Calculation of 0x59
m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) = 0x59 = 01011001 = x^6 + x^4 + x^3 + 1
m(x) = (x^2 + 1) * a(x) + (x^5 + x^2 + x)
Calculation of 0x59-1 in the finite field GF(28)00100110 = 00000001 * 100011011 + 00000101 * 1011001
00010101 = 00000010 * 100011011 + 00001011 * 1011001
00001100 = 00000101 * 100011011 + 00010011 * 1011001
00000001 = 00001101 * 100011011 +
00111110 * 1011001
00000000 = 01011001 * 100011011 + 100011011 * 1011001
a
-1(x) = x^5 + x^4 + x^3 + x^2 + x = 00111110 = 0x3e
The calculation of 0x59
-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 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(59) = 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 0 0 1
SBOX(59) = 11001011 = cb
For more information see
FIPS 197.
Implemented by bachph [at] philba [dot] com