The Spine of a Supersingular $\ell$-Isogeny graph

TL;DR

This paper investigates the structural properties of supersingular elliptic curve $ ext{l}$-isogeny graphs, focusing on the subgraph induced by $ ext{F}_p$-vertices, and compares their behavior to random regular graphs to inform cryptographic security assumptions.

Contribution

It analyzes the behavior and diameter of the spine of supersingular $ ext{l}$-isogeny graphs under certain conditions, providing new insights into their structure and randomness properties.

Findings

The diameter of the spine is characterized.

Numerical data on vertex counts over $ ext{F}_p$ and algebraic closures.

Wave-shaped pattern in vertex counts supports similarity to random graphs.

Abstract

Supersingular elliptic curve $\ell$-isogeny graphs over finite fields offer a setting for a number of quantum-resistant cryptographic protocols. The security analysis of these schemes typically assumes that these graphs behave randomly. Motivated by this debatable assertion, we explore structural properties of these graphs. We detail the behavior, governed by congruence conditions on $p$, of the $\ell$-isogeny graph over $\mathbb{F}_p$ when passing to the spine, i.e. the subgraph induced by the $\mathbb{F}_p$-vertices in the full $\ell$-isogeny graph. We describe the diameter of the spine and offer numerical data on the number of vertices, over both $\mathbb{F}_p$ and $\overline{\mathbb{F}_p}$, in the center of the $\ell$-isogeny graph. Our plots of these counts exhibit a wave-shaped pattern which supports the assertion that centers of supersingular $\ell$-isogeny graphs exhibit the same behavior as those of random $(\ell+1)$-regular graphs.