Linear regression is a basic method that models the relationship between an outcome value and some explanatory values using a linear function. Traditionally, this method is conducted on a clear dataset provided by one data owner. However, in today's ever-increasingly digital world, the data for regression analysis are likely distributed among multiple parties and even contain sensitive information about the data owners. In this case, data owners are not willing to share their data unless data privacy is guaranteed. In this paper, we propose a novel protocol for conducting privacy-preserving linear regression (PPLR) on horizontally partitioned data. Our system architecture includes multiple clients and two noncolluding servers. In our protocol, each client submits its data in encrypted form to a server, and two servers collaboratively determine the regression model on pooled data without learning its contents. We construct our protocol with Paillier homomorphic encryption and a new data masking technique. This data masking technique can perturb data by multiplying a rational number while the data are encrypted. Due to the use of the data masking technique, the efficiency of our protocol is greatly improved. We provide an error bound of the protocol and prove it rigorously. We also provide security analysis of the protocol. Finally, we implement our system in C++ and Java, and then we evaluate our protocol using real datasets provided by UCI. The experiments show our protocol is one of the most effective approaches to date and has negligible errors compared with performing linear regression on clear data.

Support vector machine (SVM) is an important technique for data classification. Traditional SVM assumes free access to data. If the data are split and held by different users, for privacy reasons, users are likely unwilling to submit their data to a third party for classification. In this paper, by using additive homomorphic encryption and random transformations (matrix transformation and vector decomposition), we design a privacy-preserving outsourcing scheme for conducting Least Squares SVM (LS-SVM) classification on vertically partitioned data. In our system, multiple data owners (users) submit their encrypted data to two non-colluding service providers, which conduct SVM algorithm on it. During the execution of our algorithm, neither service provider learns anything about the input data, the intermediate results, or the predicted result. In other words, our algorithm is encrypted in the whole process. Extensive theoretical analysis and experimental evaluation demonstrate the correctness, security, and efficiency of the method.