Privacy Amplification via Unbundled Shuffling: Exact Asymptotics and an Analytic Privacy Accountant

This paper develops a fully analytic privacy accountant for the unbundled multi-message shuffle model. We derive exact asymptotics for Gaussian differential privacy parameters, establish a provable exponential separation between bundled and unbundled shuffling, and construct a closed-form, Edgeworth-corrected accountant that maps target privacy levels directly to admissible privacy budgets. All results are non-asymptotic in structure, rely only on explicit channel moments, and avoid numerical integration or FFT-based methods, yielding a self-contained and practically usable privacy accounting framework.