A Complete Characterization of Plateaued Boolean Functions in Terms of Their Cayley Graphs

  • Constanza Riera
  • Patrick Solé
  • Pantelimon Stănică
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10831)


In this paper we find a complete characterization of plateaued Boolean functions in terms of the associated Cayley graphs. Precisely, we show that a Boolean function f is s-plateaued (of weight \(=2^{(n+s-2)/2}\)) if and only if the associated Cayley graph is a complete bipartite graph between the support of f and its complement (hence the graph is strongly regular of parameters \(e=0,d=2^{(n+s-2)/2}\)). Moreover, a Boolean function f is s-plateaued (of weight \(\ne 2^{(n+s-2)/2}\)) if and only if the associated Cayley graph is strongly 3-walk-regular (and also strongly \(\ell \)-walk-regular, for all odd \(\ell \ge 3\)) with some explicitly given parameters.


Plateaued Boolean functions Cayley graphs Strongly regular Walk regular 


  1. 1.
    Bernasconi, A., Codenotti, B.: Spectral analysis of Boolean functions as a graph eigenvalue problem. IEEE Trans. Comput. 48(3), 345–351 (1999)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bernasconi, A., Codenotti, B., VanderKam, J.M.: A characterization of bent functions in terms of strongly regular graphs. IEEE Trans. Comput. 50(9), 984–985 (2001)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Budaghyan, L.: Construction and Analysis of Cryptographic Functions. Springer, Heidelberg (2014). Scholar
  4. 4.
    Carlet, C.: Boolean models and methods in mathematics, computer science, and engineering. In: Hammer, P., Crama, Y. (eds.) Boolean Functions for Cryptography and Error Correcting Codes, pp. 257–397. Cambridge University Press, Cambridge (2010)Google Scholar
  5. 5.
    Cusick, T.W., Stănică, P.: Cryptographic Boolean Functions and Applications, 2nd edn. Academic Press, San Diego (2017). 1st edn. (2009)zbMATHGoogle Scholar
  6. 6.
    Cvetkovic, D.M., Doob, M., Sachs, H.: Spectra of Graphs. Academic Press, New York (1979)zbMATHGoogle Scholar
  7. 7.
    van Dam, E.R., Omidi, G.R.: Strongly walk-regular graphs. J. Comb. Theory Ser. A 120, 803–810 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fiol, M.A., Garriga, E.: Spectral and geometric properties of \(k\)-walk-regular graphs. Electron. Notes Discrete Math. 29, 333–337 (2007)CrossRefzbMATHGoogle Scholar
  9. 9.
    Godsil, C.D.: Bounding the diameter of distance-regular graphs. Combinatorica 8(4), 333–343 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Huang, X., Huang, Q.: On regular graphs with four distinct eigenvalues. Linear Algebra Appl. 512, 219–233 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Mesnager, S.: On semi-bent functions and related plateaued functions over the Galois field \(\mathbb{F}_{2^{n}}\). In: Koç, Ç.K. (ed.) Open Problems in Mathematics and Computational Science, pp. 243–273. Springer, Cham (2014). Scholar
  12. 12.
    Mesnager, S.: Bent Functions: Fundamentals and Results. Springer, New York (2016). Scholar
  13. 13.
    Rothaus, O.S.: On bent functions. J. Comb. Theory Ser. A 20, 300–305 (1976)CrossRefzbMATHGoogle Scholar
  14. 14.
    Tokareva, N.: Bent Functions, Results and Applications to Cryptography. Academic Press, San Diego (2015)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Computing, Mathematics, and PhysicsWestern Norway University of Applied SciencesBergenNorway
  2. 2.CNRS/LAGA, University of Paris 8Saint-DenisFrance
  3. 3.Department of Applied MathematicsNaval Postgraduate SchoolMontereyUSA

Personalised recommendations