  In this paper, we present a method to encrypt dynamic controllers that can be
implemented through most homomorphic encryption schemes, including somewhat,
leveled fully, and fully homomorphic encryption. To this end, we represent the
output of the given controller as a linear combination of a fixed number of
previous inputs and outputs. As a result, the encrypted controller involves
only a limited number of homomorphic multiplications on every encrypted data,
assuming that the output is re-encrypted and transmitted back from the
actuator. A guidance for parameter choice is also provided, ensuring that the
encrypted controller achieves predefined performance for an infinite time
horizon. Furthermore, we propose a customization of the method for
Ring-Learning With Errors (Ring-LWE) based cryptosystems, where a vector of
messages can be encrypted into a single ciphertext and operated simultaneously,
thus reducing computation and communication loads. Unlike previous results, the
proposed customization does not require extra algorithms such as rotation,
other than basic addition and multiplication. Simulation results demonstrate
the effectiveness of the proposed method.
