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BLEACH: Cleaning Errors in Discrete Computations Over CKKS

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Abstract

Approximated homomorphic encryption (HE) schemes such as CKKS are commonly used to perform computations over encrypted real numbers. It is commonly assumed that these schemes are not “exact” and thus they cannot execute circuits with unbounded depth over discrete sets, such as binary or integer numbers, without error overflows. These circuits are usually executed using BGV and B/FV for integers and TFHE for binary numbers. This artificial separation can cause users to favor one scheme over another for a given computation, without even exploring other, perhaps better, options. We show that by treating step functions as “clean-up” utilities and by leveraging the SIMD capabilities of CKKS, we can extend the homomorphic encryption toolbox with efficient tools. These tools use CKKS to run unbounded circuits that operate over binary and small-integer elements and even combine these circuits with fixed-point real numbers circuits. We demonstrate the results using the Turing-complete Conway’s Game of Life. In our evaluation, for boards of size 256\(\times \)256, these tools achieved orders of magnitude faster latency than previous implementations using other HE schemes. We argue and demonstrate that for large enough real-world inputs, performing binary circuits over CKKS, while considering it as an “exact” scheme, results in comparable or even better performance than using other schemes tailored for similar inputs.

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Notes

  1. Theorem 6 exists in the peer-reviewed paper in Appendix D but not in the ePrint version.

References

  1. L. Adam, BoringSSL (2015), https://www.imperialviolet.org/2015/10/17/boringssl.html

  2. E. Aharoni, A. Adir, M. Baruch, N. Drucker, G. Ezov, A. Farkash, L. Greenberg, R. Masalha, G. Moshkowich, D. Murik, H. Shaul, O. Soceanu, HeLayers: a tile tensors framework for large neural networks on encrypted data. PoPETs (2023), https://doi.org/10.56553/popets-2023-0020

  3. A. Akavia, M. Vald, On the privacy of protocols based on cpa-secure homomorphic encryption. IACR Cryptol. ePrint Arch. 2021,  803 (2021), https://eprint.iacr.org/2021/803

  4. S. Arita, S. Nakasato, Fully homomorphic encryption for point numbers, in Chen, K., Lin, D., Yung, M. (eds.) Information Security and Cryptology. pp. 253–270. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-54705-3_16

  5. A.A. Badawi, J. Bates, F. Bergamaschi, D.B. Cousins, S. Erabelli, N. Genise, S. Halevi, H. Hunt, A. Kim, Y. Lee, Z. Liu, D. Micciancio, I. Quah, Y. Polyakov, S. Rv, K. Rohloff, J. Saylor, D. Suponitsky, M. Triplett, V. Vaikuntanathan, V. Zucca, OpenFHE: Open-Source Fully Homomorphic Encryption Library. Cryptology ePrint Archive, Paper 2022/915 (2022), https://eprint.iacr.org/2022/915

  6. Y. Bae, J.H. Cheon, W. Cho, J. Kim, T. Kim, META-BTS: Bootstrapping Precision Beyond the Limit. Cryptology ePrint Archive, Paper 2022/1167 (2022), https://eprint.iacr.org/2022/1167

  7. F. Boemer, R. Cammarota, D. Demmler, T. Schneider, H. Yalame, MP2ML: A mixed-protocol machine learning framework for private inference. in Proceedings of the 2020 Workshop on Privacy-Preserving Machine Learning in Practice. pp. 43-45. PPMLP’20, Association for Computing Machinery, New York, NY, USA (2020). https://doi.org/10.1145/3411501.3419425

  8. F. Boemer, A. Costache, R. Cammarota, C. Wierzynski, NGraph-HE2: a high-throughput framework for neural network inference on encrypted data. in Proceedings of the 7th ACM workshop on encrypted computing and applied homomorphic cryptography. pp. 45–56. WAHC’19, Association for Computing Machinery, New York, NY, USA (2019). https://doi.org/10.1145/3338469.3358944

  9. J.P. Bossuat, C. Mouchet, J. Troncoso-Pastoriza, J.P. Hubaux, Efficient bootstrapping for approximate homomorphic encryption with non-sparse keys. in Canteaut, A., Standaert, F.X. (eds.) Advances in Cryptology—EUROCRYPT 2021. pp. 587–617. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-77870-5_21

