Exact Second-Order Jensen–Shannon Asymptotics in the Unbundled Shuffle Model

We derive a sharp second-order asymptotic expansion for the Jensen–Shannon divergence in the unbundled multi-message shuffle model. The proof is fully analytic and provides complete control of the remainder term. The results show an exponential separation between bundled and unbundled shuffling: bundling amplifies divergence exponentially in the number of messages per user, while unbundled shuffling preserves linear scaling.

Keywords

multi-message protocols, differential privacy, U-statistics, privacy amplification, Jensen–Shannon divergence, shuffle model

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