Moises Delgado

Moises Delgado, Heeralal Janwa

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 $n$, $φ(x,y,z)=f(x)+f(y)+f(z)$ 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.

Moises Delgado, Heeralal Janwa

Almost Perfect Nonlinear (APN) functions are very useful in cryptography, when they are used as S-Boxes, because of their good resistance to differential cryptanalysis. An APN function $f:\mathbb{F}_{2^n}\rightarrow\mathbb{F}_{2^n}$ is called exceptional APN if it is APN on infinitely many extensions of $\mathbb{F}_{2^n}$. Aubry, McGuire and Rodier conjectured that the only exceptional APN functions are the Gold and the Kasami-Welch monomial functions. They established that a polynomial function of odd degree is not exceptional APN provided the degree is not a Gold number $(2^k+1)$ or a Kasami-Welch number $(2^{2k}-2^k+1)$. When the degree of the polynomial function is a Gold number, several partial results have been obtained [1, 7, 8, 10, 17]. One of the results in this article is a proof of the relatively primeness of the multivariate APN polynomial conjecture, in the Gold degree case. This helps us extend substantially previous results. We prove that Gold degree polynomials of the form $x^{2^k+1}+h(x)$, where $deg(h)$ is any odd integer (with the natural exceptions), can not be exceptional APN. We also show absolute irreducibility of several classes of multivariate polynomials over finite fields and discuss their applications.