Calculation of 0x6e

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x6e =  01101110 = x^6 + x^5 + x^3 + x^2 + x

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

Calculation of 0x6e^-1 in the finite field GF(2⁸)

00010001 = 00000001 * 100011011 + 00000111 * 1101110
00001000 = 00000110 * 100011011 + 00010011 * 1101110
00000001 = 00001101 * 100011011 + 00100001 * 1101110
00000000 = 01101110 * 100011011 + 100011011 * 1101110

a^-1(x) = x^5 + 1 = 00100001 = 0x21

The calculation of 0x6e^-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      1     1     1
           1 1 0 0 0 1 1 1      0     1     1
           1 1 1 0 0 0 1 1      0     0     1
           1 1 1 1 0 0 0 1      0     0     1
SBOX(6e) = 1 1 1 1 1 0 0 0   *  0  +  0  =  1
           0 1 1 1 1 1 0 0      1     1     0
           0 0 1 1 1 1 1 0      0     1     0
           0 0 0 1 1 1 1 1      0     0     1

SBOX(6e) = 10011111 = 9f

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com