On Achieving the "Best of Both Worlds" in Secure Multiparty Computation

Two settings are typically considered for secure multiparty computation, depending on whether or not a majority of the parties are assumed to be honest. Protocols designed under this assumption provide "full security" (and, in particular, guarantee output delivery and fairness) when this assumption is correct; however, if half or more of the parties are dishonest their security is completely compromised. On the other hand, protocols tolerating arbitrarily-many faults do not provide fairness or guaranteed output delivery even if only a single party is dishonest. It is natural to wonder whether it is possible to achieve the "best of both worlds": namely, a single protocol that simultaneously achieves the best possible security in both the above settings. Isha, et al. (Crypto 2006) recently addressed this question, and ruled out constant-round protocols of this type. As our main result, we completely settle the question by ruling out protocols using any (expected) polynomial number of rounds. Given this stark negative result, we then ask what can be achieved if we are willing to assume simultaneous message transmission (or, equivalently, a non-rushing adversary). In this setting, we show that impossibility still holds for logarithmic-round protocols. We also show, for any polynomial p, a protocol (whose round complexity depends on P) that can be simulated to within closeness O(1/P).