Reversed Dickson polynomials over finite fields

AbstractReversed Dickson polynomials over finite fields are obtained from Dickson polynomials Dn(x,a) over finite fields by reversing the roles of the indeterminate x and the parameter a. We study reversed Dickson polynomials with emphasis on their permutational properties over finite fields. We show that reversed Dickson permutation polynomials (RDPPs) are closely related to almost perfect nonlinear (APN) functions. We find several families of nontrivial RDPPs over finite fields; some of them arise from known APN functions and others are new. Among RDPPs on Fq with q<200, with only one exception, all belong to the RDPP families established in this paper

Reversed Dickson polynomials

We investigate fixed points and cycle types of permutation polynomials and complete permutation polynomials arising from reversed Dickson polynomials of the first kind and second kind over $\mathbb{F}_p$. We also study the permutation behaviour of reversed Dickson polynomials of the first kind and second kind over $\mathbb{Z}_m$. Moreover, we prove two special cases of a conjecture on the permutation behaviour of reversed Dickson polynomials over $\mathbb{F}_p$.Comment: 40 page

Reversed Dickson polynomials of the (k+1)(k+1)-th kind over finite fields, II

Let $p$ be an odd prime. In this paper, we study the permutation behaviour of the reversed Dickson polynomials of the $(k+1)$-th kind $D_{n,k}(1,x)$ when $n=p^{l_1}+3$, $n=p^{l_1}+p^{l_2}+p^{l_3}$, and $n=p^{l_1}+p^{l_2}+p^{l_3}+p^{l_4}$, where $l_1, l_2$, $l_3$, and $l_4$ are non-negative integers. A generalization to $n=p^{l_1}+p^{l_2}+\cdots +p^{l_i}$ is also shown. We find some conditions under which $D_{n,k}(1,x)$ is not a permutation polynomial over finite fields for certain values of $n$ and $k$. We also present a generalization of a recent result regarding $D_{p^l-1,1}(1,x)$ and present some algebraic and arithmetic properties of $D_{n,k}(1,x)$.Comment: 30 pages, Section 7 and subsection 8.1 are adde