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Permutation polynomials of the form \(cx+\text {Tr}_{q^{l}/ q}(x^{a})\) and permutation trinomials over finite fields with even characteristic

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Abstract

Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. Particularly, permutation polynomials with few terms are more popular for their simple algebraic form and additional extraordinary properties. Very recently, G. Kyureghyan and M.E. Zieve (2016) studied permutation polynomials over \(\mathbb {F}_{q^{n}}\) of the form \(x+\gamma \text {Tr}_{q^{n}/q}(x^{k})\), where q is odd, and nine classes of permutation polynomials were constructed. In this paper, we present fifteen new classes of permutation polynomials of the form \(cx+\text {Tr}_{q^{l}/ q}(x^{a})\) over finite fields with even characteristic, which explain most of the examples with q = 2k, k > 1, k l < 14 and \(c\in \mathbb {F}_{q^{l}}^{*}\). Furthermore, we also construct four classes of permutation trinomials.

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Acknowledgments

We would like to thank the editor and the referees whose valuable comments and suggestions improve both the technical quality and the editorial quality of this paper.

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Authors and Affiliations

  1. College of Science, National University of Defense Technology, Changsha, 410073, China

    Kangquan Li, Longjiang Qu, Xi Chen & Chao Li

  2. State Key Laboratory of Cryptology, Beijing, 100878, China

    Longjiang Qu

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  1. Kangquan Li
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  2. Longjiang Qu
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  3. Xi Chen
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  4. Chao Li
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Corresponding author

Correspondence to Longjiang Qu.

Additional information

This work is supported by the National Basic Research Program of China (Grant No. 2013CB338002), the Nature Science Foundation of China (NSFC) under Grant 61272484, 11531002, 61572026, the Program for New Century Excellent Talents in University (NCET), the Basic Research Fund of National University of Defense Technology (No. CJ 13-02-01), and the Open Foundation of State Key Laboratory of Cryptology.

This article is part of the Topical Collection on Special Issue on Sequences and Their Applications

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Li, K., Qu, L., Chen, X. et al. Permutation polynomials of the form \(cx+\text {Tr}_{q^{l}/ q}(x^{a})\) and permutation trinomials over finite fields with even characteristic. Cryptogr. Commun. 10, 531–554 (2018). https://doi.org/10.1007/s12095-017-0236-7

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  • DOI: https://doi.org/10.1007/s12095-017-0236-7

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