Calculation of 0x98

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x98 =  10011000 = x^7 + x^4 + x^3

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

Calculation of 0x98^-1 in the finite field GF(2⁸)

00101011 = 00000001 * 100011011 + 00000010 * 10011000
00011111 = 00000101 * 100011011 + 00001011 * 10011000
00001010 = 00001110 * 100011011 + 00011111 * 10011000
00000001 = 00010111 * 100011011 + 00101010 * 10011000
00000000 = 10011000 * 100011011 + 100011011 * 10011000

a^-1(x) = x^5 + x^3 + x = 00101010 = 0x2a

The calculation of 0x98^-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      1     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(98) = 1 1 1 1 1 0 0 0   *  0  +  0  =  0
           0 1 1 1 1 1 0 0      1     1     0
           0 0 1 1 1 1 1 0      0     1     1
           0 0 0 1 1 1 1 1      0     0     0

SBOX(98) = 01000110 = 46

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com