NEON-SIDH: Efficient Implementation of Supersingular Isogeny Diffie-Hellman Key Exchange Protocol on ARM
We investigate the efficiency of implementing the Jao and De Feo isogeny-based post-quantum key exchange protocol (from PQCrypto 2011) on ARM-powered embedded platforms. In this work we propose new primes to speed up constant-time finite field arithmetic and perform isogenies quickly. Montgomery multiplication and reduction are employed to produce a speedup of 3 over the GNU Multiprecision Library. We analyze the recent projective isogeny formulas presented in Costello et al. (Crypto 2016) and conclude that affine isogeny formulas are much faster in ARM devices. We provide fast affine SIDH libraries over 512, 768, and 1024-bit primes. We provide timing results for emerging embedded ARM platforms using the ARMv7A architecture for the 85-, 128-, and 170-bit quantum security levels. Our assembly-optimized arithmetic cuts the computation time for the protocol by 50 % in comparison to our portable C implementation and performs approximately 3 times faster than the only other ARMv7 results found in the literature. The goal of this paper is to show that isogeny-based cryptosystems can be implemented further and be used as an alternative to classical cryptosystems on embedded devices.
KeywordsElliptic curve cryptography Post-quantum cryptography Isogeny-based cryptosystems ARM embedded processors Finite-field arithmetic Assembly implementation
The authors would like to thank the reviewers for their constructive comments. This material is based upon work supported by the National Science Foundation under grant No. CNS-1464118 awarded to Reza Azarderakhsh.
- 1.Shor, P.W.: Algorithms for quantum computation: discrete logarithms and factoring. In: 35th Annual Symposium on Foundations of Computer Science (FOCS 1994), pp. 124–134 (1994)Google Scholar
- 8.Chen, L., Jordan, S.: Report on post-quantum cryptography, NIST IR 8105 (2016)Google Scholar
- 9.Azarderakhsh, R., Jao, D., Kalach, K., Koziel, B., Leonardi, C.: Key compression for isogeny-based cryptosystems. In: Proceedings of the 3rd ACM International Workshop on ASIA Public-Key Cryptography, AsiaPKC 2016, pp. 1–10. ACM, New York (2016)Google Scholar
- 10.Azarderakhsh, R., Fishbein, D., Jao, D.: Efficient implementations of a quantum-resistant key-exchange protocol on embedded systems. Technical report, University of Waterloo (2014)Google Scholar
- 11.Silverman, J.H.: The Arithmetic of Elliptic Curves. GTM, vol. 106. Springer, New York (1992)Google Scholar
- 12.Mestre, J.F.: La méthode des graphes. Exemples et applications. In: Proceedings of the International Conference on Class Numbers and Fundamental Units of Algebraic Number Fields (Katata, 1986), pp. 217–242. Nagoya Univ., Nagoya (1986)Google Scholar
- 14.Bernstein, D.J., Lange, T.: Explicit-formulas database (2007). http://www.hyperelliptic.org/EFD/
- 15.Bernstein, D.J.: Differential addition chains. Technical report (2006). http://cr.yp.to/ecdh/diffchain-20060219.pdf
- 19.Seo, H., Liu, Z., Grobschadl, J., Kim, H.: Efficient arithmetic on ARM-NEON and its application for high-speed RSA implementation. Cryptology ePrint Archive, Report 2015/465 (2015)Google Scholar