  We provide improved differentially private algorithms for identity testing of
high-dimensional distributions. Specifically, for $d$-dimensional Gaussian
distributions with known covariance $\Sigma$, we can test whether the
distribution comes from $\mathcal{N}(\mu^*, \Sigma)$ for some fixed $\mu^*$ or
from some $\mathcal{N}(\mu, \Sigma)$ with total variation distance at least
$\alpha$ from $\mathcal{N}(\mu^*, \Sigma)$ with $(\varepsilon, 0)$-differential
privacy, using only \[\tilde{O}\left(\frac{d^{1/2}}{\alpha^2} +
\frac{d^{1/3}}{\alpha^{4/3} \cdot \varepsilon^{2/3}} + \frac{1}{\alpha \cdot
\varepsilon}\right)\]
  samples if the algorithm is allowed to be computationally inefficient, and
only \[\tilde{O}\left(\frac{d^{1/2}}{\alpha^2} + \frac{d^{1/4}}{\alpha \cdot
\varepsilon}\right)\]
  samples for a computationally efficient algorithm. We also provide a matching
lower bound showing that our computationally inefficient algorithm has optimal
sample complexity. We also extend our algorithms to various related problems,
including mean testing of Gaussians with bounded but unknown covariance,
uniformity testing of product distributions over $\{-1, 1\}^d$, and tolerant
testing.
  Our results improve over the previous best work of Canonne et
al.~\cite{CanonneKMUZ20} for both computationally efficient and inefficient
algorithms, and even our computationally efficient algorithm matches the
optimal \emph{non-private} sample complexity of
$O\left(\frac{\sqrt{d}}{\alpha^2}\right)$ in many standard parameter settings.
  In addition, our results show that, surprisingly, private identity testing of
$d$-dimensional Gaussians can be done with fewer samples than private identity
testing of discrete distributions over a domain of size $d$ \cite{AcharyaSZ18},
which refutes a conjectured lower bound of~\cite{CanonneKMUZ20}.
