An Image Encryption Algorithm Based on Hyperchaotic System and Variable-Step Josephus Problem

Abstract

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 256 ${}^{\ast }$ 256 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.