Calculation of 0x41

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x41 =  01000001 = x^6 + 1

m(x) = (x^2) * a(x) + (x^4 + x^3 + x^2 + x + 1)

Calculation of 0x41^-1 in the finite field GF(2⁸)

00011111 = 00000001 * 100011011 + 00000100 * 1000001
00000011 = 00000110 * 100011011 + 00011001 * 1000001
00000001 = 00111101 * 100011011 + 11111110 * 1000001
00000000 = 01000001 * 100011011 + 100011011 * 1000001

a^-1(x) = x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x = 11111110 = 0xfe

The calculation of 0x41^-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      1     0     0
SBOX(41) = 1 1 1 1 1 0 0 0   *  1  +  0  =  0
           0 1 1 1 1 1 0 0      1     1     0
           0 0 1 1 1 1 1 0      1     1     0
           0 0 0 1 1 1 1 1      1     0     1

SBOX(41) = 10000011 = 83

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com