Each round use its own round key into the "" 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 )
128β 11=1408 bits for AES-128
128β 13=1664 bits for AES-192
128β 15=1920 bits for AES-256
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.
As explained before, AES-128 need a total key lenght of 1408 bits ( 11β 128 ).
As each word has a size of 32 bits, ( 321408β=44 )44 words are needed.
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 )
To serve as the round key for the ith round, i must be a multiple of 4.
These will obviously serve as the round key for the i/4th round. For example :
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
Letβs say that we have the four words of the round key for the i th round:
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
wi+4β is the beginning of the 4-word group and is obtained by using :
wi+4β=wiββg(wi+3β)
The first word of the new 4-word group is obtained by XORβing the first word of the last group ( wiβ ) with the result of a function g() applied to the last word of the previous 4-word group
The g() function
The function g()consists of the following 3 steps :
Perform a one-byte left circular rotation on the argument 4-byte word.
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, XORβing with the round constant amounts to XORβing with just its leftmost byte.
The round constant for the ithround is noted Rcon[i].
Rcon[i]=(RC[i],0,0,0)
The only non-zero byte in the round constants, RC[i], obeys the following recursion:
RC[1]=0x01RC[i]=0x02β RC[iβ1]
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)]
Figure 2. AES 192 Expansion key
Perform a byte substitution for each byte of the word using the same "S-box" in the step of the encryption rounds
The multiplication applied here is the same as in when multiplying by 2.