This paper considers the problem of outsourcing the multiplication of two private and sparse matrices to untrusted workers. Secret sharing schemes can be used to tolerate stragglers and guarantee information-theoretic privacy of the matrices. However, traditional secret sharing schemes destroy all sparsity in the offloaded computational tasks. Since exploiting the sparse nature of matrices was shown to speed up the multiplication process, preserving the sparsity of the input matrices in the computational tasks sent to the workers is desirable. It was recently shown that sparsity can be guaranteed at the expense of a weaker privacy guarantee. Sparse secret sharing schemes with only two output shares were constructed. In this work, we construct sparse secret sharing schemes that generalize Shamir's secret sharing schemes for a fixed threshold $t=2$ and an arbitrarily large number of shares. We design our schemes to provide the strongest privacy guarantee given a desired sparsity of the shares under some mild assumptions. We show that increasing the number of shares, i.e., increasing straggler tolerance, incurs a degradation of the privacy guarantee. However, this degradation is negligible when the number of shares is comparably small to the cardinality of the input alphabet.

翻译：本文考虑将两个私密稀疏矩阵的乘法任务外包给不可信工作节点的问题。秘密共享方案可用于容忍掉队节点并保证矩阵的信息论隐私性。然而传统秘密共享方案会破坏所有外包计算任务中的稀疏性。鉴于利用矩阵稀疏性可加速乘法运算，在发送给工作节点的计算任务中保留输入矩阵的稀疏性具有重要价值。最新研究表明，通过降低隐私保护强度可实现稀疏性保障，目前已构造出仅包含两个输出份额的稀疏秘密共享方案。本研究构建了能泛化Shamir秘密共享方案的稀疏秘密共享新方案，该方案针对固定阈值$t=2$且支持任意数量的份额生成。我们设计的新方案在给定份额期望稀疏度与若干温和假设条件下，可提供最强隐私保护强度。研究表明：增加份额数量（即提升掉队容忍度）会导致隐私保护强度下降，但当份额数量远小于输入字母表基数时，该性能衰减可忽略不计。