- Badih Ghazi
- Noah Golowich
- Ravi Kumar
- Pasin Manurangsi
- Rasmus Pagh
- Ameya Velingker

## Abstract

The shuffled (aka anonymous) model has recently generated significant interest as a candidate dis- tributed privacy framework with trust assumptions better than the central model but with achievable error rates smaller than the local model. In this paper, we study pure differentially private protocols in the shuffled model for summation, a very basic and widely used primitive. Specifically: • For the binary summation problem where each of n users holds a bit as an input, we give a pure ε- differentially private protocol for estimating the number of ones held by the users up to an absolute error of Oε(1), and where each user sends Oε(logn) messages each consisting of a single bit. This √ is the first pure differentially private protocol in the shuffled model with error o( n) for constant values of ε. Using our binary summation protocol as a building block, we give a pure ε-differentially private protocol that performs summation of real numbers (in [0,1]) up to an absolute error of Oε(1), and where each user sends Oε(log3 n) messages each consisting of O(loglogn) bits. • In contrast, we show that for any pure ε-differentially private protocol for binary summation in the shuffled model having absolute error n0.5−Ω(1), the per user communication has to be at least Ωε( log n) bits. This implies (i) the first separation between the (bounded-communication) multi- message shuffled model and the central model, and (ii) the first separation between pure and approximate differentially private protocols in the shuffled model. Interestingly, over the course of proving our lower bound, we have to consider (a generalization of) the following question which might be of independent interest: given γ ∈ (0, 1), what is the smallest positive integer m for which there exist two random variables X0 and X1 supported on {0, . . . , m} such that (i) the total variation distance between X0 and X1 is at least 1 − γ, and (ii) the moment generating functions of X0 and X1 are within a constant factor of each other everywhere? We show that the answer to this question is m = Θ(

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