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Towards Optimized and Constant-Time CSIDH on Embedded Devices

  • Amir JalaliEmail author
  • Reza Azarderakhsh
  • Mehran Mozaffari Kermani
  • David Jao
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11421)

Abstract

We present an optimized, constant-time software library for commutative supersingular isogeny Diffie-Hellman key exchange (CSIDH) proposed by Castryck et al. which targets 64-bit ARM processors. The proposed library is implemented based on highly-optimized field arithmetic operations and computes the entire key exchange in constant-time. The proposed implementation is resistant to timing attacks. We adopt optimization techniques to evaluate the highest performance CSIDH on ARM-powered embedded devices such as cellphones, analyzing the possibility of using such a scheme in the quantum era. To the best of our knowledge, the proposed implementation is the first constant-time implementation of CSIDH and the first evaluation of this scheme on embedded devices. The benchmark result on a Google Pixel 2 smartphone equipped with 64-bit high-performance ARM Cortex-A72 core shows that it takes almost 12 s for each party to compute a commutative action operation in constant-time over the 511-bit finite field proposed by Castryck et al. However, using uniform but variable-time Montgomery ladder with security considerations improves these results significantly.

Keywords

Commutative supersingular isogeny Constant-time Embedded devices Post-quantum cryptography 

Notes

Acknowledgment

This work is supported in parts by NSF CNS-1801341, NIST-60NANB17D184, NIST-60NANB16D246, and ARO W911NF-17-1-0311, as well as NSERC, CryptoWorks21, Public Works and Government Services Canada, Canada First Research Excellence Fund, and the Royal Bank of Canada.

