  Monte-Carlo computations for highly relativistic parallel shock particle
acceleration are presented for upstream flow gamma factors,
$\Gamma=(1-V_{1}^{2}/c^{2})^{-0.5}$ with values between 5 and $10^{3}$. The
results show that the spectral shape at the shock depends on whether or not the
particle scattering is small angle with $\delta \theta < \Gamma^{-1}$ or large
angle, which is possible if $\lambda > 2r_{g} \Gamma^{2}$ where $\lambda$ is
the scattering mean free path along the field line and $r_{g}$ the gyroradius,
these quantities being measured in the plasma flow frame. The large angle
scattering case exhibits distinctive structure superimposed on the basic
power-law spectrum, largely absent in the pitch angle case. Also, both cases
yield an acceleration rate faster than estimated by the conventional,
non-relativistic expression, $t_{acc}=[c/(V_{1}-V_{2})]
[\lambda_{1}/V_{1}+\lambda_{2}/V_{2}]$ where '1' and '2' refer to upstream and
downstream and $\lambda$ is the mean free path. A $\Gamma^{2}$ energy
enhancement factor in the first shock crossing cycle and a significant energy
multiplication in the subsequent shock cycles are also observed. The results
may be important for our understanding of the production of very high energy
cosmic rays and the high energy neutrino and gamma-ray output from Gamma Ray
Bursts (GRB) and Active Galactic Nuclei (AGN).
