  We prove new lower bounds for statistical estimation tasks under the
constraint of $(\varepsilon, \delta)$-differential privacy. First, we provide
tight lower bounds for private covariance estimation of Gaussian distributions.
We show that estimating the covariance matrix in Frobenius norm requires
$\Omega(d^2)$ samples, and in spectral norm requires $\Omega(d^{3/2})$ samples,
both matching upper bounds up to logarithmic factors. The latter bound verifies
the existence of a conjectured statistical gap between the private and the
non-private sample complexities for spectral estimation of Gaussian
covariances. We prove these bounds via our main technical contribution, a broad
generalization of the fingerprinting method to exponential families.
Additionally, using the private Assouad method of Acharya, Sun, and Zhang, we
show a tight $\Omega(d/(\alpha^2 \varepsilon))$ lower bound for estimating the
mean of a distribution with bounded covariance to $\alpha$-error in
$\ell_2$-distance. Prior known lower bounds for all these problems were either
polynomially weaker or held under the stricter condition of $(\varepsilon,
0)$-differential privacy.
