On the functional graph generated by a plane curve and its quadratic twist

Bernard Mans, Min Sha, Jeffrey Smith, Daniel Sutantyo

Published 2019-06-28Version 1

Given a plane curve defined by $Y^2 = f(X)$ over a finite field $\mathbb{F}_q$ of odd characteristic and a non-square element $\lambda$ in $\mathbb{F}_q$, we define a functional graph by choosing the elements in $\mathbb{F}_q$ as vertices and drawing an edge from $x$ to $y$ if and only if $(x,y)$ is either a point on $Y^2 = f(X)$ or a point on $\lambda Y^2 = f(X)$. We show that if $f$ is a permutation polynomial over $\mathbb{F}_q$, then every connected component of the graph has a Hamiltonian cycle. Moreover, these Hamiltonian cycles can be used to construct balancing binary sequences. By making computations for permutation polynomials $f$ of low degree, it turns out that almost all these graphs are strongly connected, and there are many Hamiltonian cycles in such a graph if it is connected.