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On the Precision Loss in Approximate Homomorphic Encryption

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Selected Areas in Cryptography – SAC 2023 (SAC 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14201))

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Abstract

Since its introduction at Asiacrypt 2017, the CKKS approximate homomorphic encryption scheme has become one of the most widely used and implemented homomorphic encryption schemes. Due to the approximate nature of the scheme, application developers using CKKS must ensure that the evaluation output is within a tolerable error of the corresponding plaintext computation. Choosing appropriate parameters requires a good understanding of how the noise will grow through the computation. A strong understanding of the noise growth is also necessary to limit the performance impact of mitigations to the attacks on CKKS presented by Li and Micciancio (Eurocrypt [34]).

In this work, we present a comprehensive noise analysis of CKKS, that considers noise coming both from the encoding and homomorphic operations. Our main contribution is the first average-case analysis for CKKS noise, and we also introduce refinements to prior worst-case noise analyses. We develop noise heuristics both for the original CKKS scheme and the RNS variant presented at SAC 2018. We then evaluate these heuristics by comparing the predicted noise growth with experiments in the HEAAN and FullRNS-HEAAN libraries, and by comparing with a worst-case noise analysis as done in prior work. Our findings show mixed results: while our new analyses lead to heuristic estimates that more closely model the observed noise growth than prior approaches, the new heuristics sometimes slightly underestimate the observed noise growth. This evidences the need for implementation-specific noise analyses for CKKS, which recent work has shown to be effective for implementations of similar schemes.

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Notes

  1. 1.

    See e.g. https://ibm.github.io/fhe-toolkit-linux/html/helib/md__opt__i_b_m__f_h_e-distro__h_elib__c_k_k_s-security.html.

References

  1. Albrecht, M., et al.: Homomorphic encryption security standard. HomomorphicEncryption.org, Technical report (2018)

    Google Scholar 

  2. Al Badawi, A., et al.: Openfhe: open-source fully homomorphic encryption library. Cryptology ePrint Archive, Paper 2022/915 (2022). https://eprint.iacr.org/2022/915

  3. Biasioli, B., Marcolla, C., Calderini, M., Mono, J.: Improving and automating BFV parameters selection: an average-case approach. Cryptology ePrint Archive, Paper 2023/600 (2023). https://eprint.iacr.org/2023/600

  4. Boemer, F., Costache, A., Cammarota, R., Wierzynski, C.: ngraph-he2: a high-throughput framework for neural network inference on encrypted data. In: Brenner, M., Lepoint, T., Rohloff, K. (eds.) Proceedings of the 7th ACM Workshop on Encrypted Computing & Applied Homomorphic Cryptography, WAHC@CCS 2019, London, UK, 11–15 November 2019, pp. 45–56. ACM (2019)

    Google Scholar 

  5. Brakerski, Z., Gentry, C., Vaikuntanathan, V.: (Leveled) fully homomorphic encryption without bootstrapping. In: Goldwasser, S. (ed.) ITCS 2012, pp. 309–325. ACM (2012)

    Google Scholar 

  6. Brisebarre, N., Joldeş, M., Muller, J.-M., Naneş, A.-M., Picot, J.: Error analysis of some operations involved in the cooley-tukey fast fourier transform. ACM Trans. Math. Softw. (TOMS) 46(2), 1–27 (2020)

    Article  MathSciNet  Google Scholar 

  7. Chen, H., Dai, W., Kim, M., Song, Y.: Efficient multi-key homomorphic encryption with packed ciphertexts with application to oblivious neural network inference. In: Cavallaro, L., Kinder, J., Wang, X.F., Katz, J. (eds.) ACM CCS 2019, pp. 395–412. ACM Press (2019)

    Google Scholar 

  8. 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

    Chapter  Google Scholar 

  9. Cheon, J.H., Han, K., Kim, A., Kim, M., Song, Y.: A full RNS variant of approximate homomorphic encryption. In: Cid, C., Jacobson Jr, M.J. (eds.) SAC 2018, vol. 11349 of LNCS, pp. 347–368. Springer, Heidelberg (2019). https://doi.org/10.1007/978-3-030-10970-7_16

