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Large Message Homomorphic Secret Sharing from DCR and Applications

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Advances in Cryptology – CRYPTO 2021 (CRYPTO 2021)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 12827))

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Abstract

We present the first homomorphic secret sharing (HSS) construction that simultaneously (1) has negligible correctness error, (2) supports integers from an exponentially large range, and (3) relies on an assumption not known to imply FHE—specifically, the Decisional Composite Residuosity (DCR) assumption. This resolves an open question posed by Boyle, Gilboa, and Ishai (Crypto 2016). Homomorphic secret sharing is analogous to fully-homomorphic encryption, except the ciphertexts are shared across two non-colluding evaluators. Previous constructions of HSS either had non-negligible correctness error and polynomial-size plaintext space or were based on the stronger LWE assumption. We also present two applications of our technique: a two server ORAM with constant bandwidth overhead, and a rate-1 trapdoor hash function with negligible error rate.

L. Roy—Supported by a DoE CSGF fellowship.

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Notes

  1. 1.

    A concurrent work [OSY21] has also independently solved this problem.

  2. 2.

    Damgård–Jurik was originally defined using \(\exp (x) = (1 + N)^x\), which required \(\log \) to use Hensel lifting. We instead chose to use Taylor series because it simplifies the description of \(\log \), and only require O(s) additions and multiplications to evaluate with Horner’s rule, while the Hensel lifting algorithm took \(O(s^2)\) arithmetic operations.

  3. 3.

    This structure may seem familiar to readers interested in category theory. In fact, we hit on these definitions by thinking of circuit semantics as functors. The homomorphism property then requires that \(\mathsf{Decode} \) be a natural transformation from the homomorphic evaluation semantics to the plaintext evaluation semantics.

References

  1. Boyle, E., Gilboa, N., Ishai, Y.: Breaking the circuit size barrier for secure computation under DDH. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, Part I, vol. 9814, pp. 509–539. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53018-4_19

    Chapter  Google Scholar 

  2. Boyle, E., Gilboa, N., Ishai, Y., Lin, H., Tessaro, S.: Foundations of homomorphic secret sharing. Cryptology ePrint Archive, Report 2017/1248 (2017). https://eprint.iacr.org/2017/1248

  3. Boyle, E., Kohl, L., Scholl, P.: Homomorphic secret sharing from lattices without FHE. In: Ishai, Y., Rijmen, V. (eds.) EUROCRYPT 2019. LNCS, Part II, vol. 11477, pp. 3–33. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17656-3_1

    Chapter  Google Scholar 

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

    Google Scholar 

  5. Cleve, R.: Towards optimal simulations of formulas by bounded-width programs. In: 22nd ACM STOC, pp. 271–277. ACM Press, May 1990

    Google Scholar 

  6. Döttling, N., Garg, S., Ishai, Y., Malavolta, G., Mour, T., Ostrovsky, R.: Trapdoor hash functions and their applications. In: Boldyreva, A., Micciancio, D. (eds.) CRYPTO 2019. LNCS, Part III, vol. 11694, pp. 3–32. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-26954-8_1

    Chapter  Google Scholar 

  7. Damgård, I., Jurik, M.: A generalisation, a simplification and some applications of Paillier’s probabilistic public-key system. In: Kim, K. (ed.) PKC 2001. LNCS, vol. 1992, pp. 119–136. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44586-2_9

    Chapter  MATH  Google Scholar 

  8. Dinur, I., Keller, N., Klein, O.: An optimal distributed discrete log protocol with applications to homomorphic secret sharing. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018. Part III, vol. 10993. LNCS, pp. 213–242. Springer, Heidelberg (2018). https://doi.org/10.1007/s00145-019-09330-2

  9. Doerner, J., Shelat, A.: Scaling ORAM for secure computation. In: Proceedings of the 2017 ACM SIGSAC Conference on Computer and Communications Security, pp. 523–535 (2017)

    Google Scholar 

  10. Devadas, S., van Dijk, M., Fletcher, C.W., Ren, L., Shi, E., Wichs, D.: Onion ORAM: a constant bandwidth blowup oblivious RAM. In: Kushilevitz, E., Malkin, T. (eds.) TCC 2016. LNCS, vol. 9563, pp. 145–174. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49099-0_6

    Chapter  Google Scholar 

  11. Fazio, N., Gennaro, R., Jafarikhah, T., Skeith III, W.E.: Homomorphic secret sharing from Paillier encryption. In: Okamoto, T., Yu, Y., Au, M.H., Li, Y. (eds.) ProvSec 2017. LNCS, vol. 10592, pp. 381–399. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-68637-0_23

    Chapter  Google Scholar 

  12. Fletcher, C.W., Naveed, M., Ren, L., Shi, E., Stefanov, E.: Bucket ORAM: single online roundtrip, constant bandwidth oblivious ram. IACR Cryptol. ePrint Arch., 2015:1065 (2015)

    Google Scholar 

  13. Goldreich, O.: Towards a theory of software protection and simulation by oblivious rams. In: Proceedings of the Nineteenth Annual ACM Symposium on Theory of Computing, pp. 182–194 (1987)

    Google Scholar 

  14. Hoang, T., Ozkaptan, C.D., Yavuz, A.A., Guajardo, J., Nguyen, T.: S3ORAM: a computation-efficient and constant client bandwidth blowup ORAM with Shamir secret sharing. In: Proceedings of the 2017 ACM SIGSAC Conference on Computer and Communications Security, pp. 491–505 (2017)

    Google Scholar 

  15. Ostrovsky, R.: Software protection and simulation on oblivious RAMs. PhD thesis, Massachusetts Institute of Technology (1992)

    Google Scholar 

  16. Orlandi, C., Scholl, P., Yakoubov, S.: The rise of Paillier: homomorphic secret sharing and public-key silent OT. In: Canteaut, A., Standaert, F.-X. (eds.) EUROCRYPT 2021. LNCS, vol. 12696, pp. 678–708. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-77870-5_24

    Chapter  Google Scholar 

  17. Paillier, P.: Public-key cryptosystems based on composite degree residuosity classes. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 223–238. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-48910-X_16

    Chapter  Google Scholar 

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Authors and Affiliations

  1. Oregon State University, Corvallis, USA

    Lawrence Roy & Jaspal Singh

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  1. Lawrence Roy
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  2. Jaspal Singh
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Correspondence to Lawrence Roy or Jaspal Singh .

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Editors and Affiliations

  1. Columbia University, New York City, NY, USA

    Tal Malkin

  2. University of Michigan, Ann Arbor, MI, USA

    Chris Peikert

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Roy, L., Singh, J. (2021). Large Message Homomorphic Secret Sharing from DCR and Applications. In: Malkin, T., Peikert, C. (eds) Advances in Cryptology – CRYPTO 2021. CRYPTO 2021. Lecture Notes in Computer Science(), vol 12827. Springer, Cham. https://doi.org/10.1007/978-3-030-84252-9_23

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  • DOI: https://doi.org/10.1007/978-3-030-84252-9_23

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