  The shuffle model of differential privacy has gained significant interest as
an intermediate trust model between the standard local and central models
[EFMRTT19; CSUZZ19]. A key result in this model is that randomly shuffling
locally randomized data amplifies differential privacy guarantees. Such
amplification implies substantially stronger privacy guarantees for systems in
which data is contributed anonymously [BEMMRLRKTS17].
  In this work, we improve the state of the art privacy amplification by
shuffling results both theoretically and numerically. Our first contribution is
the first asymptotically optimal analysis of the R\'enyi differential privacy
parameters for the shuffled outputs of LDP randomizers. Our second contribution
is a new analysis of privacy amplification by shuffling. This analysis improves
on the techniques of [FMT20] and leads to tighter numerical bounds in all
parameter settings.
