Optimized csidh implementation using a 2-torsion point

Donghoe Heo, Suhri Kim, Kisoon Yoon, Young Ho Park, Seokhie Hong

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


The implementation of isogeny-based cryptography mainly use Montgomery curves, as they offer fast elliptic curve arithmetic and isogeny computation. However, although Montgomery curves have efficient 3-and 4-isogeny formula, it becomes inefficient when recovering the coefficient of the image curve for large degree isogenies. Because the Commutative Supersingular Isogeny Diffie-Hellman (CSIDH) requires odd-degree isogenies up to at least 587, this inefficiency is the main bottleneck of using a Montgomery curve for CSIDH. 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 three rational two-torsion points exist. By using the proposed parameters, the CSIDH moves around the surface. The curve coefficient of the image curve can be recovered by a two-torsion point. We also proved that the CSIDH while 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 6.4% faster than the original CSIDH. Our works show that quite higher performance of CSIDH is achieved while only using Montgomery curves.

Original languageEnglish
Article number20
Pages (from-to)1-13
Number of pages13
Issue number3
Publication statusPublished - 2020 Sept

Bibliographical note

Funding Information:
This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. NRF-2020R1A2C1011769).

Publisher Copyright:
© 2020 by the authors. Licensee MDPI, Basel, Switzerland.


  • Commutative Supersingular Isogeny Diffie-Hellman (CSIDH)
  • Isogeny
  • Montgomery curves
  • Post-quantum cryptography
  • Two-torsion points

ASJC Scopus subject areas

  • Computational Theory and Mathematics
  • Computer Networks and Communications
  • Computer Science Applications
  • Software
  • Applied Mathematics


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