Calculation of 0x15

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x15 =  00010101 = x^4 + x^2 + 1

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

Calculation of 0x15^-1 in the finite field GF(2⁸)

00001010 = 00000001 * 100011011 + 00010101 * 10101
00000001 = 00000010 * 100011011 + 00101011 * 10101
00000000 = 00010101 * 100011011 + 100011011 * 10101

a^-1(x) = x^5 + x^3 + x + 1 = 00101011 = 0x2b

The calculation of 0x15^-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      1     1     1
           1 1 0 0 0 1 1 1      1     1     0
           1 1 1 0 0 0 1 1      0     0     0
           1 1 1 1 0 0 0 1      1     0     1
SBOX(15) = 1 1 1 1 1 0 0 0   *  0  +  0  =  1
           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      0     0     0

SBOX(15) = 01011001 = 59

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com