Calculation of 0xb1
m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) = 0xb1 = 10110001 = x^7 + x^5 + x^4 + 1
m(x) = (x) * a(x) + (x^6 + x^5 + x^4 + x^3 + 1)
Calculation of 0xb1-1 in the finite field GF(28)01111001 = 00000001 * 100011011 + 00000010 * 10110001
00111010 = 00000011 * 100011011 + 00000111 * 10110001
00001101 = 00000111 * 100011011 + 00001100 * 10110001
00000011 = 00011000 * 100011011 + 00111011 * 10110001
00000001 = 01100111 * 100011011 +
11100000 * 10110001
00000000 = 10110001 * 100011011 + 100011011 * 10110001
a
-1(x) = x^7 + x^6 + x^5 = 11100000 = 0xe0
The calculation of 0xb1
-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 0
1 1 1 0 0 0 1 1 0 0 0
1 1 1 1 0 0 0 1 0 0 1
SBOX(b1) = 1 1 1 1 1 0 0 0 * 0 + 0 = 0
0 1 1 1 1 1 0 0 1 1 0
0 0 1 1 1 1 1 0 1 1 1
0 0 0 1 1 1 1 1 1 0 1
SBOX(b1) = 11001000 = c8
For more information see
FIPS 197.
Implemented by bachph [at] philba [dot] com