Calculation of 0x1f
m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) = 0x1f = 00011111 = x^4 + x^3 + x^2 + x + 1
m(x) = (x^4 + x^3 + 1) * a(x) + (x^3 + x^2)
Calculation of 0x1f-1 in the finite field GF(28)00001100 = 00000001 * 100011011 + 00011001 * 11111
00000111 = 00000010 * 100011011 + 00110011 * 11111
00000010 = 00000101 * 100011011 + 01111111 * 11111
00000001 = 00001101 * 100011011 +
10110010 * 11111
00000000 = 00011111 * 100011011 + 100011011 * 11111
a
-1(x) = x^7 + x^5 + x^4 + x = 10110010 = 0xb2
The calculation of 0x1f
-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 1 1 0
1 1 1 0 0 0 1 1 0 0 0
1 1 1 1 0 0 0 1 0 0 0
SBOX(1f) = 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 0 0 1 1
0 0 0 1 1 1 1 1 1 0 1
SBOX(1f) = 11000000 = c0
For more information see
FIPS 197.
Implemented by bachph [at] philba [dot] com