Calculation of 0x5b

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x5b =  01011011 = x^6 + x^4 + x^3 + x + 1

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

Calculation of 0x5b^-1 in the finite field GF(2⁸)

00101100 = 00000001 * 100011011 + 00000101 * 1011011
00000011 = 00000010 * 100011011 + 00001011 * 1011011
00000001 = 00110111 * 100011011 + 11110000 * 1011011
00000000 = 01011011 * 100011011 + 100011011 * 1011011

a^-1(x) = x^7 + x^6 + x^5 + x^4 = 11110000 = 0xf0

The calculation of 0x5b^-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      0     0     0
           1 1 1 1 0 0 0 1      0     0     1
SBOX(5b) = 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     0

SBOX(5b) = 00111001 = 39

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com