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.

翻译：在本文中,我们研究的是以下问题: eemph{ 私人和安全分布式矩阵乘法(PSDMM)},用户拥有私人基质 $A$ 和 $N$的非混合服务器,共享图书馆$L$(L>1美元) 的基质 $B ⁇ (0),B ⁇ (1)},\ ldots,B ⁇ (L-1)}},用户希望为此以复制的形式在服务器上存储$AB ⁇ ((Teta)}$,一些美元没有向服务器披露矩阵中$A$(0,L)的任何信息,并且将指数中$$@theta$保持为私有服务器。以前的工作仅限于共享图书馆(\ textit{i.e.} $B ⁇ (0)}, B ⁇ (1)},\\\\\\\\\\\\(L-1)$$美元),对此,用户希望以复制的形式在文献中非常稀少,但还有很大的改进余地。在本文件中,我们提议两个SDIM计划,其中的一个方案不是完全以共同的基质的存储系统上一个比共同的版本的基码系统更精细的版本 。