AES (Rijndael) uses a key schedule to expand a short key into a number of separate round keys. This is known as the Rijndael key schedule.
Common operations Edit
Rijndael's key schedule utilizes a number of operations, which will be described before describing the key schedule.
Rotate Edit
The rotate operation takes a 32-bit word like this (in hexadecimal):
- 1d2c3a4f
And rotates it eight bits to the left such that the high eight bits "wrap around" and become the low eight bits of the result.
- 2c3a4f1d
Rcon Edit
Rcon is what the Rijndael documentation calls the exponentiation of 2 to a user-specified value. Note that this operation is not performed with regular integers, but in Rijndael's finite field. In polynomial form, 2 is $ 2 = 00000010 = 0 x^7 + 0 x^6 + 0 x^5 + 0 x^4 + 0 x^3 + 0 x^2 + 1 x + 0 = x $, and we compute
- $ \textrm{rcon}(i) = x^{(254+i)} $
in $ \mathbb{F}_{2^8} $ or equivalently,
- $ \textrm{rcon}(i) = x^{(254+i)} \mod x^8 + x^4 + x^3 + x + 1 $
in $ \mathbb{F}_{2} $.
For example, the rcon(1) = 1, the rcon(2) = 2, the rcon(3) = 4, and the rcon(9) is the hexadecimal number 0x1b (27 in decimal).
Rcon[255] = { 0x8d, 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, 0x72, 0xe4, 0xd3, 0xbd, 0x61, 0xc2, 0x9f, 0x25, 0x4a, 0x94, 0x33, 0x66, 0xcc, 0x83, 0x1d, 0x3a, 0x74, 0xe8, 0xcb, 0x8d, 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, 0x72, 0xe4, 0xd3, 0xbd, 0x61, 0xc2, 0x9f, 0x25, 0x4a, 0x94, 0x33, 0x66, 0xcc, 0x83, 0x1d, 0x3a, 0x74, 0xe8, 0xcb, 0x8d, 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, 0x72, 0xe4, 0xd3, 0xbd, 0x61, 0xc2, 0x9f, 0x25, 0x4a, 0x94, 0x33, 0x66, 0xcc, 0x83, 0x1d, 0x3a, 0x74, 0xe8, 0xcb, 0x8d, 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, 0x72, 0xe4, 0xd3, 0xbd, 0x61, 0xc2, 0x9f, 0x25, 0x4a, 0x94, 0x33, 0x66, 0xcc, 0x83, 0x1d, 0x3a, 0x74, 0xe8, 0xcb, 0x8d, 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, 0x72, 0xe4, 0xd3, 0xbd, 0x61, 0xc2, 0x9f, 0x25, 0x4a, 0x94, 0x33, 0x66, 0xcc, 0x83, 0x1d, 0x3a, 0x74, 0xe8, 0xcb}
S-box Edit
The key schedule uses Rijndael's S-box.
Key schedule core Edit
This operation is used as an inner loop in the key schedule, and is done thus:
- The input is a 32-bit word and an iteration number i. The output is a 32-bit word.
- Copy the input over to the output.
- Use the above described rotate operation to rotate the output eight bits to the left
- Apply Rijndael's S-box on all four individual bytes in the output word
- On just the first (leftmost) byte of the output word, exclusive or the byte with 2 to the power of (i-1). In other words, perform the rcon operation with i as the input, and exclusive or the rcon output with the first byte of the output word
The key schedule Edit
Constants Edit
Since the key schedule for 128-bit, 192-bit, and 256-bit encryption are very similar, with only some constants changed, the following keysize constants are defined here:
- n has a value of 16 for 128-bit keys, 24 for 192-bit keys, and 32 for 256-bit keys
- b has a value of 176 for 128-bit keys, 208 for 192-bit keys, and 240 for 256-bit keys
Key schedule description Edit
Rijndael's key schedule is done as follows:
- The first n bytes of the expanded key are simply the encryption key.
- The rcon iteration value i is set to 1
- Until we have b bytes of expanded key, we do the following to generate n more bytes of expanded key:
- We do the following to create 4 bytes of expanded key:
- We create a 4-byte temporary variable, t
- We assign the value of the previous four bytes in the expanded key to t
- We perform the key schedule core (see above) on t, with i as the rcon iteration value
- We increment i by 1
- We exclusive-or t with the four-byte block n bytes before the new expanded key. This becomes the next 4 bytes in the expanded key
- We then do the following three times to create the next twelve bytes of expanded key:
- We assign the value of the previous 4 bytes in the expanded key to t
- We exclusive-or t with the four-byte block n bytes before the new expanded key. This becomes the next 4 bytes in the expanded key
- If we are generating a 256-bit key, we do the following to generate the next 4 bytes of expanded key:
- We assign the value of the previous 4 bytes in the expanded key to t
- We run each of the 4 bytes in t through Rijndael's S-box
- We exclusive-or t with the 4-byte block n bytes before the new expanded key. This becomes the next 4 bytes in the expanded key.
- If we are generating a 128-bit key, we do not perform the following steps. If we are generating a 192-bit key, we run the following steps twice. If we are generating a 256-bit key, we run the following steps three times:
- We assign the value of the previous 4 bytes in the expanded key to t
- We exclusive-or t with the four-byte block n bytes before the new expanded key. This becomes the next 4 bytes in the expanded key
- We do the following to create 4 bytes of expanded key:
References Edit
- FIPS PUB 197: the official AES standard (PDF file)
- Description of Rijndael's key schedule
- Flash Application with Rijndael's key schedule (functional; text in Portuguese)
- Flash Video Describing the AES key schedule