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We consider the problem of designing an efficient and robust distributed random number generator for peer-to-peer systems that is easy to implement and works even if all communication channels are public. A robust random number generator is crucial for avoiding adversarial join-leave attacks on peer-to-peer overlay networks. We show that our new generator together with a light-weight rule recently proposed in [4] for keeping peers well-distributed can keep various structured overlay networks in a robust state even under a constant fraction of adversarial peers.