Abstract-To solve consensus, distributed systems have to be equipped with oracles such as a failure detector, a leader capability, or a random number generator. For each oracle, various consensus algorithms have been devised. Some of these algorithms are indulgent toward their oracle in the sense that they never violate consensus safety, no matter how the underlying oracle behaves. This paper presents a simple and generic indulgent consensus algorithm that can be instantiated with any specific oracle and be as efficient as any ad hoc consensus algorithm initially devised with that oracle in mind. The key to combining genericity and efficiency is to factor out the information structure of indulgent consensus executions within a new distributed abstraction, which we call "Lambda." Interestingly, identifying this information structure also promotes a fine-grained study of the inherent complexity of indulgent consensus. We show that instantiations of our generic algorithm with specific oracles, or combinations of them, match lower bounds on oracle-efficiency, zero-degradation, and one-step-decision. We show, however, that no leader or failure detector-based consensus algorithm can be, at the same time, zero-degrading and configuration-efficient. Moreover, we show that leader-based consensus algorithms that are oracle-efficient are inherently zero-degrading, but some failure detector-based consensus algorithms can be both oracle-efficient and configuration-efficient. These results highlight some of the fundamental trade offs underlying each oracle.

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This paper introduces a deterministic Byzantine consensus algorithm that relies on a new weak coordinator. As opposed to previous algorithms that cannot terminate in the presence of a faulty or slow coordinator, our algorithm can terminate even when its coordinator is faulty, hence the name weak coordinator. The key idea is to allow processes to complete asynchronous rounds as soon as they receive a threshold of messages, instead of having to wait for a message from a coordinator that may be slow.The resulting algorithm assumes partial synchrony, is resilience optimal, time optimal and does not need signatures. Our presentation *