On a class of permutation trinomials in characteristic 2

  • Xiang-dong HouEmail author
Part of the following topical collections:
  1. Special Issue on Boolean Functions and Their Applications


Recently, Tu, Zeng, Li, and Helleseth considered trinomials of the form \(f(X)=X+aX^{q(q-1)+ 1}+bX^{2(q-1)+ 1}\in \mathbb {F}_{q^{2}}[X]\), where q is even and \(a,b\in \mathbb {F}_{q^{2}}^{*}\). They found sufficient conditions on a, b for f to be a permutation polynomial (PP) of \(\mathbb {F}_{q^{2}}\) and they conjectured that the sufficient conditions are also necessary. The conjecture has been confirmed by Bartoli using the Hasse-Weil bound. In this paper, we give an alternative solution to the question. We also use the Hasse-Weil bound, but in a different way. Moreover, the necessity and sufficiency of the conditions are proved by the same approach.


Finite field Hasse-Weil bound Permutation polynomial 

Mathematics Subject Classification (2010)

11T06 11T55 14H05 



  1. 1.
    Bartoli, D.: On a conjecture about a class of permutation trinomials. Finite Fields Appl. 52, 30–50 (2018)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Gupta, R., Sharma, R.K.: Some new classes of permutation trinomials over finite fields with even characteristic. Finite Fields Appl. 41, 89–96 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Hou, X.: A survey of permutation binomials and trinomials over finite fields. In: Proceedings of the 11th International Conference on Finite Fields and Their Applications, Magdeburg, Germany, 2013, Contemporary Mathematics, vol. 632, pp. 177–191 (2015)Google Scholar
  4. 4.
    Hou, X.: Permutation polynomials over finite fields — a survey of recent advances. Finite Fields Appl. 32, 82–119 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hou, X.: Determination of a type of permutation trinomials over finite fields, II. Finite Fields Appl. 35, 16–35 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Li, N., Helleseth, T.: Several classes of permutation trinomials from Niho exponents. Cryptogr. Commun. 9, 693–705 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Li, K., Qu, L., Chen, X.: New classes of permutation binomials and permutation trinomials over finite fields. Finite Fields Appl. 43, 69–85 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Park, Y.H., Lee, J.B.: Permutation polynomials and group permutation polynomials. Bull. Austral. Math. Soc. 63, 67–74 (2001)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Stichtenoth, H.: Algebraic Function Fields and Codes. Springer, Berlin (1993)zbMATHGoogle Scholar
  10. 10.
    Tu, Z., Zeng, X., Li, C., Helleseth, T.: A class of new permutation trinomials. Finite Fields Appl. 50, 178–195 (2018)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Wang, Q.: Cyclotomic mapping permutation polynomials over finite fields. In: Golomb, S.W., Gong, G., Helleseth, T., Song, H.-Y. (eds.) Sequences, Subsequences, and Consequences, Lecture Notes in Comput. Sci., vol. 4893, pp 119–128. Springer, Berlin (2007)Google Scholar
  12. 12.
    Williams, K.S.: Note on cubics over GF(2n) and GF(3n). J. Number Theory 7, 361–365 (1975)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Wu, D., Yuan, P., Ding, C., Ma, Y.: Permutation trinomials over \(\mathbb {F}_{2^{m}}\). Finite Fields Appl. 46, 38–56 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Zha, Z., Hu, L., Fan, S.: Further results on permutation trinomials over finite fields with even characteristic. Finite Fields Appl. 45, 43–52 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Zieve, M.E.: On some permutation polynomials over \(\mathbb {F}_{q}\) of the form x r h(x (q− 1)/d). Proc. Amer. Math. Soc. 137, 2209–2216 (2009)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Zieve, M.E.: Permutation polynomials on \(\mathbb {F}_{q}\) induced from Rédei function bijections on subgroups of \(\mathbb {F}_{q}^{*}\). arXiv:1310.0776

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of South FloridaTampaUSA

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