Analysis of Shuffling Beyond Pure Local Differential Privacy

2026-01-27

AI Summary

This work addresses the limitations of existing privacy amplification analyses under the shuffle model, which are largely confined to pure local differential privacy and yield loose upper bounds that fail to accurately capture the privacy gains of fundamental randomizers like the Gaussian mechanism. By revisiting the blanket divergence, we develop an asymptotic analytical framework applicable to a broad class of local randomization mechanisms without requiring pure local differential privacy assumptions. We introduce the “shuffle exponent” χ to uniformly quantify each local mechanism’s contribution to overall privacy. Our approach establishes asymptotically tight upper and lower bounds for (εₙ, δₙ)-privacy guarantees, provides necessary and sufficient structural conditions for their consistency, and devises an efficient FFT-based algorithm enabling near-linear time complexity and controllable numerical error in privacy evaluation. The framework yields significantly tighter practical privacy bounds for k-RR (k ≥ 3) and generalized Gaussian mechanisms.

Technology Category

• Machine Learning: Privacy
• Search and Optimization: Local Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

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

Abstract

Shuffling is a powerful way to amplify privacy of a local randomizer in private distributed data analysis, but existing analyses mostly treat the local differential privacy (DP) parameter $\varepsilon_0$ as the only knob and give generic upper bounds that can be loose and do not even characterize how shuffling amplifies privacy for basic mechanisms such as the Gaussian mechanism. We revisit the privacy blanket bound of Balle et al. (the blanket divergence) and develop an asymptotic analysis that applies to a broad class of local randomizers under mild regularity assumptions, without requiring pure local DP. Our key finding is that the leading term of the blanket divergence depends on the local mechanism only through a single scalar parameter $\chi$, which we call the shuffle index. By applying this asymptotic analysis to both upper and lower bounds, we obtain a tight band for $\delta_n$ in the shuffled mechanism's $(\varepsilon_n,\delta_n)$-DP guarantee. Moreover, we derive a simple structural necessary and sufficient condition on the local randomizer under which the blanket-divergence-based upper and lower bounds coincide asymptotically. $k$-RR families with $k\ge3$ satisfy this condition, while for generalized Gaussian mechanisms the condition may not hold but the resulting band remains tight. Finally, we complement the asymptotic theory with an FFT-based algorithm for computing the blanket divergence at finite $n$, which offers rigorously controlled relative error and near-linear running time in $n$, providing a practical numerical analysis for shuffle DP.

Problem

• shuffling
• local differential privacy
• privacy amplification
• Gaussian mechanism
• blanket divergence

Innovation

• shuffle privacy
• blanket divergence
• shuffle index
• asymptotic analysis
• FFT-based computation