  10. 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

    Chapter  Google Scholar 

  11. 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

    Chapter  Google Scholar 

  12. Chillotti, I., Gama, N., Georgieva, M., Izabachène, M.: TFHE: fast fully homomorphic encryption over the torus. J. Cryptology 33(1), 34–91 (2020)

    Article  MathSciNet  Google Scholar 

  13. Costache, A., Smart, N.P.: Which ring based somewhat homomorphic encryption scheme is best? In: Sako, K. (ed.) CT-RSA 2016. LNCS, vol. 9610, pp. 325–340. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-29485-8_19

    Chapter  Google Scholar 

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

  15. Costache, A., Laine, K., Player, R.: Evaluating the effectiveness of heuristic worst-case noise analysis in FHE. In: Chen, L., Li, N., Liang, K., Schneider, S. (eds.) ESORICS 2020. LNCS, vol. 12309, pp. 546–565. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-59013-0_27

    Chapter  Google Scholar 

  16. Costache, A., Nürnberger, L., Player, R.: Optimisations and tradeoffs for helib. In: Topics in Cryptology-CT-RSA 2023: Cryptographers’ Track at the RSA Conference 2023, San Francisco, CA, USA, 24–27 April 2023, Proceedings, pp. 29–53. Springer, Heidelberg (2023). https://doi.org/10.1007/978-3-031-30872-7_2

  17. Damgård, I., Pastro, V., Smart, N., Zakarias, S.: Multiparty computation from somewhat homomorphic encryption. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 643–662. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-32009-5_38

    Chapter  Google Scholar 

  18. Fan, J., Vercauteren, F.: Somewhat practical fully homomorphic encryption. Cryptology ePrint Archive, Report 2012/144 (2012). http://eprint.iacr.org/2012/144

  19. Gentry, C.: Fully homomorphic encryption using ideal lattices. In: Mitzenmacher, M. (ed.) 41st ACM STOC, pp. 169–178. ACM Press (2009)

    Google Scholar 

  20. 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

    Chapter  Google Scholar 

  21. Halevi, S., Shoup, V.: Design and implementation of HElib: a homomorphic encryption library. Cryptology ePrint Archive, Report 2020/1481 (2020). https://eprint.iacr.org/2020/1481

  22. Fullrns-heaan. https://github.com/KyoohyungHan/FullRNS-HEAAN. Version as at October 2018

  23. Heaan v2.1. https://github.com/snucrypto/HEAAN. Version as at September 2021

  24. Heaan v1.0. https://github.com/snucrypto/HEAAN/releases/tag/1.0. Version as at September 2018

  25. HElib. https://github.com/shaih/HElib. Version as at January 2019

  26. Iliashenko, I.: Optimisations of fully homomorphic encryption. PhD thesis, KU Leuven (2019)

    Google Scholar 

  27. Kim, A., Song, Y., Kim, M., Lee, K., Cheon, J.H.: Logistic regression model training based on the approximate homomorphic encryption. BMC Med. Genom. 11(4), 83 (2018)

    Article  Google Scholar 

  28. Kim, A., Papadimitriou, A., Polyakov, Y.: Approximate homomorphic encryption with reduced approximation error. In: Galbraith, S.D. (ed.) CT-RSA 2022. LNCS, vol. 13161, pp. 120–144. Springer, Cham (2022). https://doi.org/10.1007/978-3-030-95312-6_6

    Chapter  Google Scholar 

  29. Kim, A., Polyakov, Y., Zucca, V.: Revisiting homomorphic encryption schemes for finite fields. In: Tibouchi, M., Wang, H. (eds.) ASIACRYPT 2021. LNCS, vol. 13092, pp. 608–639. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-92078-4_21

