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On inverses of permutation polynomials of the form \(x\left( x^{s} -a\right) ^{(q^m-1)/s}\) over \(\mathbb {F}_{q^n}\)

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Abstract

The inverse of a class of permutation polynomials (PPs) of the form \(x(x^{s} -a)^{(q^m - 1)/s}\) over \(\mathbb {F}_{q^n}\) is given. More simplified expressions of some subclasses of this family are presented, such as \(x(x^{q-1} - a)^{q+1}\) and \(x(x^{q+1} - a)^{q-1}\) over \(\mathbb {F}_{q^3}\), \(x(x^2 -a)^3\) and \(x(x^3 -a)^2\) over \(\mathbb {F}_{7^n}\). These expressions and some known results solve the problem of determining the inverses of all PPs of degree 7 over finite fields. In addition, an explicit criteria for \(x(x^{s} -a)^{(q - 1)/s}\) to be an involution of \(\mathbb {F}_{q}\) is established.

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Acknowledgements

The authors are grateful to the referees for many useful suggestions. This work was partially supported by the Natural Science Foundation of Shandong (No. ZR2021MA061), the Guangdong Basic and Applied Basic Research Foundation (Nos. 2021A1515011954, 2021A1515011904), the National Key R &D Program of China (No. 2021YFB3100200), and the National Natural Science Foundation of China (No. 62072222).

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Authors and Affiliations

  1. School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, China

    Yanbin Zheng

  2. School of Computer Science, Guangdong Polytechnic Normal University, Guangzhou, 510665, China

    Yanbin Zheng

  3. School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, China

    Yuyin Yu

  4. School of Mathematical Sciences, Luoyang Normal University, Luoyang, 471022, China

    Zhengbang Zha

  5. Network and Information Center, Qufu Normal University, Qufu, 273165, China

    Xingchen Zhou

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  1. Yanbin Zheng
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Correspondence to Xingchen Zhou.

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Communicated by K.-U. Schmidt.

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Zheng, Y., Yu, Y., Zha, Z. et al. On inverses of permutation polynomials of the form \(x\left( x^{s} -a\right) ^{(q^m-1)/s}\) over \(\mathbb {F}_{q^n}\). Des. Codes Cryptogr. 91, 1165–1181 (2023). https://doi.org/10.1007/s10623-022-01142-z

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