Calculation of 0x9d

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x9d =  10011101 = x^7 + x^4 + x^3 + x^2 + 1

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

Calculation of 0x9d^-1 in the finite field GF(2⁸)

00100001 = 00000001 * 100011011 + 00000010 * 10011101
00011001 = 00000100 * 100011011 + 00001001 * 10011101
00001010 = 00001101 * 100011011 + 00011001 * 10011101
00000111 = 00010011 * 100011011 + 00100010 * 10011101
00000011 = 00111000 * 100011011 + 01111111 * 10011101
00000001 = 01100011 * 100011011 + 11011100 * 10011101
00000000 = 10011101 * 100011011 + 100011011 * 10011101

a^-1(x) = x^7 + x^6 + x^4 + x^3 + x^2 = 11011100 = 0xdc

The calculation of 0x9d^-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      1     0     1
           1 1 1 1 0 0 0 1      1     0     1
SBOX(9d) = 1 1 1 1 1 0 0 0   *  1  +  0  =  1
           0 1 1 1 1 1 0 0      0     1     0
           0 0 1 1 1 1 1 0      1     1     1
           0 0 0 1 1 1 1 1      1     0     0

SBOX(9d) = 01011110 = 5e

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com