On the absolute irreducibility of hyperplane sections of generalized Fermat varieties in $\Bbb{P}^3$ and the conjecture on exceptional APN functions: the Kasami-Welch degree case

Abstract

Let $f$ be a function on a finite field $F$. The decomposition of the generalized Fermat variety $X$ defined by the multivariate polynomial of degree $n$, $\phi(x,y,z)=f(x)+f(y)+f(z)$ in $\Bbb{P}^3(\overline{\mathbb{F}}_2)$, plays a crucial role in the study of almost perfect non-linear (APN) functions and exceptional APN functions. Their structure depends fundamentally on the Fermat varieties corresponding to the monomial functions of exceptional degrees $n=2^k+1$ and $n=2^{2k}-2^k+1$ (Gold and Kasami-Welch numbers, respectively). Very important results for these have been obtained by Janwa, McGuire and Wilson in [12,13]. In this paper we study $X$ related to the Kasami-Welch degree monomials and its decomposition into absolutely irreducible components. We show that, in this decomposition, the components intersect transversally at a singular point. This structural fact implies that the corresponding generalized Fermat hypersurfaces, related to Kasami-Welch degree polynomial families, are absolutely irreducible. In particular, we prove that if $f(x)=x^{2^{2k}-2^k+1}+h(x)$, where ${\rm deg}(h)\equiv 3{\pmod 4}$, then the corresponding APN multivariate hypersurface is absolutely irreducible, and hence $f(x)$ is not exceptional APN function. We also prove conditional result in the case when ${\rm deg}(h)\equiv 5{\pmod 8}$. Since for odd degree $f(x)$, the conjecture needs to be resolved only for the Gold degree and the Kasami-Welch degree cases our results contribute substantially to the proof of the conjecture on exceptional APN functions---in the hardest case: the Kasami-Welch degree.