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Byzantine Generals protocols enable processes to broadcast messages reliably in the presence of faulty processes. These protocols are run in a system that consists of n processes, t of which are faulty. The protocols are conducted in synchronous rounds of message exchange. It is shown that, in the absence of eavesdropping, without using cryptography, for any &egr; 0 and t = n/(3 + &egr;), there is a randomized protocol with O(log n) expected number of rounds. If cryptographic methods are allowed, then, for &egr; 0 and t = n/(2 + &egr;), there is a randomized protocol with O(log n) expected number of rounds. This is an improvement on the lower bound of t + 1 rounds required for deterministic protocols, and on a previous result of t/log n expected number of rounds for randomized noncryptographic protocols.