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.
Keywords
Richelot isogenies, superspecial abelian surfaces, reduced
group of automorphisms, genus-2 isogeny cryptography
