Optimal Communication Strategies in Networked Cyber-Physical Systems with Adversarial Elements

This paper studies optimal communication and coordination strategies in cyber-physical systems for both defender and attacker within a game-theoretic framework. We model the communication network of a cyber-physical system as a sensor network which involves one single Gaussian source observed by many sensors, subject to additive independent Gaussian observation noises. The sensors communicate with the estimator over a coherent Gaussian multiple access channel. The aim of the receiver is to reconstruct the underlying source with minimum mean squared error. The scenario of interest here is one where some of the sensors are captured by the attacker and they act as the adversary (jammer): they strive to maximize distortion. The receiver (estimator) knows the captured sensors but still cannot simply ignore them due to the multiple access channel, i.e., the outputs of all sensors are summed to generate the estimator input. We show that the ability of transmitter sensors to secretly agree on a random event, that is “coordination”, plays a key role in the analysis. Depending on the coordination capability of the sensors and the receiver, we consider three different problem settings. The first setting involves transmitters and the receiver with “coordination” capabilities. Here, all transmitters can use identical realization of randomized encoding for each transmission. In this case, the optimal strategy for the adversary sensors also exploits coordination, where they all generate the same realization of independent and identically distributed Gaussian noise. In the second setting, the transmitter sensors are restricted to use deterministic encoders, and this setting, which corresponds to a Stackelberg game, does not admit a saddle-point solution. We show that the optimal strategy for all sensors is uncoded communications where encoding functions of adversaries and transmitters are aligned in opposite directions. In the third, and last, setting where only a subset of the transmitter and/or jammer sensors can coordinate, we show that the solution radically depends on the fraction of the transmitter sensors that can coordinate. In the second half of the paper, we extend our analysis to an asymmetric scenario where we remove the assumption of identical power and noise variances for all sensors. Limiting the optimal strategies to conditionally affine mappings, we derive the optimal power scheduling over the sensors. We show that optimal power scheduling renders coordination superfluous for the attacker, when the transmitter sensors exploit coordination, as the attacker allocates all adversarial power to one sensor. In the setting where coordination is not allowed, both the attacker and the transmitter sensors distribute power among all available sensors to utilize the well-known estimation diversity in distributed settings.