Validated Byzantine Asynchronous Multidimensional Approximate Agreement

Consider an asynchronous system where each node begins with some point in $\mathbb{R}^m$. Given some fixed $\epsilon > 0$, we wish to have every nonfaulty node eventually output a point in $\mathbb{R}^m$, where all outputs are within distance $\epsilon$ of each other, and are within the convex hull of the original nonfaulty inputs. This problem, when some of the nodes are adversarial, is known as the ``Byzantine Asynchronous Multidimensional Approximate Agreement'' problem. Previous landmark work by Mendes et al. and Vaidya et al. presented two solutions to the problem. Both of these solutions require exponential computation by each node in each round. Furthermore, the work provides a lower bound showing that it is impossible to solve the task of approximate agreement if $n\leq (m+2)t$, and thus the protocols assume that $n>(m+2)t$. We present a Byzantine Asynchronous Multidimensional Approximate Agreement protocol in the validated setting of Cachin et al. Our protocol terminates after a logarithmic number of rounds, and requires only polynomial computation in each round. Furthermore, it is resilient to $t<\frac{n}{3}$ Byzantine nodes, which we prove to be optimal in the validated setting. In other words, working on the task in the validated setting allows us to significantly improve on previous works in several significant metrics. In addition, the techniques presented in this paper can easily yield a protocol in the original non-validated setting which requires exponential computation only in the first round, and polynomial computation in every subsequent round.