Abstract
In this work, we present an exposure model for the isogeny computation in the quantum-resistant supersingular isogeny Diffie-Hellman (SIDH) key exchange protocol. Notably, we propose this exposure model to characterize the severity of new attacks that force an SIDH user to divulge certain intermediate values. In our model, we show how an attacker can break SIDH by discovering an intermediate kernel point and its corresponding curve. To strengthen an SIDH-user against the exposure of intermediate values, we propose a random curve isomorphism that is performed just before the large-degree isogeny. We show that this countermeasure is computationally inexpensive compared to the whole of SIDH and can still operate with the Kirkwood et al. validation model that allows a static-key user to ensure the first round of the other party was performed honestly. The goal of this paper is to present an additional protection against future attacks for implementations of SIDH.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Azarderakhsh, R., Fishbein, D., Jao, D.: Efficient implementations of a quantum-resistant key-exchange protocol on embedded systems. Technical report, University of Waterloo (2014)
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 (2016)
Azarderakhsh, R., Jao, D., Leonardi, C.: Post-quantum static-static key agreement using multiple protocol instances. In: Adams, C., Camenisch, J. (eds.) SAC 2017. LNCS, vol. 10719, pp. 45–63. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-72565-9_3
Chen, L., Jordan, S.: Report on Post-Quantum Cryptography. NIST IR 8105 (2016)
Costello, C., Jao, D., Longa, P., Naehrig, M., Renes, J., Urbanik, D.: Efficient compression of SIDH public keys. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017, Part I. LNCS, vol. 10210, pp. 679–706. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-56620-7_24
Costello, C., Longa, P., Naehrig, M.: Efficient algorithms for supersingular isogeny Diffie-Hellman. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016, Part I. LNCS, vol. 9814, pp. 572–601. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53018-4_21
De Feo, L., Jao, D., Plût, J.: Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies. J. Math. Cryptol. 8(3), 209–247 (2014)
Edwards, H.M.: A normal form for elliptic curves. Bull. Am. Math. Soc. 44, 393–422 (2007)
Galbraith, S., Stolbunov, A.: Improved algorithm for the isogeny problem for ordinary elliptic curves. Appl. Algebra Eng. Commun. Comput. 24(2), 107–131 (2013)
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, Part I. LNCS, vol. 10031, pp. 63–91. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53887-6_3
Galbraith, S.D., Petit, C., Silva, J.: Identification protocols and signature schemes based on supersingular isogeny problems. In: Takagi, T., Peyrin, T. (eds.) ASIACRYPT 2017. LNCS, vol. 10624, pp. 3–33. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70694-8_1
Gélin, A., Wesolowski, B.: Loop-abort faults on supersingular isogeny cryptosystems. In: Lange, T., Takagi, T. (eds.) PQCrypto 2017. LNCS, vol. 10346, pp. 93–106. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-59879-6_6
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_14
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_2
Joye, M., Tymen, C.: Protections against differential analysis for elliptic curve cryptography—an algebraic approach—. In: Koç, Ç.K., Naccache, D., Paar, C. (eds.) CHES 2001. LNCS, vol. 2162, pp. 377–390. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44709-1_31
Kirkwood, D., Lackey, B.C., McVey, J., Motley, M., Solinas, J.A., Tuller, D.: Failure is not an option: standardization issues for post-quantum key agreement. Technical report, Workshop on Cybersecurity in a Post-Quantum World (2015)
Koziel, B., Azarderakhsh, R., Jao, D.: Side-channel attacks on quantum-resistant supersingular isogeny Diffie-Hellman. In: Adams, C., Camenisch, J. (eds.) SAC 2017. LNCS, vol. 10719, pp. 64–81. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-72565-9_4
Koziel, B., Azarderakhsh, R., Mozaffari-Kermani, M.: Fast hardware architectures for supersingular isogeny Diffie-Hellman key exchange on FPGA. In: Dunkelman, O., Sanadhya, S.K. (eds.) INDOCRYPT 2016. LNCS, vol. 10095, pp. 191–206. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-49890-4_11
Koziel, B., Azarderakhsh, R., Mozaffari-Kermani, M., Jao, D.: Post-quantum cryptography on FPGA based on isogenies on elliptic curves. IEEE Trans. Circuits Syst. I Regul. Pap. 64(1), 86–99 (2017)
Koziel, B., Jalali, A., Azarderakhsh, R., Jao, D., Mozaffari-Kermani, M.: NEON-SIDH: efficient implementation of supersingular isogeny Diffie-Hellman key exchange protocol on ARM. In: Foresti, S., Persiano, G. (eds.) CANS 2016. LNCS, vol. 10052, pp. 88–103. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-48965-0_6
Montgomery, P.L.: Speeding the pollard and elliptic curve methods of factorization. Math. Comput. 48, 243–264 (1987)
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
Tani, S.: Claw finding algorithms using quantum walk. Theor. Comput. Sci. 410(50), 5285–5297 (2009)
Ti, Y.B.: Fault attack on supersingular isogeny cryptosystems. In: Lange, T., Takagi, T. (eds.) PQCrypto 2017. LNCS, vol. 10346, pp. 107–122. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-59879-6_7
Vélu, J.: Isogénies Entre Courbes Elliptiques. Comptes Rendus de l’Académie des Sciences Paris Séries A-B 273, A238–A241 (1971)
Yoo, Y., Azarderakhsh, R., Jalali, A., Jao, D., Soukharev, V.: A post-quantum digital signature scheme based on supersingular isogenies. In: Kiayias, A. (ed.) FC 2017. LNCS, vol. 10322, pp. 163–181. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70972-7_9
Acknowledgement
The authors would like to thank the reviewers for their comments. Also, the authors would like to thank Dr. Luca De Feo for discussion and feedback. This work is supported in part by the grants NIST-60NANB17D184, NIST-60NANB16D246, ARO W911NF-17-1-0311, and NSF CNS-1661557, as well as CryptoWorks21, Public Works and Government Services Canada, Canada First Research Excellence Fund, and an RBC Fellowship.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Koziel, B., Azarderakhsh, R., Jao, D. (2018). An Exposure Model for Supersingular Isogeny Diffie-Hellman Key Exchange. In: Smart, N. (eds) Topics in Cryptology – CT-RSA 2018. CT-RSA 2018. Lecture Notes in Computer Science(), vol 10808. Springer, Cham. https://doi.org/10.1007/978-3-319-76953-0_24
Download citation
DOI: https://doi.org/10.1007/978-3-319-76953-0_24
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-76952-3
Online ISBN: 978-3-319-76953-0
eBook Packages: Computer ScienceComputer Science (R0)