  Encrypted control is a framework for the secure outsourcing of controller
computation using homomorphic encryption that allows to perform arithmetic
operations on encrypted data without decryption. In a previous study, the
security level of encrypted control systems was quantified based on the
difficulty and computation time of system identification. This study
investigates an optimal design of encrypted control systems when facing an
attack attempting to estimate a system parameter by the least squares method
from the perspective of the security level. This study proposes an optimal
$H_2$ controller that maximizes the difficulty of estimation and an equation to
determine the minimum security parameter that guarantee the security of an
encrypted control system as a solution to the design problem. The proposed
controller and security parameter are beneficial for reducing the computation
costs of an encrypted control system, while achieving the desired security
level. Furthermore, the proposed design method enables the systematic design of
encrypted control systems.
