We study two problems of private matrix multiplication, over a distributed computing system consisting of a master node, and multiple servers that collectively store a family of public matrices using Maximum-Distance-Separable (MDS) codes. In the first problem of Private and Secure Matrix Multiplication (PSMM) from colluding servers, the master intends to compute the product of its confidential matrix $\mathbf{A}$ with a target matrix stored on the servers, without revealing any information about $\mathbf{A}$ and the index of target matrix to some colluding servers. In the second problem of Fully Private Matrix Multiplication (FPMM) from colluding servers, the matrix $\mathbf{A}$ is also selected from another family of public matrices stored at the servers in MDS form. In this case, the indices of the two target matrices should both be kept private from colluding servers. We develop novel strategies for the two PSMM and FPMM problems, which simultaneously guarantee information-theoretic data/index privacy and computation correctness. We compare the proposed PSMM strategy with a previous PSMM strategy with a weaker privacy guarantee (non-colluding servers), and demonstrate substantial improvements over the previous strategy in terms of communication and computation overheads. Moreover, compared with a baseline FPMM strategy that uses the idea of Private Information Retrieval (PIR) to directly retrieve the desired matrix multiplication, the proposed FPMM strategy significantly reduces storage overhead, but slightly incurs large communication and computation overheads.

翻译：我们研究了两类私有矩阵乘法问题，这些研究基于一个分布式计算系统，系统包含一个主节点和多个服务器，这些服务器使用最大距离可分（MDS）编码共同存储一组公有矩阵。在第一个问题——针对合谋服务器的私密安全矩阵乘法（PSMM）中，主节点旨在计算其机密矩阵$\mathbf{A}$与服务器上存储的目标矩阵的乘积，同时不向部分合谋服务器泄露关于$\mathbf{A}$及目标矩阵索引的任何信息。在第二个问题——针对合谋服务器的完全私有矩阵乘法（FPMM）中，矩阵$\mathbf{A}$同样选自另一组以MDS形式存储在服务器上的公有矩阵。此时，两个目标矩阵的索引均需对合谋服务器保密。我们针对PSMM和FPMM这两个问题提出了新颖的策略，这些策略同时保证了信息论意义上的数据/索引隐私性和计算正确性。我们将所提出的PSMM策略与一种隐私保证较弱（非合谋服务器场景）的已有PSMM策略进行了比较，结果表明本策略在通信和计算开销方面有显著改进。此外，与一种利用私有信息检索（PIR）思想直接检索所需矩阵乘法的基线FPMM策略相比，本文提出的FPMM策略大幅降低了存储开销，但略微增加了通信和计算开销。