Calculation of 0xc3
m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) = 0xc3 = 11000011 = x^7 + x^6 + x + 1
m(x) = (x + 1) * a(x) + (x^6 + x^4 + x^3 + x^2 + x)
Calculation of 0xc3-1 in the finite field GF(28)01011110 = 00000001 * 100011011 + 00000011 * 11000011
00100001 = 00000011 * 100011011 + 00000100 * 11000011
00011100 = 00000111 * 100011011 + 00001011 * 11000011
00000101 = 00001010 * 100011011 + 00011001 * 11000011
00000010 = 00111011 * 100011011 + 01011101 * 11000011
00000001 = 01111100 * 100011011 +
10100011 * 11000011
00000000 = 11000011 * 100011011 + 100011011 * 11000011
a
-1(x) = x^7 + x^5 + x + 1 = 10100011 = 0xa3
The calculation of 0xc3
-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 1 1 1
1 1 1 0 0 0 1 1 0 0 1
1 1 1 1 0 0 0 1 0 0 1
SBOX(c3) = 1 1 1 1 1 0 0 0 * 0 + 0 = 0
0 1 1 1 1 1 0 0 1 1 1
0 0 1 1 1 1 1 0 0 1 0
0 0 0 1 1 1 1 1 1 0 0
SBOX(c3) = 00101110 = 2e
For more information see
FIPS 197.
Implemented by bachph [at] philba [dot] com