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- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula> 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 (SNR) 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 IRS elements and by the Nakagami- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula> fading parameters; 2) If the Nakagami- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula> parameter is no less than 2, the SOP of both User 1 and User 2 becomes unity, when the number of IRS elements tends to infinity; 3) The secrecy diversity orders are affected both by the number of IRS elements and by the Nakagami- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/...