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A sample identifying complexity has been introduced in the previous study tocapture an adversary's estimation error of system identification. Thecomplexity plays a crucial role in defining the security of encrypted controlsystems and designing a controller and security parameter for the systems. Thisstudy proposes a novel sample identifying complexity of encrypted controlsystems under an adversary who identifies system parameters using a leastsquares method. The proposed complexity is characterized by a controllabilityGramian and ratio of identification input variance to the noise variance. Weexamine the tightness of the proposed complexity and its changes associatedwith the Gramian and variance ratio through numerical simulations. Thesimulation results demonstrate that the proposed complexity captures a behaviorof estimation error with a sufficient level. Moreover, it confirmed that theeffect of controllability Gramian in the proposed complexity becomes larger asthe variance ratio increases.

Sample Identifying Complexity of Encrypted Control Systems Under Least Squares Identification