Update: 04-Apr-2009

I fixed two bugs in my AES implementation pointed out to me by Josiah Carlson. First, I was failing to pad properly files whose length was an even multiple of the block size. In those cases, bytes would be lost upon decrypting the file. Josiah also pointed out that I was using a static IV, which leaks information about messages which share common prefixes. This is a serious security bug and I was glad to have it pointed out.

Feel free to check out the changes I made or simply download the updated script.

Symmetric Key Cryptography

• Identical keys used to encrypt/decrypt messages
• Can be implemented as block ciphers or stream ciphers

Strengths:

• Speed
• Much less computationally intensive than public-key crypto
• Easy to implement in hardware as well as software

Weaknesses:

• Key Management
• n users require n(n-1)/2 keys for all to communicate
• secure key distribution is a challenge
• Cannot be used (directly) for authentication or non-repudiation

AES – The Advanced Encryption Standard

• Rijndael algorithm invented by Joan Daemen and Vincent Rijmen and selected as AES winner by NIST in 2001
• AES uses fixed block size of 128-bits and key sizes of 128, 192 or 256 bits (though Rijndael specification allows for variable block and key sizes)
• Most of the calculations in AES are performed within a finite field
• There are a finite number of elements within the field and all operations on those elements result in an element also contained in the field

AES Operations

• AES operates on a 4×4 matrix referred to as the state
• 16 bytes == 128 bits == block size
• All operations in a round of AES are invertible
• AddRoundKey – each byte of the round key is combined with the corresponding byte in the state using XOR
• SubBytes – each byte in the state is replaced with a different byte according to the S-Box lookup table
• ShiftRows – each row in the state table is shifted by a varying number of bytes
• MixColumns – each column in the state table is multiplied with a fixed polynomial

AES Operation – AddRoundKey

• Each byte of the round key is XORed with the corresponding byte in the state table
• Inverse operation is identical since XOR a second time returns the original values

# XOR each byte of the roundKey with the state table
def addRoundKey(state, roundKey):
    for i in range(len(state)):
        state[i] = state[i] ^ roundKey[i]

AES Operation – SubBytes

• Each byte of the state table is substituted with the value in the S-Box whose index is the value of the state table byte
• Provides non-linearity (algorithm not equal to the sum of its parts)
• Inverse operation is performed using the inverted S-Box

# do sbox transform on each of the values in the state table
def subBytes(state):
    for i in range(len(state)):
        state[i] = sbox[state[i]]

# sbox transformations are invertible
>>> sbox[237]
85
>>> sboxInv[85]
237
>>> sbox[55]
154
>>> sbox[154]
184
>>> sboxInv[184]
154
>>> sboxInv[154]
55

AES Operation – ShiftRows

• Each row in the state table is shifted left by the number of bytes represented by the row number
• Inverse operation simply shifts each row to the right by the number of bytes as the row number

# returns a copy of the word shifted n bytes (chars) positive
# values for n shift bytes left, negative values shift right
def rotate(word, n):
    return word[n:]+word[0:n]

# iterate over each "virtual" row in the state table
# and shift the bytes to the LEFT by the appropriate
# offset
def shiftRows(state):
    for i in range(4):
        state[i*4:i*4+4] = rotate(state[i*4:i*4+4],i)

AES Operation – MixColumns

• MixColumns is performed by multiplying each column (within the Galois finite field) by the following matrix:

• The inverse operation is performed by multiplying each column by the following inverse matrix:

# Galois Multiplication
def galoisMult(a, b):
    p = 0
    hiBitSet = 0
    for i in range(8):
        if b & 1 == 1:
            p ^= a
        hiBitSet = a & 0x80
        a <<= 1
        if hiBitSet == 0x80:
            a ^= 0x1b
        b >>= 1
    return p % 256

