Paper 2025/189

Experimentally studying path-finding problem between conjugates in supersingular isogeny graphs: Optimizing primes and powers to speed-up cycle finding

Abstract

We study the problem of finding a path between conjugate supersingular elliptic curves over $\mathbb{F}_{p^2}$ for a prime $p$, which is important for cycle finding in supersingular isogeny graphs. We see that for any given $p$, there is some $l$ corresponding to $p$ which accelerates the process of conjugate path-finding. Also, time-wise, the most efficient way of overviewing the graph is seeing $i(=3)$ steps at once. We have outlined methods in which the next vertex of any pseudo-random walk should be chosen to reach conjugate vertex faster. We have experimentally investigated the paths between frobenius conjugates for wide ranges of small prime $l$. We introduce sets to experimentally learn about the structure of the isogeny graphs when short cycles are present.