Rate-Efficiency and Straggler-Robustness through Partition in Distributed Two-Sided Secure Matrix Computation

Abstract

Computationally efficient matrix multiplication is a fundamental requirement in various fields, including and particularly in data analytics. To do so, the computation task of a large-scale matrix multiplication is typically outsourced to multiple servers. However, due to data misusage at the servers, security is typically of concern. In this paper, we study the two-sided secure matrix multiplication problem, where a user is interested in the matrix product $\boldsymbol{AB}$ of two finite field private matrices $\boldsymbol{A}$ and $\boldsymbol{B}$ from an information-theoretic perspective. In this problem, the user exploits the computational resources of $N$ servers to compute the matrix product, but simultaneously tries to conceal the private matrices from the servers. Our goal is twofold: (i) to maximize the communication rate, and, (ii) to minimize the effective number of server observations needed to determine $\boldsymbol{AB}$, while preserving security, where we allow for up to $\ell\leq N$ servers to collude. To this end, we propose a general aligned secret sharing scheme for which we optimize the matrix partition of matrices $\boldsymbol{A}$ and $\boldsymbol{B}$ in order to either optimize objective (i) or (ii) as a function of the system parameters (e.g., $N$ and $\ell$). A proposed inductive approach gives us analytical, close-to-optimal solutions for both (i) and (ii). With respect to (i), our scheme significantly outperforms the existing scheme of Chang and Tandon in terms of (a) communication rate, (b) maximum tolerable number of colluding servers and (c) computational complexity.