Skip to main content
SpringerLink
Log in

Explicit characterization of two classes of regular bent functions

  • Original Paper
  • Published:
Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract

This paper considers two classes of p-ary functions studied by Li et al. (IEEE Trans Inf Theory 59(3):1818–1831, 2013). The first class of p-ary functions is of the form

$$\begin{aligned} f(x)=Tr^n_1\left( a x^{l(q-1)}+b x^{\left( l+\frac{q+1}{2}\right) (q-1)}\right) +\epsilon x^{\frac{q^2-1}{2}}. \end{aligned}$$

Another class of p-ary functions is of the form

$$\begin{aligned} f(x)={\left\{ \begin{array}{ll} \sum ^{q-1}_{i=0} Tr^n_1(a x^{(ri+s)(q-1)})+\epsilon x^{\frac{q^2-1}{2}},&{} x\ne 0,\\ f(0),&{} x=0. \end{array}\right. } \end{aligned}$$

We generalize Li et al.’s results, give necessary conditions for two classes of bent functions, and present more explicit characterization of these regular bent functions for different cases.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Canteaut, A., Charpin, P., Kyureghyan, G.M.: A new class of monomial bent functions. Finite Fields Appl. 14(1), 221–241 (2008)

    Article  MathSciNet  Google Scholar 

  2. Carlet, C.: Boolean functions for cryptography and error correcting codes. In: Crama, Y., Hammer, P.L. (eds.) Boolean Models and Methods in Mathematics, Computer Science, and Engineering, pp. 257–397. Cambridge University Press, Cambridge (2010)

    Chapter  Google Scholar 

  3. Carlet, C., Mesnager, S., Tang, C., Qi, Y.: Euclidean and Hermitian LCD MDS codes. Des. Codes Cryptogr. (2018). https://doi.org/10.1007/s10623-018-0463-8

    Article  MathSciNet  MATH  Google Scholar 

  4. Charpin, P., Kyureghyan, G.M.: Cubic monomial bent functions: a subclass of \(\cal{M}\). SIAM J. Discrete Math. 22(2), 650–665 (2008)

    Article  MathSciNet  Google Scholar 

  5. Charpin, P., Pasalic, E., Tavernier, C.: On bent and semi-bent quadratic Boolean functions. IEEE Trans. Inf. Theory 51(12), 4286–4298 (2005)

    Article  MathSciNet  Google Scholar 

  6. Cohen, G., Honkala, I., Litsyn, S., Lobstein, A.: Covering Codes. North Holland, Amsterdam (1997)

    MATH  Google Scholar 

  7. Dillon J. F.: Elementary Hadamard difference sets. Ph.D. dissertation, University of Maryland, Collage Park (1974)

  8. Ding, C., Fan, C., Zhou, Z.: The dimension and minimum distance of two classes of primitive BCH codes. Finite Fields Appl. 45, 237–263 (2017)

    Article  MathSciNet  Google Scholar 

  9. Dobbertin, H., Leander, G., Canteaut, A., Carlet, C., Felke, P., Gaborit, P.: Construction of bent functions via Niho power functions. J. Combin. Theory Ser. A 113(5), 779–798 (2006)

    Article  MathSciNet  Google Scholar 

  10. Helleseth, T., Kholosha, A.: Monomial and quadratic bent functions over the finite fields of odd characteristic. IEEE Trans. Inf. Theory 52(5), 2018–2032 (2006)

    Article  MathSciNet  Google Scholar 

  11. Helleseth, T., Kholosha, A.: On generalized bent functions. In: Proceedings of IEEE Information Theory and Applications Workshop, pp. 1–6 (2010)

  12. Helleseth, T., Kholosha, A.: Sequences, Bent functions and Jacobsthal sums. SETA 6338, 416–429 (2010)

    MathSciNet  MATH  Google Scholar 

  13. 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)

    Article  MathSciNet  Google Scholar 

  14. Kononen, K.P., Rinta-Aho, M.J., Vaananen, K.O.: On integer values of Kloosterman sums. IEEE Trans. Inf. Theory 56(8), 4011–4013 (2010)

    Article  MathSciNet  Google Scholar 

  15. Kumar, P.V., Scholtz, R.A., Welch, L.R.: Generalized bent functions and their properties. J. Combin. Theory Ser. A 40(1), 90–107 (1985)

    Article  MathSciNet  Google Scholar 

  16. Leander, N.G.: Monomial bent functions. IEEE Trans. Inf. Theory 52(2), 738–743 (2006)

    Article  MathSciNet  Google Scholar 

  17. 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)

    Article  MathSciNet  Google Scholar 

  18. Lidl, R., Niederreiter, H.: Introduction to Finite Fields and Their Applications. Cambridge University Press, Cambridge (1994)

    Book  Google Scholar 

  19. 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)

    Article  MathSciNet  Google Scholar 

  20. Mesnager, S., Flori, J.P.: Hyperbent functions via Dillon-like exponents. IEEE Trans. Inf. Theory 59(5), 3215–3232 (2013)

    Article  MathSciNet  Google Scholar 

  21. Olsen, J., Scholtz, R., Welch, L.: Bent-function sequences. IEEE Trans. Inf. Theory 28(6), 858–864 (1982)

    Article  MathSciNet  Google Scholar 

  22. Tang, C., Qi, Y.: Special values of Kloosterman sums and binomial bent functions. Finite Fields Appl. 41, 113–131 (2016)

    Article  MathSciNet  Google Scholar 

  23. Wang, B., Tang, C., Qi, Y., Yang, Y., Xu, M.: A new class of hyper-bent Boolean functions in binomial forms. arxiv.org/pdf/1112.0062.pdf

  24. Yu, N.Y., Gong, G.: Constructions of quadratic bent functions in polynomial forms. IEEE Trans. Inf. Theory 52(7), 3291–3299 (2006)

    Article  MathSciNet  Google Scholar 

  25. 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)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are very grateful to the anonymous reviewers and Prof. Teo Mora for their valuable comments and suggestions. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11401480, 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).

Author information

Authors and Affiliations

  1. School of Science, Hangzhou Dianzi University, Hangzhou, 310018, Zhejiang, China

    Yanfeng Qi

  2. School of Mathematics and Information, China West Normal University, Nanchong, 637002, Sichuan, China

    Chunming Tang & Dongmei Huang

Authors
  1. Yanfeng Qi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chunming Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dongmei Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chunming Tang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qi, Y., Tang, C. & Huang, D. Explicit characterization of two classes of regular bent functions. AAECC 29, 529–544 (2018). https://doi.org/10.1007/s00200-018-0357-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00200-018-0357-z

Keywords

Mathematics Subject Classification

Search

Navigation