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.

中文翻译：

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拜占庭协议的基本限制

拜占庭协议 (BA) 是一个分布式共识问题，其中 $n$ 处理器想要就 $\ell$ 位消息或值达成协议，但多达 $t$ 的处理器是不诚实或有缺陷的。尽管存在可能任意偏离设计协议的不诚实处理器，但此 BA 问题的挑战在于达成一致。BA 协议的质量主要通过使用以下三个参数来衡量： 处理器数量 $n$ 作为 $t$ 允许的函数（弹性）；回合数（回合复杂度）；以及通信比特总数（通信复杂度）。对于任何无错误 BA 协议，这三个参数的已知下限分别为 $3t + 1$、$t + 1$ 和 $\Omega(\max\{n\ell, nt\})$，保证在所有执行中正确的协议被称为无错误。在这项工作中，通过使用编码理论，我们设计了一个编码的 BA 协议（称为 COOL），当 $\ell\ geq t\log n$，同时。提议的 COOL 是一种无错误且确定性的 BA 协议，它不依赖于诸如签名、散列、身份验证和秘密共享（无签名）之类的密码技术。它对于完全控制不诚实处理器的计算无限对手是安全的（信息理论安全）。我们表明我们的结果也可以扩展到拜占庭广播的设置，又名拜占庭将军问题，诚实的处理者希望就可能不诚实的领导者发送的消息达成一致。这项工作表明，编码是实现拜占庭协议及其变体的基本限制的有效方法。