  Recent years have witnessed growing concerns about the privacy of sensitive
data. In response to these concerns, differential privacy has emerged as a
rigorous framework for privacy protection, gaining widespread recognition in
both academic and industrial circles. While substantial progress has been made
in private data analysis, existing methods often suffer from impracticality or
a significant loss of statistical efficiency. This paper aims to alleviate
these concerns in the context of hypothesis testing by introducing
differentially private permutation tests. The proposed framework extends
classical non-private permutation tests to private settings, maintaining both
finite-sample validity and differential privacy in a rigorous manner. The power
of the proposed test depends on the choice of a test statistic, and we
establish general conditions for consistency and non-asymptotic uniform power.
To demonstrate the utility and practicality of our framework, we focus on
reproducing kernel-based test statistics and introduce differentially private
kernel tests for two-sample and independence testing: dpMMD and dpHSIC. The
proposed kernel tests are straightforward to implement, applicable to various
types of data, and attain minimax optimal power across different privacy
regimes. Our empirical evaluations further highlight their competitive power
under various synthetic and real-world scenarios, emphasizing their practical
value. The code is publicly available to facilitate the implementation of our
framework.
