This paper investigates the secrecy performance of a two-user downlink non-orthogonal multiple access systems. Both single-input and single-output and multiple-input and singleoutput systems with different transmit antenna selection (TAS) strategies are considered. Depending on whether the base station has the global channel state information of both the main and wiretap channels, the exact closed-form expressions for the secrecy outage probability (SOP) with suboptimal antenna selection and optimal antenna selection schemes are obtained and compared with the traditional space-time transmission scheme. To obtain further insights, the asymptotic analysis of the SOP in high average channel power gains regime is presented and it is found that the secrecy diversity order for all the TAS schemes with fixed power allocation is zero. Furthermore, an effective power allocation scheme is proposed to obtain the nonzero diversity order with all the TAS schemes. Monte-Carlo simulations are performed to verify the proposed analytical results.