In this study, we propose a novel all-optical Galois Field (AOGF) adder that utilizes logic all-optical XOR gates. The design is founded on the constructive and destructive interference phenomenon of optical beams and incorporates the phase shift keying (PSK) technique within a two-dimensional linear photonic crystal (2D-LPhC) structure. The suggested AOGF adder comprises eight input ports and four output ports. To obtain the electric field distribution in this structure, we employ the Finite Difference Time Domain (FDTD) procedure. The FDTD simulation results of the proposed AOGF adder demonstrate that the minimum and maximum values of the normalized power at ON and OFF states (P_(1,min), P_(0,max)) for the output ports are 95% and 1.7%, respectively. Additionally, we obtain different functional parameters, including the ON-OFF contrast ratio, rise time, fall time, and total footprint, which are measured at 17.47 dB, 0.1 ps, 0.05 ps, and 147 μm2, respectively.