Regular p-ary bent functions with five terms and Kloosterman sums

  • Chunming TangEmail author
  • Yanfeng Qi
  • Dongmei Huang
Part of the following topical collections:
  1. Special Issue on Sequences and Their Applications


Kloosterman sums are vital in the study of bent functions, including regular p-ary bent functions. In this paper, a congruence property for Kloosterman sums is presented first and is used to prove the nonexistence of a class of p-ary bent functions. Further, this paper considers p-ary functions of the form \(f(x)= \text {Tr}^{n}_{1}(a_{1}x^{r_{1}(q-1)})+\text {Tr}^{n}_{1}\left (c_{1}x^{r_{1}(q-1)+\frac {q^{2}-1}{2}}\right ) +\text {Tr}^{n}_{1}\left (a_{2}x^{r_{2}(q-1)}\right )+\text {Tr}^{n}_{1}\left (c_{2}x^{r_{2}(q-1)+\frac {q^{2}-1}{2}}\right ) +bx^{\frac {q^{2}-1}{2}}\). We use Kloosterman sums in the characterization of this class of p-ary bent functions. Finally, an open problem of Jia et al. (IEEE Trans Inf. Theory 58(9): 6054–6063, 2012) is solved and we prove the nonexistence for a class of regular p-ary bent functions.


Regular bent functions Walsh transformation Kloosterman sums p-ary functions Congruence 

Mathematics Subject Classification (2010)

06E75 94A60 11T23 



We would like to thank the anonymous reviewers and Prof. Claude Carlet for their helpful comments and suggestions. This work is supported by the National Natural Science Foundation of China (Grant No. 11871058, 11531002, 11701129). C. Tang also acknowledges support from 14E013, CXTD2014-4 and the Meritocracy Research Funds of China West Normal University. Y. Qi also acknowledges support from Zhejiang provincial Natural Science Foundation of China (LQ17A010008, LQ16A010005).


  1. 1.
    Canteaut, A., Charpin, P., Kyureghyan, G.: A new class of monomial bent functions. Finite Fields Appl. 14(1), 221–241 (2008)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Charpin, P., Kyureghyan, G.: Cubic monomial bent functions: A subclass of \(\mathcal {M}\). SIAM J. Discr. Math. 22(2), 650–665 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Charpin, P., Pasalic, E., Tavernier, C.: On bent and semi-bent quadratic Boolean functions. IEEE Trans. Inf. Theory 51(12), 4286–4298 (2005)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Dillon, J. F.: Elementary Hadamard difference sets. Ph.D. disserta-tion. Univ Maryland, Collage Park (1974)Google Scholar
  5. 5.
    Dobbertin, H., Leander, G., Canteaut, A., Carlet, C., Felke, P., Ga-borit, P.: Construction of bent functions via Niho power functions. J. Comb. Theory, Ser. A 113(5), 779–798 (2006)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Helleseth, T., Kholosha, A.: Monomial and quadratic bent functions over the finite fields of odd characteristic. IEEE Trans. Inf. Theory 52(2), 2018–2032 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Helleseth, T., Kholosha, A.: On generalized bent functions. In: Proc. IEEE Inf. Theory Appl. Workshop, pp. 1–6 (2010)Google Scholar
  8. 8.
    Helleseth, T., Kholosha, A.: Sequences, bent functions and Jacob-sthal sums. Lecture Notes Comput. Sci. 6338, 416–429 (2010)CrossRefGoogle Scholar
  9. 9.
    Jia, W., Zeng, X., Helleseth, T., Li, C.: A class of binomial bent functions over the finite fields of odd characteristic. IEEE Trans. Inf. Theory 58(9), 6054–6063 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kononen, K. P., Rinta-aho, M. J., Väänänen, K. O.: On integer values of Kloosterman sums. IEEE Trans. Inf. Theory 56(8), 4011–4013 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kumar, P. V., Scholtz, R. A., Welch, L. R.: Generalized bent functions and their properties. J. Combin. Theory Ser. A 40, 90–107 (1985)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Leander, G.: Monomial bent functions. IEEE Trans. Inf. Theory 2(52), 738–743 (2006)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Li, N., Helleseth, T., Tang, X., Kholosha, A.: Several new classes of bent functions from Dillon exponents. IEEE Trans. Inf. Theory 59(3), 1818–1831 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lidl, R., Niederreiter, H.: Introduction to finite fields and applications. Cambridge University Press, Cambridge (1994)Google Scholar
  15. 15.
    Mesnager, S.: Bent and hyper-bent functions in polynomial form and their link with some exponential sums and Dickson polynomials. IEEE Trans. Inf. Theory 57(9), 5996–6009 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Mesnager, S., Flori, J.: Hyper-bent functions via Dillon-like exponents. IEEE Trans. Inf. Theory 59(5), 3215–3232 (2013)CrossRefGoogle Scholar
  17. 17.
    Rothaus, O. S.: On bent functions. J. Combinatorial Theory Ser. A 20, 300–305 (1976)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Tang, C., Qi, Y.: Special values of Kloosterman sums and binomial bent functions. Finite Fields Appl. 41, 113–131 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Wang, B., Tang, C., Qi, Y., Yang, Y., Xu, M.: A new class of hyper-bent Boolean functions in binomial forms [Online]. Available: arXiv:pdf/1112.0062.pdf
  20. 20.
    Yu, N. Y., Gong, G.: Constructions of quadratic bent functions in polynomial forms. IEEE Trans. Inf. Theory 52(7), 3291–3299 (2006)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Zheng, D., Yu, L., Hu, L.: On a class of binomial bent functions over the finite fields of odd characteristic. Appl. Algebra Eng. Commun. Comput. 24(6), 461–475 (2013)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.School of Mathematics and InformationChina West Normal UniversitySichuanChina
  2. 2.School of ScienceHangzhou Dianzi UniversityHangzhouChina

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