Abstract
In this paper, we propose a new technique to perform several homomorphic operations in one bootstrapping call over a multi-value plaintext space. Our construction relies on the FHEW-based gate bootstrapping; we analyze its structure and propose a strategy we call multi-value bootstrapping which allows to bootstrap an arbitrary function in an efficient way.
The security of our scheme relies on the LWE assumption over the torus. We give three possible applications: we first describe how to efficiently evaluate an arbitrary boolean function (LUT) and combine LUTs in circuits. We also explain how to apply our procedure to optimize the circuit bootstrapping from (Asiacrypt’2017) which allows to compose circuits in a leveled mode. And we finally present a simple method which makes use of the multi-value bootstrapping to evaluate a encrypted neural network. We have implemented the proposed method and were able to evaluate an arbitrary 6-to-6 LUTs under 1.6 s. Our implementation is based on the TFHE library but can be easily integrated into other homomorphic libraries based on the same structure, such as FHEW (Eurocrypt’2015). The number of LUT outputs does not influence the execution time by a lot, e.g. evaluation of additional 128 outputs on the same 6 input bits takes only 0.05 more seconds.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In this paragraph only the evaluation order of an expression matters and is used for a better illustration.
- 2.
Available at https://bitbucket.org/malb/lwe-estimator. Our estimation were performed using commit 76d05ee.
References
Albrecht, M., Player, R., Scott, S.: On the concrete hardness of learning with errors. J. Math. Cryptol. 9, 169–203 (2015). ePrint Archive 2015/046
Albrecht, M.R.: On dual lattice attacks against small-secret LWE and parameter choices in HElib and SEAL. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017. LNCS, vol. 10211, pp. 103–129. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-56614-6_4
Alperin-Sheriff, J., Peikert, C.: Faster bootstrapping with polynomial error. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014. LNCS, vol. 8616, pp. 297–314. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44371-2_17
Biasse, J.-F., Ruiz, L.: FHEW with efficient multibit bootstrapping. In: Lauter, K., Rodríguez-Henríquez, F. (eds.) LATINCRYPT 2015. LNCS, vol. 9230, pp. 119–135. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-22174-8_7
Bonnoron, G., Ducas, L., Fillinger, M.: Large FHE gates from tensored homomorphic accumulator. In: Joux, A., Nitaj, A., Rachidi, T. (eds.) AFRICACRYPT 2018. LNCS, vol. 10831, pp. 217–251. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-89339-6_13
Brakerski, Z., Gentry, C., Vaikuntanathan, V.: (Leveled) fully homomorphic encryption without bootstrapping. In: ITCS, pp. 309–325 (2012)
Brakerski, Z., Langlois, A., Peikert, C., Regev, O., Stehlé, D.: Classical hardness of learning with errors. In: STOC, pp. 575–584 (2013)
Brakerski, Z., Perlman, R.: Lattice-based fully dynamic multi-key FHE with short ciphertexts. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, vol. 9814, pp. 190–213. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53018-4_8
Cheon, J.H., Stehlé, D.: Fully homomophic encryption over the integers revisited. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9056, pp. 513–536. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46800-5_20
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
Chillotti, I., Gama, N., Georgieva, M., Izabachène, M.: A homomorphic LWE based E-voting scheme. In: Takagi, T. (ed.) PQCrypto 2016. LNCS, vol. 9606, pp. 245–265. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-29360-8_16
Chillotti, I., Gama, N., Georgieva, M., Izabachène, M.: Faster packed homomorphic operations and efficient circuit bootstrapping for TFHE. In: Takagi, T., Peyrin, T. (eds.) ASIACRYPT 2017. LNCS, vol. 10624, pp. 377–408. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70694-8_14
Chillotti, I., Gama, N., Georgieva, M., Izabachène, M.: TFHE: fast fully homomorphic encryption library, August 2016. https://tfhe.github.io/tfhe/
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
Fan, J., Vercauteren, F.: Somewhat practical fully homomorphic encryption (2012). https://eprint.iacr.org/2012/144
Gentry, C.: Fully homomorphic encryption using ideal lattices. In: STOC, vol. 9, pp. 169–178 (2009)
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
Halevi, S., Shoup, I.V.: Helib - an implementation of homomorphic encryption, September 2014. https://github.com/shaih/HElib/
Lepoint, T.: FV-NFLlib: library implementing the Fan-Vercauteren homomorphic encryption scheme, May 2016 https://github.com/CryptoExperts/FV-NFLlib
Lyubashevsky, V., Peikert, C., Regev, O.: On ideal lattices and learning with errors over rings. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 1–23. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13190-5_1
Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. In: STOC, pp. 84–93 (2005)
SEAL. Simple encrypted arithmetic library. https://sealcrypto.codeplex.com/
Acknowledgements
We acknowledge the support of the french Programme d’Investissement d’Avenir under the national project RISQ.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Carpov, S., Izabachène, M., Mollimard, V. (2019). New Techniques for Multi-value Input Homomorphic Evaluation and Applications. In: Matsui, M. (eds) Topics in Cryptology – CT-RSA 2019. CT-RSA 2019. Lecture Notes in Computer Science(), vol 11405. Springer, Cham. https://doi.org/10.1007/978-3-030-12612-4_6
Download citation
DOI: https://doi.org/10.1007/978-3-030-12612-4_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-12611-7
Online ISBN: 978-3-030-12612-4
eBook Packages: Computer ScienceComputer Science (R0)