Counting Richelot isogenies between superspecial abelian surfaces

Abstract

Castryck, Decru, and Smith used superspecial genus-2 curves and their Richelot isogeny graph for basing genus-2 isogeny cryptography, and recently, Costello and Smith devised an improved isogeny path-finding algorithm in the genus-2 setting. In order to establish a firm ground for the cryptographic construction and analysis, we give a new characterization of decomposed Richelot isogenies in terms of involutive reduced automorphisms of genus-2 curves over a finite field, and explicitly count such decomposed (and nondecomposed) Richelot isogenies between superspecial principally polarized abelian surfaces. As a corollary, we give another algebraic geometric proof of Theorem 2 in the paper of Castryck et al.

An atlas of the Richelot isogeny graph

“…Specializing the kernels and quadratic splittings from §4.5, we see that the action of RA(J (C IV )) on (the indices of) the K i is given by (10,11) (12,13) (14,15) .…”

“…We now consider the impact of automorphisms on edge weights in the isogeny graph, following Katsura and Takashima [14], and recall the explicit classification of reduced automorphism groups of PPASes. In contrast with elliptic curves, where (up to isomorphism) only two curves have nontrivial reduced automorphism group, with PPASes we see much richer structures involving many more vertices.…”

“…In contrast with elliptic curves, where (up to isomorphism) only two curves have nontrivial reduced automorphism group, with PPASes we see much richer structures involving many more vertices. Proofs for all of the results in this section can be found in [12], [14], and [8].…”

“…This article is an illustrated guide to the Richelot isogeny graph. Following Katsura and Takashima [14], we present diagrams of the neighbourhoods of general vertices of each type. Going further, we also compute diagrams of neighbourhoods of general edges, which can be used to glue the vertex neighbourhoods together.…”

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

“…We recall Bolza's classification of automorphism groups of genus-2 Jacobians in §6, and apply it in the context of Richelot isogeny graphs (extending the results of Katsura and Takashima [27]). This allows us to prove that the Jacobian subgraph of the Richelot isogeny graph is connected and aperiodic, and to bound its diameter relative to the diameter of the entire superspecial graph.…”