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Wuhan University Journal of Natural Sciences

, Volume 24, Issue 6, pp 505–509 | Cite as

On Constructing Two Classes of Permutation Polynomials over Finite Fields

  • Kaimin ChengEmail author
Mathematics
  • 15 Downloads

Abstract

In this paper, we construct two classes of permutation polynomials over finite fields. First, by one well-known lemma of Zieve, we characterize one class permutation polynomials of the finite field, which generalizes the result of Marcos. Second, by using the onto property of functions related to the elementary symmetric polynomial in multivariable and the general trace function, we construct another class permutation polynomials of the finite field. This extends the results of Marcos, Zieve, Qin and Hong to the more general cases. Particularly, the latter result gives a rather more general answer to an open problem raised by Zieve in 2010.

Key words

permutation polynomial elementary symmetric polynomial finite field trace function 

CLC number

O 156.2 

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References

  1. [1]
    Laigle-Chapuy Y. Permutation polynomials and applications to coding theory [J]. Finite Fields Appl, 2007, 13(1): 58–70.CrossRefGoogle Scholar
  2. [2]
    Schwenk J, Huber K. Public key encryption and digital signatures based on permutation polynomials [J]. Electron Lett, 1998, 34(8): 759–760.CrossRefGoogle Scholar
  3. [3]
    Lidl R, Niederreiter H. Finite Fields, Encyclopedia of Mathematics and Its Applications [M]. Second Ed. Cambridge: Cambridge University Press, 1997: 20.Google Scholar
  4. [4]
    Lidl R, Mullen G L. When does a polynomial over a finite field permute the elements of the field? [J]. Amer Math Monthly, 1988, 95(3): 243–246.CrossRefGoogle Scholar
  5. [5]
    Charpin P, Kyureghyan G. When does F(x)+rTr(H(x)) permute FPN? [J]. Finite Fields Appl, 2009, 15(5): 615–632.CrossRefGoogle Scholar
  6. [6]
    Hong S F, Qin X E, Zhao W. Necessary conditions for reversed Dickson polynomials of the second kind to be permutational [J]. Finite Fields Appl, 2016, 37: 54–71.CrossRefGoogle Scholar
  7. [7]
    Cheng K M, Hong S F. The first and second moments of reversed Dickson polynomials over finite fields [J]. Journal of Number Theory, 2018, 187: 166–188.CrossRefGoogle Scholar
  8. [8]
    Qin X E, Qian G Y, Hong S F. New results on permutation polynomials over finite fields [J]. Int J Number Theory, 2015, 11(2): 437–449.CrossRefGoogle Scholar
  9. [9]
    Wan D, Lidl R. Permutation polynomials of the form x rf(x (q−1)/d) and their group structure [J]. Monatsh Math, 1991, 112(2): 149–163.CrossRefGoogle Scholar
  10. [10]
    Marcos J E. Specific permutation polynomials over finite fields [J]. Finite Fields Appl, 2011, 17(2): 105–112.CrossRefGoogle Scholar
  11. [11]
    Zieve M E. Classes of permutation polynomials based on cyclotomy and an additive analogue [C]// Additive Number Theory. Berlin: Springer-Verlag, 2010: 355–361.CrossRefGoogle Scholar
  12. [12]
    Qin X E, Hong S F. Constructing permutation polynomials over finite fields [J]. Bull Aust Math Soc, 2013, 89(3): 420–430.CrossRefGoogle Scholar
  13. [13]
    Zieve M E. Some families of permutation polynomials over finite fields [J]. Int J Number Theory, 2008, 4(5): 851–857.CrossRefGoogle Scholar

Copyright information

© Wuhan University and Springer-Verlag GmbH Germany 2019

Authors and Affiliations

  1. 1.School of Mathematics and InformationChina West Normal UniversitySichuanChina

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