Calculation of 0x10

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x10 =  00010000 = x^4

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

Calculation of 0x10^-1 in the finite field GF(2⁸)

00001011 = 00000001 * 100011011 + 00010001 * 10000
00000110 = 00000010 * 100011011 + 00100011 * 10000
00000001 = 00000111 * 100011011 + 01110100 * 10000
00000000 = 00010000 * 100011011 + 100011011 * 10000

a^-1(x) = x^6 + x^5 + x^4 + x^2 = 01110100 = 0x74

The calculation of 0x10^-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     1
           1 1 1 0 0 0 1 1      1     0     0
           1 1 1 1 0 0 0 1      0     0     1
SBOX(10) = 1 1 1 1 1 0 0 0   *  1  +  0  =  0
           0 1 1 1 1 1 0 0      1     1     0
           0 0 1 1 1 1 1 0      1     1     1
           0 0 0 1 1 1 1 1      0     0     1

SBOX(10) = 11001010 = ca

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com