Calculation of 0xa4

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0xa4 =  10100100 = x^7 + x^5 + x^2

m(x) = (x) * a(x) + (x^6 + x^4 + x + 1)

Calculation of 0xa4^-1 in the finite field GF(2⁸)

01010011 = 00000001 * 100011011 + 00000010 * 10100100
00000010 = 00000010 * 100011011 + 00000101 * 10100100
00000001 = 01010011 * 100011011 + 10001111 * 10100100
00000000 = 10100100 * 100011011 + 100011011 * 10100100

a^-1(x) = x^7 + x^3 + x^2 + x + 1 = 10001111 = 0x8f

The calculation of 0xa4^-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     0
           1 1 1 0 0 0 1 1      1     0     0
           1 1 1 1 0 0 0 1      1     0     1
SBOX(a4) = 1 1 1 1 1 0 0 0   *  0  +  0  =  0
           0 1 1 1 1 1 0 0      0     1     0
           0 0 1 1 1 1 1 0      0     1     1
           0 0 0 1 1 1 1 1      1     0     0

SBOX(a4) = 01001001 = 49

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com