## IACR News item: 19 July 2019

###### Tomoki Moriya, Hiroshi Onuki, Tsuyoshi Takagi
ePrint Report
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 $\mathbb{F}_p$-isomorphic 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. If we try to calculate the class group action on Edwards curves 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 over $\mathbb{F}_p$.

In this paper, we prove a number of theorems on the properties of Edwards curves. By using these theorems, we devise a new CSIDH algorithm that uses only Edwards curves while calculating over $\mathbb{F}_p$. This algorithm is as fast as (or a little bit faster than) the algorithm proposed by Meyer and Reith.

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