  10. C. Boura, N. Gama, M. Georgieva, D. Jetchev, CHIMERA: combining ring-LWE-based fully homomorphic encryption schemes. J. Math. Cryptol. 14(1), 316–338 (2020). https://doi.org/10.1515/jmc-2019-0026

    Article  MathSciNet  Google Scholar 

  11. Z. Brakerski, Fully Homomorphic Encryption without Modulus Switching from Classical GapSVP. in Safavi-Naini, R., Canetti, R. (eds.) Advances in Cryptology—CRYPTO 2012. vol. 7417 LNCS, pp. 868–886. Springer, Berlin (2012). https://doi.org/10.1007/978-3-642-32009-5_50

  12. Z. Brakerski, C. Gentry, V. Vaikuntanathan, (Leveled) fully homomorphic encryption without bootstrapping. ACM Trans. Comput. Theory 6(3) (2014). https://doi.org/10.1145/2633600

  13. H. Chen, I. Chillotti, Y. Song, Improved bootstrapping for approximate homomorphic encryption. in Y. Ishai, V. Rijmen (eds.) Advances in Cryptology—EUROCRYPT 2019. pp. 34–54. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17656-3_2

  14. H. Chen, K. Han, Homomorphic lower digits removal and improved FHE bootstrapping. in J.B. Nielsen, V. Rijmen (eds.) Advances in Cryptology—EUROCRYPT 2018. pp. 315–337. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-78381-9_12

  15. J. Cheon, A. Kim, M. Kim, Y. Song, Homomorphic encryption for arithmetic of approximate numbers. in Proceedings of Advances in Cryptology—ASIACRYPT 2017. pp. 409–437. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70694-8_15

  16. J.H. Cheon, D. Kim, D. Kim, Efficient homomorphic comparison methods with optimal complexity. in International Conference on the Theory and Application of Cryptology and Information Security. pp. 221–256. Springer (2020). https://doi.org/10.1007/978-3-030-64834-3_8

  17. I. Chillotti, N. Gama, M. Georgieva, M. Izabachène, Faster fully homomorphic encryption: bootstrapping in less than 0.1 seconds. in Cheon, J.H., Takagi, T. (eds.) Advances in Cryptology—ASIACRYPT 2016. pp. 3–33. Springer, Berlin (2016). https://doi.org/10.1007/978-3-662-53887-6_1

  18. I. Chillotti, N. Gama, M. Georgieva, M. Izabachène, TFHE: fast fully homomorphic encryption over the torus. J. Cryptol. 33(1), 34–91 (2020). https://doi.org/10.1007/s00145-019-09319-x

    Article  MathSciNet  Google Scholar 

  19. I. Chillotti, M. Joye, D. Ligier, J.B. Orfila, S. Tap, CONCRETE: concrete Operates oN Ciphertexts rapidly by extending TfhE. in WAHC 2020–8th Workshop on Encrypted Computing & Applied Homomorphic Cryptography. vol. 15 (2020)

  20. A. Costache, B.R. Curtis, E. Hales, S. Murphy, T. Ogilvie, R. Player, On the precision loss in approximate homomorphic encryption. Cryptology ePrint Archive, Paper 2022/162 (2022), https://eprint.iacr.org/2022/162

  21. CryptoLab: HEaaN: Homomorphic Encryption for Arithmetic of Approximate Numbers, version 0.2.0 (2022), https://www.cryptolab.co.kr/eng/product/heaan.php

  22. Cryptolab: Heaan, crypto lab’s great homomorphic encryption library, manual. Last accessed 07/09/2023 (2023), https://heaan.it/docs/heaan/namespacemembers_vars.html

  23. L. Ducas, D. Stehlé, Sanitization of fhe ciphertexts. In: Fischlin, M., Coron, J.S. (eds.) Advances in Cryptology—EUROCRYPT 2016. pp. 294–310. Springer, Berlin (2016). https://doi.org/10.1007/978-3-662-49890-3_12

