We study two problems of private matrix multiplication, over a distributed computing system consisting of a master node, and multiple servers who collectively store a family of public matrices using Maximum-Distance-Separable (MDS) codes. In the first problem of Private and Secure Matrix Multiplication from Colluding servers (MDS-C-PSMM), 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 from Colluding servers (MDS-C-FPMM), 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 MDS-C-PSMM and MDS-C-FPMM, which simultaneously guarantee information-theoretic data/index privacy and computation correctness. The key ingredient is a careful design of secret sharings of the matrix $\mathbf{A}$ and the private indices, which are tailored to matrix multiplication task and MDS storage structure, such that the computation results from the servers can be viewed as evaluations of a polynomial at distinct points, from which the intended result can be obtained through polynomial interpolation. We compare the proposed MDS-C-PSMM strategy with a previous MDS-PSMM strategy with a weaker privacy guarantee (non-colluding servers), and demonstrate substantial improvements over the previous strategy in terms of communication and computation performance.
翻译:我们研究的是私人矩阵乘法的两个问题,即由主节点组成的分布式计算系统,以及使用最大偏差可分离代码(MDS)集体存储一组公共矩阵的多个服务器。在对调服务器(MDS-C-PSMM)的私人和安全矩阵乘法的第一个问题中,主机打算用存储在服务器上的目标矩阵计算其保密矩阵$\mathbf{A}的产物,而不透露任何关于$\mathbf{A}美元的信息,以及将目标矩阵索引存储到某些串联服务器。在对调服务器(MDDS-C)的全私基矩阵乘法第二问题中,矩阵 $\mathbf{A}也是从存储服务器(MDMS-MS-MS-MS-MLMS-MRMS-MS-C-MDS-FMMMM)中存储新的战略,这可以同时保证信息-母基母体的全体变变变变变法战略, 将前MDMS-MS-MS的存储和MISMIS-deal 和Mexalalexalxx 的预算结果显示。