Calculation of 0xa8
m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) = 0xa8 = 10101000 = x^7 + x^5 + x^3
m(x) = (x) * a(x) + (x^6 + x^3 + x + 1)
Calculation of 0xa8-1 in the finite field GF(28)01001011 = 00000001 * 100011011 + 00000010 * 10101000
00111110 = 00000010 * 100011011 + 00000101 * 10101000
00001001 = 00000111 * 100011011 + 00001101 * 10101000
00000001 = 00010111 * 100011011 +
00100110 * 10101000
00000000 = 10101000 * 100011011 + 100011011 * 10101000
a
-1(x) = x^5 + x^2 + x = 00100110 = 0x26
The calculation of 0xa8
-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 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 0
SBOX(a8) = 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 1
0 0 0 1 1 1 1 1 0 0 1
SBOX(a8) = 11000010 = c2
For more information see
FIPS 197.
Implemented by bachph [at] philba [dot] com