Calculation of 0x6d

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x6d =  01101101 = x^6 + x^5 + x^3 + x^2 + 1

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

Calculation of 0x6d^-1 in the finite field GF(2⁸)

00011000 = 00000001 * 100011011 + 00000111 * 1101101
00001101 = 00000100 * 100011011 + 00011101 * 1101101
00000010 = 00001001 * 100011011 + 00111101 * 1101101
00000001 = 00110010 * 100011011 + 10010011 * 1101101
00000000 = 01101101 * 100011011 + 100011011 * 1101101

a^-1(x) = x^7 + x^4 + x + 1 = 10010011 = 0x93

The calculation of 0x6d^-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     0
           1 1 0 0 0 1 1 1      1     1     0
           1 1 1 0 0 0 1 1      0     0     1
           1 1 1 1 0 0 0 1      0     0     1
SBOX(6d) = 1 1 1 1 1 0 0 0   *  1  +  0  =  1
           0 1 1 1 1 1 0 0      0     1     1
           0 0 1 1 1 1 1 0      0     1     0
           0 0 0 1 1 1 1 1      1     0     0

SBOX(6d) = 00111100 = 3c

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com