Paper 2022/562
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.
Metadata
- Available format(s)
-
PDF
- 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
kstange @ math colorado edu
hatran1104 @ gmail com - History
- 2022-12-04: revised
- 2022-05-10: received
- See all versions
- Short URL
- https://ia.cr/2022/562
- License
-
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},
url = {https://eprint.iacr.org/2022/562}
}