  30. Lattigo v2.2.0. http://github.com/ldsec/lattigo. Version as at July 2021. EPFL-LDS

  31. Lepoint, T., Naehrig, M.: A comparison of the homomorphic encryption schemes FV and YASHE. In: Pointcheval, D., Vergnaud, D. (eds.) AFRICACRYPT 2014. LNCS, vol. 8469, pp. 318–335. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-06734-6_20

    Chapter  Google Scholar 

  32. Lee, Y., Lee, J.W., Kim, Y.S., Kim, Y., No, J.S., Kang, H.: High-Precision Bootstrapping for Approximate Homomorphic Encryption by Error Variance Minimization. In: Dunkelman, O., Dziembowski, S. (eds.) EUROCRYPT 2022. LNCS, vol. 13275, pp. 551–580. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-06944-4_19

  33. Li, B., Micciancio, D., Schultz, M., Sorrell, J.: Securing approximate homomorphic encryption using differential privacy. In: Annual International Cryptology Conference, pp. 560–589. Springer, Heidelberg (2022). https://doi.org/10.1007/978-3-031-15802-5_20

  34. Li, B., Micciancio, D.: On the security of homomorphic encryption on approximate numbers. In: Canteaut, A., Standaert, F.-X. (eds.) EUROCRYPT 2021. LNCS, vol. 12696, pp. 648–677. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-77870-5_23

  35. Murphy, S., Player, R.: A central limit framework for ring-lwe decryption. Cryptology ePrint Archive, Report 2019/452 (2019). https://eprint.iacr.org/2019/452

  36. Murphy, S., Player, R.: Discretisation and product distributions in Ring-LWE. J. Math. Cryptol. 15(1), 45–59 (2021)

    Article  MathSciNet  Google Scholar 

  37. Ogilvie, T., Player, R., Rowell, J.: Improved privacy-preserving training using fixed-hessian minimisation. In: Brenner, M., Lepoint, T. (eds.) Proceedings of the 8th Workshop on Encrypted Computing and Applied Homomorphic Cryptography (WAHC 2020) (2020). https://doi.org/10.25835/0072999

  38. PALISADE Lattice Cryptography Library (release 1.11.5). https://palisade-crypto.org/. Accessed Sept 2021

  39. Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. J. ACM (JACM) 56(6), 1–40 (2009)

    Article  MathSciNet  Google Scholar 

  40. Microsoft SEAL (release 3.6). Microsoft Research, Redmond, WA. https://github.com/Microsoft/SEAL. Version as at November 2020

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Author information

Authors and Affiliations

  1. Norwegian University of Science and Technology (NTNU), Trondheim, Norway

    Anamaria Costache

  2. Zama, Paris, France

    Benjamin R. Curtis

  3. Royal Holloway, University of London, Egham, UK

    Erin Hales, Sean Murphy, Tabitha Ogilvie & Rachel Player

Authors
  1. Anamaria Costache
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  2. Benjamin R. Curtis
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  3. Erin Hales
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  4. Sean Murphy
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  5. Tabitha Ogilvie
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  6. Rachel Player
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Correspondence to Anamaria Costache .

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Editors and Affiliations

  1. University Paris 8, Paris, France

    Claude Carlet

  2. University of New Brunswick, Fredericton, NB, Canada

    Kalikinkar Mandal

  3. University of Leuven and University of Bergen, Heverlee, Belgium

    Vincent Rijmen

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Costache, A., Curtis, B.R., Hales, E., Murphy, S., Ogilvie, T., Player, R. (2024). On the Precision Loss in Approximate Homomorphic Encryption. In: Carlet, C., Mandal, K., Rijmen, V. (eds) Selected Areas in Cryptography – SAC 2023. SAC 2023. Lecture Notes in Computer Science, vol 14201. Springer, Cham. https://doi.org/10.1007/978-3-031-53368-6_16

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  • DOI: https://doi.org/10.1007/978-3-031-53368-6_16

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