# On the nonlinearity of monotone Boolean functions

## 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.

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## Notes

1. 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].

2. 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

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8. Carlet, C., Joyner, D., Stănică, P., Tang, D.: Cryptographic properties of monotone Boolean functions. J. Math. Cryptol. 10(1), 1–14 (2016)

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## Acknowledgements

The author deeply thanks Stjepan Picek for his kind help for generating Table 3. We also thank Pante Stănică for having suggested, when we worked on [8], the problem of finding an upper bound on the nonlinearity of monotone Boolean functions.

## Author information

Authors

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Correspondence to Claude Carlet.

Claude Carlet is supported by Norwegian Research Council.

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Carlet, C. On the nonlinearity of monotone Boolean functions. Cryptogr. Commun. 10, 1051–1061 (2018). https://doi.org/10.1007/s12095-017-0262-5

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• DOI: https://doi.org/10.1007/s12095-017-0262-5

### Keywords

• Boolean functions
• Nonlinearity
• Monotone functions