An Optimal Probabilistic Protocol for Synchronous Byzantine Agreement

Broadcasting guarantees the recipient of a message that everyone else has received the same message. This guarantee no longer exists in a setting in which all communication is person-to-person and some of the people involved are untrustworthy: though he may claim to send the same message to everyone, an untrustworthy sender may send different messages to different people. In such a setting, Byzantine agreement offers the "best alternative" to broadcasting. Thus far, however, reaching Byzantine agreement has required either many rounds of communication (i.e., messages had to be sent back and forth a number of times that grew with the size of the network) or the help of some external trusted party. In this paper, for the standard communication model of synchronous networks in which each pair of processors is connected by a private communication line, we exhibit a protocol that, in probabilistic polynomial time and without relying on any external trusted party, reaches Byzantine agreement in an expected constant number of rounds and in the worst natural fault model. In fact, our protocol successfully tolerates that up to 1/3 of the processors in the network may deviate from their prescribed instructions in an arbitrary way, cooperate with each other, and perform arbitrarily long computations. Our protocol effectively demonstrates the power of randomization and zero-knowledge computation against errors. Indeed, it proves that "privacy" (a fundamental ingredient of one of our primitives), even when is not a desired goal in itself (as for the Byzantine agreement problem), can be a crucial tool for achieving correctness. Our protocol also introduces three new primitives---graded broadcast, graded verifiable secret sharing, and oblivious common coin---that are of independent interest, and may be effectively used in more practical protocols than ours.