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

Matrix multiplication computation (MMC) is one of the most important basic operations with a variety of applications in the scientific and engineering community, including linear regression, k-nearest neighbor classification and biometric identification. However, performing these tasks with large-scale datasets can result in significant computation beyond the 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 for the outsourcing MMC task and Zhao et al's scheme for matrix determinant computation. Nevertheless, Lei et al's scheme suffers from inherent security flaws that reveal the statistical information of zero elements in the original data. Additionally, Zhao et al's scheme can only be applied to specific outsourced tasks and is not suitable for more universal situations, such as MMC, where the client needs to compute the inverse matrix of the secret key. Therefore, designing an invertible matrix is a difficult task that affects privacy security, efficiency, and universality of the matrix-transformation-based privacy-protected outsourcing computing scheme. To address this challenge, we propose a subtly designed invertible matrix (SDIM) and a privacy-protected outsourcing MMC scheme based on the SDIM to remedy the inherent security flaws of Lei et al's scheme. We also propose an optimized matrix-chain multiplication method to maintain high efficiency of the SDIM-based privacy-protected scheme. This optimization also allows the SDIM to be universally applied not only to MMC tasks but also to other related outsourced tasks such as linear regression. Theoretical analyses and experiments show that our methods are more secure in terms of data privacy, with comparable efficiency to the state-of-the-art scheme based on matrix transformation. This SDIM-based scheme has achieved a well-balanced trade-off between security, efficiency and universality.