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

## TL;DR

This paper introduces a geometric method using Bezout's theorem to determine the Walsh spectra and nonlinearity of certain quadratic functions, including Taniguchi, Carlet, and Zhou-Pott classes, clarifying their spectral properties and APN conditions.

## Contribution

It presents a novel geometric approach to analyze the Walsh spectra of specific quadratic functions, providing new insights and simpler proofs for their nonlinearity and APN-ness.

## Key findings

- All Taniguchi functions have the classical spectrum regardless of APN status.
- The nonlinearity of Carlet's and Zhou-Pott's functions is explicitly determined.
- Necessary and sufficient conditions for APN-ness of Zhou-Pott functions are established.

## 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 $\mathbb{F}_{2^{2m}}$. 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.