ePrint Report: Parametrizing Maximal Orders Along Supersingular $ell$-Isogeny Paths
Laia Amorós, James Clements, Chloe Martindale
Suppose you have a supersingular $ell$-isogeny graph with vertices given by $j$-invariants defined over $mathbb{F}_{p^2}$, where $p = 4 cdot f cdot ell^e – 1$ and $ell equiv 3 pmod{4}$. We give an explicit parametrization of the maximal orders in $B_{p,infty}$ appearing as endomorphism rings of the elliptic curves in this graph that are $leq e$ steps away from a root vertex with $j$-invariant 1728. This is the first explicit parametrization of this kind and we believe it will be an aid in better understanding the structure of supersingular $ell$-isogeny graphs that are widely used in cryptography. Our method makes use of the inherent directions in the supersingular isogeny graph induced via Bruhat-Tits trees, as studied in [1]. We also discuss how in future work other interesting use cases, such as $ell=2$, could benefit from the same methodology.