Abstract: This paper develops differentially private mechanisms for $\chi^2$ test of independence. While existing works put their effort into properly controlling the type-I error, in addition to that, we investigate the type-II error of differentially private mechanisms. Based on the analysis, we present unit circle mechanism: a novel differentially private mechanism based on the geometrical property of the test statistics. Compared to existing output perturbation mechanisms, our mechanism improves the dominated term of the type-II error from $O(1)$ to $O(\exp(-\sqrt{\numSample}))$ where $\numSample$ is the sample size. Furthermore, we introduce novel procedures for multiple $\chi^2$ tests by incorporating the unit circle mechanism into the sparse vector technique and the exponential mechanism. These procedures can control the family-wise error rate (FWER) properly, which has never been attained by existing mechanisms.