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**2019/1382**

**Title:**

On the Power of Multiple Anonymous Messages

**Authors:**
Badih Ghazi, Noah Golowich, Ravi Kumar, Rasmus Pagh, Ameya Velingker

**Abstract:**

An exciting new development in differential privacy is the shuffled model, in which an anonymous channel enables non-interactive, differentially private protocols with error much smaller than what is possible in the local model, while relying on weaker trust assumptions than in the central model. In this paper, we study basic counting problems in the shuffled model and establish separations between the error that can be achieved in the single-message shuffled model and in the shuffled model with multiple messages per user. For the problem of frequency estimation for n users and a domain of size B, we obtain: - A nearly tight lower bound of \tilde{\Omega}( \min(\sqrt[4]{n}, \sqrt{B})) on the error in the single-message shuffled model. This implies that the protocols obtained from the amplification via shuffling work of Erlingsson et al.~(SODA 2019) and Balle et al.~(Crypto 2019) are essentially optimal for single-message protocols. A key ingredient in the proof is a lower bound on the error of locally-private frequency estimation in the low-privacy (aka high \epsilon) regime. For this we develop new techniques to extend the results of Duchi et al.~(FOCS 2013; JASA 2018) and Bassily & Smith~(STOC 2015), whose techniques were restricted to the high-privacy case. - Protocols in the multi-message shuffled model with \poly(\log{B}, \log{n}) bits of communication per user and \poly\log{B} error, which provide an exponential improvement on the error compared to what is possible with single-message algorithms. This implies protocols with similar error and communication guarantees for several well-studied problems such as heavy hitters, d-dimensional range counting, M-estimation of the median and quantiles, and more generally sparse non-adaptive statistical query algorithms. For the related selection problem on a domain of size B, we prove: - A nearly tight lower bound of \Omega(B) on the number of users in the single-message shuffled model. This significantly improves on the \Omega(B^{1/17}) lower bound obtained by Cheu et al.~(Eurocrypt 2019), and when combined with their \tilde{O}(\sqrt{B})-error multi-message protocol, implies the first separation between single-message and multi-message protocols for this problem.

**ePrint:**
https://eprint.iacr.org/2019/1382

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