Calculation of 0xf4

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0xf4 =  11110100 = x^7 + x^6 + x^5 + x^4 + x^2

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

Calculation of 0xf4^-1 in the finite field GF(2⁸)

00000111 = 00000001 * 100011011 + 00000011 * 11110100
00000001 = 00100111 * 100011011 + 01101000 * 11110100
00000000 = 11110100 * 100011011 + 100011011 * 11110100

a^-1(x) = x^6 + x^5 + x^3 = 01101000 = 0x68

The calculation of 0xf4^-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     1
           1 1 1 0 0 0 1 1      0     0     1
           1 1 1 1 0 0 0 1      1     0     1
SBOX(f4) = 1 1 1 1 1 0 0 0   *  0  +  0  =  1
           0 1 1 1 1 1 0 0      1     1     1
           0 0 1 1 1 1 1 0      1     1     0
           0 0 0 1 1 1 1 1      0     0     1

SBOX(f4) = 10111111 = bf

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com