Abstract: We propose the first federated correlation test framework compatible with secure aggregation, namely FED-$\chi^2$. In our protocol, the statistical computations are recast as frequency moment estimation problems, where the clients collaboratively generate a shared projection matrix and then use stable projection to encode the local information in a compact vector. As such encodings can be linearly aggregated, secure aggregation can be applied to conceal the individual updates. We formally establish the security guarantee of FED-$\chi^2$ by proving that only the minimum necessary information (i.e., the correlation statistics) is revealed to the server. We show that our protocol can be naturally extended to estimate other statistics that can be recast as frequency moment estimations. By accommodating Shannon'e Entropy in FED-$\chi^2$, we further propose the first secure federated entropy estimation protocol, FED-$H$. The evaluation results demonstrate that FED-$\chi^2$ and FED-$H$ achieve good performance with small client-side computation overhead in several real-world case studies.