Mutual Information Bounds in the Shuffle Model (2511.15051v1)

Abstract: The shuffle model enhances privacy by anonymizing users' reports through random permutation. This paper presents the first systematic study of the single-message shuffle model from an information-theoretic perspective. We analyze two regimes: the shuffle-only setting, where each user directly submits its message ($Y_i=X_i$), and the shuffle-DP setting, where each user first applies a local $\varepsilon_0$-differentially private mechanism before shuffling ($Y_i=\mathcal{R}(X_i)$). Let $\boldsymbol{Z} = (Y_{σ(i)})i$ denote the shuffled sequence produced by a uniformly random permutation $σ$, and let $K = σ^{-1}(1)$ represent the position of user 1's message after shuffling. For the shuffle-only setting, we focus on a tractable yet expressive \emph{basic configuration}, where the target user's message follows $Y_1 \sim P$ and the remaining users' messages are i.i.d.\ samples from $Q$, i.e., $Y_2,\dots,Y_n \sim Q$. We derive asymptotic expressions for the mutual information quantities $I(Y_1;\boldsymbol{Z})$ and $I(K;\boldsymbol{Z})$ as $n \to \infty$, and demonstrate how this analytical framework naturally extends to settings with heterogeneous user distributions. For the shuffle-DP setting, we establish information-theoretic upper bounds on total information leakage. When each user applies an $\varepsilon_0$-DP mechanism, the overall leakage satisfies $I(K; \boldsymbol{Z}) \le 2\varepsilon_0$ and $I(X_1; \boldsymbol{Z}\mid (X_i){i=2}^n) \le (e^{\varepsilon_0}-1)/(2n) + O(n^{-3/2})$. These results bridge shuffle differential privacy and mutual-information-based privacy.

• The paper derives asymptotic bounds on mutual information, demonstrating how shuffling minimizes privacy leakage in data transmission.
• It analyzes both shuffle-only and shuffle-DP settings, using KL-divergence and chi-squared divergence to measure differential information leakage.
• Results show that combining random shuffling with local differential privacy effectively reduces data leakage, enhancing practical privacy guarantees.

Mutual Information Bounds in the Shuffle Model

Introduction

The paper "Mutual Information Bounds in the Shuffle Model" (2511.15051) explores the information-theoretic properties of the single-message shuffle model, which is a mechanism that enhances privacy by anonymizing users' data through random permutations. This work presents the first systematic analysis of this model from the perspective of mutual information and differential privacy. The shuffle model offers a method to amplify privacy guarantees of local differential privacy (LDP) mechanisms via random shuffling, providing an effective enhancement for statistical data release.

Theoretical Framework

In the shuffle model, a centralized shuffler applies a random permutation to user-submitted messages, disguising the origin of each message to enhance anonymity. The model is divided into two primary settings: the shuffle-only setting and the shuffle-DP (differential privacy) setting. In the shuffle-only setting, each user sends their message directly ($Y_i = X_i$), whereas, in the shuffle-DP setting, users apply a local $\varepsilon_0$-LDP mechanism before shuffling ($Y_i = \mathcal{R}(X_i)$).

The paper derives asymptotic expressions for mutual information in these settings to quantify information leakage. Specifically, it focuses on the mutual information $I(Y_1; \boldsymbol{Z})$ and $I(K; \boldsymbol{Z})$, where $K$ represents the position of a user's message after shuffling, and $Y_1$ is the corresponding message content.

Shuffle-Only Setting Analysis

The shuffle-only setting is initially simplified to a basic configuration where all users' messages are identically distributed. The results show that when the message distribution $P$ equals the common distribution $Q$ of other users, the mutual information about the message position is zero, indicating perfect anonymity (Figure 1). This is represented as $I(K;\boldsymbol{Z}) = 0$ for $P=Q$.

Further exploration into cases where $P \neq Q$ reveals that differential information leakage depends on the relative support of these distributions. When $P \ll Q$, asymptotic expressions show that mutual information concerning message position and value decrease inversely with the number of users, with specific decay rates characterized by statistical divergences such as KL-divergence and chi-squared divergence.

Figure 1: Exact vs. asymptotic mutual information in the basic shuffle-only setting with $P = Q$.

Shuffle-DP Setting Analysis

For the shuffle-DP setting, where local differential privacy mechanisms precede shuffling, the paper shows that $I(K;\boldsymbol{Z}|\boldsymbol{X})$ satisfies an upper bound of $2\varepsilon_0$. This signifies a substantial reduction in information leakage, since shuffling significantly amplifies differential privacy protocols. In terms of message content, the information leakage $I(X_1;\boldsymbol{Z}|\boldsymbol{X}_{-1})$ is shown to be bounded by $(e^{\varepsilon_0} - 1)/(2n)$, plus higher-order terms, illustrating a further reduction in the adversary's ability to infer the target user's input.

Figure 2: Mutual information in the shuffle-DP setting: numerical estimates vs. asymptotic bounds.

Implications and Future Directions

The results provide a foundational understanding of how shuffling impacts privacy from an information-theoretic viewpoint, confirming that the shuffle model effectively reduces information leakage and enhances practical privacy guarantees. This paper bridges the gap between differential privacy and mutual information frameworks by demonstrating how anonymization and local randomization compound to limit private data leakage.

Future research directions could explore more realistic modeling scenarios with heterogeneous user distributions and further refinement of analytical techniques to deal with complex dependencies among user data. Additionally, developing closed-form non-asymptotic bounds for mutual information in these privacy settings could significantly advance the theoretical framework of privacy-preserving data analysis.

Conclusion

This paper offers a rigorous information-theoretic perspective on the shuffle model, uncovering critical insights into its privacy-preserving properties. By systematically analyzing the mutual information bounds, it elucidates the potential of shuffling to augment privacy in both theoretical and practical aspects, providing a robust foundation for future research and applications in privacy-aware computation.