# mixColumn does Galois multiplication on a state column
def mixColumn(column):
    temp = copy(column)
    column[0] = galoisMult(temp[0],2) ^ galoisMult(temp[3],1) ^ \
                galoisMult(temp[2],1) ^ galoisMult(temp[1],3)
    column[1] = galoisMult(temp[1],2) ^ galoisMult(temp[0],1) ^ \
                galoisMult(temp[3],1) ^ galoisMult(temp[2],3)
    column[2] = galoisMult(temp[2],2) ^ galoisMult(temp[1],1) ^ \
                galoisMult(temp[0],1) ^ galoisMult(temp[3],3)
    column[3] = galoisMult(temp[3],2) ^ galoisMult(temp[2],1) ^ \
                galoisMult(temp[1],1) ^ galoisMult(temp[0],3)

AES – Pulling It All Together

The AES Cipher operates using a varying number of rounds, based on the size of the cipher key.

• A round of AES consists of the four operations performed in succession: AddRoundKey, SubBytes, ShiftRows, and MixColumns (MixColumns is omitted in the final round)
• 128-bit key → rounds, 192-bit key → 12 rounds, 256-bit key → 14 rounds
• The AES cipher key is expanded according to the Rijndael key schedule and a different part of the expanded key is used for each round of AES
• The expanded key will be of length (block size * num rounds+1)
• 128-bit cipher key expands to 176-byte key
• 192-bit cipher key expands to 208-byte key
• 256-bit cipher key expands to 240-byte key

AES – Key Expansion Operations

AES key expansion consists of several primitive operations:

1. Rotate – takes a 4-byte word and rotates everything one byte to the left, e.g. rotate([1,2,3,4]) → [2, 3, 4, 1]
2. SubBytes – each byte of a word is substituted with the value in the S-Box whose index is the value of the original byte
3. Rcon – the first byte of a word is XORed with the round constant. Each value of the Rcon table is a member of the Rinjdael finite field.

# takes 4-byte word and iteration number
def keyScheduleCore(word, i):
    # rotate word 1 byte to the left
    word = rotate(word, 1)
    newWord = []
    # apply sbox substitution on all bytes of word
    for byte in word:
        newWord.append(sbox[byte])
    # XOR the output of the rcon[i] transformation with the first part
    # of the word
    newWord[0] = newWord[0]^rcon[i]
    return newWord

AES – Key Expansion Algorithm (256-bit)

Pseudo-code for AES Key Expansion:

1. expandedKey[0:32] → cipherKey[0:32] # copy first 32 bytes of cipher key to expanded key
2. i → 1 # Rcon iterator
3. temp = byte[4] # 4-byte container for temp storage
4. while size(expandedKey) < 240
   temp → last 4 bytes of expandedKey

   # every 32 bytes apply core schedule to temp
   if size(expandedKey)%32 == 0
   temp = keyScheduleCore(temp, i)
   i → i + 1
   # since 256-bit key -> add an extra sbox transformation to each new byte
   for j in range(4):
   temp[j] = sbox[temp[j]]
   # XOR temp with the 4-byte block 32 bytes before the end of the current expanded key.
   # These 4 bytes become the next bytes in the expanded key
   expandedKey.append( temp XOR expandedKey[size(expandedKey)-32:size(expandedKey)-28]

Another function to note…

# returns a 16-byte round key based on an expanded key and round number
def createRoundKey(expandedKey, n):
    return expandedKey[(n*16):(n*16+16)]

AES – Encrypting a Single Block

1. state → block of plaintext # 16 bytes of plaintext are copied into the state
2. expandedKey = expandKey(cipherKey) # create 240-bytes of key material to be used as round keys
3. roundNum → 0 # counter for which round number we are in
4. roundKey → createRoundKey(expandedKey, roundNum)
5. addRoundKey(state, roundKey) # each byte of state is XORed with the present roundKey
6. while roundNum < 14 # 14 rounds in AES-256
   roundKey → createRoundKey(expandedKey, roundNum)
   # round of AES consists of 1. subBytes, 2. shiftRows, 3. mixColumns, and 4. addRoundKey
   aesRound(state, roundKey)
   roundNum → roundNum + 1
7. # for the last round leave out the mixColumns operation
   roundKey = createRoundKey(expandedKey, roundNum)
   subBytes(state)
   shiftRows(state)
   addRoundKey(state)
8. return state as block of ciphertext

