Calculation of 0x3d

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x3d =  00111101 = x^5 + x^4 + x^3 + x^2 + 1

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

Calculation of 0x3d^-1 in the finite field GF(2⁸)

00000111 = 00000001 * 100011011 + 00001100 * 111101
00000010 = 00001001 * 100011011 + 01101101 * 111101
00000001 = 00011010 * 100011011 + 10111011 * 111101
00000000 = 00111101 * 100011011 + 100011011 * 111101

a^-1(x) = x^7 + x^5 + x^4 + x^3 + x + 1 = 10111011 = 0xbb

The calculation of 0x3d^-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      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(3d) = 1 1 1 1 1 0 0 0   *  1  +  0  =  0
           0 1 1 1 1 1 0 0      1     1     1
           0 0 1 1 1 1 1 0      0     1     0
           0 0 0 1 1 1 1 1      1     0     0

SBOX(3d) = 00100111 = 27

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com