Nearly-Optimal Consensus Tolerating Adaptive Omissions: Why is a Lot of Randomness Needed?
Mohammad T. Hajiaghayi, Dariusz R. Kowalski, Jan Olkowski
TL;DR
The paper tackles synchronous consensus with adaptive omission faults and an unbounded adversary, introducing a near-optimal randomized algorithm that runs in $O\left(\sqrt{n}\log^{2}{n}\right)$ rounds and uses $O\left(n^{2}\log^{3}{n}\right)$ bits of communication when $t=\Theta(n)$. It introduces an operative/inoperative partition and a biased-majority voting scheme, supported by a $\sqrt{n}$-group decomposition and fast inter-group communication via a dense, shallow random graph, achieving consensus whp. A new randomness–time lower bound shows that reducing time below certain thresholds necessarily increases randomness usage, via a one-round coin-flipping game analyzed with Talagrand inequalities. The paper also provides a tight randomness-time trade-off (and a broader interpolation) showing that higher randomness can substantially reduce time, with the total communication cost remaining near optimal, and it outlines several open directions for extending these ideas to other fault models and problems.
Abstract
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 $Θ(n)$. Our result is simultaneously tight for both these measures within polylogarithmic factors: due to the $Ω(n^2)$ lower bound on communication by Abraham et al. (PODC'19) and $Ω(\sqrt{n/\log n})$ 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 $Ω(\frac{n^2}{\max\{R,n\}\log n})$ 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.