Abstract
Approximate homomorphic encryption with the residue number system (RNS), called RNS-variant Cheon-Kim-Kim-Song (RNS-CKKS) scheme [12, 13], is a fully homomorphic encryption scheme that supports arithmetic operations for real or complex number data encrypted. Although the RNS-CKKS scheme is a fully homomorphic encryption scheme, most of the applications with the RNS-CKKS scheme use it as the only leveled homomorphic encryption scheme because of the lack of the practicality of the bootstrapping operation of the RNS-CKKS scheme. One of the crucial problems of the bootstrapping operation is its poor precision. While other basic homomorphic operations ensure sufficiently high precision for practical use, the bootstrapping operation only supports about 20-bit fixed-point precision at best, which is not high precision enough to be used for the reliable large-depth homomorphic computations until now.
In this paper, we improve the message precision in the bootstrapping operation of the RNS-CKKS scheme. Since the homomorphic modular reduction process is one of the most important steps in determining the precision of the bootstrapping, we focus on the homomorphic modular reduction process. Firstly, we propose a fast algorithm of obtaining the optimal minimax approximate polynomial of modular reduction function and the scaled sine/cosine function over the union of the approximation regions, called an improved multi-interval Remez algorithm. In fact, this algorithm derives the optimal minimax approximate polynomial of any continuous functions over any union of the finite number of intervals. Next, we propose the composite function method using the inverse sine function to reduce the difference between the scaling factor used in the bootstrapping and the default scaling factor. With these methods, we reduce the approximation error in the bootstrapping of the RNS-CKKS scheme by 1/1176–1/42 (5.4–10.2-bit precision improvement) for each parameter setting. While the bootstrapping without the composite function method has 27.2–30.3-bit precision at maximum, the bootstrapping with the composite function method has 32.6–40.5-bit precision.
This work is supported by Samsung Advanced Institute of Technology.
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
References
PALISADE Lattice Cryptography Library (release 1.10.4), September 2020. https://palisade-crypto.org/
Boemer, F., Costache, A., Cammarota, R., Wierzynski, C.: nGraph-HE2: a high-throughput framework for neural network inference on encrypted data. In: Proceedings of the 7th ACM Workshop on Encrypted Computing & Applied Homomorphic Cryptography, pp. 45–56 (2019)
Boemer, F., Lao, Y., Cammarota, R., Wierzynski, C.: nGraph-HE: a graph compiler for deep learning on homomorphically encrypted data. In: Proceedings of the 16th ACM International Conference on Computing Frontiers, pp. 3–13 (2019)
Bossuat, J.P., Mouchet, C., Troncoso-Pastoriza, J., Hubaux, J.P.: Efficient bootstrapping for approximate homomorphic encryption with non-sparse keys. Cryptology ePrint Archive, Report 2020/1203 (2020). https://eprint.iacr.org/2020/1203. Accepted to Eurocrypt 2021
Bourse, F., Minelli, M., Minihold, M., Paillier, P.: Fast homomorphic evaluation of deep discretized neural networks. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018. LNCS, vol. 10993, pp. 483–512. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96878-0_17
Brakerski, Z., Gentry, C., Vaikuntanathan, V.: (Leveled) fully homomorphic encryption without bootstrapping. ACM Trans. Comput. Theory 6(3), 13 (2014)
Brutzkus, A., Gilad-Bachrach, R., Elisha, O.: Low latency privacy preserving inference. In: International Conference on Machine Learning, pp. 812–821. PMLR (2019)
Chen, H., Chillotti, I., Song, Y.: Improved bootstrapping for approximate homomorphic encryption. In: Ishai, Y., Rijmen, V. (eds.) EUROCRYPT 2019. LNCS, vol. 11477, pp. 34–54. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17656-3_2
Chen, Y., et al.: Dadiannao: a machine-learning supercomputer. In: 2014 47th Annual IEEE/ACM International Symposium on Microarchitecture, pp. 609–622. IEEE (2014)
Cheney, E.: Introduction to Approximation Theory. McGraw-Hill, New York (1966)
