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.

中文翻译：

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来自 MDS 编码存储的信息理论上私有的矩阵乘法

我们研究了私有矩阵乘法的两个问题，分布式计算系统由一个主节点和多个服务器组成，这些服务器使用最大距离可分离 (MDS) 代码共同存储一系列公共矩阵。在来自串通服务器的私有和安全矩阵乘法 (PSMM) 的第一个问题中，master 打算计算其机密矩阵的乘积 $\mathbf {A}$将目标矩阵存储在服务器上，而不会泄露有关的任何信息 $\mathbf {A}$以及一些共谋服务器的目标矩阵索引。在来自共谋服务器的完全私有矩阵乘法 (FPMM) 的第二个问题中，矩阵 $\mathbf {A}$也是从以 MDS 形式存储在服务器中的另一组公共矩阵中选择的。在这种情况下，两个目标矩阵的索引都应该对串通服务器保密。我们针对 PSMM 和 FPMM 这两个问题开发了新的策略，同时保证了信息论数据/索引的隐私和计算的正确性。我们将所提出的 PSMM 策略与之前隐私保证较弱（非共谋服务器）的 PSMM 策略进行比较，并证明在通信和计算开销方面比之前的策略有实质性改进。此外，与使用私有信息检索 (PIR) 的思想直接检索所需矩阵乘法的基线 FPMM 策略相比，所提出的 FPMM 策略显着降低了存储开销，