Key Expansion / Key Schedule

Each round use its own round key into the "Add Key" step that is derived from the original encryption key.

So, the Key Schedule will allow creating a different key for each encryption round, each of these sub-keys being 128 bits

The entire key length is then 128 times n+1 ( where n is the amount of round, depending of the original key size )

To make the key expansion, the original key is divided into 32 bits blocks called words

Here is 4 words for AES-128, 6 for AES-192 and 8 for AES-256

The next words are calculated following these graph :

Algorithm steps

The following explanation will be based on AES-128. Some little ajustement ( such as the amount of generated words ) have to be done to make it applicable to the others key size.

The first four words group

The first four words are provided by the original key.

  • W0 = key[0:31]

  • W1 = key[32:63]

  • W2 = key[64:95]

  • W3 = key[96:127]

Here, we have the four words of the key used for the round 0 ( before the 10 loops )

  • w4, w5, w6, w7 is the round key for round 1

  • w8, w9, w10, w11 the round key for round 2,

  • and so on.

The others words groups

Let’s say that we have the four words of the round key for the i th round:

And we need to determine the words

Using the Figure 1, we can write :

Note that except for the first word in a new 4-word grouping, each word is an XOR of the previous word and the corresponding word in the previous 4-word grouping.

The first word of each groups

The g() function

  • Perform a one-byte left circular rotation on the argument 4-byte word.

  • Perform a byte substitution for each byte of the word using the same "S-box" in the SubBytes step of the encryption rounds

  • XOR the bytes obtained from the previous step with a round constant.

The round constant is a word whose three rightmost bytes are always zero.

Therefore, XOR’ing with the round constant amounts to XOR’ing with just its leftmost byte.

The multiplication applied here is the same as in Mix Column operation when multiplying by 2.

Python implementation

def expand_key(master_key):
    """
    Expands and returns a list of key matrices for the given master_key.
    """

    # Round constants https://en.wikipedia.org/wiki/AES_key_schedule#Round_constants
    r_con = (
        0x00, 0x01, 0x02, 0x04, 0x08, 0x10, 0x20, 0x40,
        0x80, 0x1B, 0x36, 0x6C, 0xD8, 0xAB, 0x4D, 0x9A,
        0x2F, 0x5E, 0xBC, 0x63, 0xC6, 0x97, 0x35, 0x6A,
        0xD4, 0xB3, 0x7D, 0xFA, 0xEF, 0xC5, 0x91, 0x39,
    )

    # Initialize round keys with raw key material.
    key_columns = bytes2matrix(master_key)
    iteration_size = len(master_key) // 4

    # Each iteration has exactly as many columns as the key material.
    i = 1
    while len(key_columns) < (N_ROUNDS + 1) * 4:
        # Copy previous word.
        word = list(key_columns[-1])

        # Perform schedule_core once every "row".
        if len(key_columns) % iteration_size == 0:
            # Circular shift.
            word.append(word.pop(0))
            # Map to S-BOX.
            word = [s_box[b] for b in word]
            # XOR with first byte of R-CON, since the others bytes of R-CON are 0.
            word[0] ^= r_con[i]
            i += 1
        elif len(master_key) == 32 and len(key_columns) % iteration_size == 4:
            # Run word through S-box in the fourth iteration when using a
            # 256-bit key.
            word = [s_box[b] for b in word]

        # XOR with equivalent word from previous iteration.
        word = bytes(i^j for i, j in zip(word, key_columns[-iteration_size]))
        key_columns.append(word)

    # Group key words in 4x4 byte matrices.
    return [key_columns[4*i : 4*(i+1)] for i in range(len(key_columns) // 4)]

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