Differentially Private Secure Multiplication: Hiding Information in the Rubble of Noise

TL;DR

This paper explores a new approach to secure multi-party multiplication that balances privacy and accuracy by allowing controlled information leakage and approximate results, especially when honest nodes are in the minority.

Contribution

It introduces a novel layered noise distribution combining differential privacy and secret-sharing to enable privacy-accuracy trade-offs with minority honest nodes.

Findings

Characterizes privacy-accuracy trade-off for N<2t+1

Develops layered noise distribution merging differential privacy and secret-sharing

Provides tight bounds on information leakage and error

Abstract

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.