Calculation of 0xce
m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) = 0xce = 11001110 = x^7 + x^6 + x^3 + x^2 + x
m(x) = (x + 1) * a(x) + (x^6 + x^3 + 1)
Calculation of 0xce-1 in the finite field GF(28)01001001 = 00000001 * 100011011 + 00000011 * 11001110
00010101 = 00000011 * 100011011 + 00000100 * 11001110
00001000 = 00001110 * 100011011 + 00010111 * 11001110
00000101 = 00011111 * 100011011 + 00101010 * 11001110
00000010 = 00110000 * 100011011 + 01000011 * 11001110
00000001 = 01111111 * 100011011 +
10101100 * 11001110
00000000 = 11001110 * 100011011 + 100011011 * 11001110
a
-1(x) = x^7 + x^5 + x^3 + x^2 = 10101100 = 0xac
The calculation of 0xce
-1 is made with the
Extended Euclidean algorithm. Instead of normal division and multiplication you need to use Polynomialdivision and Polynomialmultiplication.
1 0 0 0 1 1 1 1 0 1 1
1 1 0 0 0 1 1 1 0 1 1
1 1 1 0 0 0 1 1 1 0 0
1 1 1 1 0 0 0 1 1 0 1
SBOX(ce) = 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 0 1 0
0 0 0 1 1 1 1 1 1 0 1
SBOX(ce) = 10001011 = 8b
For more information see
FIPS 197.
Implemented by bachph [at] philba [dot] com