Calculation of 0x36

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x36 =  00110110 = x^5 + x^4 + x^2 + x

m(x) = (x^3 + x^2 + x) * a(x) + (x^4 + x^3 + x^2 + x + 1)

Calculation of 0x36^-1 in the finite field GF(2⁸)

00011111 = 00000001 * 100011011 + 00001110 * 110110
00001000 = 00000010 * 100011011 + 00011101 * 110110
00000111 = 00000111 * 100011011 + 00101001 * 110110
00000001 = 00001011 * 100011011 + 01100110 * 110110
00000000 = 00110110 * 100011011 + 100011011 * 110110

a^-1(x) = x^6 + x^5 + x^2 + x = 01100110 = 0x66

The calculation of 0x36^-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     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(36) = 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      1     1     0
           0 0 0 1 1 1 1 1      0     0     0

SBOX(36) = 00000101 = 5

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com