Calculation of 0xad

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0xad =  10101101 = x^7 + x^5 + x^3 + x^2 + 1

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

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

01000001 = 00000001 * 100011011 + 00000010 * 10101101
00101111 = 00000010 * 100011011 + 00000101 * 10101101
00011111 = 00000101 * 100011011 + 00001000 * 10101101
00001110 = 00001101 * 100011011 + 00011101 * 10101101
00000011 = 00011111 * 100011011 + 00110010 * 10101101
00000001 = 01101110 * 100011011 + 11100111 * 10101101
00000000 = 10101101 * 100011011 + 100011011 * 10101101

a^-1(x) = x^7 + x^6 + x^5 + x^2 + x + 1 = 11100111 = 0xe7

The calculation of 0xad^-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     1
           1 1 1 1 0 0 0 1      0     0     0
SBOX(ad) = 1 1 1 1 1 0 0 0   *  0  +  0  =  1
           0 1 1 1 1 1 0 0      1     1     0
           0 0 1 1 1 1 1 0      1     1     0
           0 0 0 1 1 1 1 1      1     0     1

SBOX(ad) = 10010101 = 95

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com