Calculation of 0xe8

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0xe8 =  11101000 = x^7 + x^6 + x^5 + x^3

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

Calculation of 0xe8^-1 in the finite field GF(2⁸)

00100011 = 00000001 * 100011011 + 00000011 * 11101000
00000001 = 00000111 * 100011011 + 00001000 * 11101000
00000000 = 11101000 * 100011011 + 100011011 * 11101000

a^-1(x) = x^3 = 00001000 = 0x08

The calculation of 0xe8^-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     1
           1 1 1 0 0 0 1 1      0     0     0
           1 1 1 1 0 0 0 1      1     0     1
SBOX(e8) = 1 1 1 1 1 0 0 0   *  0  +  0  =  1
           0 1 1 1 1 1 0 0      0     1     0
           0 0 1 1 1 1 1 0      0     1     0
           0 0 0 1 1 1 1 1      0     0     1

SBOX(e8) = 10011011 = 9b

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com