We study the problem of reaching agreement in a synchronous distributed system by $n$ autonomous parties, when the communication links from/to faulty parties can omit messages. The faulty parties are selected and controlled by an adaptive, full-information, computationally unbounded adversary. We design a randomized algorithm that works in $O(\sqrt{n}\log^2 n)$ rounds and sends $O(n^2\log^3 n)$ communication bits, where the number of faulty parties is $\Theta(n)$. Our result is simultaneously tight for both these measures within polylogarithmic factors: due to the $\Omega(n^2)$ lower bound on communication by Abraham et al. (PODC'19) and $\Omega(\sqrt{n/\log n})$ lower bound on the number of rounds by Bar-Joseph and Ben-Or (PODC'98). We also quantify how much randomness is necessary and sufficient to reduce time complexity to a certain value, while keeping the communication complexity (nearly) optimal. We prove that no MC algorithm can work in less than $\Omega(\frac{n^2}{\max\{R,n\}\log n})$ rounds if it uses less than $O(R)$ calls to a random source, assuming a constant fraction of faulty parties. This can be contrasted with a long line of work on consensus against an {\em adversary limited to polynomial computation time}, thus unable to break cryptographic primitives, culminating in a work by Ghinea et al. (EUROCRYPT'22), where an optimal $O(r)$-round solution with probability $1-(cr)^{-r}$ is given. Our lower bound strictly separates these two regimes, by excluding such results if the adversary is computationally unbounded. On the upper bound side, we show that for $R\in\tilde{O}(n^{3/2})$ there exists an algorithm solving consensus in $\tilde{O}(\frac{n^2}{R})$ rounds with high probability, where tilde notation hides a polylogarithmic factor. The communication complexity of the algorithm does not depend on the amount of randomness $R$ and stays optimal within polylogarithmic factor.

翻译：我们研究在同步分布式系统中，由$n$个自治方达成一致的问题，其中故障方的通信链路可能遗漏消息。故障方由自适应、全信息、计算能力无限的对手选择和操控。我们设计了一种随机算法，该算法在$O(\sqrt{n}\log^2 n)$轮内完成工作，并发送$O(n^2\log^3 n)$比特通信量，其中故障方数量为$\Theta(n)$。我们的结果在这两个指标上同时达到了对数因子内的紧致性：根据Abraham等人(PODC'19)的$\Omega(n^2)$通信下界，以及Bar-Joseph和Ben-Or(PODC'98)的$\Omega(\sqrt{n/\log n})$轮数下界。我们还量化了将时间复杂度降低到特定值所需的最小随机量，同时保持通信复杂度（近乎）最优。我们证明，如果MC算法使用的随机源调用次数少于$O(R)$，且假设故障方比例恒定，则该算法不能在少于$\Omega(\frac{n^2}{\max\{R,n\}\log n})$轮内完成工作。这与针对计算时间有限的对手（无法破解密码原语）的一致性研究工作形成对比，这些研究的成果以Ghinea等人(EUROCRYPT'22)的工作达到顶峰，他们给出了一个概率为$1-(cr)^{-r}$的最优$O(r)$轮解决方案。我们的下界严格区分了这两种情况，排除了对手计算能力无限时此类结果的可能性。在上界方面，我们证明对于$R\in\tilde{O}(n^{3/2})$，存在一种算法能在$\tilde{O}(\frac{n^2}{R})$轮内以高概率解决一致性问题，其中波浪号表示隐藏了对数因子。该算法的通信复杂度不依赖于随机量$R$，并在对数因子内保持最优。