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An Analysis of FV Parameters Impact Towards Its Hardware Acceleration

  • Joël CathébrasEmail author
  • Alexandre Carbon
  • Renaud Sirdey
  • Nicolas Ventroux
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10323)

Abstract

The development of cloud computing services is restrained by privacy concerns. Centralized medical services for instance, require a guarantee of confidentiality when using outsourced computation platforms. Fully Homomorphic Encryption is an intuitive solution to address such issue, but until 2009, existing schemes were only able to evaluate a reduced number of operations (Partially Homomorphic Encryption). In 2009, C. Gentry proposed a blueprint to construct FHE schemes from SHE schemes. However, it was not practical due to the huge data size overhead and the exponential noise growth of the initial SHE. Since then, major improvements have been made over SHE schemes and their noise management, and resulting schemes, like BGV and FV, allow to foresee small applications.

Besides scheme improvements, new practical approaches were proposed to bring homomorphic encryption closer to practice. The IV-based stream cipher trans-ciphering approach brought by Canteaut et al. in 2015 reduces the on-line latency of the trans-ciphering process to a simple homomorphic addition. The homomorphic evaluation of stream ciphers, that produces the trans-ciphering keystream, could be computed in an off-line phase, resulting in an almost transparent trans-ciphering process from the user point of view. This approach combined with hardware accelerations could bring homomorphic encryption closer to practice.

This paper deals the choice of FV parameters for efficient implementation of this scheme in the light of related works’ common approaches. At first sight, using large polynomial degree to reduce the coefficients size seemed to be advantageous, but further observations contradict it. Large polynomial degrees imply larger ciphertexts and more complex implementations, but smaller ones imply more primes to find for CRT polynomial representation. The result of this preliminary work for the choice of an adequate hardware target motivates the choice of small degree polynomials rather than small coefficients for the FV scheme.

Keywords

Homomorphic evaluation FV parameters Chinese Remainder Theorem Number Theorical Transform 

