Abstract
Very recently, Carlet, Méaux and Rotella have studied the main cryptographic features of Boolean functions when, for a given number n of variables, the input to these functions is restricted to some subset E of \(\mathbb {F}_{2}^{n}\). Their study includes the particular case when E equals the set of vectors of fixed Hamming weight, which is important in the robustness of the Boolean function involved in the FLIP stream cipher. In this paper we focus on the nonlinearity of Boolean functions with restricted input and present new results related to the analysis of this nonlinearity improving the upper bound given by Carlet et al.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Carlet, C.: Boolean functions for cryptography and error correcting codes. In: Crama, Y., Hammer, P. L. (eds.) Boolean Models and Methods in Mathematics, Computer Science, and Engineering, pp 257–397. Cambridge University Press, Cambridge (2010)
Carlet, C., Mesnager, S.: Improving the upper bounds on the covering radii of binary Reed-Muller codes. IEEE Trans. Inf. Theory 53(1), 162–173 (2007)
Carlet, C., Mesnager, S.: Four decades of research on bent functions. Des. Codes Crypt. 78(1), 5–50 (2016)
Carlet, C., Méaux, P., Rotella, Y.: Boolean functions with restricted input and their robustness; application to the flip cipher. IACR Transactions on Symmetric Cryptology (3), 192–227. https://doi.org/10.13154/tosc.v2017.i3.192-227 (2017)
Dillon, J.: Elementary Hadamard difference sets. PhD Thesis, University of Maryland (1974)
Méaux, P., Journault, A., Standaert, F.-X., Carlet, C.: Towards stream ciphers for efficient FHE with low-noise ciphertexts. In: EUROCRYPT 2016, pp. 311–343 (2016)
Mesnager, S.: Binary Bent Functions: Fundamentals and Results. Springer, Switzerland (2016)
Rothaus, O.S.: On “bent” functions. Journal of Combinatorial Theory, Series A 20, 300–305 (1976)
Wang, Q., Johansson, T.: A note on fast algebraic attacks and higher order nonlinearities. In: International Conference on Information Security and Cryptology, Inscrypt 2010, pp. 404–414 (2010)
Author information
Authors and Affiliations
Corresponding author
Additional information
This article is part of the Topical Collection on Special Issue on Boolean Functions and Their Applications
Rights and permissions
About this article
Cite this article
Mesnager, S., Zhou, Z. & Ding, C. On the nonlinearity of Boolean functions with restricted input. Cryptogr. Commun. 11, 63–76 (2019). https://doi.org/10.1007/s12095-018-0293-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12095-018-0293-6