This letter investigates the ergodic secrecy rate (ESR) of a reconfigurable intelligent surface (RIS)-assisted communication system in the presence of discrete phase shifts and multiple eavesdroppers (Eves). 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 $\log_2 N$ for both non-colluding and colluding Eves. Numerical results are provided to verify the analytical results and the obtained scaling laws.