Transmission scheduling for privacy-optimal encryption against eavesdropping attacks on remote state estimation

In remote state estimation systems, the state being estimated may be wiretapped by eavesdropping communications from the sensor to the remote estimator. A straightforward remedy for protecting the state privacy is to encrypt all the sensory data before transmission, which however may not be affordable for an energy-constrained sensor. In this paper, we address the optimal sensor transmission scheduling problem in order to protect the state privacy in an energy-efficient manner. The sensor can choose either to transmit data in plaintext or encrypted text, or to stay in silence without transmitting. We formulate the problem as the one of maximizing the eavesdropper's estimation error of the system state, while guaranteeing a certain level of the remote estimation performance under a given energy budget for the sensor in the finite time horizon. Then, we transform the problem into an optimal scheduling problem under any fixed transmission times and an energy allocation problem, and propose algorithms to solve each of them. We theoretically prove that, provided that a certain level of remote estimation performance is guaranteed, the optimal sensor schedule is to group instants of encrypted transmissions and silent states as tighter as possible and the optimal energy allocation tends to arrange more energy to transmit encrypted text. Furthermore, we extend our results over finite time horizon to infinite periodic time horizon. We construct a set of monotonic sequences to approximate the estimator's performance constraint. Then, we design an algorithm to obtain the suboptimal schedule Finally, numerical results are presented to demonstrate the performance of the proposed methods.