Calculation of 0x49

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x49 =  01001001 = x^6 + x^3 + 1

m(x) = (x^2) * a(x) + (x^5 + x^4 + x^3 + x^2 + x + 1)

Calculation of 0x49^-1 in the finite field GF(2⁸)

00111111 = 00000001 * 100011011 + 00000100 * 1001001
00001000 = 00000011 * 100011011 + 00001101 * 1001001
00000111 = 00001000 * 100011011 + 00100111 * 1001001
00000001 = 00011011 * 100011011 + 01100100 * 1001001
00000000 = 01001001 * 100011011 + 100011011 * 1001001

a^-1(x) = x^6 + x^5 + x^2 = 01100100 = 0x64

The calculation of 0x49^-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     1
           1 1 1 0 0 0 1 1      1     0     0
           1 1 1 1 0 0 0 1      0     0     1
SBOX(49) = 1 1 1 1 1 0 0 0   *  0  +  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      0     0     0

SBOX(49) = 00111011 = 3b

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com