Fixed-Composition Shuffle Asymptotics in the Full-Support Gaussian Regime

Abstract

We study privacy amplification by shuffling for binary-input local randomizers with a fixed finite output alphabet and full support. For a dataset containing exactly k ones among n users, let T_{n,k} denote the shuffled histogram law. For fixed-composition neighboring shuffled histogram laws in the interior regime, we identify the covariance and Fisher constant governing the neighboring pair (T_{n,k},T_{n,k+1}). For a composition parameter pi in [0,1], the correct covariance is Sigma_pi=(1-pi)Sigma_0+pi Sigma_1 rather than the multinomial covariance of the mixture. With v=W_1-W_0, the resulting constant is I_pi=v^T Sigma_pi^+ v. We prove exact likelihood-ratio identities and a regression decomposition with residual moments E[R^2]=O(n^{-2}) and E[R^4]=O(n^{-4}), uniformly over interior compositions. Consequently, JSD(T_{n,k}||T_{n,k+1})=I_{k/n}/(8n)+O(n^{-2}), and the same constant governs smooth divergence asymptotics. For mu_n=(I_{k/n}/n)^{1/2}, both directed hockey-stick privacy curves at epsilon=t mu_n equal mu_n{phi(t)-t Phi(-t)}+O(n^{-1}) uniformly for t in compact sets. Exact finite-n accounting formulas and fixed-message unbundled specializations are also provided.