发表机构
University of California San Diego; Clemson University; Villanova University（加州大学圣地亚哥分校; 克莱姆森大学; 维拉诺瓦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出周期间隙框架构造改进的度数表，得到SHIFT和COVER两种新方案，在多数情况下优于现有方法，并证明平衡情形下SHIFT在度数表构造中渐近最优。
AI 中文摘要
在安全分布式矩阵乘法中，用户希望在服务器的协助下计算两个矩阵的乘积，同时确保任意 $T$ 个共谋服务器无法获知任一矩阵的信息。度数表是为该问题构造多项式码的组合工具，并支撑了多种最先进的方案，包括 $\mathsf{GASP}_r$、$\mathsf{GASP}_{r,s}$ 和 $\mathsf{DOG}_{r,s}$。我们引入了一种用于构造度数表的周期间隙框架，该框架将上述三个族作为特例，并产生了两个新构造：$\mathsf{SHIFT}_{r,s}$ 和 $\mathsf{COVER}_r$。我们确定了它们的确切恢复阈值，并表明在许多情况下它们优于当前最先进的方法。我们还证明了普通整数度数表恢复阈值的一个新下界。在平衡情形下，即划分参数与安全参数相等时，我们将该下界收紧至与 $\mathsf{SHIFT}_{r,s}$ 相差低阶项，表明其在度数表构造中渐近最优。
英文摘要
In secure distributed matrix multiplication, a user wishes to compute the product of two matrices with the assistance of servers, in such a way that any $T$ colluding servers learn nothing about either matrix. Degree tables are a combinatorial tool for constructing polynomial codes for this problem and underlie several state-of-the-art schemes, including $\mathsf{GASP}_r$, $\mathsf{GASP}_{r,s}$, and $\mathsf{DOG}_{r,s}$. We introduce a periodic-gap framework for constructing degree tables that contains these three families as special cases and leads to two new constructions, $\mathsf{SHIFT}_{r,s}$ and $\mathsf{COVER}_r$. We determine their exact recovery thresholds and show that, in many cases, they outperform the current state of the art. We also prove a new lower bound on the recovery threshold of ordinary integer degree tables. In the balanced case, in which the partitioning parameters and the security parameter are equal, we sharpen this bound to match $\mathsf{SHIFT}_{r,s}$ up to lower-order terms, showing that it is asymptotically optimal among degree-table constructions.
