Paper 2007/301 Original: eprint.iacr.org/2007/301 ↗

On Asymptotic Behavior of the Ratio Between the Numbers of Binary Primitive and Irreducible Polynomials

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 $\frac{1}{2}$.