Calculation of 0x3f

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x3f =  00111111 = x^5 + x^4 + x^3 + x^2 + x + 1

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

Calculation of 0x3f^-1 in the finite field GF(2⁸)

00011111 = 00000001 * 100011011 + 00001100 * 111111
00000001 = 00000010 * 100011011 + 00011001 * 111111
00000000 = 00111111 * 100011011 + 100011011 * 111111

a^-1(x) = x^4 + x^3 + 1 = 00011001 = 0x19

The calculation of 0x3f^-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     0
           1 1 1 0 0 0 1 1      0     0     1
           1 1 1 1 0 0 0 1      1     0     0
SBOX(3f) = 1 1 1 1 1 0 0 0   *  1  +  0  =  1
           0 1 1 1 1 1 0 0      0     1     1
           0 0 1 1 1 1 1 0      0     1     1
           0 0 0 1 1 1 1 1      0     0     0

SBOX(3f) = 01110101 = 75

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com