We obtain a new protocol for binary counting in the -shuffle-DP model with error and expected communication messages per user. Previous protocols incur either an error of with messages per user (Ghazi et al., ITC 2020) or an error of with messages per user (Cheu and Yan, TPDP 2022). Using the new protocol, we obtained improved -shuffle-DP protocols for real summation and histograms.
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