Skip to main content
SpringerLink
Log in

A class of exponential sums and sequence families

  • Published:
Cryptography and Communications Aims and scope Submit manuscript

Abstract

Let m1 and m2 be two distinct positive integers with \(d=\gcd (m_{1}, m_{2})\). Let \(\mathbb {F}_{2^{m}}\) be the finite field with 2m elements, where m = m1m2/d. In this paper, we investigate the exponential sums

$$ S(a,b)=\sum\limits_{x \in \mathbb{F}_{2^{m}}^{*}}(-1)^{\text{Tr}_{m_{1}}(ax^{\frac {2^{m}-1}{2^{m_{1}}-1}})+\text{Tr}_{m_{2}}(bx^{\frac {2^{m}-1}{2^{m_{2}}-1}})}, $$

where \(a \in \mathbb {F}_{2^{m_{1}}}\), \(b \in \mathbb {F}_{2^{m_{2}}}\), and Trt denotes the trace function from \(\mathbb {F}_{2^{t}}\) to \(\mathbb {F}_{2}\). When d = 1, 2, 3, 4, we present the value distribution of the exponential sums S(a, b) explicitly. As an application, we construct three families of binary sequences with three-valued correlation.

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. Berndt, B., Evans, R., Williams, K.: Gauss and Jacobi Sums. Wiley, New York (1997)

    MATH  Google Scholar 

  2. Boztas, S., Kumar, P.V.: Binary sequences with Gold-like correlation but larger linear span. IEEE Trans. Inf. Theory 40(2), 532–537 (1994)

    Article  Google Scholar 

  3. Ding, C., Yang, J.: Hamming weights in irreducible cyclic codes. Discret. Math. 313(4), 434–446 (2013)

    Article  MathSciNet  Google Scholar 

  4. Feng, K., Luo, J.: Values distributions of exponential sums from perfect nonlinear functions and their applications. IEEE Trans. Inf. Theory 53(9), 3035–3041 (2007)

    Article  MathSciNet  Google Scholar 

  5. Gold, R.: Maximal recursive sequences with 3-valued recursive crosscorrelation functions. IEEE Trans. Inf. Theory 14(1), 154–156 (1968)

    Article  Google Scholar 

  6. Han, Y.K., Yang, K.: New M-ary sequence families with low correlation and large size. IEEE Trans. Inf. Theory 55(4), 1815–1823 (2009)

    Article  MathSciNet  Google Scholar 

  7. Helleseth, T.: Some results about the cross-correlation function between two maximal linear sequences. Discr. Math. 16, 209–232 (1976)

    Article  MathSciNet  Google Scholar 

  8. Helleseth, T., Kholosha, A., Ness, G.J.: Characterization of m-sequences of lengths 22k − 1 and 2k − 1 with three-valued cross correlations. IEEE Trans. Inf. Theory 53(6), 2236–2245 (2007)

    Article  Google Scholar 

  9. Helleseth, T., Kumar, P.V. In: Pless, V., Huffman, C. (eds.) : Sequences with Low Correlation. Handbook of Coding Theory. Elsevier, New York (1998)

  10. Kasami, T.: Weight enumerators for several classes of subcodes of the 2nd order Reed-Muller codes. Inf. Control 18, 369–394 (1971)

    Article  Google Scholar 

  11. Kim, D.S., Chae, H.J., Song, H.Y.: A generalization of the family of p-ary decimated sequences with low correlation. IEEE Trans. Inf. Theory 57(11), 7614–7617 (2011)

    Article  MathSciNet  Google Scholar 

  12. Kim, Y.S., Chung, J.S., No, J.S., Chung, H.: New families of M-ary sequences with low correlation constructed from Sidelnikov sequences. IEEE Trans. Inf. Theory 54(8), 3768–3774 (2008)

    Article  MathSciNet  Google Scholar 

  13. Kumar, P.V., Moreno, O.: Prime-phase sequences with periodic correlation properties better than binary sequences. IEEE Trans. Inf. Theory 37(3), 603–616 (1991)

    Article  Google Scholar 

  14. Li, C., Yue, Q.: A class of cyclic codes from two distinct finite fields. Finite Fields Appl. 34, 305–316 (2015)

    Article  MathSciNet  Google Scholar 

  15. Li, C., Yue, Q., Li, F.: Weight distributions of cyclic codes with respect to pairwise coprime order elements. Finite Fields Appl. 28, 94–114 (2014)

    Article  MathSciNet  Google Scholar 

  16. Li, N., Helleseth, T., Kholosha, A., Tang, X.: On the Walsh transform of a class of functions from Niho exponents. IEEE Trans. Inf. Theory 59(7), 4662–4667 (2013)

