Growing Alphabets Do Not Automatically Amplify Shuffle Privacy: Obstruction, Estimation Bounds, and Optimal Mechanism Design

Abstract: We study neighboring shuffle experiments for epsilon_0-LDP channels along growing alphabets d -> infinity, and optimal mechanism design for frequency estimation under a canonical pairwise chi-squared budget. On the privacy side, we prove an exact compression theorem: the shuffled histogram experiment depends only on the pushforward law of the pairwise likelihood ratio. We establish a sharp universal bound chi^2 <= (e^{epsilon_0}-1)^2/e^{epsilon_0}, construct explicit obstruction families for which the shuffled privacy curve equals binary randomized response for all d, and prove a sharp diluting/persistent dichotomy. On the estimation side, we prove a universal lower bound of order (d-1)/(n chi_(W)) via Cramer-Rao and Assouad arguments, and show that symmetrization to equivariant channels is WLOG. On the design side, we show calibrated GRR is not optimal. The optimal mechanism is an augmented GRR: fraction p of users applies aggressive GRR with lambda_ = sqrt(d-1), the rest sends a null symbol. This thinning principle is specific to shuffle and has no local-DP counterpart. For low budget 0 < C <= C_*(d), augmented GRR is optimal among all permutation-equivariant channels. GRR is also the unique optimizer within the subset-selection family.