Paper 2019/843

How to Construct CSIDH on Edwards Curves

Tomoki Moriya and Hiroshi Onuki and Tsuyoshi Takagi

Abstract

CSIDH is an isogeny-based key exchange protocol proposed by Castryck, Lange, Martindale, Panny, and Renes in 2018. CSIDH is based on the ideal class group action on $\mathbb{F}_p$-isomorphism classes of Montgomery curves. In order to calculate the class group action, we need to take points defined over $\mathbb{F}_{p^2}$. The original CSIDH algorithm requires a calculation over $\mathbb{F}_p$ by representing points as $x$-coordinate over Montgomery curves. Meyer and Reith proposed a faster CSIDH algorithm in 2018 which calculates isogenies on Edwards curves by using a birational map between a Montgomery curve and an Edwards curve. There is a special coordinate on Edwards curves (the $w$-coordinate) to calculate group operations and isogenies. If we try to calculate the class group action on Edwards curves by using the $w$-coordinate in a similar way on Montgomery curves, we have to consider points defined over $\mathbb{F}_{p^4}$. Therefore, it is not a trivial task to calculate the class group action on Edwards curves with $w$-coordinates over only $\mathbb{F}_p$. In this paper, we prove a number of theorems on the properties of Edwards curves. By using these theorems, we extend the CSIDH algorithm to that on Edwards curves with $w$-coordinates over $\mathbb{F}_p$. This algorithm is as fast as (or a little bit faster than) the algorithm proposed by Meyer and Reith.