Achieving independence in logarithmic number of rounds

Simultaneous broadcast [CGMA] is a fundamental tool in designing secure protocols for fault tolerant distributed computing. A system that supports it enables n processes to globally commit to independently chosen values (a significantly harder task than mere agreement). It is also a basic building block in a recent "completeness" theorem of [GMW2]. In this paper we present a new protocol for simultaneous broadcast. Building upon past work, we introduce a novel method of concurrently alternating and interleaving n executions of verifiable secret sharing protocols. This approach greatly improves the time complexity (number of communication rounds) of simultaneous broadcast. Previous protocols (combination of [CGMA] and [GMW]) required the complete serialization of the n verifiable secret sharings, resulting in n(n) communication rounds. Our protocol is constructive, and requires only logn + log log n serial executions of verifiable secret sharings. It preserves maximum fault tolerance (t < n/2 faults), and polynomial resource bounds (internal computation and communication bits). The same improvement applies to the general simulation in [GMW2]. In light of its improved performance, it is significant that our our protocol has a fairly simple correctness proof. In the slippery business of distributed cryptographic protocols, simpler proofs are important.