  Recently, $\varepsilon$-almost dual universal$_2$ hash functions has been
proposed as a new and wider class of hash functions. Using this class of hash
functions, several efficient hash functions were proposed. This paper evaluates
the security performance when we apply this kind of hash functions. We evaluate
the security in several kinds of setting based on the $L_1$ distinguishability
criterion and the modified mutual information criterion. The obtained
evaluation is based on smoothing of R\'{e}nyi entropy of order 2 and/or min
entropy. We clarify the difference between these two methods.
