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Differentially Private Secure Multiplication: Beyond Two Multiplicands

Haoyang Hu, Viveck R. Cadambe

TL;DR

A secure multiplication framework based on carefully designed encoding polynomials combined with layered noise injection is proposed that generalizes existing schemes and enables the systematic cancellation of lower-order noise terms, leading to improved estimation accuracy.

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.

Differentially Private Secure Multiplication: Beyond Two Multiplicands

TL;DR

A secure multiplication framework based on carefully designed encoding polynomials combined with layered noise injection is proposed that generalizes existing schemes and enables the systematic cancellation of lower-order noise terms, leading to improved estimation accuracy.

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.
Paper Structure (22 sections, 14 theorems, 121 equations, 3 figures)

This paper contains 22 sections, 14 theorems, 121 equations, 3 figures.

Table of Contents

  1. Introduction
  2. System Model and Main Results
  3. System Model
  4. Main Results
  5. Illustrative Examples
  6. $\mathsf{M}=2,\mathsf{N}=2, \mathsf{T}=1$
  7. $\mathsf{M}=3,\mathsf{N}=3, \mathsf{T}=1$
  8. $\mathsf{M}=3,\mathsf{N}=5, \mathsf{T}=2$
  9. Achievability: Proof of Theorem \ref{['thm::achievability']}
  10. Encoding Schemes
  11. Differential Privacy Analysis
  12. Accuracy Analysis
  13. Converse: Proof of Theorem \ref{['thm::converse']}
  14. Conclusion
  15. Achievability Proof ($\mathsf{N}= \mathsf{T} + 1, \mathsf{N} < \mathsf{M}$)
  16. ...and 7 more sections

Key Result

Theorem 1

Consider positive integers $\mathsf{N}, \mathsf{T}, \mathsf{M}$ with $(\mathsf{M}-1)\mathsf{T}+1 \leq \mathsf{N \le \mathsf{M} \mathsf{T}}$. For any $\epsilon,\xi > 0,$ there exists a coding scheme $\mathcal{C}$ that achieves $\mathsf{T}$-node $\epsilon$-DP secure multiplication with

Figures (3)

  • Figure 1: Illustration of the system model, where $\tilde{V}^{(j)} = \prod_{i \in [\mathsf{M}]} \bigl(A_i + \tilde{R}_i^{(j)}\bigr)$.
  • Figure 2: Performance of the proposed optimal scheme for $\mathsf{N}=5$, $\mathsf{M}=3$, and $\mathsf{T}=2$, compared with the baselines: (i) complex-valued Shamir secret sharing and (ii) independent noise across nodes.
  • Figure 3: Geometric interpretation of Theorem \ref{['thm::achievability']} for $\mathsf{N}=2$ and $\mathsf{T}=1$, illustrating signal and noise components in the estimation of $A_1$ and $A_1 A_2$.

Theorems & Definitions (23)

  • Definition 1: $\mathsf{T}$-node $\epsilon$-Differential Privacy ($\mathsf{T}$-node $\epsilon$-DP)
  • Definition 2: Linear Mean Square Error ($\tt LMSE$)
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Remark 1
  • Lemma 1: Theorem 7 in geng2015optimal
  • Lemma 2
  • proof
  • ...and 13 more