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Compositional inverses of permutation polynomials of the form xrh(xs) over finite fields

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Abstract

The study of computing compositional inverses of permutation polynomials over finite fields efficiently is motivated by an open problem proposed by G. L. Mullen (1991), as well as the potential applications of these permutation polynomials (Dillon 1974, Khachatrian and Kyureghyan, Discrete Appl. Math. 216, 622–626 2017, Lidl 1985, Lidl and Müller 1984, Rivest et al., ACM Commun. Comput. Algebra. 1978, 120–126 1976, Schwenk and Huber, Electron. Lett. 34, 759–760 1998). It is well known that every permutation polynomial over a finite field \(\mathbb {F}_{q}\) can be reduced to a permutation polynomial of the form xrh(xs) with s∣(q − 1) and \(h(x) \in \mathbb {F}_{q}[x]\) (Akbary et al., Finite Fields Appl. 15(2), 195–206 2009, Wang, Finite Fields Appl. 22, 57–69 2013). Recently, several explicit classes of permutation polynomials of the form xrh(xs) over \({\mathbb F}_{q}\) have been constructed. However, all the known methods to compute the compositional inverses of permutation polynomials of this form seem to be inadequately explicit, which could be a hurdle to potential applications. In this paper, for any prime power q, we introduce a new approach to explicitly compute the compositional inverse of a permutation polynomial of the form xrh(xs) over \({\mathbb F}_{q}\), where s∣(q − 1) and \(\gcd (r,q-1)= 1\). The main idea relies on transforming the problem of computing the compositional inverses of permutation polynomials over \({\mathbb F}_{q}\) into computing the compositional inverses of two restricted permutation mappings, where one of them is a monomial over \(\mathbb {F}_{q}\) and the other is the polynomial xrh(x)s over a particular subgroup of \(\mathbb {F}_{q}^{*}\) with order (q − 1)/s. This is a multiplicative analog of Tuxanidy and Wang (Finite Fields Appl. 28, 244–281 2014), Wu and Liu (Finite Fields Appl. 24, 136–147 2013). We demonstrate that the inverses of these two restricted permutations can be explicitly obtained in many cases. As consequences, many explicit compositional inverses of permutation polynomials given in Zieve (Proc. Am. Math. Soc. 137, 2209–2216 2009), Zieve (arXiv:1310.0776, 2013), Zieve (arXiv:1312.1325v3, 2013) are obtained using this method.

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Acknowledgments

We would like to thank the editor and the anonymous 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

  2. School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, K1S 5B6, Canada

    Qiang Wang

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  1. Kangquan Li
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  2. Longjiang Qu
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Correspondence to Longjiang Qu.

Additional information

The research of Longjiang Qu is partially supported by the National Key R&D Program of China (No. 2017YFB0802000), the Nature Science Foundation of China (NSFC) under Grant 61722213, 11531002, 61572026, 61672530. The research of Qiang Wang is partially supported by NSERC of Canada.

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Li, K., Qu, L. & Wang, Q. Compositional inverses of permutation polynomials of the form xrh(xs) over finite fields. Cryptogr. Commun. 11, 279–298 (2019). https://doi.org/10.1007/s12095-018-0292-7

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  • DOI: https://doi.org/10.1007/s12095-018-0292-7

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