Calculation of 0x1e

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x1e =  00011110 = x^4 + x^3 + x^2 + x

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

Calculation of 0x1e^-1 in the finite field GF(2⁸)

00001011 = 00000001 * 100011011 + 00011000 * 11110
00000011 = 00000011 * 100011011 + 00101001 * 11110
00000001 = 00001011 * 100011011 + 11101110 * 11110
00000000 = 00011110 * 100011011 + 100011011 * 11110

a^-1(x) = x^7 + x^6 + x^5 + x^3 + x^2 + x = 11101110 = 0xee

The calculation of 0x1e^-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     0
           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(1e) = 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     1
           0 0 0 1 1 1 1 1      1     0     0

SBOX(1e) = 01110010 = 72

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com