Counting Richelot isogenies between superspecial abelian surfaces
This work addresses foundational issues in genus-2 isogeny cryptography for cryptographic applications, but it is incremental as it builds on prior results.
The paper tackles the problem of characterizing and counting Richelot isogenies between superspecial abelian surfaces, providing explicit counts for decomposed and non-decomposed isogenies as a corollary to an existing theorem.
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 {\em decomposed Richelot isogenies} in terms of {\em involutive reduced automorphisms} of genus-2 curves over a finite field, and explicitly count such decomposed (and non-decomposed) Richelot isogenies between {\em superspecial} principally polarized abelian surfaces. As a corollary, we give another algebraic geometric proof of Theorem 2 in the paper of Castryck et al.