Calculation of 0x54
m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) = 0x54 = 01010100 = x^6 + x^4 + x^2
m(x) = (x^2 + 1) * a(x) + (x^4 + x^3 + x^2 + x + 1)
Calculation of 0x54-1 in the finite field GF(28)00011111 = 00000001 * 100011011 + 00000101 * 1010100
00001001 = 00000111 * 100011011 + 00011010 * 1010100
00000100 = 00001000 * 100011011 + 00101011 * 1010100
00000001 = 00010111 * 100011011 +
01001100 * 1010100
00000000 = 01010100 * 100011011 + 100011011 * 1010100
a
-1(x) = x^6 + x^3 + x^2 = 01001100 = 0x4c
The calculation of 0x54
-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 0
1 1 0 0 0 1 1 1 0 1 0
1 1 1 0 0 0 1 1 1 0 0
1 1 1 1 0 0 0 1 1 0 0
SBOX(54) = 1 1 1 1 1 0 0 0 * 0 + 0 = 0
0 1 1 1 1 1 0 0 0 1 1
0 0 1 1 1 1 1 0 1 1 0
0 0 0 1 1 1 1 1 0 0 0
SBOX(54) = 00100000 = 20
For more information see
FIPS 197.
Implemented by bachph [at] philba [dot] com