Abstract
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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Bernasconi, A., Codenotti, B.: Spectral analysis of Boolean functions as a graph eigenvalue problem. IEEE Trans. Comput. 48(3), 345–351 (1999)
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)
Budaghyan, L.: Construction and Analysis of Cryptographic Functions. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-319-12991-4
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)
Cusick, T.W., Stănică, P.: Cryptographic Boolean Functions and Applications, 2nd edn. Academic Press, San Diego (2017). 1st edn. (2009)
Cvetkovic, D.M., Doob, M., Sachs, H.: Spectra of Graphs. Academic Press, New York (1979)
van Dam, E.R., Omidi, G.R.: Strongly walk-regular graphs. J. Comb. Theory Ser. A 120, 803–810 (2013)
Fiol, M.A., Garriga, E.: Spectral and geometric properties of \(k\)-walk-regular graphs. Electron. Notes Discrete Math. 29, 333–337 (2007)
Godsil, C.D.: Bounding the diameter of distance-regular graphs. Combinatorica 8(4), 333–343 (1988)
Huang, X., Huang, Q.: On regular graphs with four distinct eigenvalues. Linear Algebra Appl. 512, 219–233 (2017)
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). https://doi.org/10.1007/978-3-319-10683-0_11
Mesnager, S.: Bent Functions: Fundamentals and Results. Springer, New York (2016). https://doi.org/10.1007/978-3-319-32595-8
Rothaus, O.S.: On bent functions. J. Comb. Theory Ser. A 20, 300–305 (1976)
Tokareva, N.: Bent Functions, Results and Applications to Cryptography. Academic Press, San Diego (2015)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Riera, C., Solé, P., Stănică, P. (2018). A Complete Characterization of Plateaued Boolean Functions in Terms of Their Cayley Graphs. In: Joux, A., Nitaj, A., Rachidi, T. (eds) Progress in Cryptology – AFRICACRYPT 2018. AFRICACRYPT 2018. Lecture Notes in Computer Science(), vol 10831. Springer, Cham. https://doi.org/10.1007/978-3-319-89339-6_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-89339-6_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-89338-9
Online ISBN: 978-3-319-89339-6
eBook Packages: Computer ScienceComputer Science (R0)