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Differentially Private Secure Multiplication with Erasures and Adversaries

Haoyang Hu, Viveck R. Cadambe

TL;DR

The paper tackles private distributed multiplication in an honest-minority setting by embedding differential privacy into a polynomial-based scheme that preserves the robustness of Reed–Solomon codes. It designs an encoding with layered real-valued noise to achieve T-node ε-DP while tolerating up to E erasures and A adversaries, and analyzes both privacy and accuracy. A key contribution is a real-domain Berlekamp–Welch–based error correction that enables adversarial detection and recovery without sacrificing the DP-utility tradeoff, with theoretical guarantees and numerical validation. The results extend DP-secure computation to scenarios with erasures and active adversaries, approaching the fundamental converse bounds and enabling practical, resilient private computation over the real field.

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.

Differentially Private Secure Multiplication with Erasures and Adversaries

TL;DR

The paper tackles private distributed multiplication in an honest-minority setting by embedding differential privacy into a polynomial-based scheme that preserves the robustness of Reed–Solomon codes. It designs an encoding with layered real-valued noise to achieve T-node ε-DP while tolerating up to E erasures and A adversaries, and analyzes both privacy and accuracy. A key contribution is a real-domain Berlekamp–Welch–based error correction that enables adversarial detection and recovery without sacrificing the DP-utility tradeoff, with theoretical guarantees and numerical validation. The results extend DP-secure computation to scenarios with erasures and active adversaries, approaching the fundamental converse bounds and enabling practical, resilient private computation over the real field.

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.
Paper Structure (17 sections, 8 theorems, 57 equations, 2 figures, 1 algorithm)

This paper contains 17 sections, 8 theorems, 57 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. System Model and Main Results
  3. System Model
  4. Main Results
  5. Proposed Coding Schemes
  6. Coding Schemes
  7. Differential Privacy Analysis
  8. Accuracy Analysis
  9. Error Correction Methods
  10. Numerical Simulations
  11. Analysis of Polynomial $\tilde{E}(x)$
  12. Analysis of Decoding Algorithm
  13. Additional Proofs
  14. Proof of Proposition \ref{['prop::safe_points']}
  15. Proof of Proposition \ref{['prop::nonzero_proposition']}
  16. ...and 2 more sections

Key Result

Theorem 1

For a positive integer $\mathsf{N}$ and non-negative integers $\mathsf{T},\mathsf{E}$ such that $\mathsf{N} \ge \mathsf{T}+\mathsf{E}+1$, there exists a secure multiplication scheme $\mathcal{C}$ that guarantees $\mathsf{T}$-node $\epsilon$-DP in presence of at most $\mathsf{E}$ erasures with the me where ${\tt SNR^*} = \frac{\eta}{\left(\sigma^*(\epsilon) \right)^2}$.

Figures (2)

  • Figure 1: Comparison of the error detection rate with different error variance.
  • Figure 2: Comparison of the mean square error with different error variance.

Theorems & Definitions (23)

  • Definition 1: Mean square error
  • Definition 2: $\mathsf{T}$-node $\epsilon$-DP
  • Theorem 1
  • Theorem 2
  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 1
  • Proposition 1
  • proof
  • ...and 13 more