  The max-relative entropy together with its smoothed version is a basic tool
in quantum information theory. In this paper, we derive the exact exponent for
the asymptotic decay of the small modification of the quantum state in
smoothing the max-relative entropy based on purified distance. We then apply
this result to the problem of privacy amplification against quantum side
information, and we obtain an upper bound for the exponent of the asymptotic
decreasing of the insecurity, measured using either purified distance or
relative entropy. Our upper bound complements the earlier lower bound
established by Hayashi, and the two bounds match when the rate of randomness
extraction is above a critical value. Thus, for the case of high rate, we have
determined the exact security exponent. Following this, we give examples and
show that in the low-rate case, neither the upper bound nor the lower bound is
tight in general. This exhibits a picture similar to that of the error exponent
in channel coding. Lastly, we investigate the asymptotics of equivocation and
its exponent under the security measure using the sandwiched R\'enyi divergence
of order $s\in (1,2]$, which has not been addressed previously in the quantum
setting.
