Calculation of 0x27

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x27 =  00100111 = x^5 + x^2 + x + 1

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

Calculation of 0x27^-1 in the finite field GF(2⁸)

00000100 = 00000001 * 100011011 + 00001001 * 100111
00000011 = 00001001 * 100011011 + 01000000 * 100111
00000001 = 00011010 * 100011011 + 11001001 * 100111
00000000 = 00100111 * 100011011 + 100011011 * 100111

a^-1(x) = x^7 + x^6 + x^3 + 1 = 11001001 = 0xc9

The calculation of 0x27^-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      0     1     0
           1 1 1 0 0 0 1 1      0     0     1
           1 1 1 1 0 0 0 1      1     0     1
SBOX(27) = 1 1 1 1 1 0 0 0   *  0  +  0  =  0
           0 1 1 1 1 1 0 0      0     1     0
           0 0 1 1 1 1 1 0      1     1     1
           0 0 0 1 1 1 1 1      1     0     1

SBOX(27) = 11001100 = cc

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com