Calculation of 0x66

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x66 =  01100110 = x^6 + x^5 + x^2 + x

m(x) = (x^2 + x + 1) * a(x) + (x^5 + x^3 + 1)

Calculation of 0x66^-1 in the finite field GF(2⁸)

00101001 = 00000001 * 100011011 + 00000111 * 1100110
00011101 = 00000011 * 100011011 + 00001000 * 1100110
00001110 = 00000100 * 100011011 + 00011111 * 1100110
00000001 = 00001011 * 100011011 + 00110110 * 1100110
00000000 = 01100110 * 100011011 + 100011011 * 1100110

a^-1(x) = x^5 + x^4 + x^2 + x = 00110110 = 0x36

The calculation of 0x66^-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      1     1     1
           1 1 1 0 0 0 1 1      1     0     0
           1 1 1 1 0 0 0 1      0     0     0
SBOX(66) = 1 1 1 1 1 0 0 0   *  1  +  0  =  1
           0 1 1 1 1 1 0 0      1     1     1
           0 0 1 1 1 1 1 0      0     1     0
           0 0 0 1 1 1 1 1      0     0     0

SBOX(66) = 00110011 = 33

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com