O grafo funcional de uma família de polinômios sobre uma extensão quadrática de um corpo finito

Abstract

Let $X$ be a finite set and $f: X \rightarrow X$ an arbitrary function. The iteration of $f$ over $X$ yields a dynamical system that can be represented as a graph. In fact, the functional graph of $f$ is the directed graph $\mathscr{G}(f)=(\mathcal{V}, \mathcal{E})$, where $\mathcal{V}=X$ and $\mathcal{E}=\{\langle x, f(x)\rangle \mid x \in X\}$. The obtained dynamical system has applications in many areas such as cryptography, communications and physics.

Let $\mathbb{F}_q$ be a finite field with $q$ elements of odd characteristic. In this work, we will fully describe the dynamics of the function $f(X)=c\left(X^{q+1} \pm a X^2\right)$ over the finite field $\mathbb{F}_{q^2}$, for $a \in\{ \pm 1\}$ and $c \in \mathbb{F}_q^*$. We will also provide partial results for $a \in \mathbb{F}_q^* \backslash\{ \pm 1\}$.