AES – Encrypting a Single Block (Demo)

>>> key = passwordToKey("s0m3_p@ssw0rD")
>>> key
[62, 142, 78, 2, 164, 231, 18, 196, 148, 177, 82, 186, 240, 44, 136, 242,
23, 13, 20, 169, 248, 69, 163, 79, 13, 155, 97, 200, 241, 15, 76, 15]
>>> plaintext = textToBlock("Hiro Protagonist")
>>> plaintext
[72, 105, 114, 111, 32, 80, 114, 111, 116, 97, 103, 111, 110, 105, 115, 116]
>>> blockToText(plaintext)
'Hiro Protagonist'
>>> ciphertext = aesEncrypt(plaintext, key)

*** aesMain ***
initial state:
[72, 105, 114, 111, 32, 80, 114, 111, 116, 97, 103, 111, 110, 105, 115, 116]
state after adding roundKey0:
[118, 231, 60, 109, 132, 183, 96, 171, 224, 208, 53, 213, 158, 69, 251, 134]

*** AES Round1 ***
state after subBytes:
[56, 148, 235, 60, 95, 169, 208, 98, 225, 112, 150, 3, 11, 110, 15, 68]
state after shiftRows:
[56, 148, 235, 60, 169, 208, 98, 95, 150, 3, 225, 112, 68, 11, 110, 15]
state after mixColumns:
[66, 80, 228, 230, 148, 33, 121, 29, 106, 95, 226, 146, 255, 98, 121, 117]
state after addRoundKey:
[85, 93, 240, 79, 108, 100, 218, 82, 103, 196, 131, 90, 14, 109, 53, 122]

<-- SNIP -->

*** AES Round 14 (final) ***
state after subBytes:
[0, 229, 171, 70, 93, 137, 135, 251, 99, 182, 88, 166, 228, 229, 251, 97]
state after shiftRows:
[0, 229, 171, 70, 137, 135, 251, 93, 88, 166, 99, 182, 97, 228, 229, 251]
state after addRoundKey:
[195, 123, 205, 183, 213, 202, 50, 223, 223, 164, 99, 86, 126, 34, 107, 142]

>>> ciphertext
[195, 123, 205, 183, 213, 202, 50, 223, 223, 164, 99, 86, 126, 34, 107, 142]

>>> blockToText(ciphertext)
'\xc3{\xcd\xb7\xd5\xca2\xdf\xdf\xa4cV~"k\x8e'
>>> cleartext = aesDecrypt(ciphertext, key)

*** aesMainInv ***
initial state:
[195, 123, 205, 183, 213, 202, 50, 223, 223, 164, 99, 86, 126, 34, 107, 142]

*** AES Round 14 ***
state after addRoundKey:
[0, 229, 171, 70, 137, 135, 251, 93, 88, 166, 99, 182, 97, 228, 229, 251]
state after shiftRowsInv:
[0, 229, 171, 70, 93, 137, 135, 251, 99, 182, 88, 166, 228, 229, 251, 97]
state after subBytesInv:
[82, 42, 14, 152, 141, 242, 234, 99, 0, 121, 94, 197, 174, 42, 99, 216]

<-- SNIP -->

*** AES Round 0 (final) ***
state after adding roundKey0:
[72, 105, 114, 111, 32, 80, 114, 111, 116, 97, 103, 111, 110, 105, 115, 116]
>>> cleartext
[72, 105, 114, 111, 32, 80, 114, 111, 116, 97, 103, 111, 110, 105, 115, 116]
>>> blockToText(cleartext)
'Hiro Protagonist'