Calculation of 0xe5

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0xe5 =  11100101 = x^7 + x^6 + x^5 + x^2 + 1

m(x) = (x + 1) * a(x) + (x^5 + x^4 + x^2)

Calculation of 0xe5^-1 in the finite field GF(2⁸)

00110100 = 00000001 * 100011011 + 00000011 * 11100101
00000001 = 00000101 * 100011011 + 00001110 * 11100101
00000000 = 11100101 * 100011011 + 100011011 * 11100101

a^-1(x) = x^3 + x^2 + x = 00001110 = 0x0e

The calculation of 0xe5^-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      1     1     0
           1 1 1 0 0 0 1 1      1     0     0
           1 1 1 1 0 0 0 1      1     0     1
SBOX(e5) = 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     1
           0 0 0 1 1 1 1 1      0     0     1

SBOX(e5) = 11011001 = d9

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com