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New results of sparse permutation polynomials with trace functions over \(\mathbb {F}_{q^n}\)

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Abstract

Permutation polynomials with sparse forms over finite fields attract researchers’ great interest and have important applications in many areas of mathematics and engineering. In this paper, by investigating the exponents (s, i) and the coefficients \(a,b\in \mathbb {F}_{q}^{*}\), we present some new results of permutation polynomials of the form \(f(x)= ax^{q^i(q^{2}-q+1)} + bx^{s} + \textrm{Tr}_{q^n/q}(x)\) over \(\mathbb {F}_{q^n}\) (\(n=2\) or 3). The permutation property of the new results is given by studying the number of solutions of special equations over \(\mathbb {F}_{q^n}\).

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Acknowledgements

The authors wish to thank the anonymous referees for valuable comments which significantly improved both the quality and presentation of this paper. This work is supported in part by the National Natural Science Foundation of China under Grants 12361107 and 62072222, and in part by Lanzhou Youth Scientific and Technological Talents Innovation Project under Grant 2023-QN-105, and in part by the Key Project of Gansu Natural Science Foundation (No.23JRRA685), and in part by the Funds for Innovative Fundamental Research Group Project of Gansu Province (No.23JRRA684).

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Authors and Affiliations

  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, China

    Yan-Ping Wang

  2. Key Laboratory of Cryptography and Data Analytics, Northwest Normal University, Lanzhou, 730070, China

    Yan-Ping Wang

  3. Gansu Provincial Research Center for Basic Disciplines of Mathematics and Statistics, Lanzhou, 730070, China

    Yan-Ping Wang

  4. College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046, China

    Zhengbang Zha

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  1. Yan-Ping Wang
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Correspondence to Zhengbang Zha.

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Wang, YP., Zha, Z. New results of sparse permutation polynomials with trace functions over \(\mathbb {F}_{q^n}\). AAECC (2024). https://doi.org/10.1007/s00200-024-00658-2

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  • DOI: https://doi.org/10.1007/s00200-024-00658-2

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