  Local differential privacy (LDP) is a variant of differential privacy (DP)
that avoids the need for a trusted central curator, at the cost of a worse
trade-off between privacy and utility. The shuffle model is a way to provide
greater anonymity to users by randomly permuting their messages, so that the
link between users and their reported values is lost to the data collector. By
combining an LDP mechanism with a shuffler, privacy can be improved at no cost
for the accuracy of operations insensitive to permutations, thereby improving
utility in many tasks. However, the privacy implications of shuffling are not
always immediately evident, and derivations of privacy bounds are made on a
case-by-case basis.
  In this paper, we analyze the combination of LDP with shuffling in the
rigorous framework of quantitative information flow (QIF), and reason about the
resulting resilience to inference attacks. QIF naturally captures randomization
mechanisms as information-theoretic channels, thus allowing for precise
modeling of a variety of inference attacks in a natural way and for measuring
the leakage of private information under these attacks. We exploit symmetries
of the particular combination of k-RR mechanisms with the shuffle model to
achieve closed formulas that express leakage exactly. In particular, we provide
formulas that show how shuffling improves protection against leaks in the local
model, and study how leakage behaves for various values of the privacy
parameter of the LDP mechanism.
  In contrast to the strong adversary from differential privacy, we focus on an
uninformed adversary, who does not know the value of any individual in the
dataset. This adversary is often more realistic as a consumer of statistical
datasets, and we show that in some situations mechanisms that are equivalent
w.r.t. the strong adversary can provide different privacy guarantees under the
uninformed one.
