Calculation of 0x72
m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) = 0x72 = 01110010 = x^6 + x^5 + x^4 + x
m(x) = (x^2 + x) * a(x) + (x^5 + x^4 + x^2 + x + 1)
Calculation of 0x72-1 in the finite field GF(28)00110111 = 00000001 * 100011011 + 00000110 * 1110010
00011100 = 00000010 * 100011011 + 00001101 * 1110010
00001111 = 00000101 * 100011011 + 00011100 * 1110010
00000010 = 00001000 * 100011011 + 00110101 * 1110010
00000001 = 00111101 * 100011011 +
10010111 * 1110010
00000000 = 01110010 * 100011011 + 100011011 * 1110010
a
-1(x) = x^7 + x^4 + x^2 + x + 1 = 10010111 = 0x97
The calculation of 0x72
-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 0
1 1 1 0 0 0 1 1 1 0 0
1 1 1 1 0 0 0 1 0 0 0
SBOX(72) = 1 1 1 1 1 0 0 0 * 1 + 0 = 0
0 1 1 1 1 1 0 0 0 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(72) = 01000000 = 40
For more information see
FIPS 197.
Implemented by bachph [at] philba [dot] com