The Gram matrix of a matrix $A$ is defined as $AA^T$ (or $A^T\!A$). Computing the Gram matrix is an important operation in many applications, such as linear regression with the least squares method, where the explicit solution formula includes the Gram matrix of the data matrix. Secure distributed matrix multiplication (SDMM) can be used to compute the product of two matrices using the help of worker servers. If a Gram matrix were computed using SDMM, the data matrix would need to be encoded twice, which causes an unnecessary overhead in the communication cost. We propose a new scheme for this purpose called secure distributed Gram matrix multiplication (SDGMM). It can leverage the advantages of computing a Gram matrix instead of a regular matrix product.

翻译：矩阵$A$的Gram矩阵定义为$AA^T$（或$A^T\!A$）。在许多应用中，例如采用最小二乘法的线性回归中，计算Gram矩阵是一项重要运算，因为其显式解公式中包含数据矩阵的Gram矩阵。安全分布式矩阵乘法可利用工作服务器的辅助计算两个矩阵的乘积。若使用安全分布式矩阵乘法计算Gram矩阵，数据矩阵需要被编码两次，这将导致通信开销不必要地增加。我们针对此问题提出一种新方案——安全分布式Gram矩阵乘法（SDGMM）。该方案能充分利用计算Gram矩阵相较于常规矩阵乘积的优势。