Advisor:  Dr. Rachel Pries

Committee:  Dr. Alexander Hulpke, Dr. Mark Shoemaker, Dr Joe Gersch

Title: An Exploration of the Genus 2 Superspecial (2,2)-Isogeny Graph

Abstract: In 2022, one of the finalists for NIST’s call for a post-quantum cryptography, SIKE, was broken by Castryck and Decru. They constructed an attack relying on the Genus 2 Superspecial Isogeny graph.

This graph represents a still relatively early area of study, and many things about its structure are relatively unknown.  In this talk we introduce the graph, its elements, and the theory required to build it.

We also discuss previous work of Florit and Smith, describing an atlas of the nodes on the graph.  Code developed by the author to produce local neighborhoods on the graph is introduced and described, along with the author’s intentions for its future application. Background for future work is also discussed. This includes a brief description of work by Arpin, et al. on the $\mathbb{F}_p$ subgraphs of the genus 1 graph.

We discuss intentions to extend this to the genus 2 setting in the future.