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A new randomized Byzantine agreement algorithm is presented. This algorithm operates in a synchronous system of n processors, at most t of which can fail. The algorithm reaches agreement in 0(t/log n) expected rounds and O(n2tf/log n) expected message bits independent of the distribution of processor failures. This performance is further improved to a constant expected number of rounds and O(n2) message bits if the distribution of processor failures is assumed to be uniform. In either event, the algorithm improves on the known lower bound on rounds for deterministic algorithms. Some other advantages of the algorithm are that it requires no cryptographic techniques, that the amount of local computation is small, and that the expected number of random bits used per processor is only one. It is argued that in many practical applications of Byzantine agreement, the randomized algorithm of this paper achieves superior performance.