Abstract
In this work, we study unconditionally-secure multi-party computation (MPC) tolerating \(t < n/3\) corruptions, where n is the total number of parties involved. In this setting, it is well known that if the underlying network is completely asynchronous, then one can achieve only statistical security; moreover it is impossible to ensure input provision and consider inputs of all the honest parties. The best known statistically-secure asynchronous MPC (AMPC) with \(t<n/3\) requires a communication of \(\varOmega (n^5)\) field elements per multiplication. We consider a partially synchronous setting, where the parties are assumed to be globally synchronized initially for few rounds and then the network becomes completely asynchronous. In such a setting, we present a MPC protocol, which requires \(\mathcal {O}(n^2)\) communication per multiplication while ensuring input provision. Our MPC protocol relies on a new four round, communication efficient statistical verifiable secret-sharing (VSS) protocol with broadcast communication complexity independent of the number of secret-shared values.
A. Choudhury—Financial support from Infosys Foundation acknowledged.
A. Patra—Work partially supported by INSPIRE Faculty Fellowship (DST/INSPIRE/04/2014/015727) from Department of Science & Technology, India.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
Notes
- 1.
The outcome of a perfectly-secure protocol is error-free, while a negligible error is allowed in a statistically-secure protocol.
- 2.
Informally such a scheme ensures that the shared value remains information-theoretically secure even if upto t shares are revealed. Shamir sharing of a secret with threshold t is done by selecting a random polynomial of degree at most t with the secret as the constant term and defining the individual shares as distinct evaluations of the polynomial.
- 3.
The amortized communication complexity is derived under the assumption that the circuit is large enough so that the terms that are independent of the circuit size can be ignored.
- 4.
We say a protocol requires \((r, r')\) (synchronous) rounds, if it requires a total of r rounds of interaction among the parties and out of these r rounds, \(r'\) rounds require broadcast by the parties, where \(r' \le r\).
- 5.
The actual complexity (communication, computation and round) of these protocols are of the form \(\mathcal {O}((\log ^k n\; \cdot \text{ poly }(\log |C|)) \cdot |C|) + \mathcal {O}(\text{ poly }(n, \log |C|, \mathcal {D}))\), where \(\mathcal {D}\) is the multiplicative depth of the underlying circuit C.
- 6.
The interpretation of a proof corresponding to a set of values will be clear later during the formal presentation of our ICPoP.
- 7.
This explains the need for two masking polynomials: one is used to preserve the privacy of the secret-encoding polynomials during the authentication phase while the other is used to maintain the privacy during the revelation phase.
References
Asharov, G., Lindell, Y.: A full proof of the BGW protocol for perfectly-secure multiparty computation. J. Cryptol. 30(1), 58–151 (2017)
Beaver, D.: Efficient multiparty protocols using circuit randomization. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 420–432. Springer, Heidelberg (1992). https://doi.org/10.1007/3-540-46766-1_34
Beerliová-TrubÃniová, Z., Hirt, M.: Efficient multi-party computation with dispute control. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 305–328. Springer, Heidelberg (2006). https://doi.org/10.1007/11681878_16
Beerliová-TrubÃniová, Z., Hirt, M.: Simple and efficient perfectly-secure asynchronous MPC. In: Kurosawa, K. (ed.) ASIACRYPT 2007. LNCS, vol. 4833, pp. 376–392. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-76900-2_23
Beerliová-TrubÃniová, Z., Hirt, M.: Perfectly-secure MPC with linear communication complexity. In: Canetti, R. (ed.) TCC 2008. LNCS, vol. 4948, pp. 213–230. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78524-8_13
Beerliová-TrubÃniová, Z., Hirt, M., Nielsen, J.B.: On the theoretical gap between synchronous and asynchronous MPC protocols. In: Proceedings of the PODC, pp. 211–218. ACM (2010)
Ben-Or, M., Canetti, R., Goldreich, O.: Asynchronous secure computation. In: Proceedings of the STOC, pp. 52–61. ACM (1993)
Ben-Or, M., Goldwasser, S., Wigderson, A.: Completeness theorems for non-cryptographic fault-tolerant distributed computation (Extended Abstract). In: Proceedings of the STOC, pp. 1–10. ACM (1988)
Ben-Or, M., Kelmer, B., Rabin, T.: Asynchronous secure computations with optimal resilience (Extended Abstract). In: Proceedings of the PODC, pp. 183–192. ACM (1994)
Ben-Sasson, E., Fehr, S., Ostrovsky, R.: Near-Linear unconditionally-secure multiparty computation with a dishonest minority. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 663–680. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-32009-5_39
Canetti, R.: Studies in Secure Multiparty Computation and Applications. Ph.D. thesis, Weizmann Institute, Israel (1995)
Chaum, D., Crépeau, C., Damgård, I.: Multiparty unconditionally secure protocols (Extended Abstract). In: STOC, pp. 11–19. ACM (1988)
