Calculation of 0x67
m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) = 0x67 = 01100111 = x^6 + x^5 + x^2 + x + 1
m(x) = (x^2 + x + 1) * a(x) + (x^5 + x^3 + x^2 + x)
Calculation of 0x67-1 in the finite field GF(28)00101110 = 00000001 * 100011011 + 00000111 * 1100111
00010101 = 00000011 * 100011011 + 00001000 * 1100111
00000100 = 00000111 * 100011011 + 00010111 * 1100111
00000001 = 00011000 * 100011011 +
01000011 * 1100111
00000000 = 01100111 * 100011011 + 100011011 * 1100111
a
-1(x) = x^6 + x + 1 = 01000011 = 0x43
The calculation of 0x67
-1 is made with the
Extended Euclidean algorithm. Instead of normal division and multiplication you need to use Polynomialdivision and Polynomialmultiplication.
1 0 0 0 1 1 1 1 1 1 1
1 1 0 0 0 1 1 1 1 1 0
1 1 1 0 0 0 1 1 0 0 1
1 1 1 1 0 0 0 1 0 0 0
SBOX(67) = 1 1 1 1 1 0 0 0 * 0 + 0 = 0
0 1 1 1 1 1 0 0 0 1 0
0 0 1 1 1 1 1 0 1 1 0
0 0 0 1 1 1 1 1 0 0 1
SBOX(67) = 10000101 = 85
For more information see
FIPS 197.
Implemented by bachph [at] philba [dot] com