In this paper, an image encryption algorithm based on a hyperchaotic system and variable-step Josephus problem is proposed. Based on an in-depth analysis of the classic Josephus problem, a new variable-step Josephus problem that combines the pseudorandom sequence with the Josephus problem is proposed. Firstly, the hash value of the plaintext image is calculated, which is converted to the initial value of the chaotic system. Secondly, the chaotic system is iterated to generate four pseudorandom sequences X, Y, Z, and W. The sequences X, Y, and Z are input as parameters into the variable-step Josephus function to scramble the positions of the rows, pixel bits, and columns of the image, respectively. Finally, the elements of the sequence W and the image pixels are used to perform the addition operation. According to the experiments, the information entropy of the encrypted image with size 256256 reaches 7.997 and the adjacent correlations in three directions are within ±0.01. The experimental results show that image encryption algorithm proposed in this paper has plaintext sensitivity and can resist the common attacks.

中文翻译：

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基于超混沌系统和变步长约瑟夫斯问题的图像加密算法

提出了一种基于超混沌系统和变步长约瑟夫斯问题的图像加密算法。在对经典约瑟夫斯问题的深入分析的基础上，提出了一种将伪随机序列与约瑟夫斯问题相结合的变步长约瑟夫斯问题。首先，计算明文图像的哈希值，将其转换为混沌系统的初始值。其次，混沌系统被迭代以产生四个伪随机序列X，ÿ，Ž，和w ^。序列X，Y和Z作为参数输入到可变步长约瑟夫斯函数中，分别对图像的行，像素位和列的位置进行加扰。最后，序列W的元素和图像像素用于执行加法运算。根据实验，大小为256 256的加密图像的信息熵达到7.997，三个方向的相邻相关度均在±0.01以内。实验结果表明，本文提出的图像加密算法具有明文敏感性，可以抵抗常见的攻击。