  We evaluate the asymptotics of equivocations, their exponents as well as
their second-order coding rates under various R\'{e}nyi information measures.
Specifically, we consider the effect of applying a hash function on a source
and we quantify the level of non-uniformity and dependence of the compressed
source from another correlated source when the number of copies of the sources
is large. Unlike previous works that use Shannon information measures to
quantify randomness, information or uniformity, we define our security measures
in terms of a more general class of information measures--the R\'{e}nyi
information measures and their Gallager-type counterparts. A special case of
these R\'{e}nyi information measure is the class of Shannon information
measures. We prove tight asymptotic results for the security measures and their
exponential rates of decay. We also prove bounds on the second-order
asymptotics and show that these bounds match when the magnitudes of the
second-order coding rates are large. We do so by establishing new classes
non-asymptotic bounds on the equivocation and evaluating these bounds using
various probabilistic limit theorems asymptotically.
