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Batch Bootstrapping I:

A New Framework for SIMD Bootstrapping in Polynomial Modulus

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Advances in Cryptology – EUROCRYPT 2023 (EUROCRYPT 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14006))

Abstract

In this series of work, we aim at improving the bootstrapping paradigm for fully homomorphic encryption (FHE). Our main goal is to show that the amortized cost of bootstrapping within a polynomial modulus only requires \(\tilde{O}(1)\) FHE multiplications.

To achieve this, we develop substantial algebraic techniques in two papers. Particularly, the first one (this work) proposes a new mathematical framework for batch homomorphic computation that is compatible with the existing bootstrapping methods of AP14/FHEW/TFHE. To show that our overall method requires only a polynomial modulus, we develop a critical algebraic analysis over noise growth, which might be of independent interest. Overall, the framework yields an amortized complexity \(\tilde{O}(\lambda ^{0.75})\) FHE multiplications, where \(\lambda \) is the security parameter. This improves the prior methods of AP14/FHEW/TFHE, which required \(O(\lambda )\) FHE multiplications in amortization.

Developing many substantial new techniques based on the foundation of this work, the sequel (Bootstrapping II, Eurocrypt 2023) shows how to further improve the recursive bootstrapping method of MS18 (Micciancio and Sorrell, ICALP 2018), yielding a substantial theoretical improvement that can potentially lead to more practical methods.

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Notes

  1. 1.

    Homomorphic computation refers to the ability to compute on ciphertexts (encrypted data). A fully homomorphic encryption supports general homomorphic computation, i.e., computation for any arbitrary function.

  2. 2.

    More precisely, the computation should be denoted as \(\textsf{Eval} (\textsf{Dec} (\textsf{ct}, \cdot ), \textsf{Enc} (\textsf{sk}))\). By correctness, the output ciphertext should belong to \(\textsf{Enc} (m)\), though perhaps distributed differently from a fresh ciphertext.

  3. 3.

    The work [4] sets \(q = \tilde{O}(\lambda )\). If we use a randomized rounding for the modulus switch, q can be further reduced to \(\tilde{O}(\sqrt{n}) = \tilde{O}(\sqrt{\lambda })\).

  4. 4.

    The cyclotomic polynomial would be of a different form if q is not a power of two. Here we use this setting for simplicity of exposition, but note that our framework works for general cyclotomic rings.

  5. 5.

    We abuse the notation in the subscribe by using the rings for simplicity. Precisely, this should be \(\textsf{Tr}_{K/\mathbb {Q}}\) where K is the number field for which \(\mathcal {R}\) is its ring of integers.

  6. 6.

    We notice that \(\mathcal {R}_2\) is used to denote a second ring in our framework. To avoid notation overloading, we use \(\mathcal {R}/2\mathcal {R}\) to denote \(\mathcal {R}\) modulo 2.

  7. 7.

    Here we use the same function name as the above, where the input type specifies which function the call refers to.

  8. 8.

    For small d’s, the Hoistng technique [22] can be used to improve efficiency.

  9. 9.

    In the full version of this work, we present how to achieve such a q.

  10. 10.

    For any \((\boldsymbol{s}, \boldsymbol{a}) \in \mathbb {Z}_q^n \times \mathbb {Z}_q^n\), \(\langle \boldsymbol{s}, \boldsymbol{a}\rangle = \langle \boldsymbol{s}', \boldsymbol{a}'\rangle \) where \(\boldsymbol{a}' \in \mathbb {Z}_q^{n\log q}\) is the power-of-two of \(\boldsymbol{a}\) and \(\boldsymbol{s}'\in \mathbb {Z}_q^{n\log q}\) is the bit-decomposition of \(\boldsymbol{s}\). Using this insight, it is without loss of generality to just consider binary secret vectors in the bootstrapping task. Some practical optimizations, e.g., [6, 13, 17, 28] use binary or ternary LWE, so that the secret vector \(\boldsymbol{s}\) is set directly to binary or ternary. In this case, there is no need to blow up the dimension of \(\boldsymbol{a}\).