Cheon, J.H., Han, K., Kim, A., Kim, M., Song, Y.: Bootstrapping for approximate homomorphic encryption. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018. LNCS, vol. 10820, pp. 360–384. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-78381-9_14
Cheon, J., Han, K., Kim, A., Kim, M., Song, Y.: A full RNS variant of approximate homomorphic encryption. In: Cid, C., Jacobson, M. (eds.) SAC 2018. LNCS, vol. 11349, pp. 347–368. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-10970-7_16
Cheon, J.H., Kim, A., Kim, M., Song, Y.: Homomorphic encryption for arithmetic of approximate numbers. In: Takagi, T., Peyrin, T. (eds.) ASIACRYPT 2017. LNCS, vol. 10624, pp. 409–437. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70694-8_15
Chillotti, I., Gama, N., Georgieva, M., Izabachène, M.: TFHE: fast fully homomorphic encryption over the torus. J. Cryptol. 33(1), 34–91 (2020)
Dathathri, R., et al.: Chet: an optimizing compiler for fully-homomorphic neural-network inferencing. In: Proceedings of the 40th ACM SIGPLAN Conference on Programming Language Design and Implementation, pp. 142–156 (2019)
Fan, J., Vercauteren, F.: Somewhat practical fully homomorphic encryption. Cryptology ePrint Archive, Report 2012/144 (2012). https://eprint.iacr.org/2012/144
Filip, S.: A robust and scalable implementation of the Parks-McClellan algorithm for designing FIR filters. ACM Trans. Math. Softw. 43(1), 1–24 (2016)
Gentry, C.: Fully homomorphic encryption using ideal lattices. In: Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing, 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
Gupta, S., Agrawal, A., Gopalakrishnan, K., Narayanan, P.: Deep learning with limited numerical precision. In: International Conference on Machine Learning, pp. 1737–1746 (2015)
Gysel, P., Motamedi, M., Ghiasi, S.: Hardware-oriented approximation of convolutional neural networks. In: International Conference on Learning Representation (2016)
Han, K., Ki, D.: Better bootstrapping for approximate homomorphic encryption. In: Jarecki, S. (ed.) CT-RSA 2020. LNCS, vol. 12006, pp. 364–390. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-40186-3_16
Jiang, X., Kim, M., Lauter, K., Song, Y.: Secure outsourced matrix computation and application to neural networks. In: Proceedings of the 2018 ACM SIGSAC Conference on Computer and Communications Security, pp. 1209–1222 (2018)
Kim, A., Papadimitriou, A., Polyakov, Y.: Approximate homomorphic encryption with reduced approximation error. Cryptology ePrint Archive, Report 2020/1118 (2020). https://eprint.iacr.org/2020/1118
Lee, J.W., Lee, E., Lee, Y., Kim, Y.S., No, J.S.: High-precision bootstrapping of RNS-CKKS homomorphic encryption using optimal minimax polynomial approximation and inverse sine function. Cryptology ePrint Archive, Report 2020/552 (2020). https://eprint.iacr.org/2020/552
McClellan, J., Parks, T.: A personal history of the Parks-McClellan algorithm. IEEE Signal Process. Mag. 22(2), 82–86 (2005)
Powell, M.: Approximation Theory and Methods. Cambridge University Press, Cambridge (1981)
Remez, E.: Sur la détermination des polynômes d’approximation de degré donnée. Commun. Kharkov Math. Soc. 10(196), 41–63 (1934)
Microsoft SEAL (release 3.5), Microsoft Research, Redmond, April 2020. https://github.com/Microsoft/SEAL
van Dijk, M., Gentry, C., Halevi, S., Vaikuntanathan, V.: Fully homomorphic encryption over the integers. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 24–43. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13190-5_2
Acknowledgement
We thank Jean-Philippe Bossuat for his help with optimizing the approximation for the inverse sine function by observing the distribution of the message polynomial coefficients.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 International Association for Cryptologic Research
About this paper
Cite this paper
Lee, JW., Lee, E., Lee, Y., Kim, YS., No, JS. (2021). High-Precision Bootstrapping of RNS-CKKS Homomorphic Encryption Using Optimal Minimax Polynomial Approximation and Inverse Sine Function. In: Canteaut, A., Standaert, FX. (eds) Advances in Cryptology – EUROCRYPT 2021. EUROCRYPT 2021. Lecture Notes in Computer Science(), vol 12696. Springer, Cham. https://doi.org/10.1007/978-3-030-77870-5_22
Download citation
DOI: https://doi.org/10.1007/978-3-030-77870-5_22
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-77869-9
Online ISBN: 978-3-030-77870-5
eBook Packages: Computer ScienceComputer Science (R0)
