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.

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