Sam Trenholme's webpage

The S-box is generated by determining the multiplicative inverse for a given number in Rijndael's Galois field. The multiplicative inverse is then transformed using the following affine transformation matrix:

```1 0 0 0 1 1 1 1
1 1 0 0 0 1 1 1
1 1 1 0 0 0 1 1
1 1 1 1 0 0 0 1
1 1 1 1 1 0 0 0
0 1 1 1 1 1 0 0
0 0 1 1 1 1 1 0
0 0 0 1 1 1 1 1
```
This affine transformation can also be calculated by the following algorithm:
1. Store the multiplicative inverse of the input number in two 8-bit unsigned temporary variables: s and x
2. Rotate the value s one bit to the left; if the value of s had a high bit (eight bit from the left) of one, make the low bit of s one; otherwise the low bit of s is zero.
3. Exclusive or the value of x with the value of s, storing the value in x
4. For three more iterations, repeat steps two and three; steps two and three are done a total of four times.
5. The value of ''x'' will now have the transformed value.
After the affine transformation is done, exclusive or the value by the decimal number 99 (the hexadecimal number 0x63).

Given the gmul_inverse function which is demonstrated on this page, here is come C code which performs the above algorithm:

```/* Calculate the s-box for a given number */
unsigned char sbox(unsigned char in) {
unsigned char c, s, x;
s = x = gmul_inverse(in);
for(c = 0; c < 4; c++) {
/* One bit circular rotate to the left */
s = (s << 1) | (s >> 7);
/* xor with x */
x ^= s;
}
x ^= 99; /* 0x63 */
return x;
}
```

This will generate the Rijndael (AES) S-box, which is represented here with hexadecimal notation:

```   | 0  1  2  3  4  5  6  7  8  9  a  b  c  d  e  f
---|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|
00 |63 7c 77 7b f2 6b 6f c5 30 01 67 2b fe d7 ab 76
10 |ca 82 c9 7d fa 59 47 f0 ad d4 a2 af 9c a4 72 c0
20 |b7 fd 93 26 36 3f f7 cc 34 a5 e5 f1 71 d8 31 15
30 |04 c7 23 c3 18 96 05 9a 07 12 80 e2 eb 27 b2 75
40 |09 83 2c 1a 1b 6e 5a a0 52 3b d6 b3 29 e3 2f 84
50 |53 d1 00 ed 20 fc b1 5b 6a cb be 39 4a 4c 58 cf
60 |d0 ef aa fb 43 4d 33 85 45 f9 02 7f 50 3c 9f a8
70 |51 a3 40 8f 92 9d 38 f5 bc b6 da 21 10 ff f3 d2
80 |cd 0c 13 ec 5f 97 44 17 c4 a7 7e 3d 64 5d 19 73
90 |60 81 4f dc 22 2a 90 88 46 ee b8 14 de 5e 0b db
a0 |e0 32 3a 0a 49 06 24 5c c2 d3 ac 62 91 95 e4 79
b0 |e7 c8 37 6d 8d d5 4e a9 6c 56 f4 ea 65 7a ae 08
c0 |ba 78 25 2e 1c a6 b4 c6 e8 dd 74 1f 4b bd 8b 8a
d0 |70 3e b5 66 48 03 f6 0e 61 35 57 b9 86 c1 1d 9e
e0 |e1 f8 98 11 69 d9 8e 94 9b 1e 87 e9 ce 55 28 df
f0 |8c a1 89 0d bf e6 42 68 41 99 2d 0f b0 54 bb 16
```
Here the column is determined by the least significant nybble, and the row is determined by the most significant nybble. For example, the value 0x9a is converted in to 0xb8 by Rijndael's S-box.

The inverse S-box is simply the S-box run in reverse. For example, the inverse S-box of 0xdb is 0x9f. The following table represents Rijndael's inverse S-box:

```   | 0  1  2  3  4  5  6  7  8  9  a  b  c  d  e  f
---|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|
00 |52 09 6a d5 30 36 a5 38 bf 40 a3 9e 81 f3 d7 fb
10 |7c e3 39 82 9b 2f ff 87 34 8e 43 44 c4 de e9 cb
20 |54 7b 94 32 a6 c2 23 3d ee 4c 95 0b 42 fa c3 4e
30 |08 2e a1 66 28 d9 24 b2 76 5b a2 49 6d 8b d1 25
40 |72 f8 f6 64 86 68 98 16 d4 a4 5c cc 5d 65 b6 92
50 |6c 70 48 50 fd ed b9 da 5e 15 46 57 a7 8d 9d 84
60 |90 d8 ab 00 8c bc d3 0a f7 e4 58 05 b8 b3 45 06
70 |d0 2c 1e 8f ca 3f 0f 02 c1 af bd 03 01 13 8a 6b
80 |3a 91 11 41 4f 67 dc ea 97 f2 cf ce f0 b4 e6 73
90 |96 ac 74 22 e7 ad 35 85 e2 f9 37 e8 1c 75 df 6e
a0 |47 f1 1a 71 1d 29 c5 89 6f b7 62 0e aa 18 be 1b
b0 |fc 56 3e 4b c6 d2 79 20 9a db c0 fe 78 cd 5a f4
c0 |1f dd a8 33 88 07 c7 31 b1 12 10 59 27 80 ec 5f
d0 |60 51 7f a9 19 b5 4a 0d 2d e5 7a 9f 93 c9 9c ef
e0 |a0 e0 3b 4d ae 2a f5 b0 c8 eb bb 3c 83 53 99 61
f0 |17 2b 04 7e ba 77 d6 26 e1 69 14 63 55 21 0c 7d
```