[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

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

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Akbary A., Wang Q.: On polynomials of the form \(x^rf(x^{(q-1)/l})\). Int. J. Math. Math. Sci. 2007, 23408 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  2. Bartoli D., Giulietti M., Zini G.: On monomial complete permutation polynomials. Finite Fields Appl. 41, 132–158 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  3. Bartoli D., Giulietti M., Quoos L., Zini G.: Complete permutation polynomials from exceptional polynomials. J. Number Theory 176, 46–66 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  4. Ding C., Qu L., Wang Q., Yuan J., Yuan P.: Permutation trinomials over finite fields with even characteristic. SIAM J. Discret. Math. 29(1), 79–92 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  5. Gupta R., Sharma R.K.: Some new classes of permutation trinomials over finite fields with even characteristic. Finite Fields Appl. 41, 89–96 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  6. Hou X.: Determination of a type of permutation trinomials over finite fields. Acta Arith. 166, 253–278 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  7. Hou X.: Determination of a type of permutation trinomials over finite fields. II. Finite Fields Appl. 35, 16–35 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  8. Laigle-Chapuy Y.: Permutation polynomials and applications to coding theory. Finite Fields Appl. 13, 58–70 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  9. Li, N., Helleseth, T.: New permutation trinomials From Niho exponents over finite fields with even characteristic. arXiv:1606.03768v1.

  10. Li, K., Qu, L., Li, C., Fu, S.: New permutation trinomials constructed from fractional polynomials. arXiv:1605.06216v1.

  11. Lidl R., Niederreiter H.: Finite Fields (Encyclopedia of Mathematics and its Applications). Cambridge University Press, Cambridge (1997).

    Google Scholar 

  12. Marcos J.E.: Specific permutation polynomials over finite fields. Finite Fields Appl. 17, 105–112 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  13. Mullen G.L., Niederreiter H.: Dickson polynomials over finite fields and complete mappings. Can. Math. Bull. 30(1), 19–27 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  14. Mullen G.L., Panario D.: Handbook of Finite Fields. Chapman and Hall/CRC, Boca Raton (2013).

    Book  MATH  Google Scholar 

  15. Park Y.H., Lee J.B.: Permutation polynomials and group permutation polynomials. Bull. Aust. Math. Soc. 63, 67–74 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  16. Yuan P., Ding C.: Permutation polynomials over finite fields from a powerful lemma. Finite Fields Appl. 17, 560–574 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  17. Zhou Y., Qu L.: Constructions of negabent functions over finite fields. Cryptogr. Commun. 9, 165–180 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  18. Zieve M.: On some permutation polynomials over \({\mathbb{F}}_q\) of the form \(x^r h(x^{(q-1)/d})\). Proc. Am. Math. Soc. 137, 2209–2216 (2009).

    Article  MATH  Google Scholar 

  19. Zieve, M.: Classes of permutation polynomials based on cyclotomy and an additive analogue. In: Additive Number Theory, pp. 355–361, Springer, New York (2010)

Download references

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.

Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli 1, 06123, Perugia, Italy

    Daniele Bartoli

  2. Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro, 21941-909, Brazil

    Luciane Quoos

Authors
  1. Daniele Bartoli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luciane Quoos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luciane Quoos.

Additional information

Communicated by D. Panario.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bartoli, D., Quoos, L. Permutation polynomials of the type \(x^rg(x^{s})\) over \({\mathbb {F}}_{q^{2n}}\) . Des. Codes Cryptogr. 86, 1589–1599 (2018). https://doi.org/10.1007/s10623-017-0415-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-017-0415-8

Keywords

Mathematics Subject Classification

Search

Navigation