Calculation of 0x0d

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0xd =  00001101 = x^3 + x^2 + 1

m(x) = (x^5 + x^4 + x^3) * a(x) + (x + 1)

Calculation of 0x0d^-1 in the finite field GF(2⁸)

00000011 = 00000001 * 100011011 + 00111000 * 1101
00000001 = 00000100 * 100011011 + 11100001 * 1101
00000000 = 00001101 * 100011011 + 100011011 * 1101

a^-1(x) = x^7 + x^6 + x^5 + 1 = 11100001 = 0xe1

The calculation of 0x0d^-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      0     1     1
           1 1 1 0 0 0 1 1      0     0     1
           1 1 1 1 0 0 0 1      0     0     0
SBOX(0d) = 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     1
           0 0 0 1 1 1 1 1      1     0     1

SBOX(0d) = 11010111 = d7

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com