Rijndael: A Successor to the Data Encryption Standard - Cryptography in C and C++ - page 241

Table 11-10. The values of the inverted S-box

52 09 6A D5 30 36 A5 38 BF 40 A3 9E 81 F3 D7 FB

7C E3 39 82 9B 2F FF 87 34 8E 43 44 C4 DE E9 CB

54 7B 94 32 A6 C2 23 3D EE 4C 95 0B 42 FA C3 4E

08 2E A1 66 28 D9 24 B2 76 5B A2 49 6D 8B D1 25

72 F8 F6 64 86 68 98 16 D4 A4 5C CC 5D 65 B6 92

6C 70 48 50 FD ED B9 DA 5E 15 46 57 A7 8D 9D 84

90 D8 AB 00 8C BC D3 0A F7 E4 58 05 B8 B3 45 06

D0 2C 1E 8F CA 3F 0F 02 C1 AF BD 03 01 13 8A 6B

3A 91 11 41 4F 67 DC EA 97 F2 CF CE F0 B4 E6 73

96 AC 74 22 E7 AD 35 85 E2 F9 37 E8 1C 75 DF 6E

47 F1 1A 71 1D 29 C5 89 6F B7 62 0E AA 18 BE 1B

FC 56 3E 4B C6 D2 79 20 9A DB C0 FE 78 CD 5A F4

1F DD A8 33 88 07 C7 31 B1 12 10 59 27 80 EC 5F

60 51 7F A9 19 B5 4A 0D 2D E5 7A 9F 93 C9 9C EF

A0 E0 3B 4D AE 2A F5 B0 C8 EB BB 3C 83 53 99 61

17 2B 04 7E BA 77 D6 26 E1 69 14 63 55 21 0C 7D

Table 11-11. ShiftRows for blocks of length 128 bits ( L b =4 )

After ShiftRows

Before ShiftRows
04 8

048 2
5

2

9 13 1

15 9 3

10 14 2 6

2 6 10 14

15 3 7 11

3 7 11 15

We have already seen (see page 244) how this step can be reduced to a matrix
multiplication

⎡
⎣

⎤
⎦ ←

⎡
⎣

⎤
⎦ ·

⎡
⎣

⎤
⎦

b 0 ,j
b 1 ,j
b 2 ,j
b 3 ,j

02 03 01 01 b 0 ,j

b 1 ,j
b 2 ,j
b 3 ,j

01 02 03 01

,

(11.3)

01 01 02 03

03 01 01 02

F 2 8 . For multiplication by '02'
(respectively x ) the function xtime() has already been defined; multiplication by
'03' (respectively x +1 ) has already been handled similarly (cf. page 247).
For inverting the MixColumns transformation every column ( b i,j ) of a block is
multiplied by the polynomial r ( x ):= r 3 x 3 + r 2 x 2 + r 1 x + r 0 with coefficients

with multiplication and addition carried out over