**1. Introduction**

With the development of information technologies, data security has aroused wide public concern. As an important data format, images occupy a large proportion of network data. Their secure transmission plays a vital role in personal and military privacy. In recent years, many chaotic image encryption algorithms have been proposed [1–10] due to the excellent properties of chaotic maps, such as initial value sensitivity and intrinsic randomness.

Researchers have improved the single chaotic map or combined multiple chaotic maps to improve chaotic properties, producing larger secret key spaces and more random chaotic sequences. Pak et al. [3] proposed a structure to modify two same chaotic maps to produce better performance than a single map [11–14]. Li et al. [15] improved the logistic map using linear delay. Zhou et al. [6] proved that cascading chaotic maps can increase the Lyapunov exponent, and many chaotic maps can be generated with the cascade model. Hua et al. [4] combined the logistic map and sine map to generate a two-dimensional (2D) map. This paper proposes a new framework that combines the cascade model and delay. This framework rationally integrates three chaotic maps to overcome the performance flaws of one-dimensional (1D) chaotic maps [16]. The experimental results showed that the chaotic maps produced by this model have initial value sensitivity and a large parameter interval.

Based on the proposed chaotic map, we constructed a new image encryption algorithm. Generally, image encryption algorithms can be divided into two steps: confusion and diffusion. Confusion involves randomly changing the position of pixels. The two commonly used confusion algorithms are: performing row and column confusion on a image, and reshaping a two-dimensional image into a vector, and then performing position confusion on it [17–19]. The basic diffusion methods are based on an XOR operation or mod operation after addition [20,21].

In this paper, a new simultaneous confusion and diffusion algorithm is proposed, which is applied in the vertical and horizontal directions based on an XOR operation. After analysis, the algorithm can resist chosen-plain text attacks and statistical attacks.

The rest of the paper is organized as follows. In Section 2, the proposed chaotic map is introduced. Section 3 shows the details of the image encryption algorithm. The proposed algorithm is analyzed and compared with other works in Section 4. Section 5 concludes the work.
