Abstract
We prove a conjecture on the nonlinearity of monotone Boolean functions in even dimension, proposed in the recent paper “Cryptographic properties of monotone Boolean functions”, by Carlet et al. (J. Math. Cryptol. 10(1), 1–14, 2016). We also prove an upper bound on such nonlinearity, which is asymptotically much stronger than the conjectured upper bound and than the upper bound proved for odd dimension in this same paper. Contrary to these two previous bounds, which were not tight enough for allowing to clarify if monotone functions can have good nonlinearity, this new bound shows that the nonlinearity of monotone functions is always very bad, which represents a fatal cryptographic weakness of monotone Boolean functions; they are too closely approximated by affine functions for being usable as nonlinear components in cryptographic applications. We deduce a necessary criterion to be satisfied by a Boolean (resp. vectorial) function for being nonlinear.
Similar content being viewed by others
Notes
Which states that for every Boolean function f over \(\mathbb {F}_{2}^{n}\), for every vector subspace E of \(\mathbb {F}_{2}^{n}\), and every elements a and b of \(\Bbb {F}_{2}^{n}\), we have \({\sum }_{\mathbf {u}\in \mathbf {a}+E}(-1)^{\mathbf {b}\cdot \mathbf {u}}\, W_{f}(\mathbf {u})= |E|\,(-1)^{\mathbf {a}\cdot \mathbf {b}}\, {\sum }_{\mathbf {x}\in \mathbf {b}+E^{\perp }}(-1)^{f(\mathbf {x})+\mathbf {a}\cdot \mathbf {x}}\), where E ⊥ denotes the orthogonal of E, see e.g. in [6].
It is rare that this formula needs to be used for Boolean functions rather than the simpler Poisson formula; it is interesting to find such situation (here and in the next section as well).
References
Alon, N., Spencer, J.H.: The Probabilistic Method, 2nd edn. Wiley-VCH, New York (2000)
Blum, A., Burch, C., Langford, J.: On learning monotone Boolean functions. In: Proceedings of the 39th FOCS, pp. 408–415. IEEE Computer Society Press (1998)
Blum, A., Furst, M., Kearns, M., Lipton, R.: Cryptographic primitives based on hard learning problems. In: Proceedings of Advances in Cryptology—CRYPTO’93, number 773 in LNCS, pp. 278–291. Springer, Berlin (1993)
Bshouty, N., Tamon, C.: On the Fourier spectrum of monotone functions. J. ACM 43(4), 747–770 (1996)
Canteaut, A., Carlet, C., Charpin, P., Fontaine, C.: On cryptographic properties of the cosets of R(1,m). IEEE Trans. Inf. Theory 47(4), 1494–1513 (2001)
Carlet, C.: Boolean functions for cryptography and error correcting codes. In: Crama, Y., Hammer, P. (eds.) Boolean Methods and Models, pp. 257–397. Cambridge University Press, Cambridge (2010). Available at: www.math.univ-paris13.fr/~carlet/pubs.html
Carlet, C.: Vectorial Boolean functions for cryptography. In: Crama, Y., Hammer, P. (eds.) Boolean Methods and Models, pp. 398–469. Cambridge University Press, Cambridge (2010). Available at: www.math.univ-paris13.fr/~carlet/pubs.html
Carlet, C., Joyner, D., Stănică, P., Tang, D.: Cryptographic properties of monotone Boolean functions. J. Math. Cryptol. 10(1), 1–14 (2016)
Crama, Y., Hammer, P.L.: Boolean Functions. Theory, Algorithms, and Applications. Cambridge University Press, Cambridge (2011)
Dalai, D.K., Maitra, S., Sarkar, S.: Basic theory in construction of Boolean functions with maximum possible annihilator immunity. Des. Codes Cryptogr. 40(1), 41–58 (2006)
Dachman-Soled, D., Lee, H.K., Malkin, T., Servedio, R.A., Wan, A., Wee, H.: Optimal cryptographic hardness of learning monotone functions. Theory Comput. 5, 257–282 (2009)
Mossel, E., O’Donnell, R.: On the noise sensitivity of monotone functions. Random Struct. Algorithms 23(3), 333–350 (2003)
Author information
Authors and Affiliations
Corresponding author
Additional information
Claude Carlet is supported by Norwegian Research Council.
This article is part of the Topical Collection on Special Issue on Sequences and Their Applications
Appendix
Appendix
Rights and permissions
About this article
Cite this article
Carlet, C. On the nonlinearity of monotone Boolean functions. Cryptogr. Commun. 10, 1051–1061 (2018). https://doi.org/10.1007/s12095-017-0262-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12095-017-0262-5