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On various nonlinearity measures for boolean functions

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Abstract

A necessary condition for the security of cryptographic functions is to be “sufficiently distant” from linear, and cryptographers have proposed several measures for this distance. In this paper, we show that six common measures, nonlinearity, algebraic degree, annihilator immunity, algebraic thickness, normality, and multiplicative complexity, are incomparable in the sense that for each pair of measures, μ 1,μ 2, there exist functions f 1,f 2 with f 1 being more nonlinear than f 2 according to μ 1, but less nonlinear according to μ 2. We also present new connections between two of these measures. Additionally, we give a lower bound on the multiplicative complexity of collision-free functions.

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Notes

  1. Unfortunately, this introduces an overloading of the word “nonlinearity” since it also refers to the more general concept of distance to linear functions. The meaning will be clear from context.

  2. In this paper we use the term “annihilator immunity” rather than “algebraic immunity”, see the remark in [17].

  3. We have experimentally verified that all functions on four bits have multiplicative complexity at most three. This is somewhat surprising, as circuit realization of random functions (e.g. x 1 x 2 x 3 x 4+x 1 x 2 x 3+x 2 x 3 x 4+x 1 x 3 x 4+x 1 x 3+x 2 x 4+x 1 x 4) would appear to need more than three AND gates. In [2] we conjectured that some function on five bits should have multiplicative complexity five. It turns out this is false ([42]). We expect that some function on six bits will have multiplicative complexity six.

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Acknowledgments

We are grateful to Meltem Sönmez Turan for many discussions on the subject of this work.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Southern Denmark, Odense, Denmark

    Joan Boyar

  2. Information Technology Laboratory, National Institute of Standards and Technology, Gaithersburg, MD, USA

    Magnus Gausdal Find & René Peralta

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  1. Joan Boyar
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  2. Magnus Gausdal Find
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  3. René Peralta
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Correspondence to Magnus Gausdal Find.

Additional information

Joan Boyar is partially supported by the Danish Council for Independent Research, Natural Sciences. Part of this work was done while visiting the University of Waterloo.

Most of this work was done while Magnus Gausdal Find was at the Department of Mathematics and Computer Science, University of Southern Denmark. Part of this work was done while he was visiting the University of Toronto.

Parts of this work appeared in [2].

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Boyar, J., Find, M.G. & Peralta, R. On various nonlinearity measures for boolean functions. Cryptogr. Commun. 8, 313–330 (2016). https://doi.org/10.1007/s12095-015-0150-9

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