This paper proposes a new design of non-orthogonal multiple access (NOMA) under secrecy considerations. We focus on a NOMA system where a transmitter sends confidential messages to multiple users in the presence of an external eavesdropper. The optimal designs of decoding order, transmission rates, and power allocated to each user are investigated. Considering the practical passive eavesdropping scenario where the instantaneous channel state of the eavesdropper is unknown, we adopt the secrecy outage probability as the secrecy metric. We first consider the problem of minimizing the transmit power subject to the secrecy outage and quality of service constraints, and derive the closed-form solution to this problem. We then explore the problem of maximizing the minimum confidential information rate among users subject to the secrecy outage and transmit power constraints, and provide an iterative algorithm to solve this problem. We find that the secrecy outage constraint in the studied problems does not change the optimal decoding order for NOMA, and one should increase the power allocated to the user whose channel is relatively bad when the secrecy constraint becomes more stringent. Finally, we show the advantage of NOMA over orthogonal multiple access in the studied problems both analytically and numerically.
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