Reconfigurable Intelligent Surface (RIS) technology is emerging as a promising performance enhancement for next-generation wireless networks in terms of the quality of service and radio connectivity. Inspired by the promising potential of RIS technology, we investigate the secrecy performance of the downlink RIS-aided non-orthogonal multiple access network. To characterize the network’s performance, the expectation of the new channel statistics for the reflected links with Nakagami-m fading is derived. Furthermore, the performance of the proposed network is evaluated according to the secrecy outage probability (SOP). The closed-form expressions of the SOP are derived. To obtain further insights, the asymptotic SOP and secrecy diversity orders are obtained. Our analytical results demonstrate that: 1) the expectation of channel gain for the reflected links is determined by the number of RISs and the Nakagami-m fading parameters; 2) Both the SOP of user Bob1 and the SOP of user Bob2 are 1 when the number of RISs is sufficiently large; 3) The secrecy diversity orders are affected by the number of RISs and Nakagami-m fading parameters.