This paper is concerned with the problem of how secure the innovation-based remote state estimation can be under linear attacks. A linear time-invariant system equipped with a smart sensor is studied. A metric based on Kullback–Leibler divergence is adopted to characterize the stealthiness of the attack. The adversary aims to maximize the state estimation error covariance while stay stealthy. The maximal performance degradations that an adversary can achieve with any linear first-order false-data injection attack under strict stealthiness for vector systems and ε-stealthiness for scalar systems are characterized. We also provide an explicit attack strategy that achieves this bound and compare this attack strategy with strategies previously proposed in the literature. Finally, some numerical examples are given to illustrate the results.