Calculation of 0x65

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x65 =  01100101 = x^6 + x^5 + x^2 + 1

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

Calculation of 0x65^-1 in the finite field GF(2⁸)

00100000 = 00000001 * 100011011 + 00000111 * 1100101
00000101 = 00000011 * 100011011 + 00001000 * 1100101
00000010 = 00011111 * 100011011 + 01010111 * 1100101
00000001 = 00111101 * 100011011 + 10100110 * 1100101
00000000 = 01100101 * 100011011 + 100011011 * 1100101

a^-1(x) = x^7 + x^5 + x^2 + x = 10100110 = 0xa6

The calculation of 0x65^-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     0
           1 1 1 0 0 0 1 1      1     0     1
           1 1 1 1 0 0 0 1      0     0     1
SBOX(65) = 1 1 1 1 1 0 0 0   *  0  +  0  =  0
           0 1 1 1 1 1 0 0      1     1     0
           0 0 1 1 1 1 1 0      0     1     1
           0 0 0 1 1 1 1 1      1     0     0

SBOX(65) = 01001101 = 4d

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com