2024/08/10 by Erbes, Mose Mizrahi, Wattenhofer, Roger
#Cryptography and Security (cs.CR) #Distributed #FOS: Computer and information sciences #Parallel #and Cluster Computing (cs.DC)
paper · doi:10.48550/arxiv.2408.05495
We consider an asynchronous network of n message-sending parties, up to t of which are byzantine. We study approximate agreement, where the parties obtain approximately equal outputs in the convex hull of their inputs. In their seminal work, Abraham, Amit and Dolev [OPODIS '04] solve this problem in ℝ with the optimal resilience t < (n)/(3) with a protocol where each party reliably broadcasts a value in every iteration. This takes Θ(n2) messages per reliable broadcast, or Θ(n3) messages per iteration. In this work, we forgo reliable broadcast to achieve asynchronous approximate agreement against t < (n)/(3) faults with a quadratic communication. In a tree with the maximum degree Δ and the centroid decomposition height h, we achieve edge agreement in at most 6h + 1 rounds with O(n2) messages of size O(log Δ+ log h) per round. We do this by designing a 6-round multivalued 2-graded consensus protocol and using it to recursively reduce the task to edge agreement in a subtree with a smaller centroid decomposition height. Then, we achieve edge agreement in the infinite path ℤ, again with the help of 2-graded consensus. Finally, we show that our edge agreement protocol enables ε-agreement in ℝ in 6log2(M)/(ε) + O(log log (M)/(ε)) rounds with O(n2 log (M)/(ε)) messages and O(n2log (M)/(ε)log log (M)/(ε)) bits of communication, where M is the maximum non-byzantine input magnitude.