Determining the Walsh spectra of Taniguchi's and related APN-functions

Abstract

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) [20], Carlet (2011) [7], and Zhou and Pott (2013) [22], 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.