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Annual International Conference on the Theory and Applications of Cryptographic Techniques

EUROCRYPT 2022: Advances in Cryptology – EUROCRYPT 2022 pp 521–550Cite as

Limits of Polynomial Packings for \(\mathbb {Z}_{p^k}\) and \(\mathbb {F}_{p^k}\)

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13275))

Abstract

We formally define polynomial packing methods and initiate a unified study of related concepts in various contexts of cryptography. This includes homomorphic encryption (HE) packing and reverse multiplication-friendly embedding (RMFE) in information-theoretically secure multi-party computation (MPC). We prove several upper bounds and impossibility results on packing methods for \(\mathbb {Z}_{p^k}\) or \(\mathbb {F}_{p^k}\)-messages into \(\mathbb {Z}_{p^t}[x]/f(x)\) in terms of (i) packing density, (ii) level-consistency, and (iii) surjectivity. These results have implications on recent development of HE-based MPC over \(\mathbb {Z}_{2^k}\) secure against actively corrupted majority and provide new proofs for upper bounds on RMFE.

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Notes

  1. 1.

    Chinese Remainder Theorem.

  2. 2.

    Single Instruction, Multiple Data.

  3. 3.

    The original method of [33] does not consider packings for \({\mathbb Z}_{p^k}\). Gentry-Halevi-Smart [24] later generalized the method to support such packing. However, this method achieves only considerably low efficiency. See Sect. 4.1.

  4. 4.

    Indeed, the number of evaluation points is bounded by the size of the field.

  5. 5.

    Zero-knowledge proof of message knowledge.

  6. 6.

    Nonetheless, this object was also previously studied in [2] to amortize oblivious linear evaluations (OLE) into a larger extension field for correlation extraction problem in MPC. However, their construction achieved only sublinear density.

  7. 7.

    Similar problems were also considered in other cryptography literature [2, 18, 29]. For more detailed discussions, see the full version [14].

  8. 8.

    In a sense that any element of \({\mathcal R}\) could be an image of \({\textsf {{Pack}}}_1(\cdot )\).

  9. 9.

    ZKPoMK was first conceptualized in MHZ2k [13], but it is also performed in Overdrive2k [31] implicitly. For detailed discussion, refer to [13].

References

  1. Baum, C., Cozzo, D., Smart, N.P.: Using TopGear in overdrive: a more efficient ZKPoK for SPDZ. In: Paterson, K.G., Stebila, D. (eds.) SAC 2019. LNCS, vol. 11959, pp. 274–302. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-38471-5_12

    Chapter  MATH  Google Scholar 

  2. Block, A.R., Maji, H.K., Nguyen, H.H.: Secure computation based on leaky correlations: high resilience setting. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017. LNCS, vol. 10402, pp. 3–32. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63715-0_1

    Chapter  Google Scholar 

  3. Block, A.R., Maji, H.K., Nguyen, H.H.: Secure computation with constant communication overhead using multiplication embeddings. In: Chakraborty, D., Iwata, T. (eds.) INDOCRYPT 2018. LNCS, vol. 11356, pp. 375–398. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-05378-9_20

    Chapter  Google Scholar 

  4. Brakerski, Z., Gentry, C., Vaikuntanathan, V.: (Leveled) fully homomorphic encryption without bootstrapping. In: Proceedings of the 3rd Innovations in Theoretical Computer Science Conference, pp. 309–325 (2012)

    Google Scholar 

  5. Cascudo, I., Cramer, R., Xing, C., Yuan, C.: Amortized complexity of information-theoretically secure MPC revisited. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018. LNCS, vol. 10993, pp. 395–426. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96878-0_14

    Chapter  Google Scholar 

  6. Cascudo, I., Giunta, E.: On interactive oracle proofs for Boolean R1CS statements. Cryptology ePrint Archive, Report 2021/694 (2021)