  24. J. Fan, F. Vercauteren, Somewhat practical fully homomorphic encryption. in Proceedings of the 15th International Conference on Practice and Theory in Public Key Cryptography pp. 1–16 (2012), https://eprint.iacr.org/2012/144

  25. M. Florent, Game of life using fully homomorphic encryption commit 04b7deebd9b96b2701c13e2d08c141b84f1c8479 (2022), https://github.com/FlorentCLMichel/homomorphic_game_of_life

  26. M. Gardner, The fantastic combinations of John Conway’s new solitaire game “life” (Oct 1970), https://www.scientificamerican.com/article/mathematical-games-1970-10/

  27. C. Gentry, A Fully Homomorphic Encryption Scheme. Ph.D. thesis, Stanford University (2009), https://crypto.stanford.edu/craig

  28. C. Gentry, S. Halevi, N.P. Smart, Better bootstrapping in fully homomorphic encryption. in International Workshop on Public Key Cryptography. pp. 1–16. Springer (2012). https://doi.org/10.1007/978-3-642-30057-8_1

  29. R. Gilad-Bachrach, N. Dowlin, K. Laine, K. Lauter, M. Naehrig, J. Wernsing, Cryptonets: applying neural networks to encrypted data with high throughput and accuracy. in International Conference on Machine Learning. pp. 201–210 (2016), http://proceedings.mlr.press/v48/gilad-bachrach16.pdf

  30. S. Halevi, V. Shoup, Bootstrapping for HElib. in: Oswald, E., Fischlin, M. (eds.) Advances in Cryptology—EUROCRYPT 2015. pp. 641–670. Springer, Berlin (2015). https://doi.org/10.1007/978-3-662-46800-5_25

  31. K. Han, D. Ki, Better bootstrapping for approximate homomorphic encryption. in Cryptographers’ Track at the RSA Conference. pp. 364–390. Springer (2020)

  32. X. Jiang, M. Kim, K. Lauter, Y. Song, 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. CCS ’18, New York, NY, USA (2018). https://doi.org/10.1145/3243734.3243837

  33. C.S. Jutla, N. Manohar, Sine series approximation of the mod function for bootstrapping of approximate HE. In: Dunkelman, O., Dziembowski, S. (eds.) Advances in Cryptology—EUROCRYPT 2022. pp. 491–520. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-06944-4_17

  34. C. Juvekar, V. Vaikuntanathan, A. Chandrakasan, GAZELLE: a low latency framework for secure neural network inference. In: 27th USENIX Security Symposium (USENIX Security 18). pp. 1651–1669. USENIX Association, Baltimore, MD (2018), https://www.usenix.org/conference/usenixsecurity18/presentation/juvekar

  35. A. Kim, A. Papadimitriou, Y. Polyakov, Approximate homomorphic encryption with reduced approximation error. in Cryptographers’ Track at the RSA Conference, pp. 120–144. Springer (2022). https://doi.org/10.1007/978-3-030-95312-6_6

  36. E. Lee, J.W. Lee, Y.S. Kim, J.S. No, Minimax approximation of sign function by composite polynomial for homomorphic comparison. IEEE Trans. Depend. Secure Comput. (2021). https://doi.org/10.1109/TDSC.2021.3105111

    Article  Google Scholar 

  37. J. Lee, E. Lee, J.W. Lee, Y. Kim, Y.S. Kim, J.S. No, Precise approximation of convolutional neural networks for homomorphically encrypted data. arXiv preprint arXiv:2105.10879 (2021)

  38. Y. Lee, J.W. Lee, Y.S. Kim, Y. Kim, J.S. No, H. Kang, High-precision bootstrapping for approximate homomorphic encryption by error variance minimization. in Dunkelman, O., Dziembowski, S. (eds.) Advances in Cryptology—EUROCRYPT 2022. pp. 551–580. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-06944-4_19

  39. R. Lehmkuhl, P. Mishra, A. Srinivasan, R.A. Popa, Muse: secure inference resilient to malicious clients. in 30th USENIX Security Symposium (USENIX Security 21). pp. 2201–2218. USENIX Association (2021), https://www.usenix.org/conference/usenixsecurity21/presentation/lehmkuhl

