Taha Hedayat | Searcharxiv

arXiv2026

The Spine: A Supersingular Highway

We consider the structure of the spine of the supersingular $\ell$-isogeny graph for one of the cases which arXiv:2502.03613 was not able to fully describe, $\ell = 2$ and $p = 71, 119\pmod{120}$. We find the distance, eccentricity, and diameter functions, of the components of the spine without the non-trivial edge not defined over $\mathbb{F}_p$. Using these functions, we find the mean diameter of the spine and show how this value distinguishes the different structures of the spine. Thus, allowing us to use explicit computations to provide heuristics on the behavior of the spine's structure as $p$ varies.

math.NT↗

arXiv2025

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

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.

math.NT↗