Multidimensional Approximate Agreement with Asynchronous Fallback

Multidimensional Approximate Agreement considers a setting of n parties, where each party holds a vector in R-D as input. The honest parties are required to obtain very close outputs in 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 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 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 l(s) corruptions when the network is synchronous, and also tolerate t(a) <= t(s) corruptions when the network is asynchronous. We provide a protocol that works as long as (D + 1) center dot t(s) + t(a) < n, and matches several existing lower bounds.