Calculation of 0xbd
m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) = 0xbd = 10111101 = x^7 + x^5 + x^4 + x^3 + x^2 + 1
m(x) = (x) * a(x) + (x^6 + x^5 + 1)
Calculation of 0xbd-1 in the finite field GF(28)01100001 = 00000001 * 100011011 + 00000010 * 10111101
00011110 = 00000011 * 100011011 + 00000111 * 10111101
00000111 = 00001110 * 100011011 + 00011001 * 10111101
00000010 = 00111011 * 100011011 + 01100011 * 10111101
00000001 = 01000011 * 100011011 +
10111100 * 10111101
00000000 = 10111101 * 100011011 + 100011011 * 10111101
a
-1(x) = x^7 + x^5 + x^4 + x^3 + x^2 = 10111100 = 0xbc
The calculation of 0xbd
-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 0 1 1
1 1 1 0 0 0 1 1 1 0 0
1 1 1 1 0 0 0 1 1 0 1
SBOX(bd) = 1 1 1 1 1 0 0 0 * 1 + 0 = 1
0 1 1 1 1 1 0 0 1 1 1
0 0 1 1 1 1 1 0 0 1 1
0 0 0 1 1 1 1 1 1 0 0
SBOX(bd) = 01111010 = 7a
For more information see
FIPS 197.
Implemented by bachph [at] philba [dot] com