In the emph{shuffle model} of differential privacy, data-holding users send randomized messages to a secure shuffler, the shuffler permutes the messages, and the resulting collection of messages must be differentially private with regard to user data. In the emph{pan-private} model, an algorithm processes a stream of data while maintaining an internal state that is differentially private with regard to the stream data. We give evidence connecting these two apparently different models. Our results focus on emph{robustly} shuffle private protocols, whose privacy guarantees are not greatly affected by malicious users. First, we give robustly shuffle private protocols and upper bounds for counting distinct elements and uniformity testing. Second, we use pan-private lower bounds to prove robustly shuffle private lower bounds for both problems. Focusing on the dependence on the domain size $k$, we find that robust approximate shuffle privacy and approximate pan-privacy have additive error $Theta(sqrt{k})$ for counting distinct elements. For uniformity testing, we give a robust approximate shuffle private protocol with sample complexity $tilde O(k^{2/3})$ and show that an $Omega(k^{2/3})$ dependence is necessary for any robust pure shuffle private tester. Finally, we show that this connection is useful in both directions: we give a pan-private adaptation of recent work on shuffle private histograms and use it to recover further separations between pan-privacy and interactive local privacy.
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