Multidimensional Approximate Agreement with Asynchronous Fallback

Abstract

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