# A Compact Multi-Identity Fully Homomorphic Encryption Scheme Without Fresh Ciphertexts
## 1 Introduction
Their work demonstrates how lightweight cryptographic primitives can address identity-related security challenges in IoT applications. Similarly, our work leverages the advantages of IBE to reduce the public key overhead in homomorphic encryption systems. Identity-Based Fully Homomorphic Encryption (IBFHE) combines the strengths of both homomorphic encryption and identity-based encryption, enabling access control and homomorphic operations on identity-based ciphertexts while also facilitating efficient key management. In 2013, [PERSON] et al. [6] introduced the method of approximate eigenvector and constructed a FHE scheme, known as the GSW scheme. The method of approximate eigenvector can transform any IBE scheme that meets specific conditions into an IBFHE scheme. Unfortunately, this IBFHE scheme only allows homomorphic operations on ciphertexts under the same identity, limiting its ability to address outsourced computation scenarios.In order to resolve this issue, [PERSON] et al. [11] in 2015 extended the GSW scheme to support multiple identities by introducing a Mask System (MS). Through the MS, they expanded \"fresh\" ciphertext matrices encrypted under different identities into multi-identity ciphertext matrices. These extended ciphertext matrices could then undergo homomorphic operations, ultimately constructing the first selectively secure multi-identity fully homomorphic encryption (MIBFHE) scheme in the oracle model. However, the ciphertext expansion process in this scheme was complex, and the noise growth occurred too rapidly. In 2017, [PERSON] et al. [12] dynamically combined Multi-Key Fully Homomorphic Encryption (MKFHE) with IBE to construct a Multi-Identity Based Fully Homomorphic Encryption scheme. However, the ciphertext expansion in this scheme depended on the number of ciphertexts, making it less compact. That same year, [PERSON] et al. [13] leveraged the MP12 trapdoor by [PERSON] [14] along with the MS to design a more efficient MIBFHE scheme. In 2019, [PERSON] et al. [15] applied the MP12 trapdoor to improve the CHKP10 [16] identity-based scheme, upon which they constructed an MIBFHE scheme. However, since the size of the public key and the length of the ID in the CHKP10 scheme grow linearly in proportion, this greatly affects the storage space and computational efficiency of the MIBFHE system. In the same year, [PERSON] et al. [17] optimized the ABB10 [18] scheme using the MP12 trapdoor, and based on this optimization, they built a highly efficient MIBFHE scheme. The optimized scheme significantly improved the size of the master secret key, identity public key, and ciphertext. In 2021, [PERSON] et al. [19] utilized a compressible ciphertext extension technique to construct a compact MIBFHE scheme, which, to some extent, addressed the issue of low bandwidth efficiency in multi-identity settings. In 2022, [PERSON] et al. [20] proposed a multi-hop MIBFHE scheme utilizing the ciphertext extension technique from PS16 [21], thereby expanding the functionality of MIBFHE schemes. That same year, [PERSON] et al. [22] proposed an improved lattice-based MIBFHE scheme by combining the MP12 trapdoor with the Dual Regev algorithm to construct an enhanced IBE scheme. Based on this, they used MS to design a highly efficient MIBFHE scheme. Compared to similar schemes, their approach significantly reduced both lattice dimensions and ciphertext size. Although the aforementioned MIBFHE schemes have achieved varying degrees of optimization, the efficiency of MIBFHE still faces bottlenecks. The core issue lies in the fact that the trapdoors used in these schemes are outdated compared to the theoretical advancements in trapdoor research. These trapdoors rely on computationally expensive matrix inversion operations, and the preimage sampling algorithms require high-precision real-number orthogonal iteration during the sampling process. This results in large public parameter sizes and a lack of compactness in the schemes.