Calculation of 0xcd

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0xcd =  11001101 = x^7 + x^6 + x^3 + x^2 + 1

m(x) = (x + 1) * a(x) + (x^6 + x^3 + x^2)

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

01001100 = 00000001 * 100011011 + 00000011 * 11001101
00011001 = 00000011 * 100011011 + 00000100 * 11001101
00000011 = 00001000 * 100011011 + 00011111 * 11001101
00000001 = 01000011 * 100011011 + 11111100 * 11001101
00000000 = 11001101 * 100011011 + 100011011 * 11001101

a^-1(x) = x^7 + x^6 + x^5 + x^4 + x^3 + x^2 = 11111100 = 0xfc

The calculation of 0xcd^-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     0
           1 1 1 0 0 0 1 1      1     0     1
           1 1 1 1 0 0 0 1      1     0     1
SBOX(cd) = 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      1     1     0
           0 0 0 1 1 1 1 1      1     0     1

SBOX(cd) = 10111101 = bd

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com