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Multiple characters transforms and generalized Boolean functions

  • Sihem MesnagerEmail author
  • Constanza Riera
  • Pantelimon Stănică
Article
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Part of the following topical collections:
  1. Special Issue on Boolean Functions and Their Applications

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.

Keywords

Generalized Boolean functions Characters Bent Plateaued 

Mathematics Subject Classification (2010)

Primary 94C10; Secondary 06E30 11A07 

Notes

Acknowledgements

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

References

  1. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 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. 3.
    Carlet, C., Gaborit, P.: Hyper-bent functions and cyclic codes. J. Combin. Theory Ser A 113, 446–482 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Carlet, C., Mesnager, S.: Four decades of research on bent functions. J. Des. Codes Crypt. 78, 5–50 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Charpin, P., Gong, G.: Hyperbent functions, Kloosterman sums, and Dickson polynomials. IEEE Trans. Inform. Theory 54(9), 4230–4238 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cusick, T.W., Stănică, P.: Cryptographic Boolean Functions and Applications, 2nd edn. Academic Press, San Diego, CA (2017)zbMATHGoogle Scholar
  7. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Martinsen, T., Meidl, W., Mesnager, S., Stănică, P.: Decomposing generalized bent and hyperbent functions. IEEE Trans. Inf. Theory 63(12), 7804–7812 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 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)MathSciNetzbMATHGoogle Scholar
  11. 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)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Martinsen, T., Meidl, W., Stănică, P.: Partial spread and vectorial generalized bent functions. Des. Codes Crypt. 85(1), 1–13 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Mesnager, S.: Bent Functions: Fundamentals and Results. Springer, Switzerland (2016)CrossRefzbMATHGoogle Scholar
  14. 14.
    Mesnager, S., Tang, C., Qi, Y.: Generalized plateaued functions and admissible (plateaued) functions. IEEE Trans Inform. Theory 63, 6139–6148 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Riera, C., Stănică, P.: Landscape Boolean functions, to appear in Advances in Math. Communication. Available at: arXiv:1806.05878 (2019)
  18. 18.
    Schmidt, B.: On (p a,p b,p a,p ab)-relative difference sets. J. Algebraic Combin. 6, 279–297 (1997)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Schmidt, K.U.: Quaternary constant-amplitude codes for multicode CDMA. IEEE Trans. Inform. Theory 55(4), 1824–1832 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 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. 21.
    Stănică, P., Martinsen, T., Gangopadhyay, S., Singh, B.K.: Bent and generalized bent Boolean functions. Des. Codes & Cryptogr. 69, 77–94 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 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)CrossRefzbMATHGoogle Scholar
  23. 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. 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. 25.
    Zheng, Y.L., Zhang, X.M.: On plateaued functions. IEEE Trans. Inf. Theory 47(9), 1215–1223 (2001)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Paris VIIISaint-DenisFrance
  2. 2.CNRS, LAGA UMR 7539, Sorbonne Paris CitéUniversity of Paris XIIIVilletaneuseFrance
  3. 3.Telecom ParisTechParisFrance
  4. 4.Department of Computing, Mathematics, and PhysicsWestern Norway University of Applied SciencesBergenNorway
  5. 5.Department of Applied MathematicsNaval Postgraduate SchoolMontereyUSA

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