) where
import Data.Complex (Complex((:+)))

dft, idft :: [Complex Float] -> [Complex Float]
dft = dft' 1
idft = dft' (-1)
           
dft' :: Float -> [Complex Float] -> [Complex Float]
dft' e x = let t = length x
               y = 1 / fromIntegral t
               dft'' k = let w = fromIntegral k * 2 * e * pi * y
                         in (sqrt y :+ 0) *
                             sum (zipWith (\n z -> z * exp (0 :+ n * w)) [0..] x)                         
           in map dft'' [0..t-1]

The DFT can be seen as a linear transformation where an real-valued N-tuple is multiplied by an N x N matrix returning a complex-valued N-tuple. This matrix operation takes N² scalar multiplications, which makes it very expensive to deal with large data sets.

Fast Fourier Transform is the generic name given to algorithms that compute the DFT (or its inverse) faster than O(N²). The most commonly used FFT algorithm is due to Cooley & Tukey (1965), a “divide and conquer” procedure that recursively breaks down a DFT of any composite size N = N1 * N2 into many smaller DFTs of sizes N1 and N2. The Cooley-Tukey and other FFTs schemes reduce the algorithm complexity from O(N²) to O(N log N).

The advent of FFT algorithms has revolutionized the use of the Discrete Fourier Transform, making it possible to use it on a variety of areas: applied mechanics, biomedical engineering, communications, electromagnetics, finance, instrumentation, numerical methods, radar, signal processing, sonics and acoustics etc.

It’s interesting to note, however, that the general idea behind the FFT algorithms was invented by Carl Friedrich Gauss (“The Prince of Mathematicians”) around 1805, when he was calculating the orbit of an asteroid. Amazing!

The Cooley–Tukey algorithm is usually employed to recursively divide a N-point transform into two N/2-point transforms and for this reason it is limited to cases where N is a power of two. The Haskell code below implements this technique, called decimation-in-time radix-2 (fft implements the forward transform and ifft the inverse one).

-------------------------------------------------------------
--
-- Author: Joao H de A Franco (jhafranco@acm.org)
--
-- Description: FFT implementation in Haskell
--
-- Date: 2013-Mar-28
--
-- License: Attribution-NonCommercial-ShareAlike 3.0 Unported
--          (CC BY-NC-SA 3.0)
--
-------------------------------------------------------------
module Aux.FFT (fft,ifft) where
import Data.Complex(Complex((:+)))

fft, ifft :: [Complex Float] -> [Complex Float]
fft = fft' 1
ifft = fft' (-1)

fft' :: Float -> [Complex Float] -> [Complex Float]
fft' e xs = if pow2 (length xs) 
            then let z = sqrt (1 / fromIntegral (length xs))
                  in map ((z :+ 0) *) $ fft'' e xs
            else error "Number of points is not a power of 2"
            where pow2 n
                     | n == 1 || n == 2 = True
                     | otherwise = n `mod` 2 == 0 && pow2 (n `div` 2)

fft'' :: Float -> [Complex Float] -> [Complex Float]
fft'' _ [] = []
fft'' _ [x] = [x]
fft'' e xs = fft'' e (evens xs) <+> t (fft'' e (odds xs))
  where (<+>) r s = zipWith (+) (r ++ r) (s ++ map negate s)
        evens [] = []
        evens [u] = [u]
        evens (v:_:vs) = v:evens vs
        odds = evens . drop 1
        n = 2 * pi / fromIntegral (length xs)
        t = zipWith (\k z -> z * exp (0 :+ k * n * e)) ([0..] :: [Float])

Besides classic references on this subject such as Brigham’s and Bracewell’s books, I recommend the excellent lectures on The Fourier Transforms and Its Applications (EE261) by Prof. Brad Osgood from Stanford University. As a matter of fact, I’ve watched only lectures 20 to 22, as they cover DFT and the basics of FFT algorithms, but most probably the remaining lectures should be equally good. Extensive course materials (syllabus, handouts, past exams, exercises & solutions) are available.

