Nearly-Optimal Consensus Tolerating Adaptive Omissions: Why is a Lot of Randomness Needed?
ACM Symposium on Principles of Distributed Computing(2024)
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(√(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.
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