  The central question studied in this paper is Renyi Differential Privacy
(RDP) guarantees for general discrete local mechanisms in the shuffle privacy
model. In the shuffle model, each of the $n$ clients randomizes its response
using a local differentially private (LDP) mechanism and the untrusted server
only receives a random permutation (shuffle) of the client responses without
association to each client. The principal result in this paper is the first
non-trivial RDP guarantee for general discrete local randomization mechanisms
in the shuffled privacy model, and we develop new analysis techniques for
deriving our results which could be of independent interest. In applications,
such an RDP guarantee is most useful when we use it for composing several
private interactions. We numerically demonstrate that, for important regimes,
with composition our bound yields an improvement in privacy guarantee by a
factor of $8\times$ over the state-of-the-art approximate Differential Privacy
(DP) guarantee (with standard composition) for shuffled models. Moreover,
combining with Poisson subsampling, our result leads to at least $10\times$
improvement over subsampled approximate DP with standard composition.
