### Orientations and cycles in supersingular isogeny graphs

##### Abstract

The paper concerns several theoretical aspects of oriented supersingular $\ell$-isogeny volcanoes and their relationship to closed walks in the supersingular $\ell$-isogeny graph. Our main result is a bijection between the rims of the union of all oriented supersingular $\ell$-isogeny volcanoes over $\overline{\mathbb{F}}_p$ (up to conjugation of the orientations), and isogeny cycles (non-backtracking closed walks which are not powers of smaller walks) of the supersingular $\ell$-isogeny graph over $\overline{\mathbb{F}}_p$. The exact proof and statement of this bijection are made more intricate by special behaviours arising from extra automorphisms and the ramification of $p$ in certain quadratic orders. We use the bijection to count isogeny cycles of given length in the supersingular $\ell$-isogeny graph exactly as a sum of class numbers of these orders, and also give an explicit upper bound by estimating the class numbers.

Available format(s)
Category
Public-key cryptography
Publication info
Preprint.
Keywords
supersingular isogeny elliptic curve orientation
Contact author(s)
S A Arpin @ math leidenuniv nl
m chen 1 @ bham ac uk
klauter @ fb com
rscheidl @ ucalgary ca
hatran1104 @ gmail com
History
2022-12-04: revised
See all versions
Short URL
https://ia.cr/2022/562

CC BY

BibTeX

@misc{cryptoeprint:2022/562,
author = {Sarah Arpin and Mingjie Chen and Kristin E.  Lauter and Renate Scheidler and Katherine Stange and Ha T.  N.  Tran},
title = {Orientations and cycles in supersingular isogeny graphs},
howpublished = {Cryptology ePrint Archive, Paper 2022/562},
year = {2022},
note = {\url{https://eprint.iacr.org/2022/562}},
url = {https://eprint.iacr.org/2022/562}
}

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