Calculation of 0x5d
m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) = 0x5d = 01011101 = x^6 + x^4 + x^3 + x^2 + 1
m(x) = (x^2 + 1) * a(x) + (x^5 + x^4 + x)
Calculation of 0x5d-1 in the finite field GF(28)00110010 = 00000001 * 100011011 + 00000101 * 1011101
00001011 = 00000011 * 100011011 + 00001110 * 1011101
00000011 = 00001000 * 100011011 + 00101111 * 1011101
00000001 = 00110011 * 100011011 +
11101100 * 1011101
00000000 = 01011101 * 100011011 + 100011011 * 1011101
a
-1(x) = x^7 + x^6 + x^5 + x^3 + x^2 = 11101100 = 0xec
The calculation of 0x5d
-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 1
1 1 1 1 0 0 0 1 1 0 1
SBOX(5d) = 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 1 1 1
0 0 0 1 1 1 1 1 1 0 0
SBOX(5d) = 01001100 = 4c
For more information see
FIPS 197.
Implemented by bachph [at] philba [dot] com