References

  1. Abla, P., Liu, F.-H., Wang, H., Wang, Z.: Ring-based identity based encryption – asymptotically shorter MPK and tighter security. In: Nissim, K., Waters, B. (eds.) TCC 2021. LNCS, vol. 13044, pp. 157–187. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-90456-2_6

    Chapter  Google Scholar 

  2. Albrecht, M.R., et al.: Estimate all the LWE, NTRU schemes! In: Catalano, D., De Prisco, R. (eds.) SCN 2018. LNCS, vol. 11035, pp. 351–367. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-98113-0_19

    Chapter  Google Scholar 

  3. Alperin-Sheriff, J., Peikert, C.: Practical bootstrapping in quasilinear time. In Canetti and Garay [10], pp. 1–20 (2013)

    Google Scholar 

  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

    Chapter  Google Scholar 

  5. 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

    Chapter  Google Scholar 

  6. Bonte, C., Iliashenko, I., Park, J., Pereira, H.V., Smart, N.P.: Smart. FINAL: Faster FHE instantiated with NTRU and LWE. Cryptology ePrint Archive, Report 2022/074 (2022). https://eprint.iacr.org/2022/074

  7. Brakerski, Z., Langlois, A., Peikert, C., Regev, O., Stehlé, D.: Classical hardness of learning with errors. In: Boneh, D., Roughgarden, T., Feigenbaum, J. (eds.) 45th ACM STOC, pp. 575–584. ACM Press (2013)

    Google Scholar 

  8. Brakerski, Z., Vaikuntanathan, V.: Efficient fully homomorphic encryption from (standard) LWE. In: Ostrovsky, R. (ed.) 52nd FOCS, pp. 97–106. IEEE Computer Society Press (2011)

    Google Scholar 

  9. Brakerski, Z., Vaikuntanathan, V.: Lattice-based FHE as secure as PKE. In: Naor, M. (ed.) ITCS 2014, pp. 1–12. ACM (2014)

    Google Scholar 

  10. Canetti, R., Garay, J.A. (eds.): LNCS, vol. 8042. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40041-4

    Book  MATH  Google Scholar 

  11. Chen, H., Dai, W., Kim, M., Song, Y.: Efficient homomorphic conversion between (ring) LWE ciphertexts. In: Sako, K., Tippenhauer, N.O. (eds.) ACNS 2021. LNCS, vol. 12726, pp. 460–479. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-78372-3_18

    Chapter  Google Scholar 

  12. Cheon, J.H., Kim, A., Kim, M., Song, Y.S.: Homomorphic encryption for arithmetic of approximate numbers. In Takagi and Peyrin [35], pp. 409–437 (2017)

    Google Scholar 

  13. 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. Part I, volume 10031 of LNCS, pp. 3–33. Springer, Heidelberg (2016)

    Chapter  Google Scholar 

  14. Chillotti, I., Gama, N., Georgieva, M., Izabachène, M.: Faster packed homomorphic operations and efficient circuit bootstrapping for TFHE. In: Takagi and Peyrin [35], pp. 377–408 (2017)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Chillotti, I., Ligier, D., Orfila, J.-B., Tap, S.: Improved programmable bootstrapping with larger precision and efficient arithmetic circuits for TFHE. In: Tibouchi, M., Wang, H. (eds.) ASIACRYPT 2021. LNCS, vol. 13092, pp. 670–699. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-92078-4_23

    Chapter  Google Scholar 

  17. 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

    Chapter  MATH  Google Scholar 

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

    Google Scholar 

  19. Gentry, C., Halevi, S., Smart, N.P.: Better bootstrapping in fully homomorphic encryption. In: Fischlin, M., Buchmann, J., Manulis, M. (eds.) PKC 2012. LNCS, vol. 7293, pp. 1–16. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-30057-8_1

    Chapter  Google Scholar 

  20. Gentry, C.,Sahai, A. , Waters, B.: Homomorphic encryption from learning with errors: Conceptually-simpler, asymptotically-faster, attribute-based. In: Canetti and Garay [10], pp. 75–92 (2013)

    Google Scholar 

  21. Halevi, S., Shoup, V.: Bootstrapping for HElib. Cryptology ePrint Archive, Report 2014/873 (2014). https://eprint.iacr.org/2014/873

  22. Halevi, S., Shoup, V.: Faster homomorphic linear transformations in HElib. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018. LNCS, vol. 10991, pp. 93–120. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96884-1_4

