# Secure Control of Linear Controllers Using Fully Homomorphic Encryption
###### Abstract  
In actual operation, there are security risks to the data of the network control system, mainly in the form of possible eavesdropping of signals in the transmission channel and parameters in the controller leading to data leakage. In this paper, we propose a scheme for encrypting linear controllers using fully homomorphic encryption, which effectively removes these security risks and substantially improves the security of networked control systems. Meanwhile, this paper uses precomputation to handle data encryption, which eliminates the encryption time and solves the drawback of fully homomorphic encryption that it is difficult to apply due to the efficiency problem. Compared to previous schemes with precomputation, for the first time, we propose two methods to mitigate the problem of the slight security degradation caused by precomputation, which makes our scheme more secure. Finally, we provide numerical simulation results to support our scheme, and the data show that the encrypted controller achieves normal control and improves safety and efficiency. encrypted controller; networked control system; fully homomorphic; BFV encryption 2023133071.1233071.1233071.133071.  
the controller and transmitted to the actuator to perform decryption. However, in this process, this conventional communication encryption not only requires two encryptions and decryptions, but the data in the controller are in plaintext as well, and thus does not protect this part of the data from eavesdropping. [PERSON] et al. [5] proposed the concept of encrypted controller in 2015 to make up for the deficiency of communication encryption. The ideal encryption controller can directly calculate the encrypted data. As shown in Figure 1b, the data exist in ciphertext throughout the network control loop, and a malicious attacker would have no way to get at it. In this way, the encryption controller greatly improves the security of the data in the NCS compared to the NCS without the encrypted controller. However, the concrete encryption controller scheme proposed by [PERSON] et al. [5] was implemented by RSA [6] and ElGamal [7] partially homomorphic encryption. Subsequent research studies on encrypted controllers have also mostly been conducted on partially homomorphic encryption controllers, including a series of research studies based on encrypted controllers [8; 9; 10; 11] with Paillier [12] encryption and ElGamal encryption. Homomorphic encryption allows direct operations on ciphertext. Partially homomorphic encryption is homomorphic encryption that supports only one operation in addition or multiplication. Therefore, it may not be possible to complete the operation in the encrypted controller using partially homomorphic encryption. This leads to the fact that operations inside the encrypted controller are often guaranteed at the expense of data security. Therefore, can we implement the encrypted controller using a fully homomorphic encryption scheme? For the first time, a fully homomorphic encrypted controller was shown by [PERSON] et al. [13]. But the scheme faced two problems. On the one hand, they suggested running multiple controllers and a catch-up mechanism to solve the problem because the controller could not function while the bootstrapping of encrypted variables was performed. On the other hand, the finite life of the encrypted variable was reduced with the operation. To solve this problem, a tree-based algorithm was introduced. But this also led to an increased complexity of the control systems. In this paper, we propose a encrypted controller scheme that is more secure and efficient while less complex. The highlights of this paper are as follows. Firstly, we adopt a fully homomorphic encryption scheme [14; 15] proposed by [PERSON], [PERSON], and [PERSON] to encrypt the controller in this paper, which we usually refer to as BFV encryption. Secondly, we use the method of generating tables by precomputation [16] to improve encryption efficiency. Thirdly, we propose to continuously update the table to improve security.