Need for Controllers Having Integer Coefficients in Homomorphically Encrypted Dynamic System

Encryption of feedback controller based on homomorphic encryption techniques allows control operation on encrypted sensor measurements, to protect data secrecy of networked control systems. In this paper, we first revisit the fact that an encrypted dynamic controller based on partially homomorphic cryptosystems cannot operate for infinite time horizon in general due to the problem of the size of encrypted controller state. Then, it is highlighted that an encrypted LTI controller can operate for infinite time horizon when the denominator of its transfer function is a monic polynomial with integer coefficients. In this case, it does not utilize bootstrapping techniques from fully homomorphic encryption for resizing the controller state, and it does not assume the decryption of controller state from time to time. In this respect, we discuss potential benefits of using PID controllers and FIR filters. Then, we introduce a way how to approximately convert controllers having non-integer coefficients to have integer coefficients without losing stability.