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 a \emph{new 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 that this scheme performs the secure distributed matrix multiplication in a significantly smaller finite field and expands security allowances compared to the existing results in the literature.

翻译：我们考虑安全分布式矩阵乘法（SDMM）问题：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的信息均不得泄露给任何服务器。本文提出了一种\emph{新方案}，该方案考虑了矩阵A和B的内积划分。我们的核心技术依赖于分别对矩阵A和B采用Hermitian码及其对偶码进行编码。我们提出了Hermitian代数（HerA）方案，该方案采用Hermitian码，并刻画了在矩阵元素属于含有$q^2$个元素的有限域情形下的划分与安全容量。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，并扩展了安全容许范围。