Calculation of 0x65
m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) = 0x65 = 01100101 = x^6 + x^5 + x^2 + 1
m(x) = (x^2 + x + 1) * a(x) + (x^5)
Calculation of 0x65-1 in the finite field GF(28)00100000 = 00000001 * 100011011 + 00000111 * 1100101
00000101 = 00000011 * 100011011 + 00001000 * 1100101
00000010 = 00011111 * 100011011 + 01010111 * 1100101
00000001 = 00111101 * 100011011 +
10100110 * 1100101
00000000 = 01100101 * 100011011 + 100011011 * 1100101
a
-1(x) = x^7 + x^5 + x^2 + x = 10100110 = 0xa6
The calculation of 0x65
-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 1
1 1 0 0 0 1 1 1 1 1 0
1 1 1 0 0 0 1 1 1 0 1
1 1 1 1 0 0 0 1 0 0 1
SBOX(65) = 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 1 0 0
SBOX(65) = 01001101 = 4d
For more information see
FIPS 197.
Implemented by bachph [at] philba [dot] com