Calculation of 0x3b

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x3b =  00111011 = x^5 + x^4 + x^3 + x + 1

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

Calculation of 0x3b^-1 in the finite field GF(2⁸)

00010100 = 00000001 * 100011011 + 00001101 * 111011
00000111 = 00000011 * 100011011 + 00010110 * 111011
00000001 = 00001000 * 100011011 + 01101111 * 111011
00000000 = 00111011 * 100011011 + 100011011 * 111011

a^-1(x) = x^6 + x^5 + x^3 + x^2 + x + 1 = 01101111 = 0x6f

The calculation of 0x3b^-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     0
           1 1 0 0 0 1 1 1      1     1     1
           1 1 1 0 0 0 1 1      1     0     0
           1 1 1 1 0 0 0 1      1     0     0
SBOX(3b) = 1 1 1 1 1 0 0 0   *  0  +  0  =  0
           0 1 1 1 1 1 0 0      1     1     1
           0 0 1 1 1 1 1 0      1     1     1
           0 0 0 1 1 1 1 1      0     0     1

SBOX(3b) = 11100010 = e2

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com