We study the problem of distribution testing when the samples can only be accessed using a locally differentially private mechanism and focus on two representative testing questions of identity (goodness-of-fit) and independence testing for discrete distributions. We are concerned with two settings: First, when we insist on using an already deployed, general-purpose locally differentially private mechanism such as the popular RAPPOR or the recently introduced Hadamard Response for collecting data, and must build our tests based on the data collected via this mechanism; and second, when no such restriction is imposed, and we can design a bespoke mechanism specifically for testing. For the latter purpose, we introduce the Randomized Aggregated Private Testing Optimal Response (RAPTOR) mechanism which is remarkably simple and requires only one bit of communication per sample. We propose tests based on these mechanisms and analyze their sample complexities. Each proposed test can be implemented efficiently. In each case (barring one), we complement our performance bounds for algorithms with information-theoretic lower bounds and establish sample optimality of our proposed algorithm. A peculiar feature that emerges is that our sample-optimal algorithm based on RAPTOR uses public-coins, and any test based on RAPPOR or Hadamard Response, which are both private-coin mechanisms, requires significantly more samples.

本研究针对分布测试问题，考虑使用本地差分隐私机制时的样本，并着重研究离散分布的两个代表性测试问题：完全匹配性检验和独立性检验。我们提出了基于RAPTOR和RAPPOR等机制的测试，并分析了它们的样本复杂度，我们的算法利用的是公共硬币且样本最优。

无信任测试：最优的本地隐私分布测试