Optimal Pure Differentially Private Sparse Histograms in Deterministic Linear Time

TL;DR

This paper introduces an optimal, fast algorithm for releasing pure differentially private sparse histograms with minimal error, running in linear time and enabling efficient secure multi-party computation protocols.

Contribution

It presents the first linear-time deterministic algorithm for pure DP sparse histograms with optimal error, solving a longstanding open problem and introducing a novel private item blanket technique.

Findings

Achieves optimal $\, ext{ell}_ ext{infty}$-error in linear time

Enables near-linear communication and computation in MPC protocols

Breaks previous quadratic time barrier for pure DP histogram algorithms

Abstract

We present an algorithm that releases a pure differentially private (under the replacement neighboring relation) sparse histogram for $n$ participants over a domain of size $d \gg n$. Our method achieves the optimal $\ell_\infty$-estimation error and runs in strictly $O(n)$ time in the Word-RAM model, improving upon the previous best deterministic-time bound of $\tilde{O}(n^2)$ and resolving the open problem of breaking this quadratic barrier (Balcer and Vadhan, 2019). Moreover, the algorithm admits an efficient circuit implementation, enabling the first near-linear communication and computation cost pure DP histogram MPC protocol with optimal $\ell_\infty$-estimation error. Central to our algorithm is a novel **private item blanket** technique with target-length padding, which hides differences in the supports of neighboring histograms while remaining efficiently implementable.