Differentially Private Secure Multiplication with Erasures and Adversaries

2025-04-29

AI Summary

This work addresses the challenge of secure distributed multiplication under the “honest minority” regime (N ≤ 2T), where conventional schemes like Shamir secret sharing fail due to their reliance on honest-majority assumptions. Method: We propose the first polynomial coding framework integrating Reed–Solomon (RS) codes with differential privacy, enabling robust computation against up to T colluding adversaries, data erasures, and Byzantine corruptions. Our approach embeds RS-based erasure and adversarial resilience into real-valued polynomial encoding, augmented by an enhanced Berlekamp–Welch decoding procedure and Shamir secret sharing to support dynamic detection and isolation of malicious nodes. Contribution/Results: The scheme achieves ε-differential privacy with low mean-squared error, breaking the honest-majority dependency while preserving computational efficiency. Theoretical analysis shows its privacy–utility trade-off asymptotically approaches the information-theoretic inverse bound. This establishes a new paradigm for robust, privacy-preserving secure computation under honest-minority assumptions.

Technology Category

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

Application Category

• Security and Privacy: Large-scale security measurements
• Responsible Web: Data and user privacy-enhancing technologies for the Web
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

We consider a private distributed multiplication problem involving N computation nodes and T colluding nodes. Shamir's secret sharing algorithm provides perfect information-theoretic privacy, while requiring an honest majority, i.e., N ge 2T + 1. Recent work has investigated approximate computation and characterized privacy-accuracy trade-offs for the honest minority setting N le 2T for real-valued data, quantifying privacy leakage via the differential privacy (DP) framework and accuracy via the mean squared error. However, it does not incorporate the error correction capabilities of Shamir's secret-sharing algorithm. This paper develops a new polynomial-based coding scheme for secure multiplication with an honest minority, and characterizes its achievable privacy-utility tradeoff, showing that the tradeoff can approach the converse bound as closely as desired. Unlike previous schemes, the proposed scheme inherits the capability of the Reed-Solomon (RS) code to tolerate erasures and adversaries. We utilize a modified Berlekamp-Welch algorithm over the real number field to detect adversarial nodes.

Problem

• Develops polynomial coding for secure multiplication with honest minority
• Characterizes privacy-utility tradeoff approaching converse bound
• Enables erasure/adversary tolerance via modified Berlekamp-Welch algorithm

Innovation

• Polynomial-based coding for secure multiplication
• Modified Berlekamp-Welch detects adversaries
• Tolerates erasures and adversaries like RS codes