Calculation of 0x9a

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x9a =  10011010 = x^7 + x^4 + x^3 + x

m(x) = (x) * a(x) + (x^5 + x^3 + x^2 + x + 1)

Calculation of 0x9a^-1 in the finite field GF(2⁸)

00101111 = 00000001 * 100011011 + 00000010 * 10011010
00001001 = 00000101 * 100011011 + 00001011 * 10011010
00000010 = 00010000 * 100011011 + 00100101 * 10011010
00000001 = 01000101 * 100011011 + 10011111 * 10011010
00000000 = 10011010 * 100011011 + 100011011 * 10011010

a^-1(x) = x^7 + x^4 + x^3 + x^2 + x + 1 = 10011111 = 0x9f

The calculation of 0x9a^-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      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(9a) = 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     0
           0 0 0 1 1 1 1 1      1     0     1

SBOX(9a) = 10111000 = b8

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com