Degree 2 is Complete for the Round-Complexity of Malicious MPC
We show, via a non-interactive reduction, that the existence of a secure multi-party computation (MPC) protocol for degree-2 functions implies the existence of a protocol with the same round complexity for general functions. Thus showing that when considering the round complexity of MPC, it is sufficient to consider very simple functions.
Our completeness theorem applies in various settings: information theoretic and computational, fully malicious and malicious with various types of aborts. In fact, we give a master theorem from which all individual settings follow as direct corollaries. Our basic transformation does not require any additional assumptions and incurs communication and computation blow-up which is polynomial in the number of players and in \(S,2^D\), where S, D are the circuit size and depth of the function to be computed. Using one-way functions as an additional assumption, the exponential dependence on the depth can be removed.
As a consequence, we are able to push the envelope on the state of the art in various settings of MPC, including the following cases.
3-round perfectly-secure protocol (with guaranteed output delivery) against an active adversary that corrupts less than 1/4 of the parties.
2-round statistically-secure protocol that achieves security with “selective abort” against an active adversary that corrupts less than half of the parties.
Assuming one-way functions, 2-round computationally-secure protocol that achieves security with (standard) abort against an active adversary that corrupts less than half of the parties. This gives a new and conceptually simpler proof to the recent result of Ananth et al. (Crypto 2018).
Technically, our non-interactive reduction draws from the encoding method of Applebaum, Brakerski and Tsabary (TCC 2018). We extend these methods to ones that can be meaningfully analyzed even in the presence of malicious adversaries.
We thank Yuval Ishai for helpful discussions, for providing us several useful pointers, and for sharing with us the full version of .
- 2.Ananth, P., Choudhuri, A.R., Goel, A., Jain, A.: Two round information-theoretic MPC with malicious security. Cryptology ePrint Archive, Report 2018/1078 (2018). https://eprint.iacr.org/2018/1078
- 4.Beaver, D., Micali, S., Rogaway, P.: The round complexity of secure protocols (extended abstract). In: Ortiz, H. (ed.) Proceedings of the 22nd Annual ACM Symposium on Theory of Computing, Baltimore, Maryland, USA, 13–17 May 1990, pp. 503–513. ACM (1990)Google Scholar
- 5.Ben-Or, M., Goldwasser, S., Wigderson, A.: Completeness theorems for non-cryptographic fault-tolerant distributed computation (extended abstract). In: Simon, J. (ed.) Proceedings of the 20th Annual ACM Symposium on Theory of Computing, Chicago, Illinois, USA, 2–4 May 1988, pp. 1–10. ACM (1988)Google Scholar
- 6.Benhamouda, F., Lin, H.: \(k\)-round multiparty computation from \(k\)-round oblivious transfer via garbled interactive circuits. In: Nielsen and Rijmen , pp. 500–532 (2018)Google Scholar
- 7.Chor, B., Goldwasser, S., Micali, S., Awerbuch, B.: Verifiable secret sharing and achieving simultaneity in the presence of faults (extended abstract). In: 26th Annual Symposium on Foundations of Computer Science, Portland, Oregon, USA, 21–23 October 1985, pp. 383–395. IEEE Computer Society (1985)Google Scholar
- 9.Garg, S., Srinivasan, A.: Garbled protocols and two-round MPC from bilinear maps. In: Umans, C. (ed.) 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, 15–17 October 2017, pp. 588–599. IEEE Computer Society (2017)Google Scholar
- 10.Garg, S., Srinivasan, A.: Two-round multiparty secure computation from minimal assumptions. In: Nielsen and Rijmen , pp. 468–499 (2018)Google Scholar
- 11.Gennaro, R., Ishai, Y., Kushilevitz, E., Rabin, T.: The round complexity of verifiable secret sharing and secure multicast. In: Vitter, J.S., Spirakis, P.G., Yannakakis, M. (eds.) Proceedings on 33rd Annual ACM Symposium on Theory of Computing, Heraklion, Crete, Greece, 6–8 July 2001, pp. 580–589. ACM (2001)Google Scholar
- 15.Ishai, Y., Kushilevitz, E.: Randomizing polynomials: a new representation with applications to round-efficient secure computation. In: 41st Annual Symposium on Foundations of Computer Science, FOCS 2000, Redondo Beach, California, USA, 12–14 November 2000, pp. 294–304. IEEE Computer Society (2000)Google Scholar
- 16.Ishai, Y., Kushilevitz, E.: Perfect constant-round secure computation via perfect randomizing polynomials. In: Widmayer, P., Eidenbenz, S., Triguero, F., Morales, R., Conejo, R., Hennessy, M. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 244–256. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-45465-9_22CrossRefGoogle Scholar
- 19.Paskin-Cherniavsky, A.: Secure computation with minimal interaction. Ph.D. thesis, Technion – Israel Institute of Technology (2012)Google Scholar
- 21.Rabin, T., Ben-Or, M.: Verifiable secret sharing and multiparty protocols with honest majority (extended abstract). In: Johnson, D.S. (ed.) Proceedings of the 21st Annual ACM Symposium on Theory of Computing, Seattle, Washigton, USA, 14–17 May 1989, pp. 73–85. ACM (1989)Google Scholar
- 22.Rogaway, P.: The round-complexity of secure protocols. Ph.D. thesis, MIT (1991)Google Scholar
- 24.Yao, A.C.-C.: How to generate and exchange secrets (extended abstract). In: FOCS, pp. 162–167 (1986)Google Scholar