Paper 2019/1056
Adventures in Supersingularland
Sarah Arpin, Catalina CamachoNavarro, Kristin Lauter, Joelle Lim, Kristina Nelson, Travis Scholl, and Jana Sotáková
Abstract
In this paper, we study isogeny graphs of supersingular elliptic curves. Supersingular isogeny graphs were introduced as a hard problem into cryptography by Charles, Goren, and Lauter for the construction of cryptographic hash functions. These are large expander graphs, and the hard problem is to find an efficient algorithm for routing, or pathfinding, between two vertices of the graph. We consider four aspects of supersingular isogeny graphs, study each thoroughly and, where appropriate, discuss how they relate to one another. First, we consider two related graphs that help us understand the structure: 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 $\mathbb{F}_p$. We show how to pass from the latter to the former. The graph $\mathcal{S}$ is relevant for cryptanalysis because routing between vertices in $\mathbb{F}_p$ is easier than in the full isogeny graph. The $\mathbb{F}_p$vertices are typically assumed to be randomly distributed in the graph, which is far from true. We provide an analysis of the distances of connected components of $\mathcal{S}$. Next, we study the involution on $\mathcal{G}_\ell(\overline{\mathbb{F}_p})$ that is given by the Frobenius of $\mathbb{F}_p$ and give heuristics on how often shortest paths between two conjugate $j$invariants are preserved by this involution (mirror paths). We also study the related question of what proportion of conjugate $j$invariants are $\ell$isogenous for $\ell = 2,3$. We conclude with experimental data on the diameters of supersingular isogeny graphs when $\ell = 2$ and compare this with previous results on diameters of LPS graphs and random Ramanujan graphs.
Metadata
 Available format(s)
 Category
 Publickey cryptography
 Publication info
 Preprint. Minor revision.
 Keywords
 number theoryisogenybased cryptography
 Contact author(s)

ja sotakova @ gmail com
Sarah Arpin @ colorado edu
krstnm nlsn @ gmail com
joellelim @ berkeley edu  History
 20190918: received
 Short URL
 https://ia.cr/2019/1056
 License

CC BY
BibTeX
@misc{cryptoeprint:2019/1056, author = {Sarah Arpin and Catalina CamachoNavarro and Kristin Lauter and Joelle Lim and Kristina Nelson and Travis Scholl and Jana Sotáková}, title = {Adventures in Supersingularland}, howpublished = {Cryptology ePrint Archive, Paper 2019/1056}, year = {2019}, note = {\url{https://eprint.iacr.org/2019/1056}}, url = {https://eprint.iacr.org/2019/1056} }