On Lions and Elligators: An Efficient Constant-Time Implementation of CSIDH
The recently proposed CSIDH primitive is a promising candidate for post quantum static-static key exchanges with very small keys. However, until now there is only a variable-time proof-of-concept implementation by Castryck, Lange, Martindale, Panny, and Renes, recently optimized by Meyer and Reith, which can leak various information about the private key. Therefore, we present an efficient constant-time implementation that samples key elements only from intervals of nonnegative numbers and uses dummy isogenies, which prevents certain kinds of side-channel attacks. We apply several optimizations, e.g. Elligator and the newly introduced SIMBA, in order to get a more efficient implementation.
KeywordsCSIDH Isogeny-based cryptography Post-quantum cryptography Constant-time implementation
This work was partially supported by Elektrobit Automotive, Erlangen, Germany. We thank Joost Renes for answering some questions during the preparation of this work, and the anonymous reviewers for their helpful and valuable comments.
- 2.Bernstein, D.J., Hamburg, M., Krasnova, A., Lange, T.: Elligator: elliptic-curve points indistinguishable from uniform random strings. In: Proceedings of the 2013 ACM SIGSAC Conference on Computer & Communications Security, pp. 967–980. ACM (2013)Google Scholar
- 3.Bernstein, D.J., Lange, T., Martindale, C., Panny, L.: Quantum circuits for the CSIDH: optimizing quantum evaluation of isogenies. Cryptology ePrint Archive, Report 2018/1059 (2018). https://eprint.iacr.org/2018/1059
- 5.Bonnetain, X., Schrottenloher, A.: Quantum security analysis of CSIDH and ordinary isogeny-based schemes. Cryptology ePrint Archive, Report 2018/537 (2018). https://eprint.iacr.org/2018/537
- 10.Couveignes, J.M.: Hard homogeneous spaces. Cryptology ePrint Archive, Report 2006/291 (2006). https://eprint.iacr.org/2006/291
- 12.Jao, D., et al.: Supersingular isogeny key encapsulation. Round 1 submission, NIST Post-Quantum Cryptography Standardization (2017)Google Scholar
- 17.National Institute of Standards and Technology (NIST): Submission requirements and evaluation criteria for the post-quantum cryptography standardization process (2016)Google Scholar
- 19.Rostovtsev, A., Stolbunov, A.: Public-key cryptosystem based on isogenies. Cryptology ePrint Archive, Report 2006/145 (2006). http://eprint.iacr.org/2006/145