Calculation of 0xe2
m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) = 0xe2 = 11100010 = x^7 + x^6 + x^5 + x
m(x) = (x + 1) * a(x) + (x^5 + x^4 + x^3 + x^2 + 1)
Calculation of 0xe2-1 in the finite field GF(28)00111101 = 00000001 * 100011011 + 00000011 * 11100010
00010110 = 00000100 * 100011011 + 00001101 * 11100010
00000111 = 00001101 * 100011011 + 00010100 * 11100010
00000011 = 00100111 * 100011011 + 01100001 * 11100010
00000001 = 01000011 * 100011011 +
11010110 * 11100010
00000000 = 11100010 * 100011011 + 100011011 * 11100010
a
-1(x) = x^7 + x^6 + x^4 + x^2 + x = 11010110 = 0xd6
The calculation of 0xe2
-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 1 0 0
1 1 1 1 0 0 0 1 0 0 1
SBOX(e2) = 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 1 1 0
0 0 0 1 1 1 1 1 1 0 1
SBOX(e2) = 10011000 = 98
For more information see
FIPS 197.
Implemented by bachph [at] philba [dot] com