Skip to main content
SpringerLink
Log in

On the Round Complexity of Fully Secure Solitary MPC with Honest Majority

  • Conference paper
  • First Online:
Theory of Cryptography (TCC 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14370))

Included in the following conference series:

  • 196 Accesses

Abstract

We study the problem of secure multiparty computation for functionalities where only one party receives the output, to which we refer as solitary MPC. Recently, Halevi et al. (TCC 2019) studied fully secure (i.e., with guaranteed output delivery) solitary MPC and showed impossibility of such protocols for certain functionalities when there is no honest majority among the parties.

In this work, we study the round complexity of fully secure solitary MPC in the honest majority setting and with computational security. We note that a broadcast channel or public key infrastructure (PKI) setup is necessary for an n-party protocol against malicious adversaries corrupting up to t parties where \(n/3 \le t < n/2\). Therefore, we study the following settings and ask the question: Can fully secure solitary MPC be achieved in fewer rounds than fully secure standard MPC in which all parties receive the output?

  • When there is a broadcast channel and no PKI:

    • We start with a negative answer to the above question. In particular, we show that the exact round complexity of fully secure solitary MPC is 3, which is the same as fully secure standard MPC.

    • We then study the minimal number of broadcast rounds needed to design round-optimal fully secure solitary MPC. We show that both the first and second rounds of broadcast are necessary when \(2 \lceil n/5 \rceil \le t < n/2\), whereas pairwise-private channels suffice in the last round. Notably, this result also applies to fully secure standard MPC in which all parties receive the output.

  • When there is a PKI and no broadcast channel, nevertheless, we show more positive results:

    • We show an upper bound of 5 rounds for any honest majority. This is superior to the super-constant lower bound for fully secure standard MPC in the exact same setting.

    • We complement this by showing a lower bound of 4 rounds when \(3\lceil n/7 \rceil \le t < n/2\).

    • For the special case of \(t=1,n=3\), when the output receiving party does not have an input to the function, we show an upper bound of 2 rounds, which is optimal. When the output receiving party has an input to the function, we show a lower bound of 3, which matches an upper bound from prior work.

    • For the special case of \(t=2,n=5\), we show a lower bound of 3 rounds (an upper bound of 4 follows from prior work).

All our results also assume the existence of a common reference string (CRS) and pairwise-private channels. Our upper bounds use a decentralized threshold fully homomorphic encryption (dTFHE) scheme (which can be built from the learning with errors (LWE) assumption) as the main building block.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 59.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 79.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Cleve’s argument shows that with dishonest majority, it is impossible for an MPC protocol to achieve fairness, which guarantees that malicious parties cannot learn the output while preventing honest parties from learning the output. Since \(\textsf {god}\) implies fairness, this impossibility also holds for standard MPC with \(\textsf {god}\). However, it doesn’t hold for solitary MPC as fairness is clearly not an issue in the solitary MPC setting.

  2. 2.

    This protocol uses a decentralized threshold fully homomorphic encryption (dTFHE) scheme. The public parameter of this dTFHE is assumed to be shared among the parties and viewed as a common reference string (refer to [28] for further details).

  3. 3.

    Fitzi et al. [21] show that converge-cast cannot be achieved when \(n/3 \le t <n/2\) in the information theoretic setting. Alon et al. [2] show a specific solitary functionality that cannot be computed by a 3-party MPC protocol with a single corruption with \(\textsf {god}\) in the plain model (with no broadcast channel and no PKI), which also extends to \(n/3 \le t <n/2\). Both arguments also work even in the presence of a CRS. We present the proof in the full version [7] for completeness.

  4. 4.

    Note that PKI setup is in fact necessary for realizing a broadcast channel when \(t \ge n/3 \) (where n is the total number of parties) [33, 37].

  5. 5.

    In these randomized broadcast protocols, the number of rounds depends on the randomness involved in the protocol. For example, the protocol by Abraham et al. [1] terminates in constant rounds except with constant probability and requires at least super-polylogarithmic rounds (in the security parameter) to terminate with all but negligible probability.

  6. 6.

    We leave it as an interesting open problem to achieve the upper bound using weaker forms of PKI setup and studying the minimal assumption required.

  7. 7.

    We use \([\![{ x }]\!]\) to denote a dTFHE encryption of x.

  8. 8.

    Generally, communication between corrupt parties need not be specified but we include it here for easier understanding of Table 13.

  9. 9.

    Let \(\mathcal {S}= \{i | \textsf{ct}_i = \bot \}\). Here, we actually homomorphically evaluate the residual function \(f_\mathcal {S}(\cdot )\) that only takes as input \(\{x_j\}_{j \notin \mathcal {S}}\) and uses the default values for all indices in the set \(\mathcal {S}\). For ease of exposition, we skip this notation in the rest of the protocol and proof.

