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Analysis of Shuffling Beyond Pure Local Differential Privacy

Shun Takagi, Seng Pei Liew

TL;DR

This work addresses how privacy amplification by shuffling behaves beyond pure local DP, introducing a single mechanism parameter called the shuffle index $\chi$ that captures the leading effect of shuffling for a broad class of local randomizers. By combining a CLT-based asymptotic analysis of the blanket divergence with an FFT-based finite-$n$ accountant, the authors derive tight $ (\varepsilon_n,\delta_n) $-DP bands and provide a practical, mechanism-aware toolkit for shuffle DP. Key contributions include a unified non-pure local-DP analysis, a necessary-and-sufficient optimality condition for the asymptotic bounds, and an efficient FFT-based algorithm with provable relative-error guarantees that runs near-linearly in the number of users. The practical impact lies in enabling precise, mechanism-aware privacy accounting for shuffled DP protocols, including generalized Gaussian mechanisms, with strong empirical validation and scalable numerical tools for real-world deployments.

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 $χ$, which we call the shuffle index. By applying this asymptotic analysis to both upper and lower bounds, we obtain a tight band for $δ_n$ in the shuffled mechanism's $(\varepsilon_n,δ_n)$-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.

Analysis of Shuffling Beyond Pure Local Differential Privacy

TL;DR

This work addresses how privacy amplification by shuffling behaves beyond pure local DP, introducing a single mechanism parameter called the shuffle index that captures the leading effect of shuffling for a broad class of local randomizers. By combining a CLT-based asymptotic analysis of the blanket divergence with an FFT-based finite- accountant, the authors derive tight -DP bands and provide a practical, mechanism-aware toolkit for shuffle DP. Key contributions include a unified non-pure local-DP analysis, a necessary-and-sufficient optimality condition for the asymptotic bounds, and an efficient FFT-based algorithm with provable relative-error guarantees that runs near-linearly in the number of users. The practical impact lies in enabling precise, mechanism-aware privacy accounting for shuffled DP protocols, including generalized Gaussian mechanisms, with strong empirical validation and scalable numerical tools for real-world deployments.

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 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 , which we call the shuffle index. By applying this asymptotic analysis to both upper and lower bounds, we obtain a tight band for in the shuffled mechanism's -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. -RR families with 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 , which offers rigorously controlled relative error and near-linear running time in , providing a practical numerical analysis for shuffle DP.
Paper Structure (39 sections, 24 theorems, 428 equations, 2 figures, 2 algorithms)

This paper contains 39 sections, 24 theorems, 428 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Limitations of existing shuffle analyses.
  3. The privacy blanket and blanket divergence.
  4. Our goal and approach.
  5. Our contributions.
  6. Preliminaries
  7. Notations
  8. Differential Privacy (DP)
  9. Local Randomizers Beyond Pure Local DP
  10. Analysis of Shuffling via the Central Limit Theorem (CLT)
  11. Analysis of Blanket Divergence
  12. Main Result
  13. Shuffle Index
  14. Analysis of Blanket Divergence via the Fast Fourier Transform (FFT)
  15. FFT Algorithm
  16. ...and 24 more sections

Key Result

lemma 1

Fix $\varepsilon \geq 0$, let $x_{1:n} \simeq x_{1:n}^\prime$ be neighboring inputs where only the first element differs (i.e., $x_1 \neq x^\prime_1$), and $\mathcal{R}$ has blanket mass $\gamma$. Then, $\mathcal{S} \circ \mathcal{R}^n$ satisfies $(\varepsilon,\delta(x_{1:n}, x_{1:n}^\prime))$-DP fo Here, $\mathrm{Bin}$ is a binomial distribution and $l_\varepsilon(y) := \frac{\mathcal{R}_{x_1}(y)

Figures (2)

  • Figure 1: Experimental validation of the FFT-based blanket-divergence accountant and the shuffle-index analysis.
  • Figure 2: Shuffle indices for generalized Gaussian mechanisms with different shape parameters $\beta$.

Theorems & Definitions (45)

  • definition 1: Hockey-stick divergence
  • definition 2: Differential privacy via hockey-stick divergence dworkAlgorithmicFoundationsDifferential2013abartheDifferentialPrivacyComposition2013
  • definition 3: Local randomizer and blanket distribution
  • definition 4: Single-message shuffling and shuffled mechanism
  • lemma 1: Lemma 5.3 of Balle et al. ballePrivacyBlanketShuffle2019
  • Theorem 2.1: Theorem 14 of Su et al. suDecompositionBasedOptimalBounds2025
  • definition 5: Privacy amplification random variable
  • definition 6: Generalized Gaussian local randomizer
  • lemma 2: Transformations for the blanket divergence
  • lemma 3: Asymptotic expansion of the blanket divergence
  • ...and 35 more