Toshiyuki Katsura

Toshiyuki Katsura, Katsuyuki Takashima

We advance previous studies on decomposed Richelot isogenies (Katsura--Takashima (ANTS 2020) and Katsura (ArXiv 2021)) which are useful for analysing superspecial Richelot isogeny graphs in cryptography. We first give a characterization of decomposed Richelot isogenies between Jacobian varieties of hyperelliptic curves of any genus. We then define generalized Howe curves, and present two theorems on their relationships with decomposed Richelot isogenies. We also give new examples including a non-hyperelliptic (resp.\,hyperelliptic) generalized Howe curve of genus 5 (resp.\,of genus 4).

Toshiyuki Katsura, Katsuyuki Takashima

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.