Calculation of 0x8a

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x8a =  10001010 = x^7 + x^3 + x

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

Calculation of 0x8a^-1 in the finite field GF(2⁸)

00001111 = 00000001 * 100011011 + 00000010 * 10001010
00000010 = 00011000 * 100011011 + 00110001 * 10001010
00000001 = 01001001 * 100011011 + 10010101 * 10001010
00000000 = 10001010 * 100011011 + 100011011 * 10001010

a^-1(x) = x^7 + x^4 + x^2 + 1 = 10010101 = 0x95

The calculation of 0x8a^-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      0     1     1
           1 1 1 0 0 0 1 1      1     0     1
           1 1 1 1 0 0 0 1      0     0     1
SBOX(8a) = 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     1
           0 0 0 1 1 1 1 1      1     0     0

SBOX(8a) = 01111110 = 7e

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com