Reed–Solomon-Based Private and Secure Distributed Matrix Multiplication From MDS-Coded Storage

The private and secure distributed matrix multiplication (PSDMM) has broad applications in fields such as finance, e-health, and machine learning. The PSDMM problem introduces a public matrix library $\mathcal {L}$ and aims to securely compute the product of a private matrix with a matrix confidentially selected from $\mathcal {L}$ . In this setting, the user leverages the assistance of d server nodes to perform this computation while ensuring that no information about the private matrix or the index of the selected matrix is disclosed to colluding servers. In most prior works, $\mathcal {L}$ is stored in a replicated form across the servers, resulting in significant storage inefficiency. This paper proposes the use of $(d,K)$ -maximum distance separable (MDS) codes to encode the library and distribute it across the servers, thus enhancing storage efficiency. Building on this, the orthogonality property of Reed-Solomon (RS) codes and their dual codes is exploited to design PSDMM schemes that consistently achieve the optimal recovery threshold. Compared to existing MDS-coded storage PSDMM schemes, the proposed schemes offer lower decoding complexity, as the user only requires a single Lagrange interpolation during the decoding phase. Furthermore, to minimize download costs during decoding phase, subspace polynomial technique from repairing RS codes is introduced, resulting in a communication-efficient PSDMM (CE-PSDMM) scheme. Theoretical analysis shows that the CE-PSDMM scheme reduces the amount of data downloaded from each server compared to conventional PSDMM schemes.