The finite battery capacity, limited spectrum resource and the transmission security are three challenges in designing mobile edge computing (MEC) systems. In this paper, a framework for wireless powered secure multiple-input single-output (MISO) cognitive-based MEC-enabled networks is proposed, which integrates several technologies: physical-layer security, cooperative relaying, cognitive radio (CR), wireless power transfer (WPT) and MEC. Two optimization problems are formulated to maximize the number of computation bits (NCB) of the mobile device (MD) and minimize the total transmission power (TTP) of the primary transmitter (PT) and the SBS, respectively. The formulated problems are non-convex and hard to solve. Two two-phase methods with block coordinate decent (BCD) and Lagrangian dual decomposition methods are proposed to jointly design the beamforming vector of the PT, beamforming matrix of the secondary base station (SBS), the central processing unit (CPU) frequency, the transmit power, the number of the offloaded bits and the offloading time of the MD. Simulation results are presented to show the effectiveness of the proposed methods. INDEX TERMS Mobile edge computing, cognitive radio, physical-layer security, wireless power transfer, MISO.