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In order to improve the unpredictability and synchronization performance of chaotic system in encryption algorithm, a hyper-chaotic Lü system with hyper-chaotic attractors has been proposed. The Lyapunov exponent spectrum and power spectrum are analyzed to demonstrate that attractors are indeed hyper-chaotic. In addition, basing on Lyapunov theory, we propose the sufficient and necessary conditions of state feedback control method to achieve hyper-chaotic synchronization in theoretically. The state feedback controller is designed for the hyper-chaotic Lü system with these conditions. The simulation results are consistent with the experimental results, which prove that the analysis is correct. Finally, the hyper-chaotic Lü system is applied to the secure communication system and shows the application potential in engineering.