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Relating three nonlinearity parameters of vectorial functions and building APN functions from bent functions

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Abstract

We survey the properties of two parameters introduced by C. Ding and the author for quantifying the balancedness of vectorial functions and of their derivatives. We give new results on the distribution of the values of the first parameter when applied to F + L, where F is a fixed function and L ranges over the set of linear functions: we show an upper bound on the nonlinearity of F by means of these values, we determine then the mean of these values and we show that their maximum is a nonlinearity parameter as well, we prove that the variance of these values is directly related to the second parameter. We briefly recall the known constructions of bent vectorial functions and introduce two new classes obtained with Gregor Leander. We show that bent functions can be used to build APN functions by concatenating the outputs of a bent (n, n/2)-function and of some other (n, n/2)-function. We obtain this way a general infinite class of quadratic APN functions. We show that this class contains the APN trinomials and hexanomials introduced in 2008 by L. Budaghyan and the author, and a class of APN functions introduced, in 2008 also, by Bracken et al.; this gives an explanation of the APNness of these functions and allows generalizing them. We also obtain this way the recently found Edel–Pott cubic function. We exhibit a large number of other sub-classes of APN functions. We eventually design with this same method classes of quadratic and non-quadratic differentially 4-uniform functions.

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Authors and Affiliations

  1. LAGA, Department of Mathematics, Universities of Paris 8 and Paris 13, CNRS, 2 rue de la liberté, 93526, Saint-Denis Cedex, France

    Claude Carlet

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  1. Claude Carlet
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Correspondence to Claude Carlet.

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The results presented in this paper have been exposed in an invited talk at WCC 2009.

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Carlet, C. Relating three nonlinearity parameters of vectorial functions and building APN functions from bent functions. Des. Codes Cryptogr. 59, 89–109 (2011). https://doi.org/10.1007/s10623-010-9468-7

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  • DOI: https://doi.org/10.1007/s10623-010-9468-7

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