Calculation of 0x11

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x11 =  00010001 = x^4 + 1

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

Calculation of 0x11^-1 in the finite field GF(2⁸)

00001011 = 00000001 * 100011011 + 00010000 * 10001
00000111 = 00000010 * 100011011 + 00100001 * 10001
00000010 = 00000111 * 100011011 + 01110011 * 10001
00000001 = 00001011 * 100011011 + 10110100 * 10001
00000000 = 00010001 * 100011011 + 100011011 * 10001

a^-1(x) = x^7 + x^5 + x^4 + x^2 = 10110100 = 0xb4

The calculation of 0x11^-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      0     1     1
           1 1 1 0 0 0 1 1      1     0     0
           1 1 1 1 0 0 0 1      0     0     0
SBOX(11) = 1 1 1 1 1 0 0 0   *  1  +  0  =  0
           0 1 1 1 1 1 0 0      1     1     0
           0 0 1 1 1 1 1 0      0     1     0
           0 0 0 1 1 1 1 1      1     0     1

SBOX(11) = 10000010 = 82

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com