Calculation of 0x68

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x68 =  01101000 = x^6 + x^5 + x^3

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

Calculation of 0x68^-1 in the finite field GF(2⁸)

00000011 = 00000001 * 100011011 + 00000111 * 1101000
00000001 = 00100111 * 100011011 + 11110100 * 1101000
00000000 = 01101000 * 100011011 + 100011011 * 1101000

a^-1(x) = x^7 + x^6 + x^5 + x^4 + x^2 = 11110100 = 0xf4

The calculation of 0x68^-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      0     1     0
           1 1 1 0 0 0 1 1      1     0     1
           1 1 1 1 0 0 0 1      0     0     0
SBOX(68) = 1 1 1 1 1 0 0 0   *  1  +  0  =  0
           0 1 1 1 1 1 0 0      1     1     0
           0 0 1 1 1 1 1 0      1     1     1
           0 0 0 1 1 1 1 1      1     0     0

SBOX(68) = 01000101 = 45

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com