Calculation of 0x57
m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) = 0x57 = 01010111 = x^6 + x^4 + x^2 + x + 1
m(x) = (x^2 + 1) * a(x) + (x^4)
Calculation of 0x57-1 in the finite field GF(28)00010000 = 00000001 * 100011011 + 00000101 * 1010111
00000111 = 00000101 * 100011011 + 00010000 * 1010111
00000010 = 00011111 * 100011011 + 01100101 * 1010111
00000001 = 00100100 * 100011011 +
10111111 * 1010111
00000000 = 01010111 * 100011011 + 100011011 * 1010111
a
-1(x) = x^7 + x^5 + x^4 + x^3 + x^2 + x + 1 = 10111111 = 0xbf
The calculation of 0x57
-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 1 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(57) = 1 1 1 1 1 0 0 0 * 1 + 0 = 1
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(57) = 01011011 = 5b
For more information see
FIPS 197.
Implemented by bachph [at] philba [dot] com