  It is known that if a 2-universal hash function $H$ is applied to elements of
a {\em block source} $(X_1,...,X_T)$, where each item $X_i$ has enough
min-entropy conditioned on the previous items, then the output distribution
$(H,H(X_1),...,H(X_T))$ will be ``close'' to the uniform distribution. We
provide improved bounds on how much min-entropy per item is required for this
to hold, both when we ask that the output be close to uniform in statistical
distance and when we only ask that it be statistically close to a distribution
with small collision probability. In both cases, we reduce the dependence of
the min-entropy on the number $T$ of items from $2\log T$ in previous work to
$\log T$, which we show to be optimal. This leads to corresponding improvements
to the recent results of Mitzenmacher and Vadhan (SODA `08) on the analysis of
hashing-based algorithms and data structures when the data items come from a
block source.
