  We study a hypothesis testing problem with a privacy constraint over a noisy
channel and derive the performance of optimal tests under the Neyman-Pearson
criterion. The fundamental limit of interest is the privacy-utility tradeoff
(PUT) between the exponent of the type-II error probability and the leakage of
the information source subject to a constant constraint on the type-I error
probability. We provide an exact characterization of the asymptotic PUT for any
non-vanishing type-I error probability. Our result implies that tolerating a
larger type-I error probability cannot improve the PUT. Such a result is known
as a strong converse or strong impossibility theorem. To prove the strong
converse theorem, we apply the recently proposed technique in (Tyagi and
Watanabe, 2020) and further demonstrate its generality. The strong converse
theorems for several problems, such as hypothesis testing against independence
over a noisy channel (Sreekumar and G\"und\"uz, 2020) and hypothesis testing
with communication and privacy constraints (Gilani \emph{et al.}, 2020), are
established or recovered as special cases of our result.