References

  1. Abraham, I., Devadas, S., Dolev, D., Nayak, K., Ren, L.: Synchronous byzantine agreement with expected O(1) rounds, expected o(n\({}^{\text{2}}\)) communication, and optimal resilience. In: FC (2019)

    Google Scholar 

  2. Alon, B., Cohen, R., Omri, E., Suad, T.: On the power of an honest majority in three-party computation without broadcast. In: TCC (2020)

    Google Scholar 

  3. Ananth, P., Choudhuri, A.R., Goel, A., Jain, A.: Round-optimal secure multiparty computation with honest majority. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018. LNCS, vol. 10992, pp. 395–424. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96881-0_14

    Chapter  Google Scholar 

  4. Asharov, G., Beimel, A., Makriyannis, N., Omri, E.: Complete characterization of fairness in secure two-party computation of Boolean functions. In: Dodis, Y., Nielsen, J.B. (eds.) TCC 2015. LNCS, vol. 9014, pp. 199–228. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46494-6_10

    Chapter  MATH  Google Scholar 

  5. Asharov, G., Jain, A., López-Alt, A., Tromer, E., Vaikuntanathan, V., Wichs, D.: Multiparty computation with low communication, computation and interaction via threshold FHE. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 483–501. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29011-4_29

    Chapter  Google Scholar 

  6. Badrinarayanan, S., Jain, A., Manohar, N., Sahai, A.: Threshold multi-key FHE and applications to round-optimal MPC. In: ASIACRYPT (2020)

    Google Scholar 

  7. Badrinarayanan, S., Miao, P., Mukherjee, P., Ravi, D.: On the round complexity of fully secure solitary mpc with honest majority. Cryptology ePrint Archive, Paper 2021/241 (2021). https://eprint.iacr.org/2021/241

  8. Bell, J.H., Bonawitz, K.A., Gascón, A., Lepoint, T., Raykova, M.: Secure single-server aggregation with (poly)logarithmic overhead. In: CCS, pp. 1253–1269. ACM (2020)

    Google Scholar 

  9. Bonawitz, K., et al. Practical secure aggregation for privacy-preserving machine learning. In: CCS (2017)

    Google Scholar 

  10. Boneh, D., Gennaro, R., Goldfeder, S., Jain, A., Kim, S., Rasmussen, P.M.R., Sahai, A.: Threshold cryptosystems from threshold fully homomorphic encryption. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018. LNCS, vol. 10991, pp. 565–596. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96884-1_19

    Chapter  Google Scholar 

  11. Canetti, R., et al.: Fiat-shamir: from practice to theory. In: STOC (2019)

    Google Scholar 

  12. Chor, B., Merritt, M., Shmoys, D.B.: Simple constant-time consensus protocols in realistic failure models. J. ACM (JACM) 36(3), 591–614 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cleve, R.: Limits on the security of coin flips when half the processors are faulty (extended abstract). In: STOC (1986)

    Google Scholar 

  14. Cohen, R., Garay, J., Zikas, V.: Broadcast-optimal two-round MPC. In: Canteaut, A., Ishai, Y. (eds.) EUROCRYPT 2020. LNCS, vol. 12106, pp. 828–858. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45724-2_28

    Chapter  Google Scholar 

  15. Damgård, I., Magri, B., Ravi, D., Siniscalchi, L., Yakoubov, S.: Broadcast-optimal two round MPC with an honest majority. In: Malkin, T., Peikert, C. (eds.) CRYPTO 2021. LNCS, vol. 12826, pp. 155–184. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-84245-1_6

    Chapter  Google Scholar 

  16. Damgård, I., Ravi, D., Siniscalchi, L., Yakoubov, S.: Minimizing setup in broadcast-optimal two round MPC. In: Hazay, C., Stam, M. (eds.) EUROCRYPT 2023. LNCS, vol. 14005, pp. 129–158. Springer, Heidelberg (2023). https://doi.org/10.1007/978-3-031-30617-4_5

    Chapter  Google Scholar 

  17. Dolev, D., Strong, H.R.: Authenticated algorithms for Byzantine agreement. SIAM J. Comput. 12(4), 656–666 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  18. Feldman, P., Micali, S.: An optimal probabilistic algorithm for synchronous Byzantine agreement. In: Ausiello, G., Dezani-Ciancaglini, M., Della Rocca, S.R. (eds.) ICALP 1989. LNCS, vol. 372, pp. 341–378. Springer, Heidelberg (1989). https://doi.org/10.1007/BFb0035770

    Chapter  Google Scholar 

  19. Fischer, M.J., Lynch, N.A.: A lower bound for the time to assure interactive consistency. Inf. Process. Lett. 14(4), 183–186 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  20. Fitzi, M., Garay, J.A.: Efficient player-optimal protocols for strong and differential consensus. In: Borowsky, E., Rajsbaum, S. (eds.) 22nd ACM PODC, pp. 211–220. ACM (2003)

    Google Scholar 

  21. Fitzi, M., Garay, J.A., Maurer, U., Ostrovsky, R.: Minimal complete primitives for secure multi-party computation. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 80–100. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44647-8_5

    Chapter  Google Scholar 

  22. Garg, S., Goel, A., Jain, A.: The broadcast message complexity of secure multiparty computation. In: Galbraith, S.D., Moriai, S. (eds.) ASIACRYPT 2019. LNCS, vol. 11921, pp. 426–455. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-34578-5_16

