Mutual Information Bounds in the Shuffle Model

TL;DR

This paper provides an information-theoretic analysis of the shuffle model in privacy, deriving asymptotic mutual information bounds for different regimes and connecting shuffle differential privacy with mutual information measures.

Contribution

It introduces the first systematic information-theoretic study of the single-message shuffle model, deriving asymptotic bounds and extending to heterogeneous user distributions.

Findings

Asymptotic expressions for mutual information in the shuffle-only setting.

Upper bounds on information leakage in the shuffle-DP setting.

Connection between shuffle differential privacy and mutual information measures.

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 $\boldsymbol{Z} = (Y_{\sigma(i)})_i$ denote the shuffled sequence produced by a uniformly random permutation $\sigma$, and let $K = \sigma^{-1}(1)$ 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.