Open Access
 Issue Wuhan Univ. J. Nat. Sci. Volume 27, Number 5, October 2022 372 - 374 https://doi.org/10.1051/wujns/2022275372 11 November 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

0 Introduction

Let be a positive integer and be the finite field having elements. -to-1 mappings (see Definition 1) of finite fields have wide applications in cryptography, finite geometry, coding theory and combinatorial design, and it has become an interesting topic in finite fields. Especially, when , -to-1 mappings become permutation polynomials.

The study of permutation polynomials has a long history in finite fields, many classes of permutation polynomials are studied, and there are a few surveys on permutation polynomials[1-3]. When , Mesnager and Qu studied 2-to-1 mappings of finite fields, and gave many explicit constructions of 2-to-1 mappings. Recently, Li et al continued to study 2-to-1 mappings, and constructed several classes of 2-to-1 trinomials and quadrinomials over finite fields. Yuan et al constructed a few classes of 2-to-1 mappings having the form over . More recently, Gao et al generalized the definition of 2-to-1 mappings to -to-1 mappings. Niu et al showed some approaches to construct -to-1 mappings of finite fields. Therefore, to find more classes of -to-1 mappings of finite fields still is an open problem. In this paper, we focus on constructing -to-1 binomials over . By using monomials and piecewise method, the authors[9-12] characterized several classes of permutation polynomials. Activated by the method, we generalized this approach to construct some -to-1 binomials with the form of .

1 Preliminaries

In Ref.[8], the authors gave the definition of -to-1 mappings as follows:

Definition 1   Let be a mapping from one finite set to another finite set . Then is called an -to-1 mapping if one of the following two cases holds:

(1) if divides , for any in , it has either or preimages in ;

(2) if does not divide , for almost in , it has either or preimages in , and for only one exception element, it has exactly preimages in .

Lemma 1   Let be a monomial over , where , and be a subset in . Then is an -to-1 mapping over if and only if .

In Ref.[8], the authors established an AGW-like criterion for -to-1 mappings in the following.

Lemma 2   Let and be finite sets with , and . Let , ,, and be maps such that , where both and are surjective.

Assume that is a bijection form to for every . There are three statements as follows:

1) is an -to-1 mapping of ;

2) is an -to-1 mapping from to ;

3) divides and that does not divide and the exception which has preimages in satisfies where .

Then, if 1) holds, so does 2). If both 2) and 3) hold, so does 1).

As a special case of Lemma 2, the authors of Ref.[8] gave the following result.

Lemma 3   Let be a prime power, be a positive integer such that and . Let , where such that if . Then is an -to-1 mapping over if and only if is an -to-1 mapping over .

2 Main Results

In this section, we will focus on constructing some -to-1 mappings over .

Theorem 1   Let be a prime power, and be positive integers having . Let in satisfy and have no roots in . Then is an -to-1 mapping over if and only if .

Proof   By using Lemma 3, we know that is an -to-1 mapping over if and only if is an -to-1 mapping over . Thus we only need to show that is an -to-1 mapping over if and only if .

By having no roots in , we can rewrite as

Since , we get that . Thus . Then by Lemma 1, it follows that is an -to-1 mapping over if and only if .

The proof of Theorem 1 is completed.

Furthermore, if we divide as and , then the -to-1 property on is translated to that on and .

Lemma 4   Let and be positive integers with , and for . For , if for . Then is an -to-1 mapping of if and only if each of following is true:

(1) , for ;

(2) for and .

Proof   By using Lemma 1 and Theorem 1.2 in Ref.[11], we can easily get the desired result.

By using Lemma 4, we can get the following result.

Theorem 2   Let be a prime power with , and be positive integers having and . Let be an even number and in satisfy . Then is an -to-1 mapping over if and only if .

Proof   By Lemma 3, we know that is an -to-1 mapping of if and only if and is an -to-1 mapping over .

In the following, we will focus on proving that is an -to-1 mapping over .

First, we consider the case of . Since , then is odd. By using , it implies from being even that for any . Thus

It follows from that . We get that is an -to-1 mapping of if and only if .

Next, for , it is also trivial to find that has no roots in . We reduce that . It is easy to conclude that is an -to-1 mapping of if and only if .

Then by Lemma 4, we get that is an -to-1 mapping over if and only if . Therefore, we can conclude that is an -to-1 mapping over if and only if . We complete the proof of Theorem 2.

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