  In this paper, we design secure multi-party computation (MPC) protocols in
the asynchronous communication setting with optimal resilience. Our protocols
are secure against a computationally-unbounded malicious adversary,
characterized by an adversary structure $\mathcal{Z}$, which enumerates all
possible subsets of potentially corrupt parties. Our protocols incur a
communication of $\mathcal{O}(|\mathcal{Z}|^2)$ and
$\mathcal{O}(|\mathcal{Z}|)$ bits per multiplication for perfect and
statistical security respectively. These are the first protocols with this
communication complexity, as such protocols were known only in the synchronous
communication setting (Hirt and Tschudi, ASIACRYPT 2013).