    Article  MathSciNet  Google Scholar 

  17. Lidl, R., Niederreiter, H.: Finite fields. Addison-Wesley Publishing Inc., Reading (1983)

    MATH  Google Scholar 

  18. Ness, G.J., Helleseth, T.: Cross correlation of m-sequences of different lengths. IEEE Trans. Inf. Theory 52(4), 1637–1648 (2006)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  20. Rothaus, O.S.: Modified Gold codes. IEEE Trans. Inf. Theory 39(2), 654–656 (1993)

    Article  MathSciNet  Google Scholar 

  21. Tang, X., Udaya, P., Fan, P.: A new family of nonbinary sequences with three-level correlation property and large linear span. IEEE Trans. Inf. Theory 51(8), 2906–2914 (2005)

    Article  MathSciNet  Google Scholar 

  22. Trachtenberg, H.M.: On the Cross-Correlation Functions of Maximal Recurring Sequences. Ph.D. Dissertation, Univ. Southern California, Log Angeles (1970)

    Google Scholar 

  23. Wu, Y., Yue, Q., Zhu, X., Yang, S.: Weight enumerators of reducible cyclic codes and their dual codes. Discr. Math. 342(3), 671–682 (2019)

    Article  MathSciNet  Google Scholar 

  24. Xia, Y., Chen, S.: A new family of p-ary sequences with low correlation constructed from decimated sequences. IEEE Trans. Inf. Theory 58(9), 6037–6046 (2012)

    Article  MathSciNet  Google Scholar 

  25. Xia, Y., Li, N., Zeng, X., Helleseth, T.: An open problem on the distribution of a Niho type cross-correlation function. IEEE Trans. Inf. Theory 62(12), 7546–7554 (2016)

    Article  MathSciNet  Google Scholar 

  26. Xia, Y., Li, N., Zeng, X., Helleseth, T.: On the correlation distribution of a Niho decimation. IEEE Trans. Inf. Theory 63(11), 7206–7217 (2017)

    Article  MathSciNet  Google Scholar 

  27. Yang, J., Xia, L.: Complete solving of the explicit evaluation of Gauss sums in the index 2 case. Sci. China Math. 53(9), 2525–2542 (2010)

    Article  MathSciNet  Google Scholar 

  28. Yang, S., Yao, Z.: Complete weight enumerators of a class of linear codes. Discr. Math. 340(4), 729–739 (2017)

    Article  MathSciNet  Google Scholar 

  29. Yang, S., Yao, Z., Zhao, C.: The weight distributions of two classes of p-ary cyclic codes with few weights. Finite Fields Appl. 44, 76–91 (2017)

    Article  MathSciNet  Google Scholar 

  30. Yang, S., Yao, Z., Zhao, C.: The weight enumerator of the duals of a class of cyclic codes with three zeros. Appl. Alg. Eng. Commun. Comput. 26(4), 347–367 (2015)

    Article  MathSciNet  Google Scholar 

  31. Yu, N.Y., Gong, G.: A new binary sequence family with low correlation and large size. IEEE Trans. Inf. Theory 52(4), 1624–1636 (2006)

    Article  MathSciNet  Google Scholar 

  32. Zeng, X., Liu, J.Q., Hu, L.: Generalized Kasami sequences: the large set. IEEE Trans. Inf. Theory 53(7), 2587–2598 (2007)

    Article  MathSciNet  Google Scholar 

  33. Zhou, Z., Helleseth, T., Parampalli, U.: A family of polyphase sequences with asymptotically optimal correlation. IEEE Trans. Inf. Theory 64(4), 2896–2900 (2018)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are very grateful to the editor and the reviewers for their detailed comments and suggestions that much improved the presentation and quality of this paper. The work was supported by the National Natural Science Foundation of China under Grant 11701179, the Shanghai Chenguang Program under Grant 18CG22, the Shanghai Sailing Program under Grant 17YF1404300, the Foundation of State Key Laboratory of Integrated Services Networks under Grant ISN20-02, the National Natural Science Foundation of China under Grants 61772015 and 61771021, the Natural Science Foundation of Hubei Province under Grant 2017CFB425, and the Shanghai Science and Technology Commission Program under Grant 18511105700.

Author information

Authors and Affiliations

  1. Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai, 200062, China

    Chengju Li

  2. State Key Laboratory of Integrated Services Networks, Xidian University, Xi’an, 710071, China

    Chengju Li

  3. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211100, People’s Republic of China

    Qin Yue

  4. Department of Mathematics and Statistics, South-Central University for Nationalities, Wuhan, 430074, China

    Yongbo Xia

  5. School of Computer Science and Software Engineering, East China Normal University, Shanghai, 200062, China

    Wei Peng

Authors
  1. Chengju Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qin Yue
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yongbo Xia
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Wei Peng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wei Peng.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the Topical Collection on Special Issue on Sequences and Their Applications

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, C., Yue, Q., Xia, Y. et al. A class of exponential sums and sequence families. Cryptogr. Commun. 12, 569–584 (2020). https://doi.org/10.1007/s12095-019-00368-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12095-019-00368-4

Keywords

Mathematics Subject Classification (2010)

Search

Navigation