Automorphisms and isogeny graphs of abelian varieties, with applications to the superspecial Richelot isogeny graph

TL;DR

This paper explores the structure and properties of isogeny graphs of abelian varieties, focusing on superspecial abelian surfaces, to enhance understanding of cryptographic security and graph connectivity.

Contribution

It provides theoretical and experimental insights into the spectral, statistical, and connectivity properties of (2, 2)-isogeny graphs of superspecial abelian surfaces, including automorphism effects.

Findings

Stationary distributions for random walks are characterized.

Bounds on eigenvalues and diameters of the graphs are established.

Connectivity of the Jacobian subgraph is proven.

Abstract

We investigate special structures due to automorphisms in isogeny graphs of principally polarized abelian varieties, and abelian surfaces in particular. We give theoretical and experimental results on the spectral and statistical properties of (2, 2)-isogeny graphs of superspecial abelian surfaces, including stationary distributions for random walks, bounds on eigenvalues and diameters, and a proof of the connectivity of the Jacobian subgraph of the (2, 2)-isogeny graph. Our results improve our understanding of the performance and security of some recently-proposed cryptosystems, and are also a concrete step towards a better understanding of general superspecial isogeny graphs in arbitrary dimension.