    Chapter  MATH  Google Scholar 

  23. Lee, Y., et al.: Efficient fhew bootstrapping with small evaluation keys, and applications to threshold homomorphic encryption. Cryptology ePrint Archive, Paper 2022/198 (2022). https://eprint.iacr.org/2022/198

  24. Liu, F.-H., Wang, H.: Batch bootstrapping II: bootstrapping in polynomial modulus only requires \(\tilde{O}(1)\) fhe multiplications in amortization. In: Eurocrypt (2023)

    Google Scholar 

  25. 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

    Chapter  Google Scholar 

  26. Lyubashevsky, V., Peikert, C., Regev, O.: A toolkit for ring-LWE cryptography. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 35–54. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38348-9_3

  27. Micciancio, D., Peikert, C.: Trapdoors for lattices: simpler, tighter, faster, smaller. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 700–718. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29011-4_41

    Chapter  Google Scholar 

  28. Micciancio, D., Polyakov, Y.: Bootstrapping in fhew-like cryptosystems. In: WAHC 2021: Proceedings of the 9th on Workshop on Encrypted Computing & Applied Homomorphic Cryptography, Virtual Event, Korea, 15 November 2021, pp. 17–28. WAHC@ACM (2021)

    Google Scholar 

  29. Micciancio, D., Sorrell, J.: Ring packing and amortized FHEW bootstrapping. In: Chatzigiannakis, I., Kaklamanis, C., Marx, D., Sannella, D. (eds.) ICALP 2018, vol. 107 of LIPIcs, pp. 100:1–100:14. Schloss Dagstuhl (2018)

    Google Scholar 

  30. Peikert, C.: How (not) to instantiate ring-LWE. In: Zikas, V., De Prisco, R. (eds.) SCN 2016. LNCS, vol. 9841, pp. 411–430. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-44618-9_22

    Chapter  Google Scholar 

  31. Peikert, C., Pepin, Z.: Algebraically structured LWE, revisited. In: Hofheinz, D., Rosen, A. (eds.) TCC 2019. LNCS, vol. 11891, pp. 1–23. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-36030-6_1

    Chapter  Google Scholar 

  32. Peikert, C., Regev, O., Stephens-Davidowitz, N.: Pseudorandomness of ring-LWE for any ring and modulus. In: Hatami, H., McKenzie, P., King, V. (eds.) 49th ACM STOC, pp. 461–473. ACM Press (2017)

    Google Scholar 

  33. Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. In: Gabow, H.N., Fagin, R. (eds.) 37th ACM STOC, pp. 84–93. ACM Press (2005)

    Google Scholar 

  34. Rivest, R.L., Adleman, L., Dertouzos, M.L.: On data banks and privacy homomorphisms. In: Foundations of Secure Computation (1978)

    Google Scholar 

  35. Takagi, T., Peyrin, T. (eds.): LNCS, vol. 10624. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70694-8

    Book  MATH  Google Scholar 

  36. Vaikuntanathan, V.: Homomorphic encryption references. https://people.csail.mit.edu/vinodv/FHE/FHE-refs.html

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Acknowledgement

The authors would like to thank anonymous reviewers for their insightful comments that significantly help improve the presentation. Feng-Hao Liu is supported by NSF CNS-1942400. Han Wang is supported by the National Key R &D Program of China under Grant 2020YFA0712303 and State Key Laboratory of Information Security under Grant TC20221013042.

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Authors and Affiliations

  1. Florida Atlantic University, Boca Raton, FL, USA

    Feng-Hao Liu

  2. State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Science, Beijing, China

    Han Wang

  3. School of Cyber Security, University of Chinese Academy of Science, Beijing, China

    Han Wang

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  1. Feng-Hao Liu
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Correspondence to Han Wang .

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Editors and Affiliations

  1. Bar-Ilan University, Ramat Gan, Israel

    Carmit Hazay

  2. Simula UiB, Bergen, Norway

    Martijn Stam

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Liu, FH., Wang, H. (2023). Batch Bootstrapping I:. In: Hazay, C., Stam, M. (eds) Advances in Cryptology – EUROCRYPT 2023. EUROCRYPT 2023. Lecture Notes in Computer Science, vol 14006. Springer, Cham. https://doi.org/10.1007/978-3-031-30620-4_11

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  • DOI: https://doi.org/10.1007/978-3-031-30620-4_11

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