## 1 Introduction  
Machine learning attempts to build models based on training data to make predictions or decisions without being explicitly programmed. It usually requires a large amount of training data to build a useful model. In many practical applications, these data may not belong to one organization but rather may be distributed across multiple sites. For example, every hospital collects the medical data of its patients. The data collected in one hospital are often insufficient to obtain a good model for developing new drugs or treatments for some specific disease. Therefore, sharing data is very important in machine learning when the data are distributed across many sites. Like medical data, almost all data contain private and sensitive personal information. The release of personal data in many countries in the world is restricted by laws and regulations. For example, since 2018, the European Data Protection Regulation has been applicable to all member states to harmonize data privacy laws across the European Union [1]. When sharing these data, security and/or personal privacy concerns must be properly addressed. Another reason for limiting data sharing is that collecting these data usually requires a lot of time and effort. Data owners usually view the data they collect as an asset. They will be more willing to share the collected data with others only if it is legal and mutually beneficial. In this paper, we consider the security and privacy preservation of a specific machine learning problem: linear regression. A linear regression models the relationship between variables by fitting a line to the observed data [2; 3]. It is the most basic model in machine learning and has many applications [4]. We propose a simple yet effective method to collaboratively build linear regression models without disclosing the underlying raw data. Furthermore, our novel collaborative learning scheme enables the removal of a user's dataset without requiring access to the user's original data, ensuring compliance with privacy protection regulations. Secure collaborative machine learning falls within the realm of secure multiparty computation, a well-explored area in the field of cryptography. In secure multiparty computing, no data can be sent directly to the other parties for security and privacy reasons. There are many techniques to achieve secrecy in multiparty computation. General method for secure multiparty computation, such as [PERSON]'s garbled circuits, may not be efficiently implementable. By exploiting the mathematical structure of linear regression, we achieve efficient secure construction of linear regression models. The primary objective in secure computation is to establish a rigorous framework that ensures an absolute lack of knowledge concerning the private data of the other parties involved. Although achieving this state of zero knowledge is highly desirable, it typically necessitates extensive computational resources, as noted by [PERSON] et al. [5]. Moreover, it is important to note that zero-knowledge guarantees are valid only against polynomial-time adversaries, meaning these schemes offer only computational security. The security of the proposed scheme does not depend on computationally hard problems, such as factoring large integers or solving discrete logarithms in a large group. We prove that the curious party does not have enough information to deduce the exact values of the other party's data. Therefore, even if the curious party has infinite computing power, he cannot learn the exact value of the other party's data. A scheme with a trusted third party can be advantageous, particularly when the third party's role in the protocol is minimal. However, in today's Internet environment, establishing and maintaining long-term trust among multiple parties can be challenging, particularly as the number of participants grows. Such a system may lack robustness, as a breach in security by the server or any participating party can disrupt the system's functionality.