A novel chaotic map constructed by geometric operations and its application

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.