On the Power of Multiple Anonymous Messages

An exciting new development in differential privacy is the shuffled model, in which an anonymous channel enables circumventing the large errors that are necessary in the local model, while relying on much 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 frequency estimation problem with $n$ users and for a domain of size $B$, we obtain: - A nearly tight lower bound of $\tilde{\Omega}( \min(n^{1/4}, \sqrt{B}))$ 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 nearly tight lower bound of $\Omega\left(\frac{\log{B}}{\log\log{B}}\right)$ on the sample complexity with constant relative error in the single-message shuffled model. This improves on the lower bound of $\Omega(\log^{1/17} B)$ obtained by Cheu et al. (Eurocrypt 2019). - Protocols in the multi-message shuffled model with $\mathrm{poly}(\log{B}, \log{n})$ bits of communication per user and $\mathrm{poly}\log{B}$ error, which provide an exponential improvement on the error compared to what is possible with single-message algorithms. For the related selection problem, we also show a nearly tight sample complexity lower bound of $\Omega(B)$ in the single-message shuffled model. This 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 algorithm, implies the first separation between single-message and multi-message protocols for this problem.