We consider the problem of secure distributed matrix multiplication (SDMM) in which a user wishes to compute the product of two matrices with the assistance of honest but curious servers. We construct polynomial codes for SDMM by studying a recently introduced combinatorial tool called the degree table. For a fixed partitioning, minimizing the total communication cost of a polynomial code for SDMM is equivalent to minimizing $N$, the number of distinct elements in the corresponding degree table. We propose new constructions of degree tables with a low number of distinct elements. These new constructions lead to a general family of polynomial codes for SDMM, which we call $\mathsf{GASP}_{r}$ (Gap Additive Secure Polynomial codes) parametrized by an integer $r$. $\mathsf{GASP}_{r}$ outperforms all previously known polynomial codes for SDMM under an outer product partitioning. We also present lower bounds on $N$ and prove the optimality or asymptotic optimality of our constructions for certain regimes. Moreover, we formulate the construction of optimal degree tables as an integer linear program and use it to prove the optimality of $\mathsf{GASP}_{r}$ for all the system parameters that we were able to test.
翻译:我们考虑了安全分布式矩阵乘法(SDMM)的问题,即用户希望在诚实但好奇的服务器的协助下计算两个矩阵的产物。我们通过研究最近推出的称为度表的组合式工具,为SDMMM构建了多式代码。对于固定分区,将SDMM多式代码的通信总成本最小化,相当于在相应度表格中将不同元素的数量降低到最低。我们建议对不同元素数量较少的度表进行新的构造。这些新的构造导致SDMMM形成一个通用的多式代码,我们称之为$\mathsf{GASP ⁇ r}(Gap Additive Secure Polynomilal code) 。对于固定分区而言,将SDMMM多式代码的通信总成本降到最低值。$\math{GASP{r}(美元) 与所有以前已知的SDMMMD的多元代码相匹配。我们还对美元进行了较低的约束,并证明我们为SDMMMMM系统的最佳或最优化的优化性优化度。此外,我们为某些制度制定了最优化的系统测试度。我们为最优化的度。