  Smooth sensitivity is one of the most commonly used techniques for designing
practical differentially private mechanisms. In this approach, one computes the
smooth sensitivity of a given query $q$ on the given input $D$ and releases
$q(D)$ with noise added proportional to this smooth sensitivity. One question
remains: what distribution should we pick the noise from?
  In this paper, we give a new class of distributions suitable for the use with
smooth sensitivity, which we name the PolyPlace distribution. This distribution
improves upon the state-of-the-art Student's T distribution in terms of
standard deviation by arbitrarily large factors, depending on a "smoothness
parameter" $\gamma$, which one has to set in the smooth sensitivity framework.
Moreover, our distribution is defined for a wider range of parameter $\gamma$,
which can lead to significantly better performance.
  Moreover, we prove that the PolyPlace distribution converges for $\gamma
\rightarrow 0$ to the Laplace distribution and so does its variance. This means
that the Laplace mechanism is a limit special case of the PolyPlace mechanism.
This implies that out mechanism is in a certain sense optimal for $\gamma \to
0$.
