This paper considers the secrecy outage performance of a multiple-relay assisted non-orthogonal multiple access (NOMA) network over Nakagami-m fading channels. Two time slots are utilized to transmit signals from the base station to destination. At the first time slot, the base station broadcasts the superposition signal of the two users to all decode-and-forward relays by message mapping strategy. Subsequently, the selected relay transmits superposition signal to the two users via power-domain NOMA technology. Three relay selection schemes, i.e., optimal single relay selection (OSRS) scheme, two-step single relay selection (TSRS) scheme, and optimal dual relay selection (ODRS) scheme are proposed and the secrecy outage performance is analyzed. As a benchmark, we also examine the secrecy outage performance of the NOMA systems with traditional multiple relay forwarding (TMRF) scheme in which all the relay that successfully decode signals from the source forward signals to the NOMA users with equal power. Considering the correlation between the secrecy capacity of two users and different secrecy requirement for two NOMA users, the analytical expressions for the security outage probability (SOP) of the proposed OSRS, TSRS, and ODRS schemes along with the TMRF scheme are derived and validated via simulations. To get more insights, we also derive the analytical expressions for the asymptotic SOP for all the schemes with fixed and dynamic power allocations. Furthermore, the secrecy diversity order (SDO) and secrecy array gain of cooperative NOMA systems are obtained. The results demonstrate that our proposed schemes can significantly enhance the secrecy performance compared to the TMRF scheme and that all the schemes with fixed power allocation obtain zero SDO and the OSRS scheme with dynamic power allocation obtains the same SDO as TMRF.