Calculation of 0x4a
m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) = 0x4a = 01001010 = x^6 + x^3 + x
m(x) = (x^2) * a(x) + (x^5 + x^4 + x + 1)
Calculation of 0x4a-1 in the finite field GF(28)00110011 = 00000001 * 100011011 + 00000100 * 1001010
00011111 = 00000011 * 100011011 + 00001101 * 1001010
00001101 = 00000111 * 100011011 + 00011110 * 1001010
00000101 = 00001101 * 100011011 + 00110001 * 1001010
00000010 = 00010000 * 100011011 + 01001101 * 1001010
00000001 = 00101101 * 100011011 +
10101011 * 1001010
00000000 = 01001010 * 100011011 + 100011011 * 1001010
a
-1(x) = x^7 + x^5 + x^3 + x + 1 = 10101011 = 0xab
The calculation of 0x4a
-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 1
1 1 1 1 0 0 0 1 1 0 0
SBOX(4a) = 1 1 1 1 1 0 0 0 * 0 + 0 = 1
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 1 0 1
SBOX(4a) = 11010110 = d6
For more information see
FIPS 197.
Implemented by bachph [at] philba [dot] com