Calculation of 0x61
m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) = 0x61 = 01100001 = x^6 + x^5 + 1
m(x) = (x^2 + x + 1) * a(x) + (x^5 + x^4 + x^3 + x^2)
Calculation of 0x61-1 in the finite field GF(28)00111100 = 00000001 * 100011011 + 00000111 * 1100001
00011001 = 00000010 * 100011011 + 00001111 * 1100001
00001110 = 00000101 * 100011011 + 00011001 * 1100001
00000101 = 00001000 * 100011011 + 00111101 * 1100001
00000001 = 00011101 * 100011011 +
01011110 * 1100001
00000000 = 01100001 * 100011011 + 100011011 * 1100001
a
-1(x) = x^6 + x^4 + x^3 + x^2 + x = 01011110 = 0x5e
The calculation of 0x61
-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 0 1 1
1 1 0 0 0 1 1 1 1 1 1
1 1 1 0 0 0 1 1 1 0 1
1 1 1 1 0 0 0 1 1 0 1
SBOX(61) = 1 1 1 1 1 0 0 0 * 1 + 0 = 0
0 1 1 1 1 1 0 0 0 1 1
0 0 1 1 1 1 1 0 1 1 1
0 0 0 1 1 1 1 1 0 0 1
SBOX(61) = 11101111 = ef
For more information see
FIPS 197.
Implemented by bachph [at] philba [dot] com