Determining the Walsh spectra of Taniguchi's and related APN-functions
Anbar Meidl, Nurdagül and Kalaycı, Tekgül and Meidl, Wilfried (2019) Determining the Walsh spectra of Taniguchi's and related APN-functions. Finite Fields and Their Applications, 60 . ISSN 1071-5797 (Print) 1090-2465 (Online)
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Official URL: http://dx.doi.org/10.1016/j.ffa.2019.101577
We introduce a method based on Bezout’s theorem on intersection points of two projective plane curves, for determining the nonlinearity of some classes of quadratic functions on F22m. Among those are the functions of Taniguchi 2019, Carlet 2011, and Zhou and Pott 2013, all of which are APN under certain conditions. This approach helps to understand why the majority of the functions in those classes have solely bent and semibent components, which in the case of APN functions is called the classical spectrum. More precisely, we show that all Taniguchi functions have the classical spectrum independent from being APN. We determine the nonlinearity of all functions belonging to Carlet’s class and to the class of Zhou and Pott, which also conﬁrms with comparatively simple proofs earlier results on the Walsh spectrum of APN-functions in these classes. Using the Hasse-Weil bound, we show that some simple suﬃcient conditions for the APN-ness of the Zhou-Pott functions, which are given in the original paper, are also necessary.
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