Verifiable secret sharing (VSS) is a fundamental cryptographic primitive, lying at the core of secure multi-party computation (MPC) and, as the distributed analogue of a commitment functionality, used in numerous applications. In this paper we focus on unconditionally secure VSS protocols with honest majority.In this setting it is typically assumed that parties are connected pairwise by authenticated, private channels, and that in addition they have access to a "broadcast" channel. Because broadcast cannot be simulated on a point-to-point network when a third or more of the parties are corrupt, it is impossible to construct VSS (and more generally, MPC) protocols in this setting without using a broadcast channel (or some equivalent addition to the model).A great deal of research has focused on increasing the efficiency of VSS, primarily in terms of round complexity. In this work we consider a refinement of the round complexity of VSS, by adding a measure we term broadcast complexity. We view the broadcast channel as an expensive resource and seek to minimize the number of rounds in which it is invoked as well.We construct a (linear) VSS protocol which uses the broadcast channel only twice in the sharing phase, while running in an overall constant number of rounds.
Byzantine broadcast is a distributed primitive that allows a specific party to consistently distribute a message among n parties in the presence of potential misbehavior of up to t of the parties. All known protocols implementing broadcast of an-bit message from point-to-point channels tolerating any t < n Byzantine corruptions have communication complexity at least Ω(n 2). In this paper we give cryptographically secure and information-theoretically secure protocols for t < n that communicate O(n) bits when is sufficiently large. This matches the optimal communication complexity bound for any protocol allowing to broadcast-bit messages. While broadcast protocols with the optimal communication complexity exist for t < n/2, this paper is the first to present such protocols for t < n.
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