Calculation of 0x63
m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) = 0x63 = 01100011 = x^6 + x^5 + x + 1
m(x) = (x^2 + x + 1) * a(x) + (x^5 + x^4 + x)
Calculation of 0x63-1 in the finite field GF(28)00110010 = 00000001 * 100011011 + 00000111 * 1100011
00000111 = 00000010 * 100011011 + 00001111 * 1100011
00000011 = 00010111 * 100011011 + 01101110 * 1100011
00000001 = 00101100 * 100011011 +
11010011 * 1100011
00000000 = 01100011 * 100011011 + 100011011 * 1100011
a
-1(x) = x^7 + x^6 + x^4 + x + 1 = 11010011 = 0xd3
The calculation of 0x63
-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 1
1 1 1 0 0 0 1 1 0 0 0
1 1 1 1 0 0 0 1 0 0 1
SBOX(63) = 1 1 1 1 1 0 0 0 * 1 + 0 = 1
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 1 0 1
SBOX(63) = 11111011 = fb
For more information see
FIPS 197.
Implemented by bachph [at] philba [dot] com