Calculation of 0x34

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x34 =  00110100 = x^5 + x^4 + x^2

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

Calculation of 0x34^-1 in the finite field GF(2⁸)

00000011 = 00000001 * 100011011 + 00001110 * 110100
00000001 = 00010011 * 100011011 + 11110011 * 110100
00000000 = 00110100 * 100011011 + 100011011 * 110100

a^-1(x) = x^7 + x^6 + x^5 + x^4 + x + 1 = 11110011 = 0xf3

The calculation of 0x34^-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     0
           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      0     0     1
SBOX(34) = 1 1 1 1 1 0 0 0   *  1  +  0  =  1
           0 1 1 1 1 1 0 0      1     1     0
           0 0 1 1 1 1 1 0      1     1     0
           0 0 0 1 1 1 1 1      1     0     0

SBOX(34) = 00011000 = 18

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com