Two classes of permutation trinomials with Niho exponents over finite fields with even characteristic

Abstract

In this paper, we consider two classes of permutation trinomials with Niho-type exponents over the finite field ${\mathbb{F}}_{2^{2m}}$. Some sufficient conditions are obtained to characterize the coefficients of the permutation trinomials. Our numerical result suggests that those sufficient conditions for one class are also necessary.

Introduction

Let ${\mathbb{F}}_q$ be the finite field with q elements. A polynomial $f\in {\mathbb{F}}_q[x]$ is called a permutation polynomial of ${\mathbb{F}}_q$ if its associated polynomial mapping $f:c\mapsto f(c)$ from ${\mathbb{F}}_q$ to itself is a bijection. Permutation polynomials over finite fields have wide applications in coding theory, cryptography, and combinatorial design theory [21], and we refer the readers to [10], [19], [29] for more details of the recent advances and contributions to the area.

Permutation polynomials with a few terms are of great interest for their simple algebraic forms and some additional properties [3], [4], [5], [6], [7], [9], [11], [12], [14], [15], [16], [17], [18], [23], [24], [25], [27], [30], [31], [32], [33]. In particular, permutation polynomials having the form $x^{r}f(x^{(q-1)/\ell })$ were intensively studied [22], [26], [28], [34]. Permutation trinomials of the form

$$f(x)=x+a{x}^{s(q-1)+1}+b{x}^{t(q-1)+1}\in {\mathbb{F}}_{q^2}[x],$$

where s and t are two integers, have attracted much attention in recent years [5], [6], [8], [9], [11], [12], [15], [17], [18], [24], [25]. Note that s, t should be interpreted as modulo $q+1$. For instance, let $q=2^m$, $(s,t)=(\frac{1}{2},\frac{3}{4})=(2^{m-1}+1,2^{m-2}+1)$. Given $(s,t)$, finding conditions on a, b that are sufficient and necessary for f to be a permutation polynomial of ${\mathbb{F}}_{q^2}$ is a hard question, in most known cases the coefficients are assumed to be trivial, that is, $(a,b)=(1,1)$, and a number of possible pairs $(s,t)$ with trivial coefficients such that f in (1) are permutations have been determined. A few years ago, Hou determined all the coefficients a, $b\in {\mathbb{F}}_{q^2}$ making f in (1) a permutation polynomial in the case $(s,t)=(1,2)$ [8], [9]. To the best of our knowledge, this excellent work is the first instance that all possible coefficients for f in (1) to be a permutation polynomial are completely determined. Recently, Tu and Zeng determined all possible coefficients a, $b\in {\mathbb{F}}_{q^2}$ making f in (1) permutations for the case $(s,t)=(-\frac{1}{2},\frac{1}{2})$ [24]. They transformed the problem to that of investigating the number of solutions in the unit circle by using the well-known result (Lemma 2.1 in Section 2). Then they can find the sufficient conditions. By a known result on the Kloosterman sum, they showed that the sufficient conditions are also necessary. In a recent work [25], Tu et al. investigated the case $(s,t)=(2,q)$ for even q. They found sufficient conditions on the coefficients a, b for f in (1) to be a permutation polynomial of ${\mathbb{F}}_{q^2}$ and due to some numerical results they conjectured that the sufficient conditions are also necessary. The conjecture has been confirmed by Bartoli in [3] and later by Hou in [11] utilizing the Hasse-Weil bound via different approaches. In [24], Tu and Zeng considered the case $(s,t)=(\frac{1}{4},\frac{3}{4})$ with q even, and a, $b\in {\mathbb{F}}_{q^2}^{⁎}$. They proved that $f(x)=x+a{x}^{s(q-1)+1}+b{x}^{t(q-1)+1}$ is a permutation of ${\mathbb{F}}_{q^2}$ if $b=a^{2-q}$ and $x^3+x+a^{-1-q}\ne 0$ for all $x\in {\mathbb{F}}_q$. Very recently, Hou showed that the sufficient condition for the case $(s,t)=(\frac{1}{4},\frac{3}{4})$ found in [24] is also necessary using a strategy similar to that of [11] through quite involved computations [12].

Let $n=2m$ be a positive integer. The purpose of this paper is to find more new permutation trinomials $f(x)$ as in (1) over ${\mathbb{F}}_{2^n}$ with general coefficients a and b. Recently, Wang, Zhang and Zha studied the permutation behavior of $x+x^{2^m}+x^d$ over ${\mathbb{F}}_{2^n}$ for m even and $d={\sum }_{i=0}^{\frac{n}{2}}{2}^{ik}$ with k being an odd positive integer satisfying $\gcd (k,\frac{m}{2})=1$ [30, Section 3]. Inspired by this work, we investigate a family of infinite classes of permutation trinomials with the form $f(x)=x+a{x}^{2^m}+b{x}^d$ defined on ${\mathbb{F}}_{2^n}$, however, for m being any positive integer, $d={\sum }_{i=0}^{\frac{n}{2}}{2}^{ik}$ with k being an odd positive integer satisfying $\gcd (k,m)=1$, and a, $b\in {\mathbb{F}}_{2^m}^{⁎}$. We find that the exponents of the family of these infinite classes are of Niho-type, and then with Lemma 2.1 we need to analyze some affine equations over ${\mathbb{F}}_{2^m}$ with degree $2^k$. We prove that the conditions on a and b are sufficient, and the numerical experiments indicate that they are also necessary. On the other hand, inspired by a recent work of Li and Helleseth [18], we consider the case of $(s,t)=(\frac{2^k}{2^k-1},\frac{-1}{2^k-1})$, where k is a positive integer satisfying $1\le k\le m-1$ and $\gcd (2^k-1,2^m+1)=1$. Assuming that a, $b\in {\mathbb{F}}_{2^m}^{⁎}$, we obtain some sufficient conditions on a and b such that $f(x)=x+a{x}^{s(2^m-1)+1}+b{x}^{t(2^m-1)+1}$ is a permutation of ${\mathbb{F}}_{2^n}$.

The remainder of this paper is organized as follows. In Section 2, we introduce some basic concepts and investigate the conditions on a, b that are sufficient and necessary for the affine equation $x^{2^k}+ax+b=0$ to have exactly two solutions (and one solution, respectively) in the unit circle, where a, $b\in {\mathbb{F}}_{2^m}^{⁎}$, and k satisfies $\gcd (k,m)=1$. In Section 3, we propose two classes of permutation trinomials having the form as in (1). Section 4 concludes the paper.