  We investigate the destruction of dust grains by sputtering in the
high-velocity interstellar shocks driven by supernovae (SNe) in the early
universe to reveal the dependence of the time-scale of dust destruction on the
gas density $n_{{\rm H}, 0}$ in the interstellar medium (ISM) as well as on the
progenitor mass $M_{\rm pr}$ and explosion energy $E_{\rm 51}$ of SN. The
sputtering yields for the combinations of dust and ion species of interest to
us are evaluated by applying the so-called universal relation with a slight
modification. The dynamics of dust grains and their destruction by sputtering
in shock are calculated by taking into account the size distribution of each
dust species, together with the time evolution of temperature and density of
gas in spherically symmetric shocks. The results of calculations show that the
efficiency of dust destruction depends not only on the sputtering yield but
also on the initial size distribution of each grain species. The efficiency of
dust destruction increases with increasing $E_{\rm 51}$ and/or increasing
$n_{{\rm H}, 0}$, but is almost independent of $M_{\rm pr}$ as long as $E_{\rm
51}$ is the same. The mass of gas swept up by shock is the increasing function
of $E_{\rm 51}$ and the decreasing function of $n_{{\rm H}, 0}$. Combining
these results, we present the approximation formula for the time-scale of
destruction for each grain species in the early universe as a function of
$E_{\rm 51}$ and $n_{{\rm H}, 0}$. This formula is applicable for investigating
the evolution of dust grains at the early epoch of the universe with the
metallicity of $Z \la 10^{-3}$ $Z_\odot$. The effects of the cooling processes
of gas on the destruction of dust are briefly discussed.
