Elliptic curves and explicit enumeration of irreducible polynomials with two coefficients prescribed

Let $F_q$ be a finite field of characteristic $p=2,3$. We give the number of irreducible polynomials x^m+a_{m-1}x^{m-1}+...+a_0\in\F_q[x] with $a_{m-1}$ and $a_{m-3}$ prescribed for any given $m$ if $p=2$, and with $a_{m-1}$ and $a_1$ prescribed for $m=1,...,10$ if $p=2,3$.Comment: 17 pages, Part of the results was presented at the Polynomials over Finite Fields and Applications workshop at Banff International Research Station, Canad

Dickson polynomials, hyperelliptic curves and hyper-bent functions

In this paper, we study the action of Dickson polynomials on subsets of finite fields of even characteristic related to the trace of the inverse of an element and provide an alternate proof of a not so well-known result. Such properties are then applied to the study of a family of Boolean functions and a characterization of their hyper-bentness in terms of exponential sums recently proposed by Wang et al. Finally, we extend previous works of Lisoněk and Flori and Mesnager to reformulate this characterization in terms of the number of points on hyperelliptic curves and present some numerical results leading to an interesting problem.