Solving the consensus problem requires in one way or another that the underlying system satisfies synchrony assumptions. Considering a system of n processes where up to t < n/3 may commit Byzantine failures, this paper investigates the synchrony assumptions that are required to solve consensus. It presents a corresponding necessary and sufficient condition. Such a condition is formulated with the notions of a synchrony property and property ambiguity. A synchrony property is a set of graphs, where each graph corresponds to a set of eventually synchronous links among correct processes. Intuitively, a property is ambiguous if it contains a graph whose connected components are such that it is impossible to distinguish a connected component that contains correct processes only from a connected component that contains faulty processes only. The paper connects then the notion of a synchrony property with the notion of eventual bi-source, and shows that the existence of a virtual 3[t + 1]bi-source is a necessary and sufficient condition to solve consensus in presence of up to t Byzantine processes in systems with message authentication.