Perfect (Parallel) Broadcast in Constant Expected Rounds via Statistical VSS

We study broadcast protocols in the information-theoretic model under optimal conditions, where the number of corruptions t is at most one-third of the parties, n. While worst-case Omega(n) round broadcast protocols are known to be impossible to achieve, protocols with an expected constant number of rounds have been demonstrated since the seminal work of Feldman and Micali [STOC'88]. Communication complexity for such protocols has gradually improved over the years, reaching O(nL) plus expected O(n(4) log n) for broadcasting a message of size L bits. This paper presents a perfectly secure broadcast protocol with expected constant rounds and communication complexity of O(nL) plus expected O(n(3) log(2) n) bits. In addition, we consider the problem of parallel broadcast, where n senders, each wish to broadcast a message of size L. We show a parallel broadcast protocol with expected constant rounds and communication complexity of O(n(2)L) plus expected O(n(3) log(2) n) bits. Our protocol is optimal (up to expectation) for messages of length L is an element of Omega(n log(2) n). Our main contribution is a framework for obtaining perfectly secure broadcast with an expected constant number of rounds from a statistically secure verifiable secret sharing. Moreover, we provide a new statistically secure verifiable secret sharing where the broadcast cost per participant is reduced from O(n log n) bits to only O(poly log n) bits. All our protocols are adaptively secure.