Calculation of 0x4d
m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) = 0x4d = 01001101 = x^6 + x^3 + x^2 + 1
m(x) = (x^2) * a(x) + (x^5 + x^3 + x^2 + x + 1)
Calculation of 0x4d-1 in the finite field GF(28)00101111 = 00000001 * 100011011 + 00000100 * 1001101
00010011 = 00000010 * 100011011 + 00001001 * 1001101
00001001 = 00000101 * 100011011 + 00010110 * 1001101
00000001 = 00001000 * 100011011 +
00100101 * 1001101
00000000 = 01001101 * 100011011 + 100011011 * 1001101
a
-1(x) = x^5 + x^2 + 1 = 00100101 = 0x25
The calculation of 0x4d
-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 1
1 1 0 0 0 1 1 1 0 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(4d) = 1 1 1 1 1 0 0 0 * 0 + 0 = 0
0 1 1 1 1 1 0 0 1 1 1
0 0 1 1 1 1 1 0 0 1 1
0 0 0 1 1 1 1 1 0 0 1
SBOX(4d) = 11100011 = e3
For more information see
FIPS 197.
Implemented by bachph [at] philba [dot] com