References

  1. 1.
    An Efficient Post-quantum Commutative Group Action. https://csidh.isogeny.org/software.html
  2. 2.
    Bernstein, D.J., Lange, T., Martindale, C., Panny, L.: Quantum Circuits for the CSIDH: Optimizing Quantum Evaluation of Isogenies. https://quantum.isogeny.org/qisog-20181031.pdf
  3. 3.
    Biasse, J.-F., Jao, D., Sankar, A.: A quantum algorithm for computing isogenies between supersingular elliptic curves. In: Meier, W., Mukhopadhyay, D. (eds.) INDOCRYPT 2014. LNCS, vol. 8885, pp. 428–442. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-13039-2_25CrossRefGoogle Scholar
  4. 4.
    Bonnetain, X., Schrottenloher, A.: Quantum security analysis of CSIDH and ordinary isogeny-based schemes. IACR Cryptology ePrint Archive (2018). https://eprint.iacr.org/2018/537
  5. 5.
    Castryck, W., Lange, T., Martindale, C., Panny, L., Renes, J.: CSIDH: an efficient post-quantum commutative group action. IACR Cryptology ePrint Archive (2018). https://eprint.iacr.org/2018/383
  6. 6.
    Charles, D.X., Lauter, K.E., Goren, E.Z.: Cryptographic hash functions from expander graphs. J. Cryptol. 22(1), 93–113 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Childs, A.M., Jao, D., Soukharev, V.: Constructing elliptic curve isogenies in quantum subexponential time. J. Math. Cryptol. 8(1), 1–29 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Coron, J.-S.: Resistance against differential power analysis for elliptic curve cryptosystems. In: Koç, Ç.K., Paar, C. (eds.) CHES 1999. LNCS, vol. 1717, pp. 292–302. Springer, Heidelberg (1999).  https://doi.org/10.1007/3-540-48059-5_25CrossRefGoogle Scholar
  9. 9.
    Costello, C., Hisil, H.: A simple and compact algorithm for SIDH with arbitrary degree isogenies. In: Takagi, T., Peyrin, T. (eds.) ASIACRYPT 2017. LNCS, vol. 10625, pp. 303–329. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-70697-9_11CrossRefGoogle Scholar
  10. 10.
    Costello, C., Longa, P., Naehrig, M.: Efficient algorithms for supersingular isogeny Diffie-Hellman. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, vol. 9814, pp. 572–601. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53018-4_21CrossRefGoogle Scholar
  11. 11.
    Couveignes, J.M.: Hard Homogeneous Spaces. IACR Cryptology ePrint Archive (2006). http://eprint.iacr.org/2006/291
  12. 12.
  13. 13.
    Feo, L.D.: Mathematics of isogeny based cryptography. CoRR abs/1711.04062 (2017). http://arxiv.org/abs/1711.04062
  14. 14.
    De Feo, L., Galbraith, S.D.: SeaSign: compact isogeny signatures from class group actions. IACR Cryptology ePrint Archive (2018). https://eprint.iacr.org/2018/824
  15. 15.
    Feo, L.D., Kieffer, J., Smith, B.: Towards practical key exchange from ordinary isogeny graphs. CoRR (2018). http://arxiv.org/abs/1809.07543
  16. 16.
    Galbraith, S.D.: Mathematics of Public Key Cryptography. Cambridge University Press, Cambridge (2012)CrossRefGoogle Scholar
  17. 17.
    Galbraith, S.D., Petit, C., Shani, B., Ti, Y.B.: On the security of supersingular isogeny cryptosystems. In: Cheon, J.H., Takagi, T. (eds.) ASIACRYPT 2016. LNCS, vol. 10031, pp. 63–91. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53887-6_3CrossRefGoogle Scholar
  18. 18.
    Jalali, A., Azarderakhsh, R., Mozaffari-Kermani, M.: Efficient post-quantum undeniable signature on 64-Bit ARM. In: Adams, C., Camenisch, J. (eds.) SAC 2017. LNCS, vol. 10719, pp. 281–298. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-72565-9_14CrossRefGoogle Scholar
  19. 19.
    Jalali, A., Azarderakhsh, R., Kermani, M.M.: NEON SIKE: supersingular isogeny key encapsulation on ARMv7. In: Security, Privacy, and Applied Cryptography Engineering - 8th International Conference, SPACE, pp. 37–51 (2018)Google Scholar
  20. 20.
    Jalali, A., Azarderakhsh, R., Kermani, M.M., Jao, D.: Supersingular isogeny Diffie-Hellman key exchange on 64-bit ARM. IEEE Trans. Depend. Secure Comput. (2017)Google Scholar
  21. 21.
    Jao, D., et al.: Supersingular isogeny key encapsulation. Submission to the NIST Post-Quantum Standardization project (2017). https://csrc.nist.gov/Projects/Post-Quantum-Cryptography/Round-1-Submissions
  22. 22.
    Jao, D., De Feo, L.: Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies. In: Yang, B.-Y. (ed.) PQCrypto 2011. LNCS, vol. 7071, pp. 19–34. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-25405-5_2CrossRefzbMATHGoogle Scholar
  23. 23.
    Koziel, B., Jalali, A., Azarderakhsh, R., Jao, D., Kermani, M.M.: NEON-SIDH: efficient implementation of supersingular isogeny Diffie-Hellman key exchange protocol on ARM. In: Cryptology and Network Security - 15th International Conference, CANS, pp. 88–103 (2016)Google Scholar
  24. 24.
    Meyer, M., Reith, S.: A faster way to the CSIDH. IACR Cryptology ePrint Archive, p. 782 (2018). https://eprint.iacr.org/2018/782
  25. 25.
    Petit, C.: Faster algorithms for isogeny problems using torsion point images. In: Takagi, T., Peyrin, T. (eds.) ASIACRYPT 2017. LNCS, vol. 10625, pp. 330–353. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-70697-9_12CrossRefGoogle Scholar
  26. 26.
    Rostovtsev, A., Stolbunov, A.: Public-key cryptosystem based on isogenies. IACR Cryptology ePrint Archive (2006). http://eprint.iacr.org/2006/145
  27. 27.
    Seo, H., Liu, Z., Longa, P., Hu, Z.: SIDH on ARM: faster modular multiplications for faster post-quantum supersingular isogeny key exchange. IACR Trans. Cryptogr. Hardw. Embed. Syst. 3, 1–20 (2018)Google Scholar
  28. 28.
    Silverman, J.H.: The Arithmetic of Elliptic Curves. GTM, vol. 106. Springer, New York (2009).  https://doi.org/10.1007/978-0-387-09494-6
  29. 29.
    Stolbunov, A.: Constructing public-key cryptographic schemes based on class group action on a set of isogenous elliptic curves. Adv. Math. Commun. 4(2), 215–235 (2010). https://doi.org/10.3934/amc.2010.4.215
  30. 30.
    Vélu, J.: Isogénies entre courbes elliptiques. CR Acad. Sci. Paris, Séries A 273, 305–347 (1971)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Amir Jalali
    • 1
    Email author
  • Reza Azarderakhsh
    • 1
  • Mehran Mozaffari Kermani
    • 2
  • David Jao
    • 3
  1. 1.Department of Computer and Electrical Engineering and Computer ScienceFlorida Atlantic UniversityBoca RatonUSA
  2. 2.Department of Computer Science and EngineeringUniversity of South FloridaTampaUSA
  3. 3.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada

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