Calculation of 0xcf
m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) = 0xcf = 11001111 = x^7 + x^6 + x^3 + x^2 + x + 1
m(x) = (x + 1) * a(x) + (x^6 + x^3 + x)
Calculation of 0xcf-1 in the finite field GF(28)01001010 = 00000001 * 100011011 + 00000011 * 11001111
00010001 = 00000011 * 100011011 + 00000100 * 11001111
00001110 = 00001101 * 100011011 + 00010011 * 11001111
00000011 = 00010100 * 100011011 + 00110001 * 11001111
00000001 = 01001001 * 100011011 +
11100110 * 11001111
00000000 = 11001111 * 100011011 + 100011011 * 11001111
a
-1(x) = x^7 + x^6 + x^5 + x^2 + x = 11100110 = 0xe6
The calculation of 0xcf
-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 0
1 1 0 0 0 1 1 1 1 1 1
1 1 1 0 0 0 1 1 1 0 0
1 1 1 1 0 0 0 1 0 0 1
SBOX(cf) = 1 1 1 1 1 0 0 0 * 0 + 0 = 0
0 1 1 1 1 1 0 0 1 1 0
0 0 1 1 1 1 1 0 1 1 0
0 0 0 1 1 1 1 1 1 0 1
SBOX(cf) = 10001010 = 8a
For more information see
FIPS 197.
Implemented by bachph [at] philba [dot] com