Bounding the communication complexity of fault-tolerant common coin tossing

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Protocols for tossing a common coin play a key role in the vast majority of implementations of consensus. Even though the common coins in the literature are usually fair (they have an equal chance of landing heads or tails), we focus on the problem of implementing a biased common coin such that the probability of landing heads is \(p \in [0,1]\) . Even though biased common coins can be implemented using fair common coins, we show that this can require significant inter-party communication. In fact, we show that there is no bound on the number of messages needed to generate a common coin of bias p in a way that tolerates even one malicious agent, even if we restrict p to an arbitrary infinite subset of \([0,1]\) (e.g., \(\lbrace 1/2^n : n \in \lbrace 0,1,2,\ldots \rbrace \rbrace\) ) and assume that the system is synchronous. By way of contrast, if we do not require the protocol to tolerate a faulty agent, we can do this. Thus, the cause of the message complexity is the requirement of fault tolerance.