Nearly-Optimal Consensus Tolerating Adaptive Omissions: Why is a Lot of Randomness Needed?

• URL: http://arxiv.org/abs/2405.04762v2
• Date: Fri, 24 May 2024 04:59:32 GMT
• Title: Nearly-Optimal Consensus Tolerating Adaptive Omissions: Why is a Lot of Randomness Needed?
• Authors: Mohammad T. Hajiaghayi, Dariusz R. Kowalski, Jan Olkowski,
• Abstract summary: 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. We design a randomized algorithm that works in $O(sqrtnlog2 n)$ rounds and sends $O(n2log3 n)$ communication bits. We prove that no MC algorithm can work in less than $Omega(fracn2maxR,nlog n)$ rounds if it uses less than $O(R)$ calls to
• Score: 4.465134753953128
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $\Theta(n)$. 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.

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