    Google Scholar 

  7. Cascudo, I., Gundersen, J.S.: A secret-sharing based MPC protocol for Boolean circuits with good amortized complexity. In: Pass, R., Pietrzak, K. (eds.) TCC 2020. LNCS, vol. 12551, pp. 652–682. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-64378-2_23

    Chapter  Google Scholar 

  8. Castryck, W., Iliashenko, I., Vercauteren, F.: Homomorphic SIM\(^2\)D operations: single instruction much more data. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018. LNCS, vol. 10820, pp. 338–359. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-78381-9_13

    Chapter  Google Scholar 

  9. Chen, H., Han, K.: Homomorphic lower digits removal and improved FHE bootstrapping. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018. LNCS, vol. 10820, pp. 315–337. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-78381-9_12

    Chapter  Google Scholar 

  10. Chen, H., Laine, K., Player, R., Xia, Y.: High-precision arithmetic in homomorphic encryption. In: Smart, N.P. (ed.) CT-RSA 2018. LNCS, vol. 10808, pp. 116–136. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-76953-0_7

    Chapter  MATH  Google Scholar 

  11. Cheon, J.H., Jeong, J., Lee, J., Lee, K.: Privacy-preserving computations of predictive medical models with minimax approximation and non-adjacent form. In: Brenner, M., et al. (eds.) FC 2017. LNCS, vol. 10323, pp. 53–74. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70278-0_4

    Chapter  Google Scholar 

  12. 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 

  13. Cheon, J.H., Kim, D., Lee, K.: MHz2k: MPC from HE over \(\mathbb{Z}_{2^k}\) with new packing, simpler Reshare, and better ZKP. In: Malkin, T., Peikert, C. (eds.) CRYPTO 2021. LNCS, vol. 12826, pp. 426–456. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-84245-1_15

    Chapter  Google Scholar 

  14. Cheon, J.H., Lee, K.: Limits of polynomial packings for \(\mathbb{Z}_{p^k}\) and \(\mathbb{F}_{p^k}\). Cryptology ePrint Archive, Report 2021/1033 (2021)

    Google Scholar 

  15. Cramer, R., Damgård, I., Escudero, D., Scholl, P., Xing, C.: SPD\(\mathbb{Z}_{2^k}\): efficient MPC mod \(2^k\) for dishonest majority. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018. LNCS, vol. 10992, pp. 769–798. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96881-0_26

    Chapter  Google Scholar 

  16. Cramer, R., Rambaud, M., Xing, C.: Asymptotically-good arithmetic secret sharing over \(\mathbb{Z}/p^{\ell }\mathbb{Z}\) with strong multiplication and its applications to efficient MPC. In: Malkin, T., Peikert, C. (eds.) CRYPTO 2021. LNCS, vol. 12827, pp. 656–686. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-84252-9_22

    Chapter  Google Scholar 

  17. Cramer, R., Xing, C., Yuan, C.: On the complexity of arithmetic secret sharing. In: Pass, R., Pietrzak, K. (eds.) TCC 2020. LNCS, vol. 12552, pp. 444–469. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-64381-2_16

    Chapter  MATH  Google Scholar 

  18. Dalskov, A., Lee, E., Soria-Vazquez, E.: Circuit amortization friendly encodings and their application to statistically secure multiparty computation. In: Moriai, S., Wang, H. (eds.) ASIACRYPT 2020. LNCS, vol. 12493, pp. 213–243. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-64840-4_8

    Chapter  Google Scholar 

  19. Damgård, I., Keller, M., Larraia, E., Pastro, V., Scholl, P., Smart, N.P.: Practical covertly secure MPC for dishonest majority – or: breaking the SPDZ limits. In: Crampton, J., Jajodia, S., Mayes, K. (eds.) ESORICS 2013. LNCS, vol. 8134, pp. 1–18. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40203-6_1

    Chapter  Google Scholar 

  20. Damgård, I., Larsen, K.G., Nielsen, J.B.: Communication lower bounds for statistically secure MPC, with or without preprocessing. In: Boldyreva, A., Micciancio, D. (eds.) CRYPTO 2019. LNCS, vol. 11693, pp. 61–84. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-26951-7_3

