  GRB 131231A was detected by the Large Area Telescope onboard Fermi Space
Gamma-ray Telescope. The high energy gamma-ray ($> 100$ MeV) afterglow emission
spectrum is $F_\nu \propto \nu^{-0.54\pm0.15}$ in the first $\sim 1300$ s after
the trigger and the most energetic photon has an energy $\sim 62$ GeV arriving
at $t\sim 520$ s. With reasonable parameters of the GRB outflow as well as the
density of the circum-burst medium, the synchrotron radiation of electrons or
protons accelerated at an external forward shock have difficulty accounting for
the data. The synchrotron self-Compton radiation of the forward
shock-accelerated electrons, instead, can account for both the spectrum and
temporal behavior of the GeV afterglow emission. We also show that the prospect
for detecting GRB 131231A$-$like GRBs with Cherenkov Telescope Array (CTA) is
promising.
