Fundamental Limits of Byzantine Agreement

Byzantine agreement (BA) is a distributed consensus problem where $n$ processors want to reach agreement on an $\ell$-bit message or value, but up to $t$ of the processors are dishonest or faulty. The challenge of this BA problem lies in achieving agreement despite the presence of dishonest processors who may arbitrarily deviate from the designed protocol. The quality of a BA protocol is measured primarily by using the following three parameters: the number of processors $n$ as a function of $t$ allowed (resilience); the number of rounds (round complexity); and the total number of communication bits (communication complexity). For any error-free BA protocol, the known lower bounds on those three parameters are $3t + 1$, $t + 1$ and $\Omega(\max\{n\ell, nt\})$, respectively, where a protocol that is guaranteed to be correct in all executions is said to be error free. In this work, by using coding theory we design a coded BA protocol (termed as COOL) that achieves consensus on an $\ell$-bit message with optimal resilience, asymptotically optimal round complexity, and asymptotically optimal communication complexity when $\ell \geq t\log n$, simultaneously. The proposed COOL is an error-free and deterministic BA protocol that does not rely on cryptographic technique such as signatures, hashing, authentication and secret sharing (signature free). It is secure against computationally unbounded adversary who takes full control over the dishonest processors (information-theoretic secure). We show that our results can also be extended to the setting of Byzantine broadcast, aka Byzantine generals problem, where the honest processors want to agree on the message sent by a leader who is potentially dishonest. This work reveals that coding is an effective approach for achieving the fundamental limits of Byzantine agreement and its variants.