New classes of complete permutation polynomials

Lisha Li,Chaoyun Li,Chunlei Li,Xiangyong Zeng

doi:10.1016/j.ffa.2018.10.001

Lisha Li, Chaoyun Li + Show 2 more

Open Access

https://doi.org/10.1016/j.ffa.2018.10.001

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Abstract

In this paper, we propose several classes of complete permutation polynomials over a finite field based on certain polynomials over its subfields or subsets. In addition, a class of complete permutation trinomials with Niho exponents is studied, and the number of these complete permutation trinomials is also determined.

Highlights

For a prime power q, let Fq be the finite field of q elements
complete permutation polynomial (CPP) were introduced in connection with the construction of orthogonal Latin squares [10]
We present two classes of new CPPs over F2m of the form axk + bx, which together with the known ones in the literature can generate many new CPPs of F2n

Summary

Introduction

For a prime power q, let Fq be the finite field of q elements. A polynomial f ∈ Fq[x] is called a permutation polynomial (PP) if the induced polynomial function f : c → f (c) from Fq to itself permutes Fq. In a recent work of [18], CPPs with fewer terms over finite fields were studied and the authors proposed three classes of monomial CPPs and a class of trinomial CPPs over F2n. We study a class of trinomial CPPs with Niho-type terms. The coefficients of such trinomial CPPs are determined by solving certain quadratic equations over the unit circle. The number of such CPPs is determined.

Preliminaries

New Niho-type CPP trinomials over Fp2m