Calculation of 0xa5

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0xa5 =  10100101 = x^7 + x^5 + x^2 + 1

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

Calculation of 0xa5^-1 in the finite field GF(2⁸)

01010001 = 00000001 * 100011011 + 00000010 * 10100101
00000111 = 00000010 * 100011011 + 00000101 * 10100101
00000010 = 00111011 * 100011011 + 01101011 * 10100101
00000001 = 01001111 * 100011011 + 10111000 * 10100101
00000000 = 10100101 * 100011011 + 100011011 * 10100101

a^-1(x) = x^7 + x^5 + x^4 + x^3 = 10111000 = 0xb8

The calculation of 0xa5^-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      0     1     0
           1 1 0 0 0 1 1 1      0     1     1
           1 1 1 0 0 0 1 1      0     0     1
           1 1 1 1 0 0 0 1      1     0     0
SBOX(a5) = 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 0      0     1     0
           0 0 0 1 1 1 1 1      1     0     0

SBOX(a5) = 00000110 = 6

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com