    Chapter  Google Scholar 

  23. Gennaro, R., Ishai, Y., Kushilevitz, E., Rabin, T.: The round complexity of verifiable secret sharing and secure multicast. In: STOC (2001)

    Google Scholar 

  24. Gennaro, R., Ishai, Y., Kushilevitz, E., Rabin, T.: On 2-round secure multiparty computation. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 178–193. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-45708-9_12

    Chapter  Google Scholar 

  25. Goldreich, O., Micali, S., Wigderson, A.: How to play any mental game or a completeness theorem for protocols with honest majority. In: STOC (1987)

    Google Scholar 

  26. Goldwasser, S., Lindell, Y.: Secure multi-party computation without agreement. J. Cryptol. 18, 247–287 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Gordon, S.D., Hazay, C., Katz, J., Lindell, Y.: Complete fairness in secure two-party computation. J. ACM 58(6), 24:1–24:37 (2011)

    Google Scholar 

  28. Dov Gordon, S., Liu, F.-H., Shi, E.: Constant-round MPC with fairness and guarantee of output delivery. In: Gennaro, R., Robshaw, M. (eds.) CRYPTO 2015. LNCS, vol. 9216, pp. 63–82. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48000-7_4

    Chapter  Google Scholar 

  29. Halevi, S., Ishai, Y., Kushilevitz, E., Makriyannis, N., Rabin, T.: On fully secure MPC with solitary output. In: Hofheinz, D., Rosen, A. (eds.) TCC 2019. LNCS, vol. 11891, pp. 312–340. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-36030-6_13

    Chapter  Google Scholar 

  30. Halevi, S., Lindell, Y., Pinkas, B.: Secure computation on the web: computing without simultaneous interaction. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 132–150. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22792-9_8

    Chapter  Google Scholar 

  31. Karlin, A., Yao, A.: Probabilistic lower bounds for byzantine agreement. Unpublished document (1986)

    Google Scholar 

  32. Katz, J., Koo, C.Y.: On expected constant-round protocols for byzantine agreement. J. Comput. Syst. Sci. 75(2), 91–112 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  33. Lamport, L., Shostak, R.E., Pease, M.C.: The byzantine generals problem. ACM Trans. Program. Lang. Syst. (1982)

    Google Scholar 

  34. Lynch, N.A.: Distributed Algorithms. Morgan Kaufmann, Burlington (1996)

    MATH  Google Scholar 

  35. Mohassel, A., Zhang, Y.: Secureml: a system for scalable privacy-preserving machine learning. In: IEEE S & P (2017)

    Google Scholar 

  36. Patra, A., Ravi, D.: On the exact round complexity of secure three-party computation. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018. LNCS, vol. 10992, pp. 425–458. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96881-0_15

    Chapter  Google Scholar 

  37. Pease, M., Shostak, R., Lamport, L.: Reaching agreement in the presence of faults. J. ACM 27(2), 228–234 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  38. Peikert, C., Shiehian, S.: Noninteractive zero knowledge for NP from (plain) learning with errors. In: Boldyreva, A., Micciancio, D. (eds.) CRYPTO 2019. LNCS, vol. 11692, pp. 89–114. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-26948-7_4

    Chapter  Google Scholar 

  39. Yao, A.C.C.: How to generate and exchange secrets (extended abstract). In: FOCS (1986)

    Google Scholar 

Download references

Acknowledgments

We would like to thank the anonymous reviewers for their helpful and constructive comments on the manuscript. P. Miao is supported in part by the NSF CNS Award 2247352, a DPI Science Team Seed Grant, a Meta Award, and a DSI Seed Grant. All the authors did part of the work while at Visa Research.

Author information

Authors and Affiliations

  1. LinkedIn, Mountain View, USA

    Saikrishna Badrinarayanan

  2. Brown University, Providence, USA

    Peihan Miao

  3. Supra Research, Kolkata, India

    Pratyay Mukherjee

  4. Aarhus University, Aarhus, Denmark

    Divya Ravi

Authors
  1. Saikrishna Badrinarayanan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peihan Miao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pratyay Mukherjee
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Divya Ravi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Saikrishna Badrinarayanan .

Editor information

Editors and Affiliations

  1. Apple, Cupertino, CA, USA

    Guy Rothblum

  2. NTT Research, Sunnyvale, CA, USA

    Hoeteck Wee

Rights and permissions

Reprints and permissions

Copyright information

© 2023 International Association for Cryptologic Research

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Badrinarayanan, S., Miao, P., Mukherjee, P., Ravi, D. (2023). On the Round Complexity of Fully Secure Solitary MPC with Honest Majority. In: Rothblum, G., Wee, H. (eds) Theory of Cryptography. TCC 2023. Lecture Notes in Computer Science, vol 14370. Springer, Cham. https://doi.org/10.1007/978-3-031-48618-0_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-48618-0_5

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-48617-3

  • Online ISBN: 978-3-031-48618-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Societies and partnerships

Search

Navigation