## 2 Related Work
### Homomorphic Encryption  
Homomorphic encryption is a cryptographic technique that allows computations to be performed on encrypted data without requiring decryption. The core concept ensures that the results of operations on ciphertexts, when decrypted, are identical to those performed directly on plaintexts, enabling computations without exposing the original data [31]. This property is a crucial safeguard for data privacy, particularly in multi-party collaborative computations and cloud computing scenarios. The fundamental principle of homomorphic encryption is based on a mathematical property, where specially designed encryption functions guarantee that operations in the ciphertext space mirror those in the plaintext space. Let the encryption and decryption functions be denoted as \(E(\cdot)\) and \(D(\cdot)\), respectively. Homomorphic encryption satisfies the following property:  
\[D(E(x_{1})\oplus E(x_{2}))=x_{1}\otimes x_{2}, \tag{11}\]  
where \(\oplus\) represents operations in the ciphertext space, and \(\otimes\) denotes corresponding operations in the plaintext space. Homomorphic encryption can be categorized based on the types of operations it supports: partially homomorphic encryption (PHE) [32], fully homomorphic encryption (FHE) [33], and somewhat homomorphic encryption (SHE) [34]. PHE typically supports either addition or multiplication. For example, the Paillier encryption scheme supports additive homomorphism. Given plaintexts \(x_{1}\) and \(x_{2}\) with ciphertexts \(E(x_{1})\) and \(E(x_{2})\), the additive homomorphism is expressed as  
\[D(E(x_{1})\cdot E(x_{2})\bmod n^{2})=x_{1}+x_{2}, \tag{12}\]  
where \(n\) is a part of the public key, and operations are performed modulo \(n^{2}\). FHE represents a significant breakthrough in homomorphic encryption as it supports arbitrary additions and multiplications on ciphertexts. This flexibility, however, comes at the cost of considerable computational complexity. For instance, in [PERSON]'s lattice-based FHE scheme, a set of special encryption functions are designed to support repeated addition and multiplication operations. To manage noise growth during these operations, FHE often requires \"bootstrapping\" to refresh ciphertexts, which significantly increases computational overhead. Specifically, in [PERSON]'s scheme, each plaintext \(x\) is encrypted as a ciphertext \(E(x)\), introducing random noise \(r\) during encryption:  
\[E(x)=x+r\pmod{q}, \tag{13}\]  
where \(q\) is a large modulus used during encryption. The encrypted data can be directly manipulated in the ciphertext space. For example, for addition:  
\[E(x_{1})+E(x_{2})=(x_{1}+r_{1})+(x_{2}+r_{2})\pmod{q}, \tag{14}\]decryption yields the correct addition result \(x_{1}+x_{2}\). Similarly, multiplication is supported, although noise accumulation during these operations necessitates bootstrapping to maintain decryption accuracy. Recent advancements have significantly improved the computational efficiency of homomorphic encryption. The CKKS (Cheon-Kim-Kim-Song) [35] scheme is a prominent example designed for approximate addition and multiplication on encrypted floating-point data. By introducing small noise and leveraging approximate decryption, CKKS enhances efficiency. Its core principle involves representing encryption operations with a polynomial \(p(x)\), such that operations on ciphertexts \(p(E(x))\) yield approximate plaintext results upon decryption:  
\[D(p(E(x_{1}))+p(E(x_{2})))\approx x_{1}+x_{2}. \tag{15}\]  
This optimization has enabled homomorphic encryption to become viable for large-scale data analysis, machine learning model training, and other computationally intensive tasks. Homomorphic encryption holds immense potential in sensitive data applications, such as finance, healthcare, and autonomous driving. In the financial sector, it enables institutions to perform encrypted customer data analysis, such as market analysis or risk prediction, without decrypting the data, thus ensuring client privacy.