  In this paper, we address which sources contributed most of the reionizing
photons. Our argument assumes that the reionization ended around z ~ 6 and that
it was a relatively quick process, i.e., that there was a non-negligible
fraction of neutral hydrogen in the Universe at somewhat earlier epochs.
Starting from our earlier estimate of the luminosity function (LF) of galaxies
at z ~ 6, we quantitatively show that the major sources of reionization are
most likely galaxies with L < L*. Our approach allows us to put stronger
constraints to the LF of galaxies at z ~ 6. To have the Universe completely
ionized at this redshift, the faint-end slope of the LF should be steeper than
$\alpha=-1.6$, which is the value measured at lower redshifts (z ~ 3), unless
either the normalization (Phi*) of the LF or the clumping factor of the ionized
hydrogen has been significantly underestimated. If Phi* is actually lower than
what we assumed by a factor of two, a steep slope close to $\alpha=-2.0$ is
required. Our LF predicts a total of 50 -- 80 z ~ 6 galaxies in the HST Ultra
Deep Field (UDF) to a depth of AB=28.4 mag, which can be used to constraint
both Phi* and $\alpha$. We conclude that the least luminous galaxies existing
at this redshift should reach as low as some critical luminosity in order to
accumulate the entire reionizing photon budget. On the other hand, the
existence of significant amounts of neutral hydrogen at slightly earlier
epochs, e.g. z ~ 7, requires that the least luminous galaxies should not be
fainter than another critical value (i.e., the LF should cut-off at this
point).
