Calculation of 0xf3

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0xf3 =  11110011 = x^7 + x^6 + x^5 + x^4 + x + 1

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

Calculation of 0xf3^-1 in the finite field GF(2⁸)

00001110 = 00000001 * 100011011 + 00000011 * 11110011
00000001 = 00010011 * 100011011 + 00110100 * 11110011
00000000 = 11110011 * 100011011 + 100011011 * 11110011

a^-1(x) = x^5 + x^4 + x^2 = 00110100 = 0x34

The calculation of 0xf3^-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     1
           1 1 0 0 0 1 1 1      0     1     0
           1 1 1 0 0 0 1 1      1     0     1
           1 1 1 1 0 0 0 1      0     0     1
SBOX(f3) = 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      0     1     0
           0 0 0 1 1 1 1 1      0     0     0

SBOX(f3) = 00001101 = d

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com