Calculation of 0x3a

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x3a =  00111010 = x^5 + x^4 + x^3 + x

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

Calculation of 0x3a^-1 in the finite field GF(2⁸)

00011001 = 00000001 * 100011011 + 00001101 * 111010
00001000 = 00000010 * 100011011 + 00011011 * 111010
00000001 = 00000111 * 100011011 + 00100000 * 111010
00000000 = 00111010 * 100011011 + 100011011 * 111010

a^-1(x) = x^5 = 00100000 = 0x20

The calculation of 0x3a^-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     0
           1 1 1 0 0 0 1 1      0     0     0
           1 1 1 1 0 0 0 1      0     0     0
SBOX(3a) = 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      0     1     0
           0 0 0 1 1 1 1 1      0     0     1

SBOX(3a) = 10000000 = 80

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com