On the Fourier spectra of new APN functions (Q2848538)

Let \({\mathbb F}_{2^n}\) be the finite field with \(2^n\) elements. A mapping \(F:{\mathbb F}_{2^n}\to {\mathbb F}_{2^n}\) is called \textit{almost perfect nonlinear} or APN if \(F(x+a)-F(x)=b\) has \(0\) or \(2\) solutions for all \(a,b\in {\mathbb F}_{2^n}\), \(a\neq 0\). Most of the known APN functions are quadratic. which means that \(F(x+a)+F(x)+F(a)+F(0)\) are linear mappings for all \(a\). The numbers \(\sum_{x\in {\mathbb F}_{2^n}} (-1)^{\langle a,x\rangle + \langle b, F(x)\rangle}\) are called the Fourier coefficients of \(F\) (here \(\langle\;,\;\rangle\) denotes any nondegenerate bilinear form on \({\mathbb F}_{2^n}\)). The set of Fourier coefficients is the \textit{Fourier spectrum} of \(F\). If \(n=2m\) is even, all except one quadratic APN functions have the same Fourier spectrum \(\{0, \pm 2^{m}, \pm 2^{m+1}\}\), and if \(n=2m+1\) is odd, all known quadratic APN functions have the same Fourier spectrum \(\{0, \pm 2^{m+1}\}\). These spectra are called \textit{classical}. One of the big open problems about APN functions is to determine possible Fourier spectra of APN functions. In this paper, the authors show that two recently constructed quadratic APN functions [\textit{Y. Zhou} and \textit{A. Pott}, Adv. Math. 234, 43--60 (2013; Zbl 1296.12007)] and [\textit{C. Carlet}, Des. Codes Cryptography 59, No. 1--3, 89--109 (2011; Zbl 1229.94041)] have the classical Fourier spectra.