Calculation of 0xec

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0xec =  11101100 = x^7 + x^6 + x^5 + x^3 + x^2

m(x) = (x + 1) * a(x) + (x^5 + x^3 + x^2 + x + 1)

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

00101111 = 00000001 * 100011011 + 00000011 * 11101100
00001110 = 00000110 * 100011011 + 00001011 * 11101100
00000101 = 00010011 * 100011011 + 00110010 * 11101100
00000001 = 00110011 * 100011011 + 01011101 * 11101100
00000000 = 11101100 * 100011011 + 100011011 * 11101100

a^-1(x) = x^6 + x^4 + x^3 + x^2 + 1 = 01011101 = 0x5d

The calculation of 0xec^-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      0     1     1
           1 1 1 0 0 0 1 1      1     0     1
           1 1 1 1 0 0 0 1      1     0     1
SBOX(ec) = 1 1 1 1 1 0 0 0   *  1  +  0  =  0
           0 1 1 1 1 1 0 0      0     1     0
           0 0 1 1 1 1 1 0      1     1     1
           0 0 0 1 1 1 1 1      0     0     1

SBOX(ec) = 11001110 = ce

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com