Calculation of 0x54

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x54 =  01010100 = x^6 + x^4 + x^2

m(x) = (x^2 + 1) * a(x) + (x^4 + x^3 + x^2 + x + 1)

Calculation of 0x54^-1 in the finite field GF(2⁸)

00011111 = 00000001 * 100011011 + 00000101 * 1010100
00001001 = 00000111 * 100011011 + 00011010 * 1010100
00000100 = 00001000 * 100011011 + 00101011 * 1010100
00000001 = 00010111 * 100011011 + 01001100 * 1010100
00000000 = 01010100 * 100011011 + 100011011 * 1010100

a^-1(x) = x^6 + x^3 + x^2 = 01001100 = 0x4c

The calculation of 0x54^-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     0
           1 1 1 0 0 0 1 1      1     0     0
           1 1 1 1 0 0 0 1      1     0     0
SBOX(54) = 1 1 1 1 1 0 0 0   *  0  +  0  =  0
           0 1 1 1 1 1 0 0      0     1     1
           0 0 1 1 1 1 1 0      1     1     0
           0 0 0 1 1 1 1 1      0     0     0

SBOX(54) = 00100000 = 20

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com