## Cryptology ePrint Archive: Report 2020/391

Optimized CSIDH Implementation Using a 2-torsion Point

Donghoe Heo and Suhri Kim and Kisoon Yoon and Young-Ho Park and Seokhie Hong

Abstract: The implementation of isogeny-based cryptography mainly use Montgomery curves as they offer fast elliptic curve arithmetic and isogeny compuation. However, although Montgomery curves have efficient 3- and 4-isogenies, it becomes inefficient when recovering the coefficient of the image curve for large degree isogenies. This is the main bottleneck of using a Montgomery curve for CSIDH as it requires odd-degree isogenies up to at least 587. In this paper, we present a new optimization method for faster CSIDH protocols entirely on Montgomery curves. To this end, we present a new parameter for CSIDH in which the rational 2-torsion points are defined over $\mathbb{F}_p$. By using the proposed parameters the CSIDH moves around the surface. The curve coefficient of the image curve can be recovered by a 2-torsion point. We also proved that the CSIDH using the proposed parameter guarantees a free and transitive group action. Additionally, we present the implementation result using our method. We demonstrated that our method is 6.1% faster than the original CSIDH. Our works show that quite higher performance of CSIDH is achieved using only Montgomery curves.

Category / Keywords: public-key cryptography / Post-quantum cryptography, Isogeny, Montgomery curves, 2-torsion points, CSIDH.

Date: received 5 Apr 2020, last revised 28 Apr 2020

Contact author: dong5641 at korea ac kr

Available format(s): PDF | BibTeX Citation

Short URL: ia.cr/2020/391

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