Calculation of 0x86
m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) = 0x86 = 10000110 = x^7 + x^2 + x
m(x) = (x) * a(x) + (x^4 + x^2 + x + 1)
Calculation of 0x86-1 in the finite field GF(28)00010111 = 00000001 * 100011011 + 00000010 * 10000110
00000111 = 00001011 * 100011011 + 00010111 * 10000110
00000010 = 00110000 * 100011011 + 01100111 * 10000110
00000001 = 01011011 * 100011011 +
10111110 * 10000110
00000000 = 10000110 * 100011011 + 100011011 * 10000110
a
-1(x) = x^7 + x^5 + x^4 + x^3 + x^2 + x = 10111110 = 0xbe
The calculation of 0x86
-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 1
1 1 1 1 0 0 0 1 1 0 0
SBOX(86) = 1 1 1 1 1 0 0 0 * 1 + 0 = 0
0 1 1 1 1 1 0 0 1 1 0
0 0 1 1 1 1 1 0 0 1 1
0 0 0 1 1 1 1 1 1 0 0
SBOX(86) = 01000100 = 44
For more information see
FIPS 197.
Implemented by bachph [at] philba [dot] com