From Crash to Byzantine Consensus with 2 f + 1 Processes ∗

Introduction Consensus is an important distributed computing problem that consists in making a set of processes agree on one of the values that each one of them proposes. Consensus in the asynchronous Byzantine message-passing model (the model we consider in the abstract) has been shown to require n ≥ 3f + 1 processes to be solvable in several variations of the basic system model, where f is the maximum number of faulty processes. Reducing the ratio n/f is important both theoretically, since achieving lower bounds has been always a goal in distributed computing, and in practice, as reducing the number of processes reduces the cost of a real system. Recently a few solutions to implement Byzantine fault-tolerant state machine replication, which involves solving consensus, with only 2f + 1 replicas have appeared [3, 2]. This reduction from 3f + 1 to 2f + 1 is possible by extending the system model with a trusted/trustworthy component that constrains the power of faulty processes to have certain behaviors. These components have been called wormholes [6]. State machine replication consists in replicating a service in a set of n servers, f of which may be faulty. Correia et al. use a wormhole called TTCB to help define an order for the execution of the clients’ requests with only 2f + 1 servers [3]. More recently, Chun et al. used an attested append-only memory (A2M) abstraction with the same purpose [2]. The main objective of this work is to contribute to a better understanding of the problem of consensus with only 2f + 1 processes. Despite these important results, the problem of solving asynchronous Byzantine consensus with only 2f + 1 processes is still far from being well understood. There are several reasons for this state of affairs: the related works we cited are only two and very recent; they solve consensus but have the solution for this problem submerged in the complications of a larger problem (state machine replication); they are based on special-purpose, reasonably complex, abstractions/wormholes (TTCB, A2M)