We consider the problem of secure distributed matrix multiplication in which a user wishes to compute the product of two matrices with the assistance of honest but curious servers. In this paper, we answer the following question: Is it beneficial to offload the computations if security is a concern? We answer this question in the affirmative by showing that by adjusting the parameters in a polynomial code we can obtain a trade-off between the user's and the servers' computational time. Indeed, we show that if the computational time complexity of an operation in $\mathbb{F}_q$ is at most $\mathcal{Z}_q$ and the computational time complexity of multiplying two $n\times n$ matrices is $\mathcal{O}(n^\omega \mathcal{Z}_q)$ then, by optimizing the trade-off, the user together with the servers can compute the multiplication in $\mathcal{O}(n^{4-\frac{6}{\omega+1}} \mathcal{Z}_q)$ time. We also show that if the user is only concerned in optimizing the download rate, a common assumption in the literature, then the problem can be converted into a simple private information retrieval problem by means of a scheme we call Private Oracle Querying. However, this comes at large upload and computational costs for both the user and the servers.

中文翻译：

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安全分布式矩阵乘法中的通信与计算注意事项

我们考虑安全分布式矩阵乘法的问题，其中用户希望在诚实但好奇的服务器的帮助下计算两个矩阵的乘积。在本文中，我们回答以下问题：如果考虑安全性，卸载计算是否有益？我们肯定地回答了这个问题，表明通过调整多项式代码中的参数，我们可以获得用户和服务器计算时间之间的权衡。事实上，我们证明如果 $\mathbb{F}_q$ 中的运算的计算时间复杂度至多为 $\mathcal{Z}_q$ 并且两个 $n\times n$ 矩阵相乘的计算时间复杂度为 $ \mathcal{O}(n^\omega \mathcal{Z}_q)$ 然后，通过优化权衡，用户和服务器可以在 $\mathcal{O}(n^{4-\frac{6}{\omega+1}} \mathcal{Z}_q)$ 时间内计算乘法。我们还表明，如果用户只关心优化下载速率，这是文献中的一个常见假设，那么该问题可以通过我们称为私有 Oracle 查询的方案转换为简单的私有信息检索问题。然而，这对用户和服务器来说都需要大量的上传和计算成本。