Periodic characteristics of the Josephus ring and its application in image scrambling

Abstract

Abstract This paper proposes a new image scrambling algorithm based on the periodic characteristics of the Josephus ring. The algorithm composes the pretreatment part of the entire image encryption system and scrambles the rows and columns of the plain image. The Josephus scrambling algorithm is adjustable by using three kinds of parameters: step, m 0, and n. Different values affect the size of the periodic value of the Josephus ring. In this paper, we focus on the method of determining the period of the Josephus cycle when the parameters are set and the Josephus rule space under arbitrary parameters. Because the Josephus ring is a mathematical problem, we analyze it using the group theory of modern algebra. After the Josephus scrambling, the plain image is encrypted. Because a CA is suitable for image encryption, the encryption part adopts a CA encryption algorithm using a one-dimensional, four-neighbor CA, which has chaotic behavior at the rules of 9d62 (hex). Finally, the number of pixels change rate, the unified averaged changed intensity test, and correlation detection are carried out on the experimental results. The results show that the use of the Josephus scrambling algorithm greatly improves the security of the entire encryption system.

Keywords

• CA
• Image scrambling
• Josephus problem
• Periodic characteristics