Differentially Private Secure Multiplication: Beyond Two Multiplicands

2026-03-09

AI Summary

This work addresses the challenge of computing the product of arbitrary M private inputs across a distributed system of N nodes while tolerating up to T colluding adversaries and guaranteeing ε-differential privacy, where perfect privacy and accuracy are inherently incompatible. The authors propose a differentially private secure multiplication framework based on coded polynomials and hierarchical noise injection, extending the privacy–accuracy trade-off theory from the two-input case to arbitrary M. By systematically canceling lower-order noise terms, the framework significantly improves estimation accuracy. The study characterizes the optimal privacy–accuracy trade-off for node counts satisfying (M−1)T+1 ≤ N ≤ MT, and for the minimal setting N = T+1, establishes asymptotically tight achievability and converse bounds in the high-privacy regime.

Technology Category

• Machine Learning: Privacy
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search

Application Category

• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Responsible Web: Data and user privacy-enhancing technologies for the Web

Abstract

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.

Problem

• differentially private
• secure multiplication
• distributed computing
• privacy-accuracy trade-off
• multiplicands

Innovation

• differentially private secure multiplication
• arbitrary multiplicands
• encoding polynomials
• layered noise injection
• privacy-accuracy trade-off