Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 341729, 10 pages

http://dx.doi.org/10.1155/2015/341729

## Image Encryption Algorithm Based on Chaotic Economic Model

^{1}Department of Statistics and Operations Researches, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia^{2}Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Received 18 November 2014; Revised 23 December 2014; Accepted 24 December 2014

Academic Editor: Wang Xing-yuan

Copyright © 2015 S. S. Askar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In literature, chaotic economic systems have got much attention because of their complex dynamic behaviors such as bifurcation and chaos. Recently, a few researches on the usage of these systems in cryptographic algorithms have been conducted. In this paper, a new image encryption algorithm based on a chaotic economic map is proposed. An implementation of the proposed algorithm on a plain image based on the chaotic map is performed. The obtained results show that the proposed algorithm can successfully encrypt and decrypt the images with the same security keys. The security analysis is encouraging and shows that the encrypted images have good information entropy and very low correlation coefficients and the distribution of the gray values of the encrypted image has random-like behavior.

#### 1. Introduction

Up-to-date development and progress in the means of multimedia industry and the ways of communications have made researches focus on creating new schemes to enhance security of transmission and storing multimedia data over open channels including the internet and wireless networks. One of the challenges that researchers face nowadays is how to protect in a confidential manner a secure route for the transmission of multimedia data through digital networks. Due to the spread of the advances of new technologies