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 (RLWE)-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.

翻译：本文提出一种可加密动态控制器的方法，该方法可通过大多数同态加密方案实现，包括部分同态加密、层级全同态加密和全同态加密。为此，我们将给定控制器的输出表示为有限个历史输入与输出的线性组合。结果表明，在假设输出被重新加密并从执行器传回的前提下，加密控制器对每个加密数据仅需进行有限次同态乘法运算。本文还提供了参数选择指导原则，确保加密控制器能在无限时间范围内达到预设性能指标。进一步地，我们针对基于环上容错学习（RLWE）的密码系统提出了定制化方案，该方案可将消息向量加密为单个密文并进行并行运算，从而降低计算与通信开销。与已有成果不同，所提出的定制化方案除基本加法和乘法运算外，无需额外算法（如旋转操作）。仿真结果验证了所提方法的有效性。