Paper 2020/1109

Karatsuba-based square-root Vélu’s formulas applied to two isogeny-based protocols

Gora Adj, Jesús-Javier Chi-Domínguez, and Francisco Rodríguez-Henríquez


At a combined computational expense of about $6{\ell}$ field operations, Vélu's formulas are used to construct and evaluate degree-$\ell$ isogenies in the vast majority of isogeny-based cryptographic schemes. By adapting to Vélu's formulas a baby-step giant-step approach, Bernstein, De Feo, Leroux, and Smith presented a procedure that can computes isogeny operations at a reduced cost of just $\tilde{O}(\sqrt{\ell})$ field operations. In this paper, we present a concrete computational analysis of these novel procedure along with several algorithmic tricks that helped us to further decrease its computational cost. We also report an optimized Python3-code implementation of several instantiations of two isogeny-based key-exchange protocols, namely, CSIDH and B-SIDH. Our software library uses a combination of the modified Vélu's formulas and an adaptation of the optimal strategies commonly used in the SIDH/SIKE protocols to produce significant speedups. Compared to a traditional Vélu constant-time implementation of CSIDH, our experimental results report a saving of 5.357\%, 13.68\% and 25.938\% base field operations for CSIDH-512, CSIDH-1024, and CSIDH-1792, respectively. Additionally, we present the first optimized implementation of B-SIDH ever reported in the open literature.

Note: This version includes a comparative between Schonage-FFT and Karatsuba-style polynomial multiplication (see appendix B).

Available format(s)
Public-key cryptography
Publication info
Preprint. MINOR revision.
isogeny-based cryptographyVélu's formulasresultantsB-SIDHSIDHCSIDHSIKE
Contact author(s)
gora adj @ udl cat
jesus chidominguez @ tuni fi
francisco @ cs cinvestav mx
2021-09-05: last of 12 revisions
2020-09-15: received
See all versions
Short URL
Creative Commons Attribution


      author = {Gora Adj and Jesús-Javier Chi-Domínguez and Francisco Rodríguez-Henríquez},
      title = {Karatsuba-based square-root Vélu’s formulas applied to two isogeny-based protocols},
      howpublished = {Cryptology ePrint Archive, Paper 2020/1109},
      year = {2020},
      note = {\url{}},
      url = {}
Note: In order to protect the privacy of readers, does not use cookies or embedded third party content.