We consider the problem of secure distributed matrix multiplication (SDMM), where a user has two matrices and wishes to compute their product with the help of $N$ honest but curious servers under the security constraint that any information about either $A$ or $B$ is not leaked to any server. This paper presents anew scheme that considers the inner product partition for matrices $A$ and $B$. Our central technique relies on encoding matrices $A$ and $B$ in a Hermitian Code and its dual code, respectively. We present the Hermitian Algebraic (HerA) scheme, which employs Hermitian Codes and characterizes the partitioning and security capacities given entries of matrices belonging to a finite field with $q^2$ elements. We showcase this scheme performs the secure distributed matrix multiplication in a significantly smaller finite field than the existing results in the literature.

翻译：本文考虑安全分布式矩阵乘法（SDMM）问题，其中用户拥有两个矩阵，并希望在安全约束下借助$N$个诚实但好奇的服务器计算它们的乘积，该约束要求任何关于矩阵$A$或$B$的信息均不会泄露给任何服务器。本文提出一种新方案，该方案采用矩阵$A$和$B$的内积划分方法。我们的核心技巧在于分别基于Hermitian码及其对偶码对矩阵$A$和$B$进行编码。我们提出Hermitian代数（HerA）方案，该方案利用Hermitian码，并刻画了当矩阵元素属于含$q^2$个元素的有限域时的划分与安全容量。我们证明，与现有文献结果相比，该方案能在显著更小的有限域上实现安全分布式矩阵乘法。