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Title:
Orientations and cycles in supersingular isogeny graphs
Authors: Sarah Arpin, Mingjie Chen, Kristin E. Lauter, Renate Scheidler, Katherine E. Stange, Ha T. N. Tran
Abstract:The paper concerns several theoretical aspects of oriented supersingular l-isogeny volcanoes and their relationship to closed walks in the supersingular l-isogeny graph. Our main result is a bijection between the rims of the union of all oriented supersingular l-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 l-isogeny graph modulo 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 l-isogeny graph exactly as a sum of class numbers, and also give an explicit upper bound by estimating the class numbers.
ePrint: https://eprint.iacr.org/2022/562
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