Several classes of complete permutation polynomials over finite fields of even characteristic

Abstract

In this paper, we find three classes of complete permutation polynomials over finite fields of even characteristic. The first class of quadrinomials is complete in the sense of addition. The second and third classes of binomials and trinomials are complete in multiplication. Moreover, a result related to the complete property in multiplication of a special class of polynomials is also given.

Introduction

The concept of complete mappings that introduced by Mann [10] was originally defined on group. Afterwards considerable attentions were paid to the problem what kinds of groups admit the existence of such mappings. Niedrreiter and Robinson raised this problem over finite fields [12]. For a prime power q, let ${\mathbb{F}}_q$ be a finite field with q elements and ${\mathbb{F}}_q^{⁎}$ be its multiplicative group. A polynomial $f\in {\mathbb{F}}_q[x]$ is called a permutation polynomial (PP) if its induced mapping $c⟼f(c)$ is bijective over ${\mathbb{F}}_q$. If both f and $f+x$ are permutation polynomials over ${\mathbb{F}}_q$, f is called a complete permutation polynomial (CPP) over ${\mathbb{F}}_q$. Compared with the definitions based on groups, the CPP relevant problems over ${\mathbb{F}}_q$ are more intuitive and popular partially due to the fact that all mappings over finite fields have univariable polynomial representations. Besides closed connections with combinatorics and coding theory, the applications of CPPs in cryptography also begin to receive attentions [15], [16], [17]. Niedrreiter and Robinson gave a more detailed analysis on CPPs over finite fields [13], and some CPPs with low degree and several complete permutation binomials were obtained. However, since the construction of CPP actually can be seen as a sieve processing on known classes of PPs, the discovery of new CPPs is not easy, and known constructions are relatively rare. The construction of CPPs is usually linked with certain subfields or subsets of ${\mathbb{F}}_q$, and several methods including the recursive method, AGW criteria (see [1]), etc., have been proposed [9], [30], [31]. PPs with fewer terms attract people's attention due to the simple algebraic expressions. Thus, it is natural to find CPPs with fewer terms, among which the monomial CPPs are mostly investigated so far. If $\alpha x^d$ is a CPP over ${\mathbb{F}}_q$, the exponent d is called a CPP exponent. In [22], by the introduction of the additive character and the representation of elements in finite fields by subfield and polar-coordination, the authors transformed the discussions about their monomial CPPs into the determination of the number of roots of some equations with low degree, and they found three CPP exponents. Subsequently, a number of CPP exponents were discussed in [3], [4], [5], [11], [27], [28], [29], and some trinomial CPPs have been obtained in [9], [11], [22]. In [26], Wu and Lin generalized the trinomial CPP in [18] to the form $xL(x)+vx$, where $L(x)$ is a linearized polynomial linked with relative trace function ${\mathrm{Tr}}_k^{sk}(\cdot )$.

The purpose of this paper is to find more CPPs over ${\mathbb{F}}_{2^n}$ with fewer terms. The first subclass of CPPs comes from a class of quadrinomials, which was firstly proposed in [19] and then characterized more completely in [20]. Recently, the sufficient condition yields permutation quadrinomials obtained in [20] is proved to be necessary [8]. These polynomials are the first class of complete permutation quadrinomials so far as we know. Except for the complete permutation polynomials over finite fields in ordinary sense, we are also interested in the permutations which are multiplicatively complete, i.e., both $f(x)$ and $xf(x)$ are permutations of ${\mathbb{F}}_q$. We find that a class of binomials determined in [6] and a subclass of trinomials in [18] are multiplicatively complete. The main techniques used here include special transforms which can be applied to induce and process certain low-degree equations over finite fields. Further, by the characterization of the coefficients, we exactly count the number of these obtained CPPs over ${\mathbb{F}}_{2^n}$. In addition, when the polynomial $h{(x)}^{\frac{q-1}{d}}$ is a monomial or constant on the d-th roots of unity in ${\mathbb{F}}_q$, we give a proposition on CPPs in multiplication having the form $x^{r}h(x^{\frac{q-1}{d}})$.

The rest of this paper is organized as follows. In Section 2 we recall some useful definitions and preliminaries. In Section 3, we study three classes of CPPs, of which the first class is complete in the sense of addition, and the last two classes are complete in multiplication, as well as a result related to CPPs in multiplication for a special class of polynomials. Section 4 concludes the study.