Abstract: An image encryption algorithm based on quantum chaos and fractional Fourier transform is proposed for the traditional security problem of natural chaotic system. By introducing the quantum chaotic map, the periodic window, pseudo-random and aperiodic problems of Logistic map are solved, and the problem of floating-point arithmetic loss is improved. At the same time, the chaotic system and the fractional Fourier transform are combined to realize the scrambling of the fractional domain between the spatial domain and the frequency domain, it overcomes the shortcomings of the traditional methods only in a single domain transformation and the simple use of a program, such as the small number of parameters, the simple structure of the system, the uneven histogram and so on. The experimental results and the simulation results show that the proposed algorithm has a large key space, low computational complexity and strong sensitivity, and can effectively resist the statistics.

Key words: quantum mapping, image encryption, fractional Fourier transform, chaotic sequence

摘要： 针对传统的自然混沌系统安全性低的问题，提出了量子混沌和分数阶Fourier变换的图像加密算法。通过引入量子Logistic混沌映射，解决了Logistic映射存在的周期窗口、伪随机和非周期性不好等缺陷，还改善了计算机进行浮点数运算丢失精度的问题。同时将混沌系统和分数阶Fourier变换相结合，实现了介于空间域和频域的分数域置乱，克服了传统一些方法只在单一域变换和单纯使用某一种方案而导致参数变量少，系统结构简单，直方图不均匀等缺点。实验和仿真结果表明，该算法具有密钥空间大，计算复杂度低，敏感性强等优点，能够有效地抵御统计分析攻击。

关键词: 量子映射, 图像加密, 分数阶Fourier变换, 混沌序列