Calculation of 0xe0

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0xe0 =  11100000 = x^7 + x^6 + x^5

m(x) = (x + 1) * a(x) + (x^5 + x^4 + x^3 + x + 1)

Calculation of 0xe0^-1 in the finite field GF(2⁸)

00111011 = 00000001 * 100011011 + 00000011 * 11100000
00001100 = 00000100 * 100011011 + 00001101 * 11100000
00000111 = 00010101 * 100011011 + 00111010 * 11100000
00000010 = 00101110 * 100011011 + 01111001 * 11100000
00000001 = 01100111 * 100011011 + 10110001 * 11100000
00000000 = 11100000 * 100011011 + 100011011 * 11100000

a^-1(x) = x^7 + x^5 + x^4 + 1 = 10110001 = 0xb1

The calculation of 0xe0^-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      0     1     0
           1 1 1 0 0 0 1 1      0     0     0
           1 1 1 1 0 0 0 1      0     0     0
SBOX(e0) = 1 1 1 1 1 0 0 0   *  1  +  0  =  0
           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      1     0     1

SBOX(e0) = 11100001 = e1

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com