Calculation of 0x12

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x12 =  00010010 = x^4 + x

m(x) = (x^4 + x + 1) * a(x) + (x^3 + x^2 + 1)

Calculation of 0x12^-1 in the finite field GF(2⁸)

00001101 = 00000001 * 100011011 + 00010011 * 10010
00000101 = 00000011 * 100011011 + 00110100 * 10010
00000010 = 00000100 * 100011011 + 01001111 * 10010
00000001 = 00001011 * 100011011 + 10101010 * 10010
00000000 = 00010010 * 100011011 + 100011011 * 10010

a^-1(x) = x^7 + x^5 + x^3 + x = 10101010 = 0xaa

The calculation of 0x12^-1 is made with the Extended Euclidean algorithm. Instead of normal division and multiplication you need to use Polynomialdivision and Polynomialmultiplication.

Affine transformation over GF(2)

           1 0 0 0 1 1 1 1      0     1     1
           1 1 0 0 0 1 1 1      1     1     0
           1 1 1 0 0 0 1 1      0     0     0
           1 1 1 1 0 0 0 1      1     0     1
SBOX(12) = 1 1 1 1 1 0 0 0   *  0  +  0  =  0
           0 1 1 1 1 1 0 0      1     1     0
           0 0 1 1 1 1 1 0      0     1     1
           0 0 0 1 1 1 1 1      1     0     1

SBOX(12) = 11001001 = c9

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com