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 F 2 2m . 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.
Pott et al. (2018) showed that F(x) = x 2 r Tr n m (x), n = 2m, r ≥ 1, is a nontrivial example of a vectorial function with the maximal possible number 2 n − 2 m of bent components. Mesnager et al. (2019) generalized this result by showing conditions on Λ(x) = x+ σ j=1 αjx 2 t j , αj ∈ F2m , under whichTr n m (Λ(x)) has the maximal possible number of bent components. We simplify these conditions and further analyse this class of functions. For all related vectorial bent functions F (x) = Tr n m (γF(x)), γ ∈ F2n \ F2m , which as we will point out belong to the Maiorana-McFarland class, we describe the collection of the solution spaces for the linear equations DaF (x) = F (x) + F (x + a) + F (a) = 0, which forms a spread of F2n . Analysing these spreads, we can infer neat conditions for functions H(x) = (F (x), G(x)) from F2n to F2m × F2m to exhibit small differential uniformity (for instance for Λ(x) = x and r = 0 this fact is used in the construction of Carlet's, Pott-Zhou's, Taniguchi's APN-function). For some classes of H(x) we determine differential uniformity and with a method based on Bezout's theorem nonlineariy.
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