The box below shows some simple tests to check the correctness of the above implementations.

-------------------------------------------------------------
--
-- Author: Joao H de A Franco (jhafranco@acm.org)
--
-- Description: Test module for FFT implementation in Haskell
--
-- Date: 2013-Mar-28
--
-- License: Attribution-NonCommercial-ShareAlike 3.0 Unported
--          (CC BY-NC-SA 3.0)
--
-------------------------------------------------------------
module Main where
import Data.Complex (Complex((:+)),realPart,imagPart)
import Aux.DFT (dft,idft)
import Aux.FFT (fft,ifft)

main :: IO ()
main = print (if test1 && test2 && test3 then "ok" else "nok")

-- 7-point DFT (prime number) compared to Mathematica's evaluation
test1 :: Bool
test1 = let mathematica1 = [3.40168 :+ 0, (-0.566947) :+ 0.564962,
                            (-0.566947) :+ (-1.41899), (-0.566947) :+ 0.645967,
                            (-0.566947) :+ (-0.645967), (-0.566947) :+ 1.41899,
                            (-0.566947) :+ (-0.564962)]
        in map (roundComplex 3 0.001) (dft [0,1,2,3,0,1,2]) == 
           map (roundComplex 3 0.001) mathematica1

-- 8-point FFT (power of 2) compared to Mathematica's evaluation
test2 :: Bool
test2 = let mathematica2 = [4.24264 :+ 0, 0 :+ 0,
                            (-1.41421) :+ (-1.41421), 0 :+ 0, 
                            (-1.41421) :+ 0, 0 :+ 0,
                            (-1.41421) :+ 1.41421, 0 :+ 0]
        in map (roundComplex 3 0.001) (fft [0,1,2,3,0,1,2,3]) == 
           map (roundComplex 3 0.001) mathematica2

-- 512-point DFT compared to FFT
test3 :: Bool
test3 = let list = take 512 $ cycle ([0,1,2,3] :: [Complex Float])
            z1 = map (roundComplex 2 0.001) $ dft list
            z2 = map (roundComplex 2 0.001) $ fft list
         in z1 == z2

-- 128-point time-shifted impulse forward & inverse DFT (auto-test)
test4 :: Bool
test4 = all (==True) t4
        where t4 = do
              n <- [0..127]
              let x = impulse n 128 :: [Complex Float]
              return $ map (roundComplex 3 0.001) (idft (dft x)) == x

-- 1024-point time-shifted impulse forward & inverse FFT (auto-test)         
test5 :: Bool
test5 = all (==True) t5
        where t4 = do
              n <- [0..1023]
              let x = impulse n 1024 :: [Complex Float]
              return $ map (roundComplex 3 0.001) (ifft (fft x)) == x

-- Discrete Dirac delta generator
impulse :: Num a => Int -> Int -> [a]
impulse n m = replicate n 0 ++ 1:replicate (m - n - 1) 0

-- Rounding function
roundComplex :: Int -> Float -> Complex Float -> Complex Float
roundComplex n e x = let zeroFloat e' f = if abs f < e' then 0 else f
                         trim = roundFloat n . zeroFloat e
                         roundFloat m y = fromIntegral (round $ y * 10 ^ m :: Int) / 10 ^^ m
                     in trim (realPart x) :+ trim (imagPart x)

You can run the above FFT script to evaluate the forward and inverse FFT for a small data set using the facilities provided by Codepad.org, an online compiler/interpreter for Haskell and other languages.

A quick benchmark shows the speed-up achieved by the radix-2 FFT over the classic DFT: 18 ms vs. 2.5 s for 4,096 points (two orders of magnitude!). FFTW, a C subroutine library for computing the DFT in one or more dimensions, is much faster, as it takes only 0.7 ms for processing 4,096 points on the same Linux system.