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Consider a stochastic process being controlled across a communication channel. The control signal that is transmitted across the control channel can be replaced by a malicious attacker. The controller is allowed to implement any arbitrary detection algorithm to detect if an attacker is present. This work characterizes some fundamental limitations of when such an attack can be detected, and quantifies the performance degradation that an attacker that seeks to be undetected or stealthy can introduce.

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Consider a scalar linear time-invariant system whose state is being estimated by an estimator using measurements from a single sensor. The sensor may be compromised by an attacker. The attacker is allowed to replace the measurement sequence by an arbitrary sequence. When the estimator uses this sequence, its estimate is degraded in the sense that the mean square error of this estimate is higher. The estimator monitors the received data to detect if an attack is in progress. The aim of the attacker is to degrade the estimate to the maximal possible amount while remaining undetected. By defining a suitable notion of stealthiness of the attacker, we characterize the trade-off between the fundamental limits of performance degradation that an attacker can induce versus its level of stealthiness. For various information patterns that characterize the information available at every time step to the attacker, we provide information theoretic bounds on the worst mean squared error of the state estimate that is possible and provide attacks that can achieve these bounds while allowing the attacker to remain stealthy even if the estimator uses arbitrary statistical ergodicity based tests on the received data.

We study the problem of binary sequential hypothesis testing using multiple sensors with associated observation costs. An off-line randomized sensor selection strategy, in which a sensor is chosen at every time step with a given probability, is considered. The objective of this work is to find a sequential detection rule and a sensor selection probability vector such that the expected total observation cost is minimized subject to constraints on reliability and sensor usage. First, the sequential probability ratio test is shown to be the optimal sequential detection rule in this framework as well. Efficient algorithms for obtaining the optimal sensor selection probability vector are then derived. In particular, a special class of problems in which the algorithm has complexity that is linear in the number of sensors is identified. An upper bound for the average sensor usage to estimate the error incurred due to Wald's approximations is also presented. This bound can be used to set a safety margin for guaranteed satisfaction of the constraints on the sensor usage.