Calculation of 0x26

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x26 =  00100110 = x^5 + x^2 + x

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

Calculation of 0x26^-1 in the finite field GF(2⁸)

00001101 = 00000001 * 100011011 + 00001001 * 100110
00000101 = 00000111 * 100011011 + 00111110 * 100110
00000010 = 00001000 * 100011011 + 01001011 * 100110
00000001 = 00010111 * 100011011 + 10101000 * 100110
00000000 = 00100110 * 100011011 + 100011011 * 100110

a^-1(x) = x^7 + x^5 + x^3 = 10101000 = 0xa8

The calculation of 0x26^-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      0     0     1
           1 1 1 1 0 0 0 1      1     0     0
SBOX(26) = 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      0     1     1
           0 0 0 1 1 1 1 1      1     0     1

SBOX(26) = 11110111 = f7

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com