Paper 2025/155

Cycles and Cuts in Supersingular L-Isogeny Graphs

Abstract

Supersingular elliptic curve isogeny graphs underlie isogeny-based cryptography. For isogenies of a single prime degree $$, their structure has been investigated graph-theoretically. We generalise the notion of $$-isogeny graphs to $$-isogeny graphs (studied in the prime field case by Delfs and Galbraith), where $$ is a set of small primes dictating the allowed isogeny degrees in the graph. We analyse the graph-theoretic structure of $$-isogeny graphs. Our approaches may be put into two categories: cycles and graph cuts. On the topic of cycles, we provide: a count for the number of non-backtracking cycles in the $$-isogeny graph using traces of Brandt matrices; an efficiently computable estimate based on this approach; and a third ideal-theoretic count for a certain subclass of $$-isogeny cycles. We provide code to compute each of these three counts. On the topic of graph cuts, we compare several algorithms to compute graph cuts which minimise a measure called the edge expansion, outlining a cryptographic motivation for doing so. Our results show that a greedy neighbour algorithm out-performs standard spectral algorithms for computing optimal graph cuts. We provide code and study explicit examples. Furthermore, we describe several directions of active and future research.