# A Compact Multi-Identity Fully Homomorphic Encryption Scheme Without Fresh Ciphertexts
## 1 Introduction  
With the advancement of technology and the popularization of digitization, people are in a data-centric era, where the volume of data is growing rapidly and the circulation and interaction between data have become more frequent. Data owners, due to their own resource constraints, usually do not store or process the data directly, but entrust it to non-trusted third parties for storage or processing. We refer to this data processing model as the outsourced computing model. The rapid development of outsourced computing has brought convenience for people to solve more complex problems, but also introduced many new issues, privacy security being the first one. How to solve the contradiction between outsourced data storage and computation and privacy security is a new issue emerging from the outsourced computing model. Homomorphic encryption technology can perfectly solve the conflict between outsourced data storage and computation and privacy security in the outsourced computing model. In 1978, [PERSON] et al. [1] first introduced the concept of homomorphic encryption, i.e., encryption methods that allow data to be encrypted in an encrypted state for specific computations. Based on different mathematical problems, cryptographers have constructed various homomorphic encryption schemes, such as the RSA scheme [2] based on the problem of integer factorization, the ElGamal scheme [3] based on the discrete logarithm problem, and the Paillier scheme [4] based on the composite residuosity assumption. However, with the rapid development of quantum computers, the limitations of these traditional number-theoretic homomorphic encryption schemes in resisting quantum algorithm attacks have been greatly amplified. Consequently, ideal lattice problems, which can withstand quantum attacks, have attracted the attention of researchers. In 2009, [PERSON] [5] constructed the first fully homomorphic encryption scheme based on ideal lattices. Since then, cryptographers have engaged in extensive and in-depth research on constructing homomorphic encryption schemes based on ideal lattices. Homomorphic encryption has different branches based on different foundational schemes. According to these foundational schemes, homomorphic encryption is divided into four generations: the first generation represented by GSW [6], the second generation represented by BGV [7], and the third generation represented by TFHE [8], Finally, there is the fourth generation of homomorphic encryption schemes, represented by CKKS [9]. Fully homomorphic encryption extends the functionality of traditional public key encryption systems, but like traditional public key encryption, it requires a complex certificate management system. In large-scale networks, managing a vast number of public keys and certificates becomes increasingly complex, affecting the system's efficiency. Identity-Based Encryption (IBE) schemes, on the other hand, can generate public keys and public parameters directly from a user's unique identity, eliminating the need for public key certificates for authentication. In an IBE system, a trusted Private Key Generator (PKG) creates the user's secret key and securely sends it to the user. The distribution mechanism is as follows: the user submits their identity information to the PKG, which generates the master key and the corresponding user private key based on the user's identity. These keys are then securely distributed to the user via encrypted transmission or similar methods. This approach avoids the overhead of public key certificates and allows for more efficient key management. The integration of identity-based mechanisms with cryptographic schemes has been explored in various contexts to address application-specific challenges. For example, [PERSON] and [PERSON] [10] proposed a keyed hash function based on Latin squares and error-correcting codes, aiming to enhance user authentication in smart home environments.