Calculation of 0x77

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x77 =  01110111 = x^6 + x^5 + x^4 + x^2 + x + 1

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

Calculation of 0x77^-1 in the finite field GF(2⁸)

00101001 = 00000001 * 100011011 + 00000110 * 1110111
00001100 = 00000011 * 100011011 + 00001011 * 1110111
00000001 = 00001011 * 100011011 + 00111100 * 1110111
00000000 = 01110111 * 100011011 + 100011011 * 1110111

a^-1(x) = x^5 + x^4 + x^3 + x^2 = 00111100 = 0x3c

The calculation of 0x77^-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     1
           1 1 0 0 0 1 1 1      0     1     0
           1 1 1 0 0 0 1 1      1     0     1
           1 1 1 1 0 0 0 1      1     0     0
SBOX(77) = 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     1
           0 0 0 1 1 1 1 1      0     0     1

SBOX(77) = 11110101 = f5

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com