## 7 Conclusions and Discussion  
We propose a simple and yet effective secure multiparty computation scheme for linear regression in a distributed environment. Our approach leverages the inherent mathematical properties of linear regression to allow participating parties to transmit only aggregated data to the server or other parties. This strategic approach enhances security and privacy protection without the need for encryption, especially homomorphic encryption. Securityand privacy play important roles in collaborative learning. By allowing data owners to protect their valuable data while still contributing to model improvement, our approach encourages greater data sharing and collaboration among participants.  
Our proposed scheme offers the flexibility of incremental dataset management within the linear regression models. The seamless integration of new datasets into the system facilitates the acquisition of updated and more precise model parameters, all without the need for computation from the beginning. Additionally, the secure removal of user datasets is supported, which is required by laws and regulations when a party departs from the group and requests the deletion of his data.  
The security of our proposed collaborated computation scheme for linear regression does not depend on computationally hard problems, such as factoring large integer and solving discrete logarithm in large finite group. These computationally hard problems usually require heavy computation to achieve a certain level of security. Furthermore, these computationally hard problems may be broken if the attacker has enough computing power, such as by using quantum computers. Due to the lack of sufficient information, we proved that the curious party cannot obtain exact values of the other party's data, even with unlimited computing power.  
Our proposed method can also accommodate the inclusion of varying weights in collaborative learning. This feature holds particular promise in scenarios requiring customized machine learning, where all parties share commonalities but possess unique characteristics. A prime example is handwritten character recognition for smartphone input processing, for which personalized models are imperative. Our approach provides a customization process that enables users to build custom models by adding additional training datasets of their own to the generic model.  
In the design of our linear regression protocol, we make no assumptions about how the dataset is collected and distributed within each site. In particular, the dataset collected at each site may contain different features. When applying our protocol, users must agree in advance on the features used in the model's calculations. Our approach belongs to horizontal federated learning as classified by [PERSON] et al. citeYLCT19. In many applications, such as medical applications, the medical data collected by each hospital must contain a fixed set of common features that can be used to develop new drugs or treatments. Therefore, this is not a serious limitation.  
A.G.: contributed significantly to the analysis and manuscript preparation; C.-H.L.: contributed to the review and editing; P.-W.C.: contributed to the analysis with constructive discussions. All authors have read and agreed to the published version of the manuscript.  
The research of the first author is supported in part by the MOST project 107-2218-E-110-017-MY3, and in part by NTNU special grant for new faculties. The research of the second author is supported in part by the MOST project 108-2622-E-110-009-CC3. The research of the third author is supported in part by the MOST project 110-2221-E-003-002-MY3.  
Not applicable.  
Data Availability Statement: Data is randomly generated and contained within the article.  
We would like to thank [PERSON] for his contribution to the design of the experiments.  
The authors declare no conflicts of interest.