Permutation Trinomials Over Finite Fields with Even Characteristic

Abstract

Abstract. Permutation polynomials have been a subject of study for a long time and have applications in many areas of science and engineering. However, only a small number of specific classes of permutation polynomials are described in the literature so far. In this paper we present a number of permutation trinomials over finite fields, which are of different forms.Key words. Cryptography, difference set, linear code, permutation polynomial, trinomial AMS subject classifications. 11C08, 05A051. Introduction. A permutation polynomial f (x) over a finite field is a polynomial that acts as a permutation of the elements of the field, i.e. the map x → f (x) is one-to-one. Permutation polynomials are closely related to highly nonlinear functions [13,14] and other areas of mathematics. They have applications in combinatorial designs [13,14], coding theory [8,21,31] and cryptography [28,29,30]. For instance, Dickson permutation polynomials of order five, i.e., D 5 (x, a) = x 5 + ax 3 − a 2 x over F 3 m , led to a 70-year research breakthrough in combinatorics [13], gave a family of perfect nonlinear functions for cryptography [13], generated good linear codes [5] for data communication and storage, and produced optimal signal sets for CDMA communications [12], to mention only a few applications of these Dickson permutation polynomials. Information about properties, constructions, and applications of permutation polynomials can be found in Cohen [7], Lidl and Niederreiter [24], and Mullen [26]. Some recent progress on permutation polynomials can be found in [1,4,18,20,22,23,27,32,33,34,35,36].Permutation polynomials with fewer terms over finite fields with even characteristics are in particular interesting. For example, in the study of Almost perfect nonlinear (APN) mappings which are of interest for their applications in cryptography, Dobbertin first proved a well-known conjecture of Welch stating that for odd n = 2m + 1, the power function x 2m+3 is even maximally nonlinear [15] or, in other terms, that the crosscorrelation function between a binary maximum-length linear shift register sequence of degree n and a decimation of that sequence by 2m + 3 takes on precisely the three values −1, −1 ± 2 m+1 . The method in fact relies on the discovery of a class of permutation trinomials. Around the same time period, Dobbertin proved Niho's conjecture similarly using a class of permutation pentanomials [16]. Another example of demonstrating the application of permutation polynomials with fewer terms in constructing cyclic codes can be found in [10]. We note that these interesting

Several classes of permutation trinomials from Niho exponents

“…1) The polynomials f 3 (x), f 4 (x), f 6 (x) can also be permutations over certain finite fields with odd characteristic p, see [15,25,4] respectively.…”

“…Since x ∈ U and x = 1, taking γ = 1/x ∈ F 2 n \F 2 m and multiplying (z + γ) 4 on both sides of the above equation, it gives…”

“…The construction of permutation polynomials with few terms attracts researchers' interest in recent years due to their simple form [4,10,12,16,20,21,23,24,25] and some additional nice properties, for example, their close relations with the Helleseth's −1 conjecture [7], Welch's conjecture [5] and…”

“…Currently, only a small number of specific classes of permutation binomials and trinomials are described in the literature, see the survey paper [9] and two recent papers [4,16] on permutation binomials and trinomials over finite fields. Tu et al in [20] investigated the permutation property of a class of polynomials over F 2 n of the form…”

A note on permutation polynomials over finite fields

“…In the same paper, they also proposed the following two conjectures, which can be used to obtain two new classes of permutation trinomials with the form (1). More recent progress on permutation trinomials can be found in [2,6,7,8,9,10,12,13,14,15,16,20].…”