This letter investigates the ergodic secrecy rate (ESR) of a reconfigurable intelligent surface (RIS)-assisted communication system in the presence multiple eavesdroppers (Eves), and by assuming discrete phase shifts at the RIS. In particular, a closed-form approximation of the ESR is derived for both non-colluding and colluding Eves. The analytical results are shown to be accurate when the number of reflecting elements of the RIS N is large. Asymptotic analysis is provided to investigate the impact of N on the ESR, and it is proved that the ESR scales with log2N for both non-colluding and colluding Eves. Numerical results are provided to verify the analytical results and the obtained scaling laws.