Skip to main content
SpringerLink
Log in

Supersingular Isogeny Diffie–Hellman Authenticated Key Exchange

  • Conference paper
  • First Online:
Book cover Information Security and Cryptology – ICISC 2018 (ICISC 2018)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11396))

Included in the following conference series:

Abstract

We propose two authenticated key exchange protocols from supersingular isogenies. Our protocols are the first post-quantum one-round Diffie–Hellman type authenticated key exchange ones in the following points: one is secure under the quantum random oracle model and the other resists against maximum exposure where a non-trivial combination of secret keys is revealed. The security of the former and the latter is proven under isogeny versions of the decisional and gap Diffie–Hellman assumptions, respectively. We also propose a new approach for invalidating the Galbraith–Vercauteren-type attack for the gap problem.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Galbraith claims that the protocol is one-round however the description shows that it is two-round as the responder generates the response after receiving the first message [14].

  2. 2.

    Static public keys must be known to both parties in advance. They can be obtained by exchanging them before starting the protocol or by receiving them from a certificate authority. This situation is common for all PKI-based AKE protocols.

References

  1. Ambainis, A., Rosmanis, A., Unruh, D.: Quantum attacks on classical proof systems: the hardness of quantum rewinding. In: FOCS 2014, pp. 474–483 (2014)

    Google Scholar 

  2. Azarderakhsh, R., Jao, D., Kalach, K., Koziel, B., Leonardi, C.: Key compression for isogeny-based cryptosystems. In: AsiaPKC 2016, pp. 1–10 (2016)

    Google Scholar 

  3. Boneh, D., Dagdelen, Ö., Fischlin, M., Lehmann, A., Schaffner, C., Zhandry, M.: Random oracles in a quantum world. In: Lee, D.H., Wang, X. (eds.) ASIACRYPT 2011. LNCS, vol. 7073, pp. 41–69. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-25385-0_3

    Chapter  MATH  Google Scholar 

  4. Bos, J.W., Friedberger, S.: Fast arithmetic modulo \(2^{\text{x}}\)\({\text{ p }}^{\text{ y }} \pm 1\). In: ARITH 2017, pp. 148–155 (2017)

    Google Scholar 

  5. Canetti, R., Krawczyk, H.: Analysis of key-exchange protocols and their use for building secure channels. In: Pfitzmann, B. (ed.) EUROCRYPT 2001. LNCS, vol. 2045, pp. 453–474. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44987-6_28

    Chapter  Google Scholar 

  6. Charles, D., Lauter, K., Goren, E.: Cryptographic hash functions from expander graphs. J. Crypt. 22(1), 93–113 (2009)

    Article  MathSciNet  Google Scholar 

  7. Childs, A., Jao, D., Soukharev, V.: Constructing elliptic curve isogenies in quantum subexponential time. J. Math. Crypt. 8(1), 1–29 (2014)

    Article  MathSciNet  Google Scholar 

  8. 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. LNCS, vol. 10210, pp. 679–706. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-56620-7_24

    Chapter  Google Scholar 

  9. 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_21

    Chapter  Google Scholar 

  10. Dagdelen, Ö., Fischlin, M., Gagliardoni, T.: The fiat–shamir transformation in a quantum world. In: Sako, K., Sarkar, P. (eds.) ASIACRYPT 2013. LNCS, vol. 8270, pp. 62–81. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-42045-0_4

    Chapter  MATH  Google Scholar 

  11. De Feo, L., Jao, D., Plût, J.: Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies. J. Math. Crypt. 8(3), 209–247 (2014)

    MathSciNet  MATH  Google Scholar 

  12. Fujioka, A., Suzuki, K., Xagawa, K., Yoneyama, K.: Practical and post-quantum authenticated key exchange from one-way secure key encapsulation mechanism. In: ASIACCS 2013, pp. 83–94 (2013)

    Google Scholar 

  13. Fujioka, A., Suzuki, K., Xagawa, K., Yoneyama, K.: Strongly secure authenticated key exchange from factoring, codes, and lattices. Des. Codes Crypt. 76(3), 469–504 (2015). A preliminary version appeared in PKC 2012 (2012)

    Article  MathSciNet  Google Scholar 

  14. Galbraith, S.D.: Authenticated key exchange for SIDH. IACR Cryptology ePrint Archive 2018, 266 (2018). http://eprint.iacr.org/2018/266

  15. 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_3

    Chapter  Google Scholar 

  16. Galbraith, S.D., Vercauteren, F.: Computational problems in supersingular elliptic curve isogenies. IACR Cryptology ePrint Archive 2017, 774 (2017). http://eprint.iacr.org/2017/774

