Designing high-dimensional chaotic maps with expected dynamic properties is an attractive but challenging task. The dynamic properties of a chaotic system can be reflected by the Lyapunov exponents (LEs). Using the inherent relationship between the parameters of a chaotic map and its LEs, this paper proposes an $n$ -dimensional polynomial chaotic system ( $n\text{D}$ -PCS) that can generate $n\text{D}$ chaotic maps with any desired LEs. The $n\text{D}$ -PCS is constructed from $n$ parametric polynomials with arbitrary orders, and its parameter matrix is configured using the preliminaries in linear algebra. Theoretical analysis proves that the $n\text{D}$ -PCS can produce high-dimensional chaotic maps with any desired LEs. To show the effects of the $n\text{D}$ -PCS, two high-dimensional chaotic maps with hyperchaotic behaviors were generated. A microcontroller-based hardware platform was developed to implement the two chaotic maps, and the test results demonstrated the randomness properties of their chaotic signals. Performance evaluations indicate that the high-dimensional chaotic maps generated from $n\text{D}$ -PCS have the desired LEs and more complicated dynamic behaviors compared with other high-dimensional chaotic maps. In addition, to demonstrate the applications of $n\text{D}$ -PCS, we developed a chaos-based secure communication scheme. Simulation results show that $n\text{D}$ -PCS has a stronger ability to resist channel noise than other high-dimensional chaotic maps.