Calculation of 0xef

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0xef =  11101111 = x^7 + x^6 + x^5 + x^3 + x^2 + x + 1

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

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

00101010 = 00000001 * 100011011 + 00000011 * 11101111
00010011 = 00000110 * 100011011 + 00001011 * 11101111
00001100 = 00001101 * 100011011 + 00010101 * 11101111
00000111 = 00010001 * 100011011 + 00110100 * 11101111
00000010 = 00101111 * 100011011 + 01111101 * 11101111
00000001 = 01100000 * 100011011 + 10110011 * 11101111
00000000 = 11101111 * 100011011 + 100011011 * 11101111

a^-1(x) = x^7 + x^5 + x^4 + x + 1 = 10110011 = 0xb3

The calculation of 0xef^-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      1     1     1
           1 1 1 0 0 0 1 1      0     0     1
           1 1 1 1 0 0 0 1      0     0     1
SBOX(ef) = 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      0     1     1
           0 0 0 1 1 1 1 1      1     0     1

SBOX(ef) = 11011111 = df

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com