Issue 
Wuhan Univ. J. Nat. Sci.
Volume 27, Number 5, October 2022



Page(s)  372  374  
DOI  https://doi.org/10.1051/wujns/2022275372  
Published online  11 November 2022 
Mathematics
CLC number: O 156.1
A Class of nto1 Binomials over Finite Fields
^{1}
School of Mathematics and Big Data, Chongqing University of Education, Chongqing 400065, China
^{2}
School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China
^{†} To whom correspondence should be addressed. Email: yanl930@163.com
Received:
22
June
2022
to1 mappings have many applications in combinatorial design, coding theory and cryptography. In this paper, by using piecewise method and monomials on subsets of th roots of unity, we show a class of to1 binomials having the form over .
Key words: finite field / nto1 mapping / binomial
Biography: QIN Xiaoer, male, Ph. D., Associate professor, research direction: finite fields. Email: qincn328@sina.com
Fundation item: Supported by the National Natural Science Foundation of China (11926344)
© Wuhan University 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. to1 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 , to1 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^{[13]}. When , Mesnager and Qu studied 2to1 mappings of finite fields, and gave many explicit constructions of 2to1 mappings. Recently, Li et al continued to study 2to1 mappings, and constructed several classes of 2to1 trinomials and quadrinomials over finite fields. Yuan et al constructed a few classes of 2to1 mappings having the form over . More recently, Gao et al generalized the definition of 2to1 mappings to to1 mappings. Niu et al showed some approaches to construct to1 mappings of finite fields. Therefore, to find more classes of to1 mappings of finite fields still is an open problem. In this paper, we focus on constructing to1 binomials over . By using monomials and piecewise method, the authors^{[912]} characterized several classes of permutation polynomials. Activated by the method, we generalized this approach to construct some to1 binomials with the form of .
1 Preliminaries
In Ref.[8], the authors gave the definition of to1 mappings as follows:
Definition 1 Let be a mapping from one finite set to another finite set . Then is called an to1 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 to1 mapping over if and only if .
In Ref.[8], the authors established an AGWlike criterion for to1 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 to1 mapping of ;
2) is an to1 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 to1 mapping over if and only if is an to1 mapping over .
2 Main Results
In this section, we will focus on constructing some to1 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 to1 mapping over if and only if .
Proof By using Lemma 3, we know that is an to1 mapping over if and only if is an to1 mapping over . Thus we only need to show that is an to1 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 to1 mapping over if and only if .
The proof of Theorem 1 is completed.
Furthermore, if we divide as and , then the to1 property on is translated to that on and .
Lemma 4 Let and be positive integers with , and for . For , if for . Then is an to1 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 to1 mapping over if and only if .
Proof By Lemma 3, we know that is an to1 mapping of if and only if and is an to1 mapping over .
In the following, we will focus on proving that is an to1 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 to1 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 to1 mapping of if and only if .
Then by Lemma 4, we get that is an to1 mapping over if and only if . Therefore, we can conclude that is an to1 mapping over if and only if . We complete the proof of Theorem 2.
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