Designing Optimal Key Lengths and Control Laws for Encrypted Control Systems Based on Sample Identifying Complexity and Deciphering Time

Abstract

In the state-of-the-art literature on cryptography and control theory, there has been no systematic methodology of constructing cyber–physical systems that can achieve the desired control performance while being protected against eavesdropping attacks. In this article, we tackle this challenging problem. We first propose two novel notions referred to as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sample identifying complexity</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sample deciphering time</i> in an encrypted control framework. The former explicitly captures the relation between the dynamical characteristics of control systems and the level of identifiability of the systems while the latter shows the relation between the computation time for the identification and the key length of a cryptosystem. Based on these two tractable new notions, we propose a systematic method for designing both of an optimal key length to prevent system identification with a given precision within a given life span of systems and of an optimal controller to maximize both of the control performance and the difficulty of the identification. The efficiency of the proposed method in terms of security level and realtime-ness is investigated through numerical simulations. To the best of our knowledge, this article first connects the relationship between the security of cryptography and dynamical systems from a control-theoretic perspective.