OBJECTIVE: Linear and logistic regression are widely used statistical techniques in population genetics for analyzing genetic data and uncovering patterns and associations in large genetic datasets, such as identifying genetic variations linked to specific diseases or traits. However, obtaining statistically significant results from these studies requires large amounts of sensitive genotype and phenotype information from thousands of patients, which raises privacy concerns. Although cryptographic techniques such as homomorphic encryption offers a potential solution to the privacy concerns as it allows computations on encrypted data, previous methods leveraging homomorphic encryption have not addressed the confidentiality of shared models, which can leak information about the training data. METHODS: In this work, we present a secure model evaluation method for linear and logistic regression using homomorphic encryption for six prediction tasks, where input genotypes, output phenotypes, and model parameters are all encrypted. RESULTS: Our method ensures no private information leakage during inference and achieves high accuracy (≥93% for all outcomes) with each inference taking less than ten seconds for ∼200 genomes. CONCLUSION: Our study demonstrates that it is possible to perform linear and logistic regression model evaluation while protecting patient confidentiality with theoretical security guarantees. Our implementation and test data are available at https://github.com/G2Lab/privateML/.

In this study, we propose an efficient sorting method for encrypted data using fully homomorphic encryption (FHE). The proposed method extends the existing 2-way sorting method by applying the k-way sorting network for any prime k to reduce the depth in terms of comparison operation from O(log22n) to O(klogk2n), thereby improving performance for k slightly larger than 2, such as k=5. We apply this method to approximate FHE which is widely used due to its efficiency of homomorphic arithmetic operations. In order to build up the k-way sorting network, the k-sorter, which sorts k-numbers with a minimal comparison depth, is used as a building block. The approximate homomorphic comparison, which is the only type of comparison working on approximate FHE, cannot be used for the construction of the k-sorter as it is because the result of the comparison is not binary, unlike the comparison in conventional bit-wise FHEs. To overcome this problem, we propose an efficient k-sorter construction utilizing the features of approximate homomorphic comparison. Also, we propose an efficient construction of a k-way sorting network using cryptographic SIMD operations. To use the proposed method most efficiently, we propose an estimation formula that finds the appropriate k that is expected to reduce the total time cost when the parameters of the approximating comparisons and the performance of the operations provided by the approximate FHE are given. We also show the implementation results of the proposed method, and it shows that sorting 56= 15625 data using 5-way sorting network can be about 23.3% faster than sorting 214= 16384 data using 2-way.

Machine learning on (homomorphic) encrypted data is a cryptographic method for analyzing private and/or sensitive data while keeping privacy. In the training phase, it takes as input an encrypted training data and outputs an encrypted model without ever decrypting. In the prediction phase, it uses the encrypted model to predict results on new encrypted data. In each phase, no decryption key is needed, and thus the data privacy is ultimately guaranteed. It has many applications in various areas such as finance, education, genomics, and medical field that have sensitive private data. While several studies have been reported on the prediction phase, few studies have been conducted on the training phase.In this paper, we present an efficient algorithm for logistic regression on homomorphic encrypted data, and evaluate our algorithm on real financial data consisting of 422,108 samples over 200 features. Our experiment shows that an encrypted model with a sufficient Kolmogorov Smirnow statistic value can be obtained in ∼17 hours in a single machine. We also evaluate our algorithm on the public MNIST dataset, and it takes ∼2 hours to learn an encrypted model with 96.4% accuracy. Considering the inefficiency of homomorphic encryption, our result is encouraging and demonstrates the practical feasibility of the logistic regression training on large encrypted data, for the first time to the best of our knowledge.