Calculation of 0x28

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x28 =  00101000 = x^5 + x^3

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

Calculation of 0x28^-1 in the finite field GF(2⁸)

00001011 = 00000001 * 100011011 + 00001010 * 101000
00000100 = 00000100 * 100011011 + 00101001 * 101000
00000011 = 00001001 * 100011011 + 01011000 * 101000
00000001 = 00011111 * 100011011 + 11000001 * 101000
00000000 = 00101000 * 100011011 + 100011011 * 101000

a^-1(x) = x^7 + x^6 + 1 = 11000001 = 0xc1

The calculation of 0x28^-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     0
           1 1 1 0 0 0 1 1      0     0     1
           1 1 1 1 0 0 0 1      0     0     0
SBOX(28) = 1 1 1 1 1 0 0 0   *  0  +  0  =  1
           0 1 1 1 1 1 0 0      0     1     1
           0 0 1 1 1 1 1 0      1     1     0
           0 0 0 1 1 1 1 1      1     0     0

SBOX(28) = 00110100 = 34

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com