  We show new lower bounds on the sample complexity of $(\varepsilon,
\delta)$-differentially private algorithms that accurately answer large sets of
counting queries. A counting query on a database $D \in (\{0,1\}^d)^n$ has the
form "What fraction of the individual records in the database satisfy the
property $q$?" We show that in order to answer an arbitrary set $\mathcal{Q}$
of $\gg nd$ counting queries on $D$ to within error $\pm \alpha$ it is
necessary that $$ n \geq \tilde{\Omega}\Bigg(\frac{\sqrt{d} \log
|\mathcal{Q}|}{\alpha^2 \varepsilon} \Bigg). $$ This bound is optimal up to
poly-logarithmic factors, as demonstrated by the Private Multiplicative Weights
algorithm (Hardt and Rothblum, FOCS'10). In particular, our lower bound is the
first to show that the sample complexity required for accuracy and
$(\varepsilon, \delta)$-differential privacy is asymptotically larger than what
is required merely for accuracy, which is $O(\log |\mathcal{Q}| / \alpha^2)$.
In addition, we show that our lower bound holds for the specific case of
$k$-way marginal queries (where $|\mathcal{Q}| = 2^k \binom{d}{k}$) when
$\alpha$ is not too small compared to $d$ (e.g. when $\alpha$ is any fixed
constant).
  Our results rely on the existence of short \emph{fingerprinting codes} (Boneh
and Shaw, CRYPTO'95, Tardos, STOC'03), which we show are closely connected to
the sample complexity of differentially private data release. We also give a
new method for combining certain types of sample complexity lower bounds into
stronger lower bounds.
