Skip to main content
SpringerLink
Log in

New open problems related to old conjectures by Helleseth

  • Published:
Cryptography and Communications Aims and scope Submit manuscript

Abstract

Recently, very interesting results have been obtained concerning the Fourier spectra of power permutations over a finite field. In this note we survey the recent ideas of Aubry, Feng, Katz, and Langevin, and we pose new open problems related to old conjectures proposed by Helleseth in the middle of the seventies.

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. Aubry, Y., Katz, D.J., Langevin, P.: Cyclotomy of Weil sums of binomials (2013) arXiv:1312.3889 [math.NT]

  2. Aubry, Y., Langevin, P.: On a conjecture of Helleseth. In: Algebraic informatics, volume 8080 of Lecture Notes in Comput. Sci., pp 113–118. Springer, Heidelberg (2013)

  3. Blokhuis, A., Calderbank, A.R.: Unpublished note

  4. Calderbank, A.R., McGuire, G., Poonen, B., Rubinstein, M.: On a conjecture of Helleseth regarding pairs of binary m-sequences. IEEE Trans. Inform. Theory 42(3), 988–990 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Charpin, P.: Cyclic codes with few weights and Niho exponents. J. Combin. Theory Ser. A 108(2), 247–259 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Feng, T.: On cyclic codes of length \(2^{2^{r}-1}\) with two zeros whose dual codes have three weights. Des. Codes Cryptogr. 62(3), 253–258 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Helleseth, T.: Some results about the cross-correlation function between two maximal linear sequences. Discrete Math. 16(3), 209–232 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  8. Katz, D.J.: Weil sums of binomials, three-level cross-correlation, and a conjecture of Helleseth. J. Combin. Theory Ser. A 119(8), 1644–1659 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Katz, D.J.: Divisibility of Weil sums of binomials (2014) arXiv:1407.7923 [math.NT]

  10. Katz, N., Livné, R.: Sommes de Kloosterman et courbes elliptiques universelles en caractéristiques 2 et 3. C. R. Acad. Sci. Paris Sér. I Math. 309(11), 723–726 (1989)

    MATH  Google Scholar 

  11. Kononen, K.P., Rinta-aho, M.J., Väänänen, K.O.: On integer values of Kloosterman sums. IEEE Trans. Inform. Theory 56(8), 4011–4013 (2010)

    Article  MathSciNet  Google Scholar 

  12. Lachaud, G., Wolfmann, J.: Sommes de Kloosterman, courbes elliptiques et codes cycliques en caractéristique 2. C. R. Acad. Sci. Paris Sér. I Math. 305(20), 881–883 (1987)

    MathSciNet  MATH  Google Scholar 

  13. Langevin, P.: Numerical projects page: spectra of power maps (2007). http://langevin.univ-tln.fr/project/spectrum

  14. Langevin, P.: Numerical projects page : nice exponents, 2013 and 2015. http://langevin.univ-tln.fr/project/expo

Download references

Acknowledgments

The first author was supported in part by a Research, Scholarship, and Creative Activity Award from California State University, Northridge. The first author was also supported in part by the Institut de Mathématiques de Toulon at Université de Toulon as a visiting professor. The authors thank anonymous reviewers for helpful corrections and suggestions.

Author information

Authors and Affiliations

  1. California State University, Northridge, CA, USA

    Daniel J. Katz

  2. Université de Toulon, La Garde, France

    Philippe Langevin

Authors
  1. Daniel J. Katz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Philippe Langevin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Philippe Langevin.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Katz, D.J., Langevin, P. New open problems related to old conjectures by Helleseth. Cryptogr. Commun. 8, 175–189 (2016). https://doi.org/10.1007/s12095-015-0137-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12095-015-0137-6

Keywords

Search

Navigation