Calculation of 0xfa

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0xfa =  11111010 = x^7 + x^6 + x^5 + x^4 + x^3 + x

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

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

00010101 = 00000001 * 100011011 + 00000011 * 11111010
00000110 = 00001100 * 100011011 + 00010101 * 11111010
00000001 = 00101001 * 100011011 + 01111101 * 11111010
00000000 = 11111010 * 100011011 + 100011011 * 11111010

a^-1(x) = x^6 + x^5 + x^4 + x^3 + x^2 + 1 = 01111101 = 0x7d

The calculation of 0xfa^-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     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      1     0     1
SBOX(fa) = 1 1 1 1 1 0 0 0   *  1  +  0  =  0
           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     0

SBOX(fa) = 00101101 = 2d

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com