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Title:
On Asymptotic Behavior of the Ratio Between the Numbers of Binary Primitive and Irreducible Polynomials
Authors: Yuri Borissov, Moon Ho Lee, Svetla Nikova
Abstract:In this paper we study the ratio \theta(n) = \frac{\lambda_2(n)}{\psi_2(n)}, where {\lambda_2(n)} is the number of primitive polynomials and {\psi_2(n)} is the number of irreducible polynomials in GF(2)[x] of degree n. %and 2n, for an arbitrary odd number n. Let n=\prod_{i=1}^{\ell} p_i^{r_i} be the prime factorization of n, where p_i are odd primes. We show that \theta(n) tends to 1 and \theta(2n) is asymptotically not less than 2/3 when r_i are fixed and p_i tend to infinity. We also, describe an infinite series of values n_{s} such that \theta(n_{s}) is strictly less than \frac{1}{2}.
ePrint: https://eprint.iacr.org/2007/301
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