Discrete chaotic system, as a pseudo-random signal source, plays a very important role in secure communication. However, many low-dimensional chaotic systems are prone to chaos degradation. Therefore, many scholars have studied the construction of high-dimensional chaotic systems. However, many existing algorithms for constructing high-dimensional chaotic systems have relatively high time complexity and relatively complex structures. To solve this problem, this paper explores an n-dimensional discrete hyperchaotic system with a simple structure. Firstly, the n-dimensional discrete hyperchaotic system is constructed by using sine function and power function and simple operations. Then, it is theoretically analyzed that the system can be through parameter settings the positive Lyapunov exponents based on Jacobian matrix method. Next, the algorithm time complexity, sample entropy, correlation dimension and other indexes are compared with the existing methods. The experimental results show that our system has a simple structure, high complexity and good algorithm time complexity. Therewith, a six-dimensional chaotic system is chosen as an example, the phase diagram, bifurcation diagram, Lyapunov expnonents, complexity and other characteristics of the system are analyzed, and the results show that the proposed system has good chaotic characteristics. Moreover, to show the application of the proposed system, we apply it to audio encryption. Based on this system, we combine it with the XOR operation and true random numbers to explore a novel method of one-cipher audio encryption. Through experimental simulation, compared with some existing audio encryption algorithms, this algorithm can satisfy various statistical tests and resists various common attacks. It also proves that the proposed system can be effectively applied in the field of audio encryption.