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    <p><strong>You simply can't do it this way with a polynomial like x^321 + ...;</strong></p> <p>If the polynomial is chosen well, you get a period length of 2^231 -1, and this is approximately 4.27 *10^96. If I'm not mistaken, this number is believed to exceed the number of atoms in the universe... (Strictly speaking, I'm referring to the posted C-code since I do not know php, but that certainly makes no difference.)</p> <p>However, there is a <strong>mathematical</strong> method to <strong>calculate the length of the period without doing a brute-force</strong> attack. Unfortunately, this can't be explained in a few lines. If you have a solid background in math (especially calculations in finite fields), I'll be glad to look for a helpful reference for you.</p> <p><strong>EDIT:</strong> The first step in calculating the period of the LFSR obtained by using a polynomial p(x) is to obtain a factorization of p(x) mod 2, i.e. in GF(2). To do this, I recommend using software like Mathematica or Maple if available. You could also try the freely available Sage, see e.g. <a href="http://www.sagemath.org/doc/constructions/polynomials.html" rel="nofollow">http://www.sagemath.org/doc/constructions/polynomials.html</a> for usage details.</p> <p>The period of p(x) is given by its order e, that means the smallest number such that p(x) divedes x^e+1. Unfortunately, I can't provide more information at the moment, it will take me several days to look for the lecture notes of a course I took several years ago...</p> <p>A small example: p(x) = (x^5+x^4+1) = (x^3+x+1)*(x^2+x+1), the individual periods are 2^3-1=7 and 2^2-1=3, and since all polynomial factors are different, the period of p(x) is 3*7=21, which I also verified in C++.</p>
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    1. COI suppose my experience in math isn't that great as it's expected to solve it, nevertheless i'd like to try if you would be so kind to help. Thank you.
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    2. COWhat is the intention behind your question? If you just want guarantee a long period, look for an irreducible polynomial (in GF(2), i.e. all calculations are performed mod 2). There are tables in the internet and documented ways to generate them. An irreducible polynomial of degree n results in a period of 2^n-1. On the other hand, if you are interested in the period of some specific polynomial, like the one you gave above, you will have to factorize it...
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    3. COThe **period** is exactly what i'm trying to find. Could show example for some polynomial? <br /> Sorry for late answer. Got issue with internet connection.
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