We consider the problem of private distributed multi-party multiplication. It is well-established that Shamir secret-sharing coding strategies can enable perfect information-theoretic privacy in distributed computation via the celebrated algorithm of Ben Or, Goldwasser and Wigderson (the "BGW algorithm"). However, perfect privacy and accuracy require an honest majority, that is, $N \geq 2t+1$ compute nodes are required to ensure privacy against any $t$ colluding adversarial nodes. By allowing for some controlled amount of information leakage and approximate multiplication instead of exact multiplication, we study coding schemes for the setting where the number of honest nodes can be a minority, that is $N< 2t+1.$ We develop a tight characterization privacy-accuracy trade-off for cases where $N < 2t+1$ by measuring information leakage using {differential} privacy instead of perfect privacy, and using the mean squared error metric for accuracy. A novel technical aspect is an intricately layered noise distribution that merges ideas from differential privacy and Shamir secret-sharing at different layers.

翻译：我们考虑分布式多方私有乘法问题。众所周知，Shamir秘密共享编码策略能够通过Ben Or、Goldwasser和Wigderson的著名算法（“BGW算法”）实现分布式计算中的完美信息论隐私。然而，完美的隐私和准确性需要诚实多数，即需要$N \geq 2t+1$个计算节点来确保对任何$t$个合谋对抗节点的隐私保护。通过允许一定可控的信息泄露以及近似乘法而非精确乘法，我们研究了在诚实节点可能成为少数（即$N < 2t+1$）情况下的编码方案。我们通过使用差分隐私而非完美隐私来衡量信息泄露，并采用均方误差度量准确性，针对$N < 2t+1$的情形建立了隐私-准确性权衡的紧致刻画。一个新颖的技术方面是精心设计的分层噪声分布，它在不同层次上融合了差分隐私和Shamir秘密共享的思想。