  17. Jao, D., et al.: Supersingular Isogeny Key Encapsulation (SIKE). Submission to NIST Post-Quantum Cryptography Standardization (2017)

    Google Scholar 

  18. Jeong, I.R., Katz, J., Lee, D.H.: One-round protocols for two-party authenticated key exchange. In: Jakobsson, M., Yung, M., Zhou, J. (eds.) ACNS 2004. LNCS, vol. 3089, pp. 220–232. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24852-1_16

    Chapter  Google Scholar 

  19. Koziel, B., Azarderakhsh, R., Kermani, M.M., Jao, D.: Post-quantum cryptography on FPGA based on isogenies on elliptic curves. IEEE Trans. Circuits Syst. 64–I(1), 86–99 (2017)

    Article  Google Scholar 

  20. 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

    Chapter  Google Scholar 

  21. Krawczyk, H.: HMQV: a high-performance secure Diffie-Hellman protocol. In: Shoup, V. (ed.) CRYPTO 2005. LNCS, vol. 3621, pp. 546–566. Springer, Heidelberg (2005). https://doi.org/10.1007/11535218_33

    Chapter  Google Scholar 

  22. LeGrow, J., Jao, D., Azarderakhsh, R.: Modeling quantum-safe authenticated key establishment, and an isogeny-based protocol. IACR Cryptology ePrint Archive 2018, 282 (2018). http://eprint.iacr.org/2018/282

  23. Longa, P.: A note on post-quantum authenticated key exchange from supersingular isogenies. IACR Cryptology ePrint Archive 2018, 267 (2018). http://eprint.iacr.org/2018/267

  24. National Institute of Standards and Technology: Post-Quantum crypto standardization: Call for Proposals Announcement, December 2016. http://csrc.nist.gov/groups/ST/post-quantum-crypto/cfp-announce-dec2016.html

  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_12

    Chapter  Google Scholar 

  26. Rostovtsev, A., Stolbunov, A.: Public-key cryptosystem based on isogenies. IACR Cryptology ePrint Archive 2006, 145 (2006). http://eprint.iacr.org/2006/145

  27. Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26(5), 1484–1509 (1997)

    Article  MathSciNet  Google Scholar 

  28. Sutherland, A.: Identifying supersingular elliptic curves. LMS J. Comput. Math. 15, 317–325 (2012)

    Article  MathSciNet  Google Scholar 

  29. Thormarker, E.: Post-quantum cryptography: supersingular isogeny Diffie-Hellman key exchange. Master’s thesis, Stockholm University (2017)

    Google Scholar 

  30. Urbanik, D., Jao, D.: SoK: the problem landscape of SIDH. In: APKC 2018, pp. 53–60 (2018)

    Google Scholar 

  31. Xu, X., Xue, H., Wang, K., Tian, S., Liang, B., Yu, W.: Strongly secure authenticated key exchange from supersingular isogeny. IACR Cryptology ePrint Archive 2018, 760 (2018). http://eprint.iacr.org/2018/760

  32. Zhandry, M.: How to construct quantum random functions. In: FOCS 2012, pp. 679–687 (2012)

    Google Scholar 

  33. Zhandry, M.: Secure identity-based encryption in the quantum random oracle model. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 758–775. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-32009-5_44

    Chapter  MATH  Google Scholar 

  34. Zhandry, M.: How to record quantum queries, and applications to quantum indifferentiability. IACR Cryptology ePrint Archive 2018, 276 (2018). http://eprint.iacr.org/2018/276

Download references

Author information

Authors and Affiliations

  1. Kanagawa University, Kanagawa, Japan

    Atsushi Fujioka

  2. Mitsubishi Electric, Kanagawa, Japan

    Katsuyuki Takashima

  3. Ibaraki University, Ibaraki, Japan

    Shintaro Terada & Kazuki Yoneyama

Authors
  1. Atsushi Fujioka
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Katsuyuki Takashima
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shintaro Terada
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Kazuki Yoneyama
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Atsushi Fujioka .

Editor information

Editors and Affiliations

  1. Sejong University, Seoul, Korea (Republic of)

    Kwangsu Lee

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Fujioka, A., Takashima, K., Terada, S., Yoneyama, K. (2019). Supersingular Isogeny Diffie–Hellman Authenticated Key Exchange. In: Lee, K. (eds) Information Security and Cryptology – ICISC 2018. ICISC 2018. Lecture Notes in Computer Science(), vol 11396. Springer, Cham. https://doi.org/10.1007/978-3-030-12146-4_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-12146-4_12

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-12145-7

  • Online ISBN: 978-3-030-12146-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation