## 1 Introduction
The method proposed in this paper for collaboratively constructing a linear model operates without the need for trusted third parties. Our approach is based on the assumption that all parties are semi-honest, meaning they adhere to the protocol but maintain a curiosity about the data of other parties. This design ensures that the system remains functional even in the absence of unwavering trust among participants. Encryption can be used to protect sensitive information in machine learning. However, in machine learning, data must be processed efficiently, and traditional encryption methods are not suitable for this purpose. Fully homomorphic encryption, such as the Paillier cryptosystem [6], has shown potential for computing model parameters in machine learning. Nonetheless, homomorphic encryption often requires substantial computational resources to achieve the required level of security. For example, the Halevi-Polyakov-Shoup fully homomorphic encryption scheme takes 51 ms for 128-bit security implemented by using a GPU [7]. Although this speed may be practical for cloud computing, the security of these cryptosystems is often based on computationally hard problems, so they can only provide computational security. The proposed scheme does not need encryption to protect users' raw data. We show that our proposed scheme can achieve information-theoretic security. A critical consideration is that in machine learning, data frequently consist of real numbers, necessitating their representation in a standard format like IEEE Standard 754 floating-point numbers, which enables efficient computation using currently available computer hardware. To mitigate rounding errors, the plaintext and ciphertext of a cryptographic system is typically designed in finite groups, finite rings, or finite fields. Consequently,the majority of these cryptographic systems, even when fully homomorphic, may not align seamlessly with the requirements of machine learning. To address this issue, recently, [PERSON] et al. developed the CKKS cryptosystem [8], which is designed to accommodate a specialized form of fixed-point arithmetic known as block floating-point arithmetic. While fixed-point arithmetic can approximate floating-point numbers within a finite memory size, it is worth noting that the current version of CKKS has limitations. Specifically, it supports only a limited number of multiplications before necessitating a time-consuming bootstrapping step. Another drawback of using the current version of CKKS is its lack of direct support for floating-point division operations, which are essential for tasks such as linear regression and for almost all machine learning models. Addressing this concern, [PERSON] et al. proposed a two-stage method for Euclidean division within the CKKS scheme and conducted a study of its properties [9]. Datasets owned by a user may also be required to be efficiently deleted from computations. This is useful when a party leaves the group and wants to remove its information. This is called machine unlearning. Machine unlearning is necessary in order to comply with privacy protection laws and regulations, such as GDPR in European countries. In our proposed scheme, both the addition and deletion of datasets can be done efficiently without recomputing the model parameters from the beginning. However, our scheme needs to invert a matrix of size \((m+1)\times(m+1)\), where \(m\) is the number of features used in the linear regression model. This is not a problem in practice, since the value of \(m\) is usually small compared to the amount of data \(n\). The main contributions of this research are summarized as follows. 1. We propose a secure and privacy-preserving machine learning scheme for linear regression without the requirement of trusted servers or encryption: notably, homomorphic encryption. 2. The security of the proposed scheme does not depend on computationally hard problems.