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