SDIM: A Subtly Designed Invertible Matrix for Enhanced Privacy-Preserving Outsourcing Matrix Multiplication and Related Tasks

Matrix multiplication computation (MMC) is a fundamental operation with various applications, including linear regression, k-nearest neighbor classification and biometric identification. However, performing these tasks with large-scale datasets surpasses the computation capabilities of resource-constrained clients. As outsourcing intensive tasks to cloud server has become a promising method, many matrix-transformation-based privacy-protected schemes have been presented for certain outsourcing tasks, such as Lei et al.'s scheme addresses MMC outsourcing, and Zhao et al.'s scheme focuses on matrix determinant computation. Nevertheless, Lei et al.'s scheme exhibits inherent security flaws, leaking statistical information about zero elements in the original data. Additionally, Zhao et al.'s scheme is task-specific, limiting its application to universal scenarios such as MMC, where clients must compute the inverse matrix of the secret key. Therefore, designing an invertible matrix presents a challenge that affects privacy security, efficiency, and universality of matrix-transformation-based privacy-protected outsourcing computing schemes. To address this issue, we propose a subtly designed invertible matrix (SDIM) and a privacy-protected outsourcing MMC scheme based on SDIM, remedying the security flaws of Lei et al.'s scheme. We also propose an optimized matrix-chain multiplication method to ensure high efficiency of the SDIM-based privacy-protected scheme. This optimization also allows SDIM to be universally applied not only to MMC but also to other related outsourced tasks such as linear regression. Theoretical analyses and experiments have demonstrated that our methods provide enhanced data privacy security while maintaining efficiency comparable to the state-of-the-art scheme based on matrix transformation, i.e., achieving a well-balanced trade-off between security, efficiency, and universality.