Private and Secure Distributed Matrix Multiplication Schemes for Replicated or MDS-Coded Servers

TL;DR

This paper introduces two novel private and secure distributed matrix multiplication schemes, one optimized for replicated storage and the other for MDS-coded storage, improving efficiency and storage requirements while preserving privacy.

Contribution

The paper presents two new PSDMM schemes tailored for replicated and MDS-coded server storage, enhancing performance and reducing storage compared to existing methods.

Findings

The replicated storage scheme achieves a smaller recovery threshold.

The MDS-coded scheme reduces server storage needs.

Both schemes maintain privacy of the matrix and index.

Abstract

In this paper, we study the problem of \emph{private and secure distributed matrix multiplication (PSDMM)}, where a user having a private matrix $A$ and $N$ non-colluding servers sharing a library of $L$ ($L>1$) matrices $B^{(0)}, B^{(1)},\ldots,B^{(L-1)}$, for which the user wishes to compute $AB^{(\theta)}$ for some $\theta\in [0, L)$ without revealing any information of the matrix $A$ to the servers, and keeping the index $\theta$ private to the servers. Previous work is limited to the case that the shared library (\textit{i.e.,} the matrices $B^{(0)}, B^{(1)},\ldots,B^{(L-1)}$) is stored across the servers in a replicated form and schemes are very scarce in the literature, there is still much room for improvement. In this paper, we propose two PSDMM schemes, where one is limited to the case that the shared library is stored across the servers in a replicated form but has a better performance than state-of-the-art schemes in that it can achieve a smaller recovery threshold and download cost. The other one focuses on the case that the shared library is stored across the servers in an MDS-coded form, which requires less storage in the servers. The second PSDMM code does not subsume the first one even if the underlying MDS code is degraded to a repetition code as they are totally two different schemes.