We study the problem of differentially private (DP) secure multiplication in distributed computing systems, focusing on regimes where perfect privacy and perfect accuracy cannot be simultaneously achieved. Specifically, N nodes collaboratively compute the product of M private inputs while guaranteeing epsilon-DP against any collusion of up to T nodes. Prior work has characterized the fundamental privacy-accuracy trade-off for the multiplication of two multiplicands. In this paper, we extend these results to the more general setting of computing the product of an arbitrary number M of multiplicands. We propose a secure multiplication framework based on carefully designed encoding polynomials combined with layered noise injection. The proposed construction generalizes existing schemes and enables the systematic cancellation of lower-order noise terms, leading to improved estimation accuracy. We explore two regimes: (M-1)T+1 <= N <= MT and N = T+1. For (M-1)T+1 <= N <= MT, we characterize the optimal privacy--accuracy trade-off. When N = T+1, we derive nontrivial achievability and converse bounds that are asymptotically tight in the high-privacy regime.

翻译：本研究探讨分布式计算系统中差分隐私（DP）安全乘法问题，重点关注完美隐私与完美精度无法同时实现的机制。具体而言，N个节点在保证ε-DP抵御最多T个节点共谋的前提下，协作计算M个私有输入的乘积。先前工作已对两个乘数相乘的基本隐私-精度权衡进行了理论刻画。本文将这些结果推广至计算任意数量M个乘数乘积的更一般场景。我们提出基于精心设计的编码多项式与分层噪声注入的安全乘法框架，该构建泛化了现有方案，并能实现低阶噪声项的系统性抵消，从而提升估计精度。我们探索两种机制：(M-1)T+1 ≤ N ≤ MT 与 N = T+1。对于(M-1)T+1 ≤ N ≤ MT的情形，我们刻画了最优隐私-精度权衡关系；当N = T+1时，我们推导出非平凡的可行性界与逆界，这些界限在高隐私机制下具有渐近紧性。