Abstract: In the setting of $f$-DP, we propose the concept \emph{canonical noise distribution} (CND) which captures whether an additive privacy mechanism is tailored for a given $f$, and give a construction of a CND for an arbitrary tradeoff function $f$. We show that private hypothesis tests are intimately related to CNDs, allowing for the release of private $p$-values at no additional privacy cost as well as the construction of uniformly most powerful (UMP) tests for binary data. We apply our techniques to difference of proportions testing.