Functional Graphs of Polynomials over Finite Fields

TL;DR

This paper studies the structure of functional graphs generated by polynomials over finite fields, providing theoretical estimates, an efficient isomorphism testing algorithm, and numerical comparisons with random maps.

Contribution

It introduces a practical linear-memory algorithm for polynomial graph isomorphism testing and extends it to general functions, with theoretical bounds and numerical analysis.

Findings

Number of non-isomorphic polynomial graphs estimated

Linear-memory algorithm for quadratic polynomial isomorphism testing

Comparison of polynomial graphs with random maps

Abstract

Given a function $f$ in a finite field ${\mathbb F}_q$ of $q$ elements, we define the functional graph of $f$ as a directed graph on $q$ nodes labelled by the elements of ${\mathbb F}_q$ where there is an edge from $u$ to $v$ if and only if $f(u) = v$. We obtain some theoretic estimates on the number of non-isomorphic graphs generated by all polynomials of a given degree. We then develop a simple and practical algorithm to test the isomorphism of quadratic polynomials that has linear memory and time complexities. Furthermore, we extend this isomorphism testing algorithm to the general case of functional graphs, and prove that, while its time complexity increases only slightly, its memory complexity remains linear. We exploit this algorithm to provide an upper bound on the number of functional graphs corresponding to polynomials of degree $d$ over ${\mathbb F}_q$. Finally, we present some numerical results and compare function graphs of quadratic polynomials with those generated by random maps and pose interesting new problems.