Designs, Codes and Cryptography

, Volume 86, Issue 8, pp 1589–1599 | Cite as

Permutation polynomials of the type \(x^rg(x^{s})\) over \({\mathbb {F}}_{q^{2n}}\)

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Abstract

We provide some new families of permutation polynomials of \({\mathbb {F}}_{q^{2n}}\) of the type \(x^rg(x^{s})\), where the integers r, s and the polynomial \(g \in {\mathbb {F}}_q[x]\) satisfy particular restrictions. Some generalizations of known permutation binomials and trinomials that involve a sort of symmetric polynomials are given. Other constructions are based on the study of algebraic curves associated to certain polynomials. In particular we generalize families of permutation polynomials constructed by Gupta–Sharma, Li–Helleseth, Li–Qu–Li–Fu.

Keywords

Finite fields Permutation polynomials Permutation trinomials 

Mathematics Subject Classification

11T06 05A05 
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Notes

Acknowledgements

The first author was partially supported by the Italian Ministero dell’Istruzione, dell’Università e della Ricerca (MIUR) and by the Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni (GNSAGA-INdAM). The second author was supported by CNPq, PDE Grant number 200434/2015-2. This work was done while the author enjoyed a sabbatical at the Università degli Studi di Perugia leave from Universidade Federal do Rio de Janeiro. We would like to thank the referees for providing us with useful comments which served to improve the paper.

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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversità degli Studi di PerugiaPerugiaItaly
  2. 2.Instituto de MatemáticaUniversidade Federal do Rio de JaneiroRio de JaneiroBrazil

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