Improved Private and Secure Distributed (Batch) Matrix Multiplication

In this paper, we study the problem of distributed matrix multiplication under various scenarios. Specifically, we focus on a scenario where a user has two matrices A and B and wishes to compute their product with the assistance of several distributed colluding servers, which is referred to as secure distributed matrix multiplication (SDMM). Meanwhile, we also consider its generalization by assuming that the user has two matrix batches, which is referred to as secure distributed batch matrix multiplication (SDBMM) or distributed batch matrix multiplication (DBMM) if the distributed servers are incurious. In a variant of this problem, we consider the problem of 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 $B^{(0)}, B^{(1)},...,B^{(L-1)}$, 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. Distributed matrix multiplication under these scenarios is supposed to have wide application potentials, such as machine learning and cloud computing. However, studies of distributed matrix multiplication are still scarce in the literature and there is much room for improvement. In this paper, we propose several new coding schemes for the various distributed matrix multiplication models, including two classes of SDMM codes, a DBMM code, an SDBMM code, and finally a PSDMM code. The proposed codes have a better performance than state-of-the-art schemes in that they can achieve a smaller recovery threshold and download cost as well as providing a more flexible tradeoff between the upload and download costs.