Chor, B., Goldwasser, S., Micali, S., Awerbuch, B.: Verifiable secret sharing and achieving simultaneity in the presence of faults. In: FOCS, pp. 383–395. IEEE Computer Society (1985)
Choudhury, A., Hirt, M., Patra, A.: Asynchronous multiparty computation with linear communication complexity. In: Afek, Y. (ed.) DISC 2013. LNCS, vol. 8205, pp. 388–402. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-41527-2_27
Choudhury, A., Patra, A.: An efficient framework for unconditionally secure multiparty computation. IEEE Trans. Inf. Theor. 63(1), 428–468 (2017)
Cramer, R., Damgård, I., Dziembowski, S., Hirt, M., Rabin, T.: Efficient multiparty computations secure against an adaptive adversary. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 311–326. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-48910-X_22
Cramer, R., Damgård, I., Maurer, U.: General secure multi-party computation from any linear secret-sharing scheme. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 316–334. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-45539-6_22
Damgård, I., Ishai, Y., Krøigaard, M.: Perfectly secure multiparty computation and the computational overhead of cryptography. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 445–465. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13190-5_23
Damgård, I., Ishai, Y., Krøigaard, M., Nielsen, J.B., Smith, A.: Scalable multiparty computation with nearly optimal work and resilience. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 241–261. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-85174-5_14
Damgård, I., Nielsen, J.B.: Scalable and unconditionally secure multiparty computation. In: Menezes, A. (ed.) CRYPTO 2007. LNCS, vol. 4622, pp. 572–590. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-74143-5_32
Fitzi, M., Garay, J., Gollakota, S., Rangan, C.P., Srinathan, K.: Round-optimal and efficient verifiable secret sharing. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 329–342. Springer, Heidelberg (2006). https://doi.org/10.1007/11681878_17
Fitzi, M., Hirt, M.: Optimally efficient multi-valued byzantine agreement. In: PODC, pp. 163–168. ACM Press (2006)
Fitzi, M., Nielsen, J.B.: On the number of synchronous rounds sufficient for authenticated Byzantine agreement. In: Keidar, I. (ed.) DISC 2009. LNCS, vol. 5805, pp. 449–463. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-04355-0_46
Franklin, M.K., Yung, M.: Communication complexity of secure computation (Extended Abstract). In: STOC, pp. 699–710. ACM (1992)
Gennaro, R., Ishai, Y., Kushilevitz, E., Rabin, T.: The round complexity of verifiable secret sharing and secure multicast. In: STOC, pp. 580–589. ACM (2001)
Goldreich, O., Micali, S., Wigderson, A.: How to play any mental game or a completeness theorem for protocols with honest majority. In: STOC, pp. 218–229. ACM (1987)
Hirt, M., Maurer, U., Przydatek, B.: Efficient secure multi-party computation. In: Okamoto, T. (ed.) ASIACRYPT 2000. LNCS, vol. 1976, pp. 143–161. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-44448-3_12
Katz, J., Koo, C.Y., Kumaresan, R.: Improving the round complexity of VSS in point-to-point networks. Inf. Comput. 207(8), 889–899 (2009)
Kushilevitz, E., Lindell, Y., Rabin, T.: Information-theoretically secure protocols and security under composition. SIAM J. Comput. 39(5), 2090–2112 (2010)
Lynch, N.A.: Distributed Algorithms. Morgan Kaufmann, San Francisco (1996)
McEliece, R.J., Sarwate, D.V.: On sharing secrets and reed-solomon codes. Commun. ACM 24(9), 583–584 (1981)
Patra, A., Choudhary, A., Pandu Rangan, C.: Efficient statistical asynchronous verifiable secret sharing and multiparty computation with optimal resilience. IACR Cryptology ePrint Archive, 2009:492 (2009)
Patra, A., Choudhury, A., Pandu Rangan, C.: Asynchronous Byzantine agreement with optimal resilience. Distrib. Comput. 27(2), 111–146 (2014)
Patra, A., Choudhury, A., Pandu Rangan, C.: Efficient asynchronous verifiable secret sharing and multiparty computation. J. Cryptology 28(1), 49–109 (2015)
Patra, A.: Error-free multi-valued broadcast and Byzantine agreement with optimal communication complexity. In: Fernà ndez Anta, A., Lipari, G., Roy, M. (eds.) OPODIS 2011. LNCS, vol. 7109, pp. 34–49. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-25873-2_4
Rabin, T., Ben-Or, M.: Verifiable secret sharing and multiparty protocols with honest majority (Extended Abstract). In: STOC, pp. 73–85. ACM (1989)
Shamir, A.: How to share a secret. Commun. ACM 22(11), 612–613 (1979)
Yao, A.C.: Protocols for secure computations. In: FOCS, pp. 160–164. IEEE Computer Society (1982)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Choudhury, A., Patra, A., Ravi, D. (2017). Round and Communication Efficient Unconditionally-Secure MPC with \(t<n/3\) in Partially Synchronous Network. In: Shikata, J. (eds) Information Theoretic Security. ICITS 2017. Lecture Notes in Computer Science(), vol 10681. Springer, Cham. https://doi.org/10.1007/978-3-319-72089-0_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-72089-0_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-72088-3
Online ISBN: 978-3-319-72089-0
eBook Packages: Computer ScienceComputer Science (R0)