Assuming millimeter wave (mmWave) systems, we consider an analog beamforming-based physical layer security (PLS) technique, which has considerably lower complexity compared to the digital beamforming-based PLS. Motivated by the limitations of the related prior arts assuming a uniform linear array (ULA), we employ a uniform planar array (UPA) to take advantage of both azimuth and elevation domains to realize three-dimensional (3D) beamforming. Also, in contrast to the approximate analysis on the secrecy rate (SR) in the existing studies, we derive the exact SR without any approximation. Based on this analysis, we analyze the optimal size of random antenna subset, which is the key parameter to maximize the SR.