References

  1. 1.
    Aguilar-Melchor, C., Barrier, J., Guelton, S., Guinet, A., Killijian, M.-O., Lepoint, T.: NFLlib: NTT-based fast lattice library. In: Sako, K. (ed.) CT-RSA 2016. LNCS, vol. 9610, pp. 341–356. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-29485-8_20 CrossRefGoogle Scholar
  2. 2.
    Bajard, J.C., Eynard, J., Hasan, A.M., Zucca, V.: A full RNS variant of FV like somewhat homomorphic encryption schemes. In: Avanzi, R., Heys, H. (eds.) SAC 2016. LNCS, vol. 10532, pp. 423–442. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-69453-5_23. http://hal.upmc.fr/hal-01371941 CrossRefGoogle Scholar
  3. 3.
    Brakerski, Z.: Fully homomorphic encryption without modulus switching from classical GapSVP. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 868–886. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-32009-5_50 CrossRefGoogle Scholar
  4. 4.
    Brakerski, Z., Vaikuntanathan, V.: Efficient fully homomorphic encryption from (standard) LWE. In: IEEE ASFC 2011 (2) (2011)Google Scholar
  5. 5.
    Canteaut, A., Carpov, S., Fontaine, C., Lepoint, T., Naya-Plasencia, M., Paillier, P., Sirdey, R.: Stream ciphers: a practical solution for efficient homomorphic-ciphertext compression. In: Peyrin, T. (ed.) FSE 2016. LNCS, vol. 9783, pp. 313–333. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-52993-5_16 CrossRefGoogle Scholar
  6. 6.
    Carpov, S., Dubrulle, P., Sirdey, R.: Armadillo: a compilation chain for privacy preserving applications. In: Proceedings of the 3rd International Workshop on Security in Cloud Computing. Association for Computing Machinery (ACM) (2015)Google Scholar
  7. 7.
    Chillotti, I., Gama, N., Georgieva, M., Izabachène, M.: Faster fully homomorphic encryption: bootstrapping in less than 0.1 seconds. In: Cheon, J.H., Takagi, T. (eds.) ASIACRYPT 2016. LNCS, vol. 10031, pp. 3–33. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53887-6_1 CrossRefGoogle Scholar
  8. 8.
    Dai, W., Doroz, Y., Sunar, B.: Accelerating NTRU based homomorphic encryption using GPUS. In: 2014 IEEE High Performance Extreme Computing Conference (HPEC), pp. 1–6. IEEE (2014)Google Scholar
  9. 9.
    Dai, W., Sunar, B.: cuHE: a homomorphic encryption accelerator library. In: Pasalic, E., Knudsen, L.R. (eds.) BalkanCryptSec 2015. LNCS, vol. 9540, pp. 169–186. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-29172-7_11 CrossRefGoogle Scholar
  10. 10.
    Ducas, L., Micciancio, D.: FHEW: bootstrapping homomorphic encryption in less than a second. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9056, pp. 617–640. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-46800-5_24 Google Scholar
  11. 11.
    Fan, J., Vercauteren, F.: Somewhat practical fully homomorphic encryption. IACR Cryptology ePrint Archive 2012, p. 144 (2012)Google Scholar
  12. 12.
    Gentry, C., Halevi, S.: Implementing gentry’s fully-homomorphic encryption scheme. In: Paterson, K.G. (ed.) EUROCRYPT 2011. LNCS, vol. 6632, pp. 129–148. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-20465-4_9 CrossRefGoogle Scholar
  13. 13.
    Gentry, C., Halevi, S., Smart, N.P.: Homomorphic evaluation of the AES circuit. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 850–867. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-32009-5_49 CrossRefGoogle Scholar
  14. 14.
    Gentry, C., Sahai, A., Waters, B.: Homomorphic encryption from learning with errors: conceptually-simpler, asymptotically-faster, attribute-based. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8042, pp. 75–92. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-40041-4_5 CrossRefGoogle Scholar
  15. 15.
    Gentry, C., et al.: Fully homomorphic encryption using ideal lattices. In: STOC, vol. 9, pp. 169–178 (2009)Google Scholar
  16. 16.
    Migliore, V., Real, M.M., Lapotre, V., Tisserand, A., Fontaine, C., Gogniat, G.: Hardware/software co-design of an accelerator for FV homomorphic encryption scheme using Karatsuba algorithm. IEEE Trans. Comput. 1 (2016).  https://doi.org/10.1109/TC.2016.2645204
  17. 17.
    Naehrig, M., Lauter, K., Vaikuntanathan, V.: Can homomorphic encryption be practical? In: Proceedings of the 3rd ACM workshop on Cloud Computing Security Workshop, pp. 113–124. ACM (2011)Google Scholar
  18. 18.
    Nethercote, N., Walsh, R., Fitzhardinge, J.: Building workload characterization tools with valgrind. In: 2006 IEEE International Symposium on Workload Characterization. Institute of Electrical and Electronics Engineers (IEEE), October 2006Google Scholar
  19. 19.
    Öztürk, E., Doröz, Y., Sunar, B., Savas, E.: Accelerating somewhat homomorphic evaluation using FPGAs. IACR Cryptology ePrint Archive 2015, p. 294 (2015)Google Scholar
  20. 20.
    Pollard, J.M.: The fast fourier transform in a finite field. Math. Comput. 25(114), 365–374 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Pöppelmann, T., Güneysu, T.: Towards practical lattice-based public-key encryption on reconfigurable hardware. In: Lange, T., Lauter, K., Lisoněk, P. (eds.) SAC 2013. LNCS, vol. 8282, pp. 68–85. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-662-43414-7_4 CrossRefGoogle Scholar
  22. 22.
    Rivest, R.L., Adleman, L., Dertouzos, M.L.: On data banks and privacy homomorphisms. Found. Secure Comput. 4(11), 169–180 (1978)MathSciNetGoogle Scholar
  23. 23.
    Sinha Roy, S., Järvinen, K., Vercauteren, F., Dimitrov, V., Verbauwhede, I.: Modular hardware architecture for somewhat homomorphic function evaluation. In: Güneysu, T., Handschuh, H. (eds.) CHES 2015. LNCS, vol. 9293, pp. 164–184. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-48324-4_9 CrossRefGoogle Scholar
  24. 24.
    Smart, N.P., Vercauteren, F.: Fully homomorphic encryption with relatively small key and ciphertext sizes. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 420–443. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-13013-7_25 CrossRefGoogle Scholar
  25. 25.
    Wang, W., Chen, Z., Huang, X.: Accelerating leveled fully homomorphic encryption using GPU. In: 2014 IEEE International Symposium on Circuits and Systems (ISCAS), pp. 2800–2803. IEEE (2014)Google Scholar

Copyright information

© International Financial Cryptography Association 2017

Authors and Affiliations

  • Joël Cathébras
    • 1
    Email author
  • Alexandre Carbon
    • 1
  • Renaud Sirdey
    • 1
  • Nicolas Ventroux
    • 1
  1. 1.CEA, LISTGif-sur-YvetteFrance

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