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Recent work in differential privacy has highlighted the shuffled model as a promising avenue to compute accurate statistics while keeping raw data in users' hands. We present a protocol in this model that estimates histograms with error independent of the domain size. This implies an arbitrarily large gap in sample complexity between the shuffled and local models. On the other hand, we show that the models are equivalent when we impose the constraints of pure differential privacy and single-message randomizers.
Balcer et al. (Fri,) studied this question.