  40. B. Li, D. Micciancio, On the security of homomorphic encryption on approximate numbers. in Canteaut, A., Standaert, F.X. (eds.) Advances in Cryptology—EUROCRYPT 2021. pp. 648–677. Springer, Cham (2021)

  41. J. Liu, M. Juuti, Y. Lu, N. Asokan, Oblivious neural network predictions via MiniONN transformations. in Proceedings of the 2017 ACM SIGSAC Conference on Computer and Communications Security. pp. 619–631. CCS ’17, Association for Computing Machinery, New York, NY, USA (2017). https://doi.org/10.1145/3133956.3134056

  42. Q. Lou, L. Jiang, HEMET: a homomorphic-encryption-friendly privacy-preserving mobile neural network architecture. in Meila, M., Zhang, T. (eds.) Proceedings of the 38th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 139, pp. 7102–7110 (2021), https://proceedings.mlr.press/v139/lou21a.html

  43. W.J. Lu, Z. Huang, C. Hong, Y. Ma, H. Qu, PEGASUS: bridging polynomial and non-polynomial evaluations in homomorphic encryption. in 2021 IEEE Symposium on Security and Privacy (SP). pp. 1057–1073 (2021). https://doi.org/10.1109/SP40001.2021.00043

  44. D. Micciancio, Y. Polyakov, Bootstrapping in FHEW-like cryptosystems. in Proceedings of the 9th on Workshop on Encrypted Computing and Applied Homomorphic Cryptography. p. 17-28. WAHC ’21, Association for Computing Machinery, New York, NY, USA (2021). https://doi.org/10.1145/3474366.3486924

  45. F. Michel, J. Wilson, E. Cottle, Concrete Boolean and Conway’s Game of Life: A Tutorial (2021), https://medium.com/zama-ai/concrete-boolean-and-conways-game-of-life-a-tutorial-f2bcfd614131

  46. F. Michel, J. Wilson, E. Cottle, Fully homomorphic encryption and the game of life (2021), https://medium.com/optalysys/fully-homomorphic-encryption-and-the-game-of-life-d7c37d74bbaf

  47. P. Mishra, R. Lehmkuhl, A. Srinivasan, W. Zheng, R.A. Popa, Delphi: a cryptographic inference service for neural networks. in 29th USENIX Security Symposium (USENIX Security 20). pp. 2505–2522. USENIX Association (aug 2020). https://doi.org/10.1145/3411501.3419418, https://www.usenix.org/conference/usenixsecurity20/presentation/mishra

  48. P. Mohassel, P. Rindal, ABY3: A mixed protocol framework for machine learning. in Proceedings of the 2018 ACM SIGSAC Conference on Computer and Communications Security. pp. 35–52. CCS ’18, Association for Computing Machinery, New York, NY, USA (2018). https://doi.org/10.1145/3243734.3243760

  49. P. Mohassel, Y. Zhang, M.L. Secure, A system for scalable privacy-preserving machine learning. in 2017 IEEE Symposium on Security and Privacy (SP). pp. 19–38 (2017). https://doi.org/10.1109/SP.2017.12

  50. D. Rathee, M. Rathee, N. Kumar, N. Chandran, D. Gupta, A. Rastogi, R. Sharma, CrypTFlow2: practical 2-party secure inference. in Proceedings of the 2020 ACM SIGSAC Conference on Computer and Communications Security, pp. 325–342. Association for Computing Machinery, New York, NY, USA (2020), https://doi.org/10.1145/3372297.3417274

  51. P. Rendell, Turing Universality of the Game of Life, pp. 513–539. Springer, London (2002). https://doi.org/10.1007/978-1-4471-0129-1_18

  52. E. Rescorla, The Transport Layer Security (TLS) Protocol Version 1.3. RFC 8446 (aug 2018). https://doi.org/10.17487/RFC8446

  53. E. Rescorla, T. Dierks, The Transport Layer Security (TLS) Protocol Version 1.2. RFC 5246 (2008). https://doi.org/10.17487/RFC5246

