Reversed Dickson polynomials over finite fields

Abstract

Reversed Dickson polynomials over finite fields are obtained from Dickson polynomials D n (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 F q with q < 200, with only one exception, all belong to the RDPP families established in this paper.

Necessary conditions for reversed Dickson polynomials of the second kind to be permutational

“…In [3], Hou et al discussed some connections between reversed Dickson PPs of F q and APN functions of F q , and obtained several families of reversed Dickson PPs. Throughout the reminder of this article, unless specified, p is always assumed to be an odd prime.…”

“…and E n (x, a) := On the other hand, Hou, Mullen, Sellers and Yucas [3] introduced the definition of the reversed Dickson polynomial of the first kind, denoted by D n (a, x), as follows…”

“…Hou, Mullen, Sellers and Yucas [3] studied the permutation properties of reversed Dickson polynomial D n (a, x) of the first kind. In fact, they showed that D n (a, x) is closely related to almost perfect nonlinear (APN) functions, and got several families of permutation polynomials from reversed Dickson polynomials of the first kind.…”

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

“…In [5], it was shown that the reversed Dickson polynomials of the first kind are closely related to Almost Perfect Nonlinear (APN) functions which have applications in cryptography. Hou and Ly found more properties of the reversed Dickson polynomials of the first kind and necessary conditions for them to be a permutation of F q ; see [4].…”

“…Motivated by [1], [2], [3], [4], and [5], we fix k and study the general properties and permutation property of the reversed Dickson polynomials of the (k + 1)-th kind over finite fields. The results obtained in this paper unify and generalize many existing results on reversed Dickson polynomials.…”