Calculation of 0xab

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0xab =  10101011 = x^7 + x^5 + x^3 + x + 1

m(x) = (x) * a(x) + (x^6 + x^3 + x^2 + 1)

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

01001101 = 00000001 * 100011011 + 00000010 * 10101011
00110001 = 00000010 * 100011011 + 00000101 * 10101011
00011110 = 00000111 * 100011011 + 00001101 * 10101011
00001101 = 00001100 * 100011011 + 00011111 * 10101011
00000100 = 00011111 * 100011011 + 00110011 * 10101011
00000001 = 00101101 * 100011011 + 01001010 * 10101011
00000000 = 10101011 * 100011011 + 100011011 * 10101011

a^-1(x) = x^6 + x^3 + x = 01001010 = 0x4a

The calculation of 0xab^-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     0
           1 1 0 0 0 1 1 1      1     1     1
           1 1 1 0 0 0 1 1      0     0     0
           1 1 1 1 0 0 0 1      1     0     0
SBOX(ab) = 1 1 1 1 1 0 0 0   *  0  +  0  =  0
           0 1 1 1 1 1 0 0      0     1     1
           0 0 1 1 1 1 1 0      1     1     1
           0 0 0 1 1 1 1 1      0     0     0

SBOX(ab) = 01100010 = 62

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com