Calculation of 0x78

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x78 =  01111000 = x^6 + x^5 + x^4 + x^3

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

Calculation of 0x78^-1 in the finite field GF(2⁸)

00001011 = 00000001 * 100011011 + 00000110 * 1111000
00000111 = 00001101 * 100011011 + 00101111 * 1111000
00000010 = 00010110 * 100011011 + 01110111 * 1111000
00000001 = 00110111 * 100011011 + 10110110 * 1111000
00000000 = 01111000 * 100011011 + 100011011 * 1111000

a^-1(x) = x^7 + x^5 + x^4 + x^2 + x = 10110110 = 0xb6

The calculation of 0x78^-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      1     1     0
           1 1 1 0 0 0 1 1      1     0     1
           1 1 1 1 0 0 0 1      0     0     1
SBOX(78) = 1 1 1 1 1 0 0 0   *  1  +  0  =  1
           0 1 1 1 1 1 0 0      1     1     1
           0 0 1 1 1 1 1 0      0     1     0
           0 0 0 1 1 1 1 1      1     0     1

SBOX(78) = 10111100 = bc

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com