Calculation of 0xc4
m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) = 0xc4 = 11000100 = x^7 + x^6 + x^2
m(x) = (x + 1) * a(x) + (x^6 + x^4 + x^2 + x + 1)
Calculation of 0xc4-1 in the finite field GF(28)01010111 = 00000001 * 100011011 + 00000011 * 11000100
00111101 = 00000011 * 100011011 + 00000100 * 11000100
00010000 = 00000100 * 100011011 + 00001111 * 11000100
00001101 = 00001111 * 100011011 + 00010101 * 11000100
00000111 = 00010101 * 100011011 + 00110000 * 11000100
00000011 = 00100101 * 100011011 + 01110101 * 11000100
00000001 = 01011111 * 100011011 +
11011010 * 11000100
00000000 = 11000100 * 100011011 + 100011011 * 11000100
a
-1(x) = x^7 + x^6 + x^4 + x^3 + x = 11011010 = 0xda
The calculation of 0xc4
-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 0
1 1 1 0 0 0 1 1 0 0 1
1 1 1 1 0 0 0 1 1 0 1
SBOX(c4) = 1 1 1 1 1 0 0 0 * 1 + 0 = 1
0 1 1 1 1 1 0 0 0 1 0
0 0 1 1 1 1 1 0 1 1 0
0 0 0 1 1 1 1 1 1 0 0
SBOX(c4) = 00011100 = 1c
For more information see
FIPS 197.
Implemented by bachph [at] philba [dot] com