    Chapter  Google Scholar 

  21. 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 

  22. Fan, J., Vercauteren, F.: Somewhat practical fully homomorphic encryption. Cryptology ePrint Archive, Report 2012/144 (2012)

    Google Scholar 

  23. 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)

    Google Scholar 

  24. 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 

  25. Halevi, S., Shoup, V.: Helib. Retrieved from HELib (2014). https://github.com.homenc/HElib

  26. Halevi, S., Shoup, V.: Bootstrapping for HElib. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9056, pp. 641–670. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46800-5_25

    Chapter  Google Scholar 

  27. Keller, M., Pastro, V., Rotaru, D.: Overdrive: making SPDZ great again. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018. LNCS, vol. 10822, pp. 158–189. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-78372-7_6

    Chapter  Google Scholar 

  28. Kim, A., Song, Y., Kim, M., Lee, K., Cheon, J.H.: Logistic regression model training based on the approximate homomorphic encryption. BMC Med. Genomics 11(4), 23–31 (2018)

    Google Scholar 

  29. Lipmaa, H.: Progression-free sets and sublinear pairing-based non-interactive zero-knowledge arguments. In: Cramer, R. (ed.) TCC 2012. LNCS, vol. 7194, pp. 169–189. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-28914-9_10

    Chapter  Google Scholar 

  30. 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 

  31. Orsini, E., Smart, N.P., Vercauteren, F.: Overdrive2k: efficient secure MPC over \(\mathbb{Z}_{2^k}\) from somewhat homomorphic encryption. In: Jarecki, S. (ed.) CT-RSA 2020. LNCS, vol. 12006, pp. 254–283. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-40186-3_12

    Chapter  Google Scholar 

  32. Polychroniadou, A., Song, Y.: Constant-overhead unconditionally secure multiparty computation over binary fields. In: Canteaut, A., Standaert, F.-X. (eds.) EUROCRYPT 2021. LNCS, vol. 12697, pp. 812–841. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-77886-6_28

    Chapter  Google Scholar 

  33. Smart, N.P., Vercauteren, F.: Fully homomorphic encryption with relatively small key and ciphertext sizes. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 420–443. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13013-7_25

    Chapter  MATH  Google Scholar 

  34. Smart, N.P., Vercauteren, F.: Fully homomorphic SIMD operations. Des. Codes Cryptogr. 71(1), 57–81 (2014). https://doi.org/10.1007/s10623-012-9720-4

    Article  MATH  Google Scholar 

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Acknowledgement

The authors thank Dongwoo Kim for insightful discussions on packing methods, Donggeon Yhee for discussions on Proposition 7, and Minki Hhan for constructive comments on an earlier version of this work. The authors also thank the reviewers of Eurocrypt 2022 who provided thoughtful suggestions to improve the earlier version of this paper. This work was supported by Institute of Information & Communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No. 2020-0-00840, Development and Library Implementation of Fully Homomorphic Machine Learning Algorithms supporting Neural Network Learning over Encrypted Data).

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Authors and Affiliations

  1. Seoul National University, Seoul, Republic of Korea

    Jung Hee Cheon & Keewoo Lee

  2. Crypto Lab Inc., Seoul, Republic of Korea

    Jung Hee Cheon

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  1. Jung Hee Cheon
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  1. University of Haifa, Haifa, Haifa, Israel

    Orr Dunkelman

  2. University of Warsaw, Warsaw, Poland

    Stefan Dziembowski

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Cheon, J.H., Lee, K. (2022). Limits of Polynomial Packings for \(\mathbb {Z}_{p^k}\) and \(\mathbb {F}_{p^k}\). In: Dunkelman, O., Dziembowski, S. (eds) Advances in Cryptology – EUROCRYPT 2022. EUROCRYPT 2022. Lecture Notes in Computer Science, vol 13275. Springer, Cham. https://doi.org/10.1007/978-3-031-06944-4_18

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