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 $F_ u propto u^{-0.54pm0.15}$ in the first $sim 1300$ s after the trigger and the most energetic photon has an energy $sim 62$ GeV arriving at $tsim 520$ 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.
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