We study the secrecy transmission of uplink non-orthogonal multiple access (NOMA) with single antenna and multi-antenna users in presence of an eavesdropper. Two phases are required for communications during each time frame between the users and the base station in cellular networks. We study the case where an eavesdropper overhears the relay and direct links from the users to the base stations. In terms of the secure performance analysis, we focus on two main metrics including secrecy outage probability (SOP) and strictly positive secrecy capacity (SPSC) with the assumption that the eavesdropper is able to detect the signals. Analytical closed-form expressions for the SOP and SPSC are derived to evaluate the system secure performance achieved by the proposed schemes. Furthermore, the asymptotic analysis is presented to gain further insights. The analytical and numerical results indicate that the proposed schemes can realize better secrecy performance once we improve the channel condition and signal-to-noise ratio (SNR) at the base station. Our results confirms that the secrecy performance gaps exist among the two users since different power allocation factors are assigned to these users.