Analysis of Shuffling Beyond Pure Local Differential Privacy

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.