AES key schedule

AES uses a key schedule to expand a short key into a number of separate round keys. The three AES variants have a different number of rounds. Each variant requires a separate 128-bit round key for each round plus one more.^{[note 1]} The key schedule produces the needed round keys from the initial key.

Round constants

Values of $rc i$ in hexadecimal
$i$	1	2	3	4	5	6	7	8	9	10
$rc i$	01	02	04	08	10	20	40	80	1B	36

The round constant $rcon i$ for round $i$ of the key expansion is the 32-bit word:^{[note 2]}

rcon_{i}={\begin{bmatrix}rc_{i}&00_{16}&00_{16}&00_{16}\end{bmatrix}}

where $rc i$ is an eight-bit value defined as :

rc_{i}={\begin{cases}1&{\text{if }}i=1\\2\cdot rc_{i-1}&{\text{if }}i>1{\text{ and }}rc_{i-1}<80_{16}\\(2\cdot rc_{i-1})\oplus {\text{11B}}_{16}&{\text{if }}i>1{\text{ and }}rc_{i-1}\geq 80_{16}\end{cases}}

where $\oplus$ is the bitwise XOR operator and constants such as $0016$ and $11B 16$ are given in hexadecimal. Equivalently:

rc_{i}=x^{i-1}

where the bits of $rc i$ are treated as the coefficients of an element of the finite field ${\rm {{GF}(2)[x]/(x^{8}+x^{4}+x^{3}+x+1)}}$ , so that e.g. $rc_{10}=36_{16}=00110110_{2}$ represents the polynomial $x^{5}+x^{4}+x^{2}+x$ .

AES uses up to $rcon 10$ for AES-128 (as 11 round keys are needed), up to $rcon 8$ for AES-192, and up to $rcon 7$ for AES-256.^{[note 3]}

The key schedule

AES key schedule for a 128-bit key.

Define:

$N$ as the length of the key in 32-bit words: 4 words for AES-128, 6 words for AES-192, and 8 words for AES-256
$K 0$ , $K 1$ , ... $K N -1$ as the 32-bit words of the original key
$R$ as the number of round keys needed: 11 round keys for AES-128, 13 keys for AES-192, and 15 keys for AES-256^{[note 4]}
$W 0$ , $W 1$ , ... $W 4 R -1$ as the 32-bit words of the expanded key^{[note 5]}

Also define $RotWord$ as a one-byte left circular shift:^{[note 6]}

\operatorname {RotWord} ({\begin{bmatrix}b_{0}&b_{1}&b_{2}&b_{3}\end{bmatrix}})={\begin{bmatrix}b_{1}&b_{2}&b_{3}&b_{0}\end{bmatrix}}

and $SubWord$ as an application of the AES S-box to each of the four bytes of the word:

\operatorname {SubWord} ({\begin{bmatrix}b_{0}&b_{1}&b_{2}&b_{3}\end{bmatrix}})={\begin{bmatrix}\operatorname {S} (b_{0})&\operatorname {S} (b_{1})&\operatorname {S} (b_{2})&\operatorname {S} (b_{3})\end{bmatrix}}

Then for $i=0\ldots 4R-1$ :

W_{i}={\begin{cases}K_{i}&{\text{if }}i<N\\W_{i-N}\oplus \operatorname {SubWord} (\operatorname {RotWord} (W_{i-1}))\oplus rcon_{i/N}&{\text{if }}i\geq N{\text{ and }}i\equiv 0{\pmod {N}}\\W_{i-N}\oplus \operatorname {SubWord} (W_{i-1})&{\text{if }}i\geq N{\text{, }}N>6{\text{, and }}i\equiv 4{\pmod {N}}\\W_{i-N}\oplus W_{i-1}&{\text{otherwise.}}\\\end{cases}}

Notes

↑ Non-AES Rijndael variants require up to 256 bits of expanded key per round
↑ In FIPS-197 the $rc_{i}$ value is the least significant byte at index 0
↑ The Rijndael variants with larger block sizes use more of these constants, up to $rcon 29$ for Rijndael with 128-bit keys and 256 bit blocks (needs 15 round keys of each 256 bit, which means 30 full rounds of key expansion, which means 29 calls to the key schedule core using the round constants). The remaining constants for $i \geq 11$ are: 6C, D8, AB, 4D, 9A, 2F, 5E, BC, 63, C6, 97, 35, 6A, D4, B3, 7D, FA, EF and C5
↑ Other Rijndael variants require $max(N, B) + 7$ round keys, where $B$ is the block size in words
↑ Other Rijndael variants require $BR$ words of expanded key, where $B$ is the block size in words
↑ Rotation is opposite of byte order direction. FIPS-197 byte addresses in arrays are increasing from left to right^{[ref 1]} in little endian but rotation is from right to left. In AES-NI^{[ref 2]} and in the Linux kernel's lib/crypto/aes.c^{[ref 3]}, the byte ordering is increasing from right to left in little endian but rotation is from left to right.

References

FIPS PUB 197: the official AES standard (PDF file)

↑ "Federal Information Processing Standards Publication 197 November 26, 2001 Announcing the ADVANCED ENCRYPTION STANDARD (AES)" (PDF). p. 8. Retrieved 2020-06-16.
↑ "Intel® Advanced Encryption Standard (AES) New Instructions Set" (PDF). p. 13.
↑ "aes.c". GitHub. Retrieved 2020-06-15.

External links

Description of Rijndael's key schedule
schematic view of the key schedule for 128 and 256 bit keys for 160-bit keys on Cryptography Stack Exchange

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[1] Non-AES Rijndael variants require up to 256 bits of expanded key per round

[2] In FIPS-197 the $rc_{i}$ value is the least significant byte at index 0

[3] The Rijndael variants with larger block sizes use more of these constants, up to $rcon 29$ for Rijndael with 128-bit keys and 256 bit blocks (needs 15 round keys of each 256 bit, which means 30 full rounds of key expansion, which means 29 calls to the key schedule core using the round constants). The remaining constants for $i \geq 11$ are: 6C, D8, AB, 4D, 9A, 2F, 5E, BC, 63, C6, 97, 35, 6A, D4, B3, 7D, FA, EF and C5

[4] Other Rijndael variants require $max(N, B) + 7$ round keys, where $B$ is the block size in words

[5] Other Rijndael variants require $BR$ words of expanded key, where $B$ is the block size in words

[9] Rotation is opposite of byte order direction. FIPS-197 byte addresses in arrays are increasing from left to right^{[ref 1]} in little endian but rotation is from right to left. In AES-NI^{[ref 2]} and in the Linux kernel's lib/crypto/aes.c^{[ref 3]}, the byte ordering is increasing from right to left in little endian but rotation is from left to right.

[6] "Federal Information Processing Standards Publication 197 November 26, 2001 Announcing the ADVANCED ENCRYPTION STANDARD (AES)" (PDF). p. 8. Retrieved 2020-06-16.

[7] "Intel® Advanced Encryption Standard (AES) New Instructions Set" (PDF). p. 13.

[8] "aes.c". GitHub. Retrieved 2020-06-15.

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