Calculation of 0x19

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x19 =  00011001 = x^4 + x^3 + 1

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

Calculation of 0x19^-1 in the finite field GF(2⁸)

00001100 = 00000001 * 100011011 + 00011111 * 11001
00000001 = 00000010 * 100011011 + 00111111 * 11001
00000000 = 00011001 * 100011011 + 100011011 * 11001

a^-1(x) = x^5 + x^4 + x^3 + x^2 + x + 1 = 00111111 = 0x3f

The calculation of 0x19^-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     0
           1 1 1 0 0 0 1 1      1     0     1
           1 1 1 1 0 0 0 1      1     0     0
SBOX(19) = 1 1 1 1 1 0 0 0   *  1  +  0  =  1
           0 1 1 1 1 1 0 0      1     1     0
           0 0 1 1 1 1 1 0      0     1     1
           0 0 0 1 1 1 1 1      0     0     1

SBOX(19) = 11010100 = d4

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com