  This paper presents a new method of Monte-Carlo simulations of test particle
Fermi acceleration at relativistic shocks. The particle trajectories in tangled
magnetic fields are integrated out exactly from entry to exit through the
shock, and the conditional probability of return as a function of ingress and
egress pitch angles is constructed by Monte-Carlo iteration. These upstream and
downstream probability laws are then used in conjunction with the energy gain
formula at shock crossing to reproduce Fermi acceleration. For pure Kolmogorov
magnetic turbulence upstream and downstream, the spectral index is found to
evolve smoothly from s=2.09 +/- 0.02 for mildly relativistic shocks with
Lorentz factor Gamma=2 to s=2.26 +/- 0.04 in the ultra-relativistic limit Gamma
>> 1. The energy gain is ~Gamma^2 at first shock crossing, and ~2 in all
subsequent cycles as anticipated by Gallant & Achterberg (1999). The
acceleration timescale is found to be as short as a fraction of Larmor time
when Gamma >> 1.
