Two classes of permutation trinomials with Niho exponents over finite fields with even characteristic
Introduction
Let be the finite field with q elements. A polynomial is called a permutation polynomial of if its associated polynomial mapping from 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 were intensively studied [22], [26], [28], [34]. Permutation trinomials of the form 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 . For instance, let , . Given , finding conditions on a, b that are sufficient and necessary for f to be a permutation polynomial of is a hard question, in most known cases the coefficients are assumed to be trivial, that is, , and a number of possible pairs with trivial coefficients such that f in (1) are permutations have been determined. A few years ago, Hou determined all the coefficients a, making f in (1) a permutation polynomial in the case [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, making f in (1) permutations for the case [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 for even q. They found sufficient conditions on the coefficients a, b for f in (1) to be a permutation polynomial of 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 with q even, and a, . They proved that is a permutation of if and for all . Very recently, Hou showed that the sufficient condition for the case found in [24] is also necessary using a strategy similar to that of [11] through quite involved computations [12].
Let be a positive integer. The purpose of this paper is to find more new permutation trinomials as in (1) over with general coefficients a and b. Recently, Wang, Zhang and Zha studied the permutation behavior of over for m even and with k being an odd positive integer satisfying [30, Section 3]. Inspired by this work, we investigate a family of infinite classes of permutation trinomials with the form defined on , however, for m being any positive integer, with k being an odd positive integer satisfying , and a, . 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 with degree . 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 , where k is a positive integer satisfying and . Assuming that a, , we obtain some sufficient conditions on a and b such that is a permutation of .
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 to have exactly two solutions (and one solution, respectively) in the unit circle, where a, , and k satisfies . In Section 3, we propose two classes of permutation trinomials having the form as in (1). Section 4 concludes the paper.
Section snippets
Preliminaries
Let m and n be two positive integers satisfying , we use to denote the trace function form to , i.e.,
The following result was discovered independently by several authors. It is also worth pointing out that it can also be derived as a special case of the AGW criterion ([1, Lemma 1]).
Lemma 2.1 [2], [22], [26], [28], [34] Let ℓ, with , and let . Then permutes if and only if both and
Two classes of permutation trinomials from Niho exponents
In this section, we consider two classes of permutation trinomials from Niho exponents with the form (1) over for q even. It can be easily seen that if is a permutation of , where , then . Thus, the assumption that is needed when we consider the permutation property of f. We have the following theorem.
Theorem 3.1 Let k be an odd positive integer, and m, n and d be positive integers such that , , and . Let a,
Conclusion
In this paper, we investigate the possible coefficients a, , for which (1) is a permutation over , in the two cases: 1) , where , and k is an odd integer such that and ; 2) , where k is a positive integer with such that . The conditions in Theorem 3.1 are proved to be sufficient, and numerical result suggests that they are also necessary. It is a question worthy to be investigated in the
CRediT authorship contribution statement
Lijing Zheng: Conceptualization, Methodology, Software, Investigation, Writing - original draft. Haibin Kan: Resources, Supervision. Jie Peng: Resources, Writing - review & editing, Data curation.
Acknowledgements
The authors would like to thank Professor Xiang-dong Hou for his helpful suggestions which improved this paper. The authors thank the anonymous reviewers for their valuable suggestions which significantly improved both the quality and the presentation of this paper. This research is supported by National Key R & D Program of China (No. 2019YFB2101703) and National Natural Science Foundation of China (Grant Nos. 61672166, 11701488, 61972258 and U19A2066), and Scientific Research Fund of Hunan
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