Sarah Arpin (CU Boulder) Adventures in Supersingularland: An Exploration of Supersingular Elliptic Curve Isogeny Graphs
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Supersingular isogeny graphs were introduced as a hard problem into cryptography by Charles, Goren, and Lauter for the construction of cryptographic hash functions (CGL06). These are large expander graphs, and the hard problem is to find an efficient algorithm for routing between two vertices of the graph. In particular, we consider two related graphs that help us understand the big picture of the full supersingular $\ell$-isogeny graph: the `spine' $\mathcal{S}$, which is the subgraph of ${\mathcal{G}}_{\ell}(\overline{{\mathbb{F}}_{p}})$ given by the j-invariants in $\mathbb{F}}_{p$, and the graph ${\mathcal{G}}_{\ell}({\mathbb{F}}_{p})$, in which both curves and isogenies must be defined over $\mathbb{F}}_{p$. This is joint work with Catalina Camacho-Navarro, Kristin Lauter, Joelle Lim, Kristina Nelson, Travis Scholl, Jana Sotáková. https://arxiv.org/abs/1909.07779 No pre-requisites are required, but a willingness to take things on faith is requested. "Why, sometimes I've believed as many as six impossible things before breakfast." ~Lewis Carroll