Calculation of 0xa3
m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) = 0xa3 = 10100011 = x^7 + x^5 + x + 1
m(x) = (x) * a(x) + (x^6 + x^4 + x^3 + x^2 + 1)
Calculation of 0xa3-1 in the finite field GF(28)01011101 = 00000001 * 100011011 + 00000010 * 10100011
00011001 = 00000010 * 100011011 + 00000101 * 10100011
00001011 = 00001101 * 100011011 + 00011100 * 10100011
00000100 = 00010101 * 100011011 + 00100001 * 10100011
00000011 = 00100111 * 100011011 + 01011110 * 10100011
00000001 = 01111100 * 100011011 +
11000011 * 10100011
00000000 = 10100011 * 100011011 + 100011011 * 10100011
a
-1(x) = x^7 + x^6 + x + 1 = 11000011 = 0xc3
The calculation of 0xa3
-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 1 1 0
1 1 0 0 0 1 1 1 1 1 1
1 1 1 0 0 0 1 1 0 0 0
1 1 1 1 0 0 0 1 0 0 1
SBOX(a3) = 1 1 1 1 1 0 0 0 * 0 + 0 = 0
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(a3) = 00001010 = a
For more information see
FIPS 197.
Implemented by bachph [at] philba [dot] com