Pseudorandom Graphs from Elliptic Curves

Abstract

Most of the constructions of pseudorandom graphs are based on additive or multiplicative groups of elements of finite fields. As a result the number of vertices of such graphs is limited to values of prime powers or some simple polynomial expressions involving prime powers. We show that elliptic curves over finite fields lead to new constructions of pseudorandom graphs with a new series of parameters. Accordingly, the number of vertices of such graphs can take most of positive integer values (in fact, any positive value under some classical conjectures about the gaps between prime numbers).