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Regular complete permutation polynomials over \({\mathbb {F}}_{2^{n}}\)

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Abstract

Applications of permutation polynomials in cryptography are closely related to their cycle structures. For example, many block ciphers use permutation polynomials defined over \({\mathbb {F}}_{2^{n}}\) with few fixed points and nontrivial cycles of length 2 as their S-boxes. In addition, permutation polynomials with long cycles have been widely used to generate keystream sequences. Complete permutation polynomials over \({\mathbb {F}}_{2^{n}}\) have a single fixed point and good bit independence, which can provide good choices for S-boxes in block ciphers. However, there are only a limited number of known infinite families of complete permutation polynomials with explicit cycle structures. In this paper, we construct two classes of r-regular complete permutation polynomials g(x) over \({\mathbb {F}}_{2^{n}}\), i.e., g(x) is a complete permutation polynomial over \({\mathbb {F}}_{2^n}\) with a single fixed point and all the other cycles of the same length r. The first class of regular complete permutation polynomials over \({\mathbb {F}}_{2^{km}}\) is based on some permutation polynomials over \({\mathbb {F}}_{2^{m}}\) for positive integers k, m, and the second class of regular complete permutation polynomials over \({\mathbb {F}}_{2^{n}}\) for a positive integer n is obtained from piecewise functions. In addition, when g(x) is a r-regular complete permutation polynomial, we also consider the r-regular complete permutation property of \(g(x)+x\).

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Acknowledgements

The authors would like to thank the anonymous reviewers for their valuable comments and helpful suggestions which improved both the quality and presentation of this paper. The work was supported by the Application Foundation Frontier Project of Wuhan Science and Technology Bureau under Grant 2020010601012189, by the National Nature Science Foundation of China (NSFC) under Grants 61761166010 and 12101209, and by the National Key Research and Development Program of China (2021YFA1000604). The work of S. Zhang was also supported by the Research Foundation of Education Bureau of Hubei Province, China under Grant D2020104. The work of X. Xu was also supported by the Research Project of Department of Education of Hubei Province of China under Grant D20214501 and the Research Project of Hubei Polytechnic University under Grants 20xjz01R and 15xjz01Y.

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Authors and Affiliations

  1. School of Mathematics and Physics, Hubei Polytechnic University, Huangshi, 435003, China

    Xiaofang Xu

  2. Faculty of Mathematics and Statistics, Hubei Key Laboratory of Applied Mathematics, Hubei University, Wuhan, 430062, China

    Xiangyong Zeng & Shasha Zhang

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  1. Xiaofang Xu
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  2. Xiangyong Zeng
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Correspondence to Xiangyong Zeng.

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Communicated by P. Charpin.

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Xu, X., Zeng, X. & Zhang, S. Regular complete permutation polynomials over \({\mathbb {F}}_{2^{n}}\). Des. Codes Cryptogr. 90, 545–575 (2022). https://doi.org/10.1007/s10623-021-00992-3

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  • DOI: https://doi.org/10.1007/s10623-021-00992-3

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