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Title:
Parametrizing Maximal Orders Along Supersingular \ell-Isogeny Paths
Authors: Laia Amorós, James Clements, Chloe Martindale
Abstract: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.
ePrint: https://eprint.iacr.org/2025/033
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