Calculation of 0xde
m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) = 0xde = 11011110 = x^7 + x^6 + x^4 + x^3 + x^2 + x
m(x) = (x + 1) * a(x) + (x^6 + x^5 + x^4 + x^3 + 1)
Calculation of 0xde-1 in the finite field GF(28)01111001 = 00000001 * 100011011 + 00000011 * 11011110
00101100 = 00000010 * 100011011 + 00000111 * 11011110
00001101 = 00000111 * 100011011 + 00001010 * 11011110
00000010 = 00010000 * 100011011 + 00111011 * 11011110
00000001 = 01100111 * 100011011 +
10010000 * 11011110
00000000 = 11011110 * 100011011 + 100011011 * 11011110
a
-1(x) = x^7 + x^4 = 10010000 = 0x90
The calculation of 0xde
-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 1
1 1 0 0 0 1 1 1 0 1 0
1 1 1 0 0 0 1 1 0 0 1
1 1 1 1 0 0 0 1 0 0 1
SBOX(de) = 1 1 1 1 1 0 0 0 * 1 + 0 = 1
0 1 1 1 1 1 0 0 0 1 0
0 0 1 1 1 1 1 0 0 1 0
0 0 0 1 1 1 1 1 1 0 0
SBOX(de) = 00011101 = 1d
For more information see
FIPS 197.
Implemented by bachph [at] philba [dot] com