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Intelligent reflective surface (IRS) technology is emerging as a promising performance enhancement technique for next-generation wireless networks. Hence, we investigate the physical layer security of the downlink in IRS-aided non-orthogonal multiple access networks in the presence of an eavesdropper, where an IRS is deployed for enhancing the quality by assisting the cell-edge user to communicate with the base station. To characterize the network's performance, the expected value of the new channel statistics is derived for the reflected links in the case of Nakagami-m fading. Furthermore, the performance of the proposed network is evaluated both in terms of the secrecy outage probability (SOP) and the average secrecy capacity (ASC). The closed-form expressions of the SOP and the ASC are derived. We also study the impact of various network parameters on the overall performance of the network considered. To obtain further insights, the secrecy diversity orders and the high signal-to-noise ratio slopes are obtained. We finally show that: 1) the expectation of the channel gain in the reflected links is determined both by the number of IRSs and by the Nakagami-m fading parameters; 2) The SOP of both receiver 1 and receiver 2 becomes unity, when the number of IRSs is sufficiently high; 3) The secrecy diversity orders are affected both by the number of IRSs and by the Nakagami-m fading parameters, whereas the high-SNR slopes are not affected by these parameters. Our Monte-Carlo simulations perfectly demonstrate the analytical results.