This work considers the problem of privately outsourcing the computation of a matrix product over a finite field $\mathbb{F}_q$ to $N$ helper servers. These servers are considered to be honest but curious, i.e., they behave according to the protocol but will try to deduce information about the user's data. Furthermore, any set of up to $X$ servers is allowed to share their data. Previous works considered this collusion a hindrance and the download cost of the schemes increases with growing $X$. We propose to utilize such linkage between servers to the user's advantage by allowing servers to cooperate in the computational task. This leads to a significant gain in the download cost for the proposed schemes. The gain naturally comes at the cost of increased communication load between the servers. Hence, the proposed cooperative scheme can be understood as outsourcing both computational cost and communication cost. While the present work exemplifies the proposed server cooperation in the case of a specific secure distributed matrix multiplication (SDMM) scheme, the same idea applies to many other use cases as well. For instance, other SDMM schemes as well as linear private information retrieval (PIR) as a special case of SDMM are instantly covered.

中文翻译：

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通过合作有效恢复共享秘密：对 SDMM 和 PIR 的应用

这项工作考虑了将有限域 $\mathbb{F}_q$ 上的矩阵乘积的计算私下外包给 $N$ 辅助服务器的问题。这些服务器被认为是诚实但好奇的，即它们按照协议行事，但会尝试推断有关用户数据的信息。此外，任何一组最多 $X$ 的服务器都可以共享其数据。以前的作品认为这种勾结是一种障碍，并且计划的下载成本随着 $X$ 的增长而增加。我们建议通过允许服务器在计算任务中合作来利用服务器之间的这种联系来为用户带来优势。这导致所提议方案的下载成本显着增加。收益自然是以增加服务器之间的通信负载为代价的。因此，所提出的合作方案可以理解为外包计算成本和通信成本。虽然目前的工作举例说明了在特定安全分布式矩阵乘法 (SDMM) 方案的情况下提议的服务器合作，但同样的想法也适用于许多其他用例。例如，其他 SDMM 方案以及作为 SDMM 特例的线性私人信息检索 (PIR) 立即涵盖。