  54. M.S. Riazi, M. Samragh, H. Chen, K. Laine, K. Lauter, F. Koushanfar, XONN: XNOR-based oblivious deep neural network inference. in 28th USENIX Security Symposium (USENIX Security 19). pp. 1501–1518. USENIX Association, Santa Clara, CA (2019), https://www.usenix.org/conference/usenixsecurity19/presentation/riazi

  55. Zama: fhe_game_of_life commit 6d15153ac234482f8b70841e5151a1a98cfc2775 (2022), https://github.com/zama-ai/fhe_game_of_life

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  1. IBM Research - Israel, Haifa, Israel

    Nir Drucker, Guy Moshkowich, Tomer Pelleg & Hayim Shaul

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Appendix A: CKKS Experiment Parameters

Appendix A: CKKS Experiment Parameters

We considered the following sets of parameters for the different experiments.

  • Gates Evaluation we used HEaaN’s FGa parameters preset which uses ring dimension of \(2^{16}\), \(\log _2(PQ)=1,547\), Hamming weight of secret key of 192 bits, security level of 128, scale of 48 bits, initial multiplication depth of 9 and depth of 6 after bootstrapping. The modulus chain primes are

    $$\begin{aligned}{} & {} {} \texttt {[ 0x7ffffffffcc0001}, \texttt {0x3ffffffd20001,} \texttt {0x3fffffcc0001,}\\{} & {} \quad \texttt {0x3fffffb20001,} \texttt { 0x3fffffa80001,}{} \texttt {0x3fffff960001,}\\{} & {} \quad \texttt {0x3fffff8e0001,} \texttt { 0x3fffff520001,} \texttt {0x3ffffec20001,} \\{} & {} \quad \texttt {0x3ffffeaa0001,} \texttt {0x7fffffff9840001,} \texttt {0x7ffffffff240001,}\\{} & {} \quad \texttt { 0x7fffffffe900001,} \texttt { 0x7fffffffe3c0001,} \texttt {0x7fffffffe240001,}\\{} & {} \quad \texttt { 0x7fffffffddc0001,} \texttt { 0x7fffffffd740001,} \texttt {0x7fffffffd640001,}\\{} & {} \quad \texttt { 0x7fffffffd080001,} \texttt { 0x7fffffffccc0001,} \texttt {0x7fffffffcbc0001,}\\{} & {} \quad \texttt { 0x3fffffffffe80001,} \texttt {0x3fffffffffb80001,} \texttt {0x1fffffffffe00001,} \\{} & {} \quad \texttt {0x7fffffffb3c0001,} \texttt {0x7fffffffadc0001,} \texttt {0x7fffffffac40001,}\\{} & {} \quad \texttt {0xcccccccc433333]}.\\ \end{aligned}$$

    In this setup, bootstrapping takes 14 s and multiplying 2 ciphertexts (and then relinearizing and rescaling the result) takes less than 1 s.

  • Game of Life we used HEaaN’s parameters preset which uses ring dimension of \(2^{17}\), \(\log _2(PQ)=2,043\), Hamming weight of secret key of 192 bits, security level of 128, scale of 24 bits, initial multiplication depth of 29 and depth of 13 after bootstrapping. The modulus chain primes are

