In this paper, we characterize and analyze the set of strategic stealthy false-data injection attacks on discrete-time linear systems. In particular, the threat scenarios tackled in the paper consider adversaries that aim at deteriorating the system's performance by maximizing the corresponding quadratic cost function, while remaining stealthy with respect to anomaly detectors. As opposed to other work in the literature, the effect of the adversary's actions on the anomaly detector's output is not constrained to be zero at all times. Moreover, scenarios where the adversary has uncertain model knowledge are also addressed. The set of strategic attack policies is formulated as a non-convex constrained optimization problem, leading to a sensitivity metric denoted as the output-to-output ℓ2-gain. Using the framework of dissipative systems, the output-to-output gain is computed through an equivalent convex optimization problem. Additionally, we derive necessary and sufficient conditions for the output-to-output gain to be unbounded, with and without model uncertainties, which are tightly related to the invariant zeros of the system.