Calculation of 0x50

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x50 =  01010000 = x^6 + x^4

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

Calculation of 0x50^-1 in the finite field GF(2⁸)

00001011 = 00000001 * 100011011 + 00000101 * 1010000
00000011 = 00001001 * 100011011 + 00101100 * 1010000
00000001 = 00110111 * 100011011 + 11101101 * 1010000
00000000 = 01010000 * 100011011 + 100011011 * 1010000

a^-1(x) = x^7 + x^6 + x^5 + x^3 + x^2 + 1 = 11101101 = 0xed

The calculation of 0x50^-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      0     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(50) = 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      1     1     1
           0 0 0 1 1 1 1 1      1     0     0

SBOX(50) = 01010011 = 53

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com