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Designs, Codes and Cryptography

, Volume 83, Issue 2, pp 425–443 | Cite as

Some new results on permutation polynomials over finite fields

  • Jingxue Ma
  • Tao Zhang
  • Tao Feng
  • Gennian GeEmail author
Article

Abstract

Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, four classes of monomial complete permutation polynomials and one class of trinomial complete permutation polynomials are presented, one of which confirms a conjecture proposed by Wu et al. (Sci China Math 58:2081–2094, 2015). Furthermore, we give two classes of permutation trinomial, and make some progress on a conjecture about the differential uniformity of power permutation polynomials proposed by Blondeau et al. (Int J Inf Coding Theory 1:149–170, 2010).

Keywords

Permutation polynomials Complete permutation polynomials Trace function Differential uniformity 

Mathematics Subject Classification

11T06 11T55 05A05 

Notes

Acknowledgments

The authors express their gratitude to the anonymous reviewers for their detailed and constructive comments which are very helpful to the improvement of the presentation of this paper. The research of T. Feng was supported by Fundamental Research Fund for the Central Universities of China, the National Natural Science Foundation of China under Grant Nos. 11201418 and 11422112, and the Research Fund for Doctoral Programs from the Ministry of Education of China under Grant 20120101120089. The research of G. Ge was supported by the National Natural Science Foundation of China under Grant Nos. 11431003 and 61571310.

References

  1. 1.
    Berlekamp E.R., Rumsey H., Solomon G.: On the solution of algebraic equations over finite fields. Inform. Control 10, 553–564 (1967).Google Scholar
  2. 2.
    Blondeau C., Canteaut A., Charpin P.: Differential properties of power functions. Int. J. Inf. Coding Theory 1, 149–170 (2010).Google Scholar
  3. 3.
    Charpin P., Kyureghyan G.M.: Cubic monomial bent functions: a subclass of \({\cal M}\). SIAM J. Discret Math. 22, 650–665 (2008).Google Scholar
  4. 4.
    Ding C., Yuan J.: A family of skew Hadamard difference sets. J. Comb. Theory Ser. A 113, 1526–1535 (2006).Google Scholar
  5. 5.
    Ding C., Qu L., Wang Q., Yuan J., Yuan P.: Permutation trinomials over finite fields with even characteristic. SIAM J. Discret Math. 29, 79–92 (2015).Google Scholar
  6. 6.
    Ding C., Xiang Q., Yuan J., Yuan P.: Explicit classes of permutation polynomials of \(\mathbb{F}_{3^{3m}}\). Sci. China Ser. A 52, 639–647 (2009).Google Scholar
  7. 7.
    Akbary A., Ghioca D., Wang Q.: On constructing permutations of finite fields. Finite Fields Appl. 17, 51–67 (2011).Google Scholar
  8. 8.
    Dobbertin H.: Almost perfect nonlinear power functions on \({\rm GF}(2^n)\): the Welch case. IEEE Trans. Inf. Theory 45, 1271–1275 (1999).Google Scholar
  9. 9.
    Dobbertin H.: Kasami power functions, permutation polynomials and cyclic difference sets. In: Difference Sets, Sequences and Their Correlation Properties (Bad Windsheim, 1998). Nato Advanced Science Institutes Series C Mathematical and Physical Sciences, vol. 542, pp. 133–158. Kluwer, Dordrecht (1999).Google Scholar
  10. 10.
    Fernando N., Hou X.: A piecewise construction of permutation polynomials over finite fields. Finite Fields Appl. 18, 1184–1194 (2012).Google Scholar
  11. 11.
    Helleseth T.: Some results about the cross-correlation function between two maximal linear sequences. Discret Math. 16, 209–232 (1976).Google Scholar
  12. 12.
    Hollmann H.D.L., Xiang Q.: A class of permutation polynomials of \({\bf F}_{2^m}\) related to Dickson polynomials. Finite Fields Appl. 11, 111–122 (2005).Google Scholar
  13. 13.
    Hou X.: Two classes of permutation polynomials over finite fields. J. Comb. Theory Ser. A 118, 448–454 (2011).Google Scholar
  14. 14.
    Hou X.: A new approach to permutation polynomials over finite fields. Finite Fields Appl. 18, 492–521 (2012).Google Scholar
  15. 15.
    Hou X.: Permutation polynomials over finite fields—a survey of recent advances. Finite Fields Appl. 32, 82–119 (2015).Google Scholar
  16. 16.
    Laigle-Chapuy Y.: Permutation polynomials and applications to coding theory. Finite Fields Appl. 13, 58–70 (2007).Google Scholar
  17. 17.
    Li N., Helleseth T., Tang X.: Further results on a class of permutation polynomials over finite fields. Finite Fields Appl. 22, 16–23 (2013).Google Scholar
  18. 18.
    Lidl R., Niederreiter H.: Finite Fields. Encyclopedia of Mathematics and Its Applications, vol. 20. Addison-Wesley Publishing Company Advanced Book Program, Reading, MA (1983).Google Scholar
  19. 19.
    Niederreiter H., Robinson K.H.: Complete mappings of finite fields. J. Aust. Math. Soc. Ser. A 33, 197–212 (1982).Google Scholar
  20. 20.
    Qu L., Tan Y., Tan C.H., Li C.: Constructing differentially 4-uniform permutations over \({\mathbb{F}}_{2^{2k}}\) via the switching method. IEEE Trans. Inf. Theory 59, 4675–4686 (2013).Google Scholar
  21. 21.
    Sun J., Takeshita O.Y.: Interleavers for turbo codes using permutation polynomials over integer rings. IEEE Trans. Inf. Theory 51, 101–119 (2005).Google Scholar
  22. 22.
    Tu Z., Zeng X., Hu L.: Several classes of complete permutation polynomials. Finite Fields Appl. 25, 182–193 (2014).Google Scholar
  23. 23.
    Tu Z., Zeng X., Hu L., Li C.: A class of binomial permutation polynomials. arXiv:1310.0337 (2013).
  24. 24.
    Wu G., Li N., Helleseth T., Zhang Y.: Some classes of monomial complete permutation polynomials over finite fields of characteristic two. Finite Fields Appl. 28, 148–165 (2014).Google Scholar
  25. 25.
    Wu G., Li N., Helleseth T., Zhang Y.: Some classes of complete permutation polynomials over \({{\mathbb{F}}_q}\). Sci. China Math. 58, 2081–2094 (2015).Google Scholar
  26. 26.
    Yuan J., Ding C.: Four classes of permutation polynomials of \({\mathbb{F}}_{2^m}\). Finite Fields Appl. 13, 869–876 (2007).Google Scholar
  27. 27.
    Yuan J., Ding C., Wang H., Pieprzyk J.: Permutation polynomials of the form \((x^p-x+\delta )^s+L(x)\). Finite Fields Appl. 14, 482–493 (2008).Google Scholar
  28. 28.
    Zha Z., Hu L.: Two classes of permutation polynomials over finite fields. Finite Fields Appl. 18, 781–790 (2012).Google Scholar
  29. 29.
    Zieve M.E.: Permutation polynomials induced from permutations of subfields, and some complete sets of mutually orthogonal latin squares. arXiv:1312.1325 (2013).

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.School of Mathematical SciencesZhejiang UniversityHangzhouChina
  2. 2.School of Mathematical SciencesCapital Normal UniversityBeijingChina
  3. 3.Beijing Center for Mathematics and Information Interdisciplinary SciencesBeijingChina

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