Calculation of 0xeb
m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) = 0xeb = 11101011 = x^7 + x^6 + x^5 + x^3 + x + 1
m(x) = (x + 1) * a(x) + (x^5 + x^2 + x)
Calculation of 0xeb-1 in the finite field GF(28)00100110 = 00000001 * 100011011 + 00000011 * 11101011
00011001 = 00000111 * 100011011 + 00001000 * 11101011
00001101 = 00001000 * 100011011 + 00011011 * 11101011
00000011 = 00010111 * 100011011 + 00111110 * 11101011
00000001 = 01010100 * 100011011 +
11100011 * 11101011
00000000 = 11101011 * 100011011 + 100011011 * 11101011
a
-1(x) = x^7 + x^6 + x^5 + x + 1 = 11100011 = 0xe3
The calculation of 0xeb
-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 1 1 1
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 1
SBOX(eb) = 1 1 1 1 1 0 0 0 * 0 + 0 = 0
0 1 1 1 1 1 0 0 1 1 1
0 0 1 1 1 1 1 0 1 1 1
0 0 0 1 1 1 1 1 1 0 1
SBOX(eb) = 11101001 = e9
For more information see
FIPS 197.
Implemented by bachph [at] philba [dot] com