Calculation of 0xa7
m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) = 0xa7 = 10100111 = x^7 + x^5 + x^2 + x + 1
m(x) = (x) * a(x) + (x^6 + x^4 + x^2 + 1)
Calculation of 0xa7-1 in the finite field GF(28)01010101 = 00000001 * 100011011 + 00000010 * 10100111
00001101 = 00000010 * 100011011 + 00000101 * 10100111
00000100 = 00011011 * 100011011 + 00111011 * 10100111
00000001 = 00101111 * 100011011 +
01001000 * 10100111
00000000 = 10100111 * 100011011 + 100011011 * 10100111
a
-1(x) = x^6 + x^3 = 01001000 = 0x48
The calculation of 0xa7
-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 1
1 1 1 1 0 0 0 1 1 0 1
SBOX(a7) = 1 1 1 1 1 0 0 0 * 0 + 0 = 1
0 1 1 1 1 1 0 0 0 1 0
0 0 1 1 1 1 1 0 1 1 1
0 0 0 1 1 1 1 1 0 0 0
SBOX(a7) = 01011100 = 5c
For more information see
FIPS 197.
Implemented by bachph [at] philba [dot] com