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In this work, we study the secrecy performance of a reconfigurable intelligent surfaces (RIS)-aided wireless communication system in the presence of an eavesdropping user. Specifically, we assume that the RIS is placed between the source and the legitimate user and is used to improve the link security. In particular, analytical results for the secrecy outage probability (SOP) are derived. We also provide an asymptotic analysis to investigate the effect of the main parameters on the secrecy performance of our proposed system, such as the number of tunable elements of the RIS and the average signal-to-noise ratios. Finally, we verify our analytical results via simulations. Results show the positive effect of utilizing RISs for enhancing the secrecy performance in wireless systems.