On the Capacity and Straggler-Robustness of Distributed Secure Matrix Multiplication

Computationally efficient matrix multiplication is a fundamental requirement in various fields, including and particularly in data analytics. To do so, the computation task of large-scale matrix multiplication is typically outsourced to multiple servers. However, due to data misusage at the servers, security is typical of concern. In this paper, we first study the two-sided secure matrix multiplication problem, where a user is interested in the matrix product \boldsymbol A \boldsymbol B 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 downlink communication rate, and (ii) to minimize the effective number of server observations needed to determine \boldsymbol A \boldsymbol B , while preserving security, where we allow for up to \ell \leq N servers to collude. To this end, we propose two schemes - an aligned secret sharing scheme (A3S) and a secure cross subspace alignment (SCSA) scheme. For A3S, we optimize the partitioning 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). The SCSA, on the other hand, is shown to be (rate) capacity-optimal for the general J -sided distributed secure matrix multiplication problem \prod _{j=1}^{J} \boldsymbol M_{j}