Loops, Multi-Edges and Collisions in Supersingular Isogeny Graphs

TL;DR

This paper investigates the structural properties of supersingular isogeny graphs, providing bounds on loops and multi-edges, and exploring conditions for simplicity, with applications to collision analysis and graph comparisons.

Contribution

It formalizes bounds on loops and multi-edges in supersingular isogeny graphs and introduces new concepts like bi-route number for graph analysis.

Findings

Supersingular isogeny graphs have very few loops and multi-edges.

Conditions for the graphs to be simple are identified.

Bounds for bi-route number and common edges between graphs are established.

Abstract

Supersingular isogeny graphs are known to have very few loops and multi-edges. We formalize this idea by studying and finding bounds for the number of loops and multi-edges in such graphs. We also find conditions under which the supersingular isogeny graph $\Lambda_p(\ell)$ is simple. The methods presented in this paper can be used to study many kinds of collisions in supersingular isogeny graphs. As an application, we introduce the notion of bi-route number for two graphs $\Lambda_p(\ell_1),\Lambda_p(\ell_2)$ and compute bounds for it. We also study the number of edges in common between the graphs $\Lambda_p(\ell_1),\Lambda_p(\ell_2)$.