Skip to main content
SpringerLink
Log in

Multiple characters transforms and generalized Boolean functions

  • Published:
Cryptography and Communications Aims and scope Submit manuscript

Abstract

In this paper we investigate generalized Boolean functions whose spectrum is flat with respect to a set of Walsh-Hadamard transforms defined using various complex primitive roots of 1. We also study some differential properties of the generalized Boolean functions in even dimension defined in terms of these different characters. We show that those functions have similar properties to the vectorial bent functions. We next clarify the case of gbent functions in odd dimension. As a by-product of our proofs, more generally, we also provide several results about plateaued functions. Furthermore, we find characterizations of plateaued functions with respect to different characters in terms of second derivatives and fourth moments.

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. Bey, C., Kyureghyan, G.: On Boolean functions with the sum of every two of them being bent Des. Codes Cryptogr. 49, 341–346 (2008)

    Article  MathSciNet  Google Scholar 

  2. Carlet, C.: Boolean functions for cryptography and error correcting codes. In: Crama, Y., Hammer, P. (eds.) Boolean Methods and Models. Available at: https://www.math.univ-paris13.fr/~carlet/pubs.html, pp 257–397. Cambridge Univ. Press, Cambridge (2010)

  3. Carlet, C., Gaborit, P.: Hyper-bent functions and cyclic codes. J. Combin. Theory Ser A 113, 446–482 (2006)

    Article  MathSciNet  Google Scholar 

  4. Carlet, C., Mesnager, S.: Four decades of research on bent functions. J. Des. Codes Crypt. 78, 5–50 (2016)

    Article  MathSciNet  Google Scholar 

  5. Charpin, P., Gong, G.: Hyperbent functions, Kloosterman sums, and Dickson polynomials. IEEE Trans. Inform. Theory 54(9), 4230–4238 (2008)

    Article  MathSciNet  Google Scholar 

  6. Cusick, T.W., Stănică, P.: Cryptographic Boolean Functions and Applications, 2nd edn. Academic Press, San Diego, CA (2017)

    MATH  Google Scholar 

  7. Hodžić, S., Meidl, W., Pasalic, E.: Full characterization of generalized bent functions as (semi)-bent spaces, their dual and the Gray image. IEEE Trans. Inf. Theory 64(7), 5432–5440 (2018)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Martinsen, T., Meidl, W., Mesnager, S., Stănică, P.: Decomposing generalized bent and hyperbent functions. IEEE Trans. Inf. Theory 63(12), 7804–7812 (2017)

    Article  MathSciNet  Google Scholar 

  10. Martinsen, T., Meidl, W., Pott, A., Stănică, P.: On symmetry and differential properties of generalized Boolean functions. Proc. of WAIFI 2018:, Arithmetic of Finite Fields, LNCS 11321, 207–223 (2018)

    MathSciNet  MATH  Google Scholar 

  11. Martinsen, T., Meidl, W., Stănică, P.: Generalized bent functions and their Gray images. Proc. of WAIFI 2016:, Arithmetic of Finite Fields, LNCS 10064, 160–173 (2017)

    MathSciNet  MATH  Google Scholar 

  12. Martinsen, T., Meidl, W., Stănică, P.: Partial spread and vectorial generalized bent functions. Des. Codes Crypt. 85(1), 1–13 (2017)

    Article  MathSciNet  Google Scholar 

  13. Mesnager, S.: Bent Functions: Fundamentals and Results. Springer, Switzerland (2016)

    Book  Google Scholar 

  14. Mesnager, S., Tang, C., Qi, Y.: Generalized plateaued functions and admissible (plateaued) functions. IEEE Trans Inform. Theory 63, 6139–6148 (2017)

    Article  MathSciNet  Google Scholar 

  15. Mesnager, S., Tang, C., Qi, Y., Wang, L., Wu, B., Feng, K.: Further results on generalized bent functions and their complete characterization. IEEE Trans. Inform. Theory 64(7), 5441–5452 (2018)

    Article  MathSciNet  Google Scholar 

  16. Pott, A.: Nonlinear functions in abelian groups and relative difference sets, Optimal discrete structures and algorithms (ODSA 2000). Discrete Appl. Math. 138, 177–193 (2004)

    Article  MathSciNet  Google Scholar 

  17. Riera, C., Stănică, P.: Landscape Boolean functions, to appear in Advances in Math. Communication. Available at: arXiv:1806.05878 (2019)

  18. Schmidt, B.: On (pa,pb,pa,pa−b)-relative difference sets. J. Algebraic Combin. 6, 279–297 (1997)

    Article  MathSciNet  Google Scholar 

  19. Schmidt, K.U.: Quaternary constant-amplitude codes for multicode CDMA. IEEE Trans. Inform. Theory 55(4), 1824–1832 (2009)

    Article  MathSciNet  Google Scholar 

  20. Solé, P., Tokareva, N.: Connections between quaternary and binary bent functions. Prikl. Diskr. Mat. 1, 16–18 (2009). (see also, http://eprint.iacr.org/2009/544.pdf)

    Google Scholar 

  21. Stănică, P., Martinsen, T., Gangopadhyay, S., Singh, B.K.: Bent and generalized bent Boolean functions. Des. Codes & Cryptogr. 69, 77–94 (2013)

    Article  MathSciNet  Google Scholar 

  22. Tang, C., Xiang, C., Qi, Y., Feng, K.: Complete characterization of generalized bent and 2k-bent Boolean functions. IEEE Trans. Inf. Theory 63(7), 4668–4674 (2017)

    Article  Google Scholar 

  23. Youssef, A.M., Gong, G.: Hyper-Bent Functions. In: Adv. Crypt. – EUROCRYPT 2001, LNCS, vol. 2045, pp 406–419. Springer, Berlin (2001)

    Google Scholar 

  24. Zhang, F., Xia, S., Stănică, P., Zhou, Y.: Further results on constructions of generalized bent Boolean functions. Inf. Sciences - China. 59, 1–3 (2016)

    Google Scholar 

  25. Zheng, Y.L., Zhang, X.M.: On plateaued functions. IEEE Trans. Inf. Theory 47(9), 1215–1223 (2001)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors deeply thank the Assoc. Edit. and the anonymous reviewers for their valuable comments, which have highly improved the manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Paris VIII, 93526, Saint-Denis, France

    Sihem Mesnager

  2. CNRS, LAGA UMR 7539, Sorbonne Paris Cité, University of Paris XIII, 93430, Villetaneuse, France

    Sihem Mesnager

  3. Telecom ParisTech, 75013, Paris, France

    Sihem Mesnager

  4. Department of Computing, Mathematics, and Physics, Western Norway University of Applied Sciences, 5020, Bergen, Norway

    Constanza Riera

  5. Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA, 93943–5216, USA

    Pantelimon Stănică

Authors
  1. Sihem Mesnager
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Constanza Riera
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pantelimon Stănică
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sihem Mesnager.

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 Boolean Functions and Their Applications

Guest Editors: Lilya Budaghyan, Claude Carlet, and Tor Helleseth

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mesnager, S., Riera, C. & Stănică, P. Multiple characters transforms and generalized Boolean functions. Cryptogr. Commun. 11, 1247–1260 (2019). https://doi.org/10.1007/s12095-019-00383-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12095-019-00383-5

Keywords

Mathematics Subject Classification (2010)

Search

Navigation