Calculation of 0xe7

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0xe7 =  11100111 = x^7 + x^6 + x^5 + x^2 + x + 1

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

Calculation of 0xe7^-1 in the finite field GF(2⁸)

00110010 = 00000001 * 100011011 + 00000011 * 11100111
00011101 = 00000101 * 100011011 + 00001110 * 11100111
00001000 = 00001011 * 100011011 + 00011111 * 11100111
00000101 = 00011000 * 100011011 + 00101111 * 11100111
00000010 = 00111011 * 100011011 + 01000001 * 11100111
00000001 = 01101110 * 100011011 + 10101101 * 11100111
00000000 = 11100111 * 100011011 + 100011011 * 11100111

a^-1(x) = x^7 + x^5 + x^3 + x^2 + 1 = 10101101 = 0xad

The calculation of 0xe7^-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     0
           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(e7) = 1 1 1 1 1 0 0 0   *  0  +  0  =  1
           0 1 1 1 1 1 0 0      1     1     0
           0 0 1 1 1 1 1 0      0     1     0
           0 0 0 1 1 1 1 1      1     0     1

SBOX(e7) = 10010100 = 94

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com