Chaotic maps are widely used in the designs of different security applications due to their series of attractive features, but many commonly used chaotic maps have several weaknesses such as limited chaotic range, small Lyapunov exponent and heavy computational cost. Although there have been many methods to overcome these shortcomings, most of them lack theoretical analysis and the improvement effect is quite limited. In order to better solve these problems, from the perspective of geometry, this paper constructs a novel chaotic map with complicated dynamic behavior, which is called piecewise cubic map. We mathematically derive its Lyapunov exponent and some associated theorems and corollaries, which reflects that the new map owns good chaotic performance. Meanwhile, it still retains the advantage of high iteration efficiency. These characteristics provide good theoretical guarantee for the security and efficiency of encryption applications based on it. Moreover, a novel pseudo-random number generator based on piecewise cubic map is designed to investigate its application in cryptography. Performance evaluation shows that the proposed generator can efficiently produce random sequences with high quality.

混沌图由于具有一系列吸引人的特性而被广泛用于各种安全应用的设计中，但是许多常用的混沌图具有一些不足，例如混沌范围有限，Lyapunov指数小和计算成本高。尽管有许多方法可以克服这些缺点，但是大多数方法都缺乏理论分析，并且改进效果非常有限。为了更好地解决这些问题，从几何学的角度出发，构造了一种具有复杂动态行为的新型混沌图，称为分段三次图。我们从数学上推导了其Lyapunov指数以及​​一些相关的定理和推论，这表明新地图具有良好的混沌性能。同时，它仍然保留了高迭代效率的优点。这些特性为基于加密的应用程序的安全性和效率提供了良好的理论保证。此外，设计了一种新颖的基于分段三次映射的伪随机数发生器，以研究其在密码学中的应用。性能评估表明，所提出的生成器可以有效地产生高质量的随机序列。