This paper investigates the problem of Secure Multi-party Batch Matrix Multiplication (SMBMM), where a user aims to compute the pairwise products $\mathbf {A}\divideontimes \mathbf {B}\triangleq (\mathbf {A}^{(1)}\mathbf {B}^{(1)},\ldots,\mathbf {A}^{(M)}\mathbf {B}^{(M)})$ of two batch of massive matrices $\mathbf {A}$ and $\mathbf {B}$ that are generated from two sources, through $N$ honest but curious servers which share some common randomness. The matrices $\mathbf {A}$ (resp. $\mathbf {B}$ ) must be kept secure from any subset of up to $X_{\mathbf {A}}$ (resp. $X_{\mathbf {B}}$ ) servers even if they collude, and the user must not obtain any information about $(\mathbf {A},\mathbf {B})$ beyond the products $\mathbf {A}\divideontimes \mathbf {B}$ . A novel computation strategy for single secure matrix multiplication problem (i.e., the case $M=1$ ) is first proposed, and then is generalized to the strategy for SMBMM by means of cross subspace alignment. The SMBMM strategy focuses on the tradeoff between recovery threshold (the number of successful computing servers that the user needs to wait for), system cost (upload cost, the amount of common randomness, and download cost) and system complexity (encoding, computing, and decoding complexities). Notably, compared with the known result by Chen et al. , the strategy for the degraded case $X= X_{\mathbf {A}}=X_{\mathbf {B}}$ achieves better recovery threshold, amount of common randomness, download cost and decoding complexity when $X$ is less than some parameter threshold, while the performance with respect to other measures remain identical.

中文翻译：

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安全多方批量矩阵乘法的改进构造

本文研究了安全多方批量矩阵乘法 (SMBMM) 的问题，其中用户旨在计算成对产品 $\mathbf {A}\divideontimes \mathbf {B}\triangleq (\mathbf {A}^{(1)}\mathbf {B}^{(1)},\ldots,\mathbf {A}^{( M)}\mathbf {B}^{(M)})$ 两批大规模矩阵 $\mathbf {A}$ 和 $\mathbf {B}$ 由两个来源产生，通过 $N$ 诚实但好奇的服务器共享一些共同的随机性。矩阵 $\mathbf {A}$ （分别 $\mathbf {B}$ ) 必须保持安全，不受任何子集的影响 $X_{\mathbf {A}}$ （分别 $X_{\mathbf {B}}$ ) 服务器，即使他们串通，用户也不得获取任何有关 $(\mathbf {A},\mathbf {B})$ 超越产品 $\mathbf {A}\divideontimes \mathbf {B}$ . 单安全矩阵乘法问题的一种新计算策略（即情况 $M=1$ )首先被提出，然后通过跨子空间对齐的方式推广到SMBMM的策略。SMBMM 策略侧重于恢复阈值（用户需要等待的成功计算服务器的数量）、系统成本（上传成本、常见随机数和下载成本）和系统复杂度（编码、计算、和解码复杂性）。值得注意的是，与陈的已知结果相比等。, 降级情况的策略 $X= X_{\mathbf {A}}=X_{\mathbf {B}}$ 实现更好的恢复阈值、公共随机数、下载成本和解码复杂度 $X$ 小于某个参数阈值，而其他度量的性​​能保持不变。