# Secure Control of Linear Controllers Using Fully Homomorphic Encryption
## 2 Related Work  
Homomorphic encryption is a form of encryption that is capable of performing computations on encrypted data, and its research can be traced back to the idea of homomorphic encryption proposed by [PERSON] in [20]. The idea can be described as the ability to directly perform functions on the ciphertext without knowing the private key under an encryption scheme with homomorphic properties. For a long time thereafter, partially homomorphic encryption, where only one of the operations of addition and multiplication can be performed on the ciphertext, developed by leaps and bounds, e.g., RSA and ElGamal are multiplicative homomorphic encryption schemes, and Paillier is an additive homomorphic encryption scheme. But there has been no breakthrough in homomorphic encryption schemes that can support both additive and multiplicative operations. It was not until 2009 that [PERSON] [21] first proposed a fully homomorphic encryption scheme based on an ideal lattice, which allowed anyone without a private key to perform any valid computable function on the encrypted data. According to different construction ideas, fully homomorphic encryption can be roughly classified into three categories: the first category is the fully homomorphic schemes constructed based on the hard problem on an ideal lattice, which is represented by the scheme proposed by [PERSON] in [21] and its improvement [22]; the second category is the fully homomorphic scheme constructed based on the (R)LWE problem, which is represented by the scheme proposed by [PERSON] et al. [23; 24], which has improved efficiency compared to the first category, and the fully homomorphic encryption proposed in [14; 15] used in this paper belongs to this category; the third category is the fully homomorphic encryption scheme that does not require any key exchange, and this category of schemes is represented by the scheme proposed by [PERSON] et al. in [25]. As we all know, fully homomorphic encryption is more secure but inefficient compared to partially homomorphic encryption. The efficiency of fully homomorphic encryption has been greatly improved in recent years, such as GSW encryption [25] and BGV encryption [24] in the second and third categories; both of them are more efficient fully homomorphic encryption schemes, with an encryption time reaching the _ms_ level. However, this time-consuming aspect of the control system cannot be ignored. The two methods we propose improve efficiency and ensure that the security of the program is not compromised. In terms of efficiency, the time spent on encryption is completely eliminated compared to existing encryption schemes. This results in a significant increase in efficiency compared to existing fully homomorphic encryption schemes. In terms of security, the table used for encryption is constantly updated, so the scheme still maintains the high security of homomorphic encryption. The above is the research work on homomorphic encryption, and with the rise of NCSs, there has been a focus on using homomorphic encryption as a tool to improve the security of networked control systems. Homomorphic encryption was first used for NCSs in [5], where two partially homomorphic encryption schemes, RSA and ElGamal, were used in the method. Paillier was subsequently proposed to be used for NCSs. Recent studies [26; 27; 28; 29] have proposed many ways to further improve and optimize these schemes, such as maintaining stability and performance. Among them, ref. [26] proposed to update the key pair and ciphertext by simple update rules and modulo operations at each sampling cycle, which brought some inspiration and reference to this paper. In this paper, we apply in-cycle updating of plaintext-ciphertext pairs in a fully homomorphic encrypted controller scheme to improve the security of the scheme. The application of fully homomorphic encryption to NCSs has been late and rare because of efficiency issues in real-world applications. Ref.