    $$\begin{aligned}{} & {} \texttt {[0x1ffffffffc000001,} \texttt { 0x3ffffffff040001,} \texttt { 0x3fffffff000001,}\\{} & {} {} \texttt {0x3ffffffef40001,} \texttt { 0x3ffffffe800001, } \texttt { 0x3ffffffe080001,}\\{} & {} {} \texttt {0x3ffffffdcc0001, }{} \texttt { 0x3ffffffdb40001, } \texttt { 0x3ffffffd480001,}\\{} & {} {} \texttt {0x3ffffffcfc0001, }{} \texttt { 0x3ffffffc3c0001,} \texttt { 0x3ffffffbc40001,}\\{} & {} {} \texttt {0x3ffffffb580001, }{} \texttt { 0x3ffffffb340001,} \texttt { 0x1fffffffed440001,}\\{} & {} {} \texttt {0x1ffffffffb300001,}{} \texttt { 0x1ffffffffb1c0001, } \texttt { 0x1ffffffffadc0001,}\\{} & {} {} \texttt {0x1ffffffffa400001,}{} \texttt { 0x1ffffffffa140001,} \texttt { 0x1ffffffff9d80001,}\\{} & {} {} \texttt {0x1ffffffff9140001,}{} \texttt { 0x1ffffffff8ac0001,} \texttt { 0x1ffffffff8a80001,}\\{} & {} {} \texttt {0x1ffffffff81c0001, }{} \texttt { 0x1ffffffff7800001,} \texttt { 0x1fffffffef600001,}\\{} & {} {} \texttt {0x1fffffffffc0001,}{} \texttt { 0x1ffffffff8c0001, } \texttt { 0x1ffffffff840001,}\\{} & {} {} \texttt {0x1fffffffebac0001,}{} \texttt { 0x1fffffffeba40001,} \texttt { 0x1fffffffeb4c0001,}\\{} & {} {} \texttt {0x1fffffffeb280001,}{} \texttt { 0x1fffffffea780001, } \texttt { 0x1fffffffea440001,}\\{} & {} {} \texttt {0x1fffffffe9f40001,}{} \texttt { 0x1fffffffe97c0001, }{} \texttt { 0x1fffffffe9300001,} \\{} & {} {} \texttt {0x1fffffffe8d00001].} \\ \end{aligned}$$

    In this setup bootstrapping takes 14 s and multiplying 2 ciphertexts (and then relinearizing and rescaling the result) takes less than 1 s.

  • Decompose Integers we used HEaaN’s parameters preset which uses ring dimension of \(2^{17}\), \(\log _2(PQ)=2,043\), Hamming weight of secret key of 192 bits, security level of 128, scale of 24 bits, initial multiplication depth of 29 and depth of 13 after bootstrapping. The modulus chain primes are

    $$\begin{aligned}{} & {} \texttt {[0x1ffffffffc000001,} \texttt { 0x7fffffc900001,} \texttt {0x7fffffe780001,}\\{} & {} \quad \texttt { 0x8000020240001,} \texttt { 0x7fffffe0c0001,} \texttt { 0x7fffff7400001,}\\{} & {} \quad \texttt { 0x7fffff7f80001,} \texttt { 0x7fffff9000001,} \texttt { 0x8000010040001,}\\{} & {} \quad \texttt { 0x7fffffdc40001,} \texttt { 0x8000006e00001,} \texttt { 0x7fffffdb00001,}\\{} & {} \quad \texttt { 0x8000002200001,} \texttt { 0x8000004240001,} \texttt {0x7fffffddc0001,}\\{} & {} \quad \texttt { 0x8000003ec0001,} \texttt { 0x7fffffd880001,} \texttt { 0x7fffffcfc0001,}\\{} & {} \quad \texttt { 0x8000001d00001,} \texttt {0x8000000940001,} \texttt { 0x8000002480001,}\\{} & {} \quad \texttt { 0x7fffffe900001,} \texttt { 0x7ffffff080001,} \texttt { 0x7ffffff240001,}\\{} & {} \quad \texttt { 0x8000001600001,} \texttt { 0x7ffffff900001,} \texttt { 0x7ffffff9c0001,}\\{} & {} \quad \texttt { 0x8000000500001,} \texttt { 0x80000002c0001,} \texttt { 0x80000001c0001,}\\{} & {} \quad \texttt { 0x1fffffffd80001,} \texttt {0x1fffffff900001,} \texttt {0x1fffffff8c0001,}\\{} & {} \quad \texttt { 0x1fffffff200001,} \texttt {0x1ffffffe880001,} \texttt {0x1ffffffe1c0001,}\\{} & {} \quad \texttt { 0x1ffffffcfc0001,} \texttt {0x1ffffffb880001,} \texttt {0x1ffffffb100001,} \\{} & {} \quad \texttt { 0x1ffffff6000001].} \\ \end{aligned}$$

    In this setup bootstrapping takes 14 s and multiplying 2 ciphertexts (and then relinearizing and rescaling the result) takes less than 1 s.

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Drucker, N., Moshkowich, G., Pelleg, T. et al. BLEACH: Cleaning Errors in Discrete Computations Over CKKS. J Cryptol 37, 3 (2024). https://doi.org/10.1007/s00145-023-09483-1

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