Counting superspecial Richelot isogenies and its cryptographic application

03/02/2020

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by   Toshiyuki Katsura, et al.

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We give a 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 non-decomposed) superspecial Richelot isogenies. The characterization implies that the decomposed principally polarized superspecial abelian surfaces are adjacent to 1-small curves in the superspecial Richelot isogeny graph, where the smallness is defined as in a similar manner to Love–Boneh's one. We then obtain an improved isogeny path-finding algorithm in genus 2 as an application by using M-small genus-2 curves for some threshold M.