Automorphisms and isogeny graphs of abelian varieties, with applications to the superspecial Richelot isogeny graph
This research improves the understanding of performance and security for recently-proposed cryptosystems that rely on these mathematical structures, offering a concrete step towards understanding general superspecial isogeny graphs.
This paper investigates the structures within isogeny graphs of principally polarized abelian varieties, particularly focusing on (2,2)-isogeny graphs of superspecial abelian surfaces. It provides theoretical and experimental results on spectral and statistical properties, including stationary distributions for random walks, bounds on eigenvalues and diameters, and a proof of connectivity for the Jacobian subgraph.
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.