Fast Hardware Architectures for Supersingular Isogeny Diffie-Hellman Key Exchange on FPGA

  • Brian Koziel
  • Reza Azarderakhsh
  • Mehran Mozaffari-Kermani
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10095)

Abstract

In this paper, we present a constant-time hardware implementation that achieves new speed records for the supersingular isogeny Diffie-Hellman (SIDH), even when compared to highly optimized Haswell computer architectures. We employ inversion-free projective isogeny formulas presented by Costello et al. at CRYPTO 2016 on an FPGA. Modern FPGA’s can take advantage of heavily parallelized arithmetic in \(\mathbb {F}_{p^{2}}\), which lies at the foundation of supersingular isogeny arithmetic. Further, by utilizing many arithmetic units, we parallelize isogeny evaluations to accelerate the computations of large-degree isogenies by approximately 57%. On a constant-time implementation of 124-bit quantum security SIDH on a Virtex-7, we generate ephemeral public keys in 10.6 and 11.6 ms and generate the shared secret key in 9.5 and 10.8 ms for Alice and Bob, respectively. This improves upon the previous best time in the literature for 768-bit implementations by a factor of 1.48. Our 83-bit quantum security implementation improves upon the only other implementation in the literature by a speedup of 1.74 featuring fewer resources and constant-time.

Keywords

Post-quantum cryptography Elliptic curve cryptography Isogeny-based cryptography Field programmable gate array 

Notes

Acknowledgment

This material is based upon work supported by the NSF CNS-1464118 and NIST 60NANB16D246 grants awarded to Reza Azarderakhsh.

References

  1. 1.
    Chen, L., Jordan, S.: Report on Post-Quantum Cryptography. NIST IR 8105 (2016)Google Scholar
  2. 2.
    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
  3. 3.
    Jao, D., 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). doi:10.1007/978-3-642-25405-5_2 CrossRefGoogle Scholar
  4. 4.
    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). doi:10.1007/978-3-662-53018-4_21 CrossRefGoogle Scholar
  5. 5.
    Rostovtsev, A., Stolbunov, A.: Public-Key Cryptosystem Based on Isogenies. IACR Cryptology ePrint Archive 2006, 145 (2006)Google Scholar
  6. 6.
    Childs, A., Jao, D., Soukharev, V.: Constructing Elliptic Curve Isogenies in Quantum Subexponential Time (2010)Google Scholar
  7. 7.
    De Feo, L., Jao, D., Plut, J.: Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies. J. Math. Crypt. 8(3), 209–247 (2014)MathSciNetMATHGoogle Scholar
  8. 8.
    Silverman, J.H.: The Arithmetic of Elliptic Curves. GTM, vol. 106. Springer, New York (1992)Google Scholar
  9. 9.
    Montgomery, P.L.: Speeding the pollard and elliptic curve methods of factorization. Math. Comput. 48, 243–264 (1987)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Couveignes, J.-M.: Hard homogeneous spaces. Cryptology ePrint Archive, Report 2006, 291 (2006)Google Scholar
  11. 11.
    Koziel, B., Azarderakhsh, R., Kermani, M.M., Jao, D.: Post-Quantum Cryptography on FPGA Based on Isogenies on Elliptic Curves. Cryptology ePrint Archive, Report 2016, 672 (2016). http://eprint.iacr.org/2016/672
  12. 12.
    Karmakar, A., Roy, S., Vercauteren, F., Verbauwhede, I.: Efficient finite field multiplication for isogeny based post quantum cryptography. In: International Workshop on the Arithmetic of Finite Fields, WAIFI 2016, to appearGoogle Scholar
  13. 13.
    Montgomery, P.L.: Modular multiplication without trial division. Math. Comput. 44(170), 519–521 (1985)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    McIvor, C., McLoone, M., McCanny, J.V.: High-radix systolic modular multiplication on reconfigurable hardware. In: IEEE International Conference on Field-Programmable Technology, pp. 13–18, December 2005Google Scholar
  15. 15.
    Orup, H.: Simplifying quotient determination in high-radix modular multiplication. In: Proceedings of the 12th Symposium on Computer Arithmetic, ARITH 1995, pp. 193–199. IEEE Computer Society, Washington (1995)Google Scholar
  16. 16.
    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
  17. 17.
    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: 15th International Conference on Cryptology and Network Security, CANS (2016)Google Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Brian Koziel
    • 1
  • Reza Azarderakhsh
    • 2
  • Mehran Mozaffari-Kermani
    • 3
  1. 1.Texas InstrumentsDallasUSA
  2. 2.CEECS DepartmentI-SENSE FAUBoca RatonUSA
  3. 3.EME DepartmentRITRochesterUSA

Personalised recommendations