Functional graphs of polynomials over finite fields

Given a function f in a finite field IFq we define the functional graph of f as a directed graph on q nodes labelled by elements of IFq 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 an 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 IFq. Finally we present some numerical results and compare function graphs of quadratic polynomials with those generated by random maps and pose interesting new problems.