Optimal Security Investment Problem for Secure State Estimation on Cyber-Physical Systems

Abstract

In this article, we deal with the optimal security investment problem for secure state estimation on cyber-physical systems. Toward effective secure state estimation, we aim to determine an optimal security investment that maximizes cost-effectiveness between the cost of the countermeasures and the effect of reinforcing the system resilience. Specifically, for each sensor, we decide whether to take costly security-aware action or inexpensive nonsecurity-aware action. In this article, we call the problem the optimal security investment problem, and we show that it is coNP-hard. Therefore, unless $ \mathrm{P} = \text{coNP}$, it is generally impossible to compute the optimal security investment in polynomial time. However, we also show a positive result that the optimal security investment can be solved in polynomial time if all the eigenvalues of the system matrix have geometric multiplicity 1. This article also presents the effect of improving the system resilience by adopting security-aware actions.