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

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(√(n)log^2 n) rounds and sends O(n^2log^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(√(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(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 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∈Õ(n^3/2) there exists an algorithm solving consensus in Õ(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.