Permutations polynomials of the form G(X)kL(X) and curves over finite fields

Abstract

For a positive integer k and a linearized polynomial L(X), polynomials of the form \(P(X)=G(X)^{k}-L(X) \in {\mathbb F}_{q^{n}}[X]\) are investigated. It is shown that when L has a non-trivial kernel and G is a permutation of \(\mathbb {F}_{q^{n}}\), then P(X) cannot be a permutation if \(\gcd (k,q^{n}-1)>1\). Further, necessary conditions for P(X) to be a permutation of \(\mathbb {F}_{q^{n}}\) are given for the case that G(X) is an arbitrary linearized polynomial. The method uses plane curves, which are obtained via the multiplicative and the additive structure of \(\mathbb {F}_{q^{n}}\), and their number of rational affine points.

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Acknowledgements

N.A. is supported by B.A.CF-19-01967.

We would like to thank Wilfried Meidl for his useful comments, which helped to improve the presentation of the manuscript considerably.

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Correspondence to Nurdagül Anbar.

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Anbar, N., Kaşıkcı, C. Permutations polynomials of the form G(X)kL(X) and curves over finite fields. Cryptogr. Commun. (2021). https://doi.org/10.1007/s12095-020-00465-9

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Keywords

  • Curves/function fields
  • Permutation polynomials
  • Rational points/places

Mathematics Subject Classification (2010)

  • 11T06
  • 14H05