Abstract
We propose a notion of stability of the Walsh–Hadamard spectrum of Boolean functions when inputs are independent and identically distributed Bernoulli random variables. We study the stability spectrum of bent Boolean functions and obtain a bound for it. We also derive the formula for the stability transform of Maiorana–McFarland type bent functions. We analyze the stability spectrum of symmetric Boolean functions and characterize it for symmetric bent Boolean functions and symmetric Boolean functions in an odd number of variables with maximum nonlinearity. We show that the stability spectrum is not, in general, invariant under extended affine transformations. Further, we display some non-bent symmetric Boolean functions whose stability spectra are flatter than that of symmetric bent Boolean functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arfken, G.B., Weber, H.J., Harris, F.E.: Mathematical Methods for Physicists. Elsevier, Amsterdam (2012)
Carlet, C.: Boolean Functions for Cryptography and Coding Theory. Cambridge University Press, Cambridge (2021)
Cusick, T., Stănică, P.: Cryptographic Boolean Functions and Applications, 2nd edn. Elsevier, Amsterdam (2017)
Carlet, C., Méaux, P., Rotella, Y.: Boolean functions with restricted input and their robustness; application to the FLIP cipher. IACR Trans. Symmetric Cryptol. 3, 192–227 (2017). https://doi.org/10.13154/tosc.v2017.i3.192-227
Gangopadhyay, S., Gangopadhyay, A.K., Pollatos, S., Stănică, P.: Cryptographic Boolean functions with biased inputs. Cryptogr. Commun. 9(2), 301–314 (2017). https://doi.org/10.1007/s12095-015-0174-1
Gangopadhyay, S., Paul, G., Sinha, N., Stănică, P.: Generalized nonlinearity of \(S\)-boxes. Adv. Math. Commun. 12(1), 115–122 (2018). https://doi.org/10.3934/amc.2018007
Maitra, S., Mandal, B., Martinsen, T., Roy, D., Stănică, P.: Analysis on Boolean function in a restricted (biased) domain. IEEE Trans. Inf. Theory 66(2), 1219–1231 (2020). https://doi.org/10.1109/TIT.2019.2932739
Maitra, S., Sarkar, P.: Maximum nonlinearity of symmetric Boolean functions on odd number of variables. IEEE Trans. Inf. Theory 48(9), 2626–2630 (2002). https://doi.org/10.1109/TIT.2002.801482
O’Donnell, R.: Analysis of Boolean Functions. Cambridge University Press, Cambridge (2014)
Savický, P.: On the bent Boolean functions that are symmetric. Eur. J. Comb. 15(4), 407–410 (1994). https://doi.org/10.1006/eujc.1994.1044
Funding
Not applicable.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declared that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Gangopadhyay, A.K., Kumar, V., Stănică, P. et al. Stability of the Walsh–Hadamard spectrum of cryptographic Boolean functions with biased inputs. J. Appl. Math. Comput. 69, 3337–3357 (2023). https://doi.org/10.1007/s12190-023-01887-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-023-01887-3