## 2 Related Work
### Secure Multi-Party Computation  
Secure multi-party computation (MPC) is a cryptographic technique designed to allow multiple participants to collaboratively perform a global computation task without revealing their private data while ensuring that the input data remain confidential throughout the process. This technique has broad applications in scenarios with high data sensitivity, such as collaborative modeling, joint analysis, and distributed data processing [29]. The primary objective of MPC is to enable secure collaboration through cryptographic protocols, ensuring that participants can only access the computation results without inferring the private inputs of other parties [30]. The fundamental concept of MPC can be formalized as a function computation problem. Consider \(n\) participants, each represented by \(P_{i}\) and holding private input \(x_{i}\). The goal is to jointly compute a public function \(f(x_{1},x_{2},\ldots,x_{n})\) while ensuring that no participant learns the inputs of others. This problem is expressed as  
\[y=f(x_{1},x_{2},\ldots,x_{n}), \tag{5}\]  
where \(y\) represents the output of the computation. In an ideal scenario, each participant knows only their own input and the final result \(y\), with no knowledge of the inputs from other participants. To achieve this, MPC typically relies on cryptographic protocols, primarily secret sharing and homomorphic encryption. In the secret sharing scheme, each participant splits their private data into multiple \"shares\" and distributes these shares to other participants. Let \(x_{i}\) be the private input of the \(i\)-th participant. Secret sharing divides \(x_{i}\) into shares \(s_{i1},s_{i2},\ldots,s_{in}\) such that  
\[x_{i}=\sum_{j=1}^{n}s_{ij}\pmod{p}, \tag{6}\]  
where \(p\) is a large prime number used for modular arithmetic to ensure security. Each participant only holds a portion of the shares from other participants. As a result, even if one participant is compromised, the full input of others cannot be reconstructed. The core steps of MPC based on secret sharing include distributed computation and result reconstruction. During distributed computation, participants perform computations on their shares. For instance, for the addition operation \(f(x_{1},x_{2})=x_{1}+x_{2}\), the computation can be performed directly on the shares as  
\[s^{\prime}_{ij}=s_{ij}+s_{kj}\pmod{p}, \tag{7}\]  
where \(s^{\prime}_{ij}\) represents the intermediate share of the addition result. For multiplication, distributed computation becomes more complex and often requires a trusted third party or additional protocols to ensure correctness. Finally, participants exchange and aggregate the shares to reconstruct the global computation result:  
\[y=\sum_{j=1}^{n}s^{\prime}_{ij}\pmod{p}. \tag{8}\]  
Apart from secret sharing, homomorphic encryption is another widely used approach for implementing MPC. Homomorphic encryption allows arithmetic operations to be performed directly on encrypted data, with the decrypted result matching the outcome of the operations performed on the plaintext. Let \(E\) and \(D\) denote the encryption and decryption functions, respectively. Homomorphic encryption satisfies the following properties:  
\[D(E(x_{1})+E(x_{2}))=x_{1}+x_{2}, \tag{9}\]  
\[D(E(x_{1})\cdot E(x_{2}))=x_{1}\cdot x_{2}. \tag{10}\]  
Using this method, participants can encrypt their private data and share them with others, allowing computations to be performed on the encrypted data. The data owner then decrypts the final result. This approach ensures higher security as the entire computation process is conducted in the encrypted domain. MPC has significant applications in domains such as finance, healthcare, and autonomous driving. In financial scenarios, MPC enables institutions to collaboratively train market prediction models, such as stock price forecasting or portfolio optimization, without sharing customer data or transaction records.