Calculation of 0x8b
m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) = 0x8b = 10001011 = x^7 + x^3 + x + 1
m(x) = (x) * a(x) + (x^3 + x^2 + 1)
Calculation of 0x8b-1 in the finite field GF(28)00001101 = 00000001 * 100011011 + 00000010 * 10001011
00000111 = 00011100 * 100011011 + 00111001 * 10001011
00000011 = 00111001 * 100011011 + 01110000 * 10001011
00000001 = 01101110 * 100011011 +
11011001 * 10001011
00000000 = 10001011 * 100011011 + 100011011 * 10001011
a
-1(x) = x^7 + x^6 + x^4 + x^3 + 1 = 11011001 = 0xd9
The calculation of 0x8b
-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 0 1 0
1 1 1 0 0 0 1 1 0 0 1
1 1 1 1 0 0 0 1 1 0 1
SBOX(8b) = 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 0
0 0 0 1 1 1 1 1 1 0 0
SBOX(8b) = 00111101 = 3d
For more information see
FIPS 197.
Implemented by bachph [at] philba [dot] com