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Measures of Efficiency for Secure Multiparty Computation

Multiparty computation (MPC) is a powerful and generic cryptographic framework capable of realizing essentially any cryptographic task ("functionality") while preserving privacy and robustness. In the MPC model, a group of mutually distrustful parties seek to jointly compute a public function f of their privately held inputs x1, ..., xn, while revealing no information about those inputs beyond what is entailed by the function output(s). Some portion of these players may be corrupted, controlled by a centralized adversary who directs their (mis)behavior and seeks to gain information on private inputs or to disrupt the protocol execution. 
In this work, we examine the resource requirements of information-theoretically secure multiparty computation. In particular, we look at trade-offs involving what we call "network-model assumptions," by which we mean two aspects of the model's setup: (1) the extent to which parties are connected by secure, pairwise communication channels; and (2) the usage (as a black box) of a "broadcast" channel—a primitive which enforces consistency among views by allowing a single party to send a message to all other parties which is guaranteed to be consistent, even if the broadcaster is corrupted. 
Part I of this work focuses on the first network-model assumption, that of the graph's connectivity by pairwise channels. We show upper and lower bounds on communication complexity in the Secure Message Transmission with Public Discussion (SMT-PD) model, as a function of the threshold of corrupted parties. The SMT-PD model, which abstracts the problem of secure communication over a partially connected network, includes two components: a "public channel" and a set of simple channels. We show significant improvements in both areas: on the public channel, we reduce communication to logarithmic in terms of the message's bit-length m, where the previous best result was linear in m; on the simple channels, we reduce from O(mn) to O(mn/(n−t)), where n is the number of simple channels and t is a bound on the number of corrupted channels. Using tools from information theory, we show this amount of simple-wire communication is essentially optimal. Finally, we give an amortized solution which drastically improves communication over public channel when the protocol is repeated many times sequentially. 
Part II shifts focus to the second network-model assumption, the use of an atomic broadcast primitive. Here we show that three broadcast rounds suffice for general secure computation, while two broadcast rounds are necessary. In particular we show that even for the natural and important functionality of Weak Secret Sharing—a type of distributed commitment—there is no protocol which uses only a single broadcast and achieves better than an explicit constant probability of success (so the protocol's success cannot be a function of a security parameter).