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We investigate a secrecy-enhancing design for cooperative downlink and uplink non-orthogonal multiple access (NOMA) transmissions with an untrusted relay. A source uses the NOMA principle to have downlink and uplink transmissions with a near user and a far user, and the communications between the source and the far user are aided by an untrusted relay. To minimize information leakage at the untrusted relay and achieve secure NOMA communications, adaptive downlink and uplink cooperative jamming schemes are proposed, where the far user in downlink and the near user in uplink adaptively emit a jamming signal to the untrusted relay to impair its eavesdropping capability. Both downlink and uplink jamming power are optimized to fully exploit the benefits of the proposed schemes for security enhancement. Then, for each scheme, we quantify the secrecy performance by deriving the ergodic secrecy …