Calculation of 0x6b

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x6b =  01101011 = x^6 + x^5 + x^3 + x + 1

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

Calculation of 0x6b^-1 in the finite field GF(2⁸)

00001010 = 00000001 * 100011011 + 00000111 * 1101011
00000111 = 00001110 * 100011011 + 00101011 * 1101011
00000011 = 00010011 * 100011011 + 01111010 * 1101011
00000001 = 00101000 * 100011011 + 11011111 * 1101011
00000000 = 01101011 * 100011011 + 100011011 * 1101011

a^-1(x) = x^7 + x^6 + x^4 + x^3 + x^2 + x + 1 = 11011111 = 0xdf

The calculation of 0x6b^-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      1     1     1
           1 1 1 0 0 0 1 1      1     0     1
           1 1 1 1 0 0 0 1      1     0     1
SBOX(6b) = 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      1     1     1
           0 0 0 1 1 1 1 1      1     0     0

SBOX(6b) = 01111111 = 7f

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com