  We obtain a new protocol for binary counting in the $\varepsilon$-shuffle-DP
model with error $O(1/\varepsilon)$ and expected communication
$\tilde{O}\left(\frac{\log n}{\varepsilon}\right)$ messages per user. Previous
protocols incur either an error of $O(1/\varepsilon^{1.5})$ with
$O_\varepsilon(\log{n})$ messages per user (Ghazi et al., ITC 2020) or an error
of $O(1/\varepsilon)$ with $O_\varepsilon(n^{2.5})$ messages per user (Cheu and
Yan, TPDP 2022). Using the new protocol, we obtained improved
$\varepsilon$-shuffle-DP protocols for real summation and histograms.
