This paper investigates physical layer security (PLS) in cognitive radio inspired non-orthogonal multiple access (CR-NOMA) networks with multiple primary and secondary users. To manage the interferences among the users and guarantee the quality of services of primary users, a new secure NOMA transmission strategy is designed, where the primary and secondary users are paired according to their channel gains, respectively, and power-domain NOMA is employed to transmit the signal. Then, the closed-form expressions for connection outage probability, secrecy outage probability, and effective secrecy throughput are derived for the primary users over Nakagami-<inline-formula><tex-math notation="LaTeX">m</tex-math></inline-formula> fading channels when the secondary users are considered as eavesdroppers. Typically, the secrecy performance can be improved by pairing the primary users with best channel gains or reducing the number of secondary users. In addition, we also investigate the performances of secondary users by deriving the closed-form expressions for throughput of secondary network. Furthermore, simulations are conducted to verify our analysis results and provide insights into the impact of the parameters on system performance.