Expander graphs based on GRH with an application to elliptic curve cryptography

TL;DR

This paper constructs expander graphs using Cayley graphs of narrow ray class groups under GRH, demonstrating their properties and applications to elliptic curve cryptography and the security of discrete logarithm problems.

Contribution

It introduces a new construction of expander graphs based on number theory assumptions and explores their implications for elliptic curve cryptography.

Findings

Cayley graphs of (Z/qZ)* with small prime generators are expanders.

Graph of small prime degree isogenies between elliptic curves has eigenvalue separation.

Expansion properties relate to elliptic curve discrete logarithm security.

Abstract

We present a construction of expander graphs obtained from Cayley graphs of narrow ray class groups, whose eigenvalue bounds follow from the Generalized Riemann Hypothesis. Our result implies that the Cayley graph of (Z/qZ)* with respect to small prime generators is an expander. As another application, we show that the graph of small prime degree isogenies between ordinary elliptic curves achieves non-negligible eigenvalue separation, and explain the relationship between the expansion properties of these graphs and the security of the elliptic curve discrete logarithm problem.