Multidimensional Approximate Agreement with Asynchronous Fallback

Diana Ghinea, Chen-Da Liu-Zhang, Roger Wattenhofer

Abstract: Multidimensional Approximate Agreement considers a setting of n parties, where each party holds a vector in ℝD as input. The honest parties are required to obtain very close outputs in ℝD that lie inside the convex hull of their inputs. Existing Multidimensional Approximate Agreement protocols achieve resilience against ts < n / (D + 1) corruptions under a synchronous network where messages are delivered within some time Δ, but become completely insecure as soon as a single message sent by an honest party is further delayed. On the other hand, asynchronous solutions do not rely on any delay upper bound, but only achieve resilience up to ta < n / (D + 2) corruptions. We investigate the feasibility of achieving Multidimensional Approximate Agreement protocols that achieve simultaneously guarantees in both network settings: We want to tolerate ts corruptions when the network is synchronous, and also tolerate ta ≤ ts corruptions when the network is asynchronous. We provide a protocol that works as long as (D + 1) · ts + ta < n, and matches several existing lower bounds.