  Adding random noise to database query results is an important tool for
achieving privacy. A challenge is to minimize this noise while still meeting
privacy requirements. Recently, a sufficient and necessary condition for
$(\epsilon, \delta)$-differential privacy for Gaussian noise was published.
This condition allows the computation of the minimum privacy-preserving scale
for this distribution. We extend this work and provide a sufficient and
necessary condition for $(\epsilon, \delta)$-differential privacy for all
symmetric and log-concave noise densities. Our results allow fine-grained
tailoring of the noise distribution to the dimensionality of the query result.
We demonstrate that this can yield significantly lower mean squared errors than
those incurred by the currently used Laplace and Gaussian mechanisms for the
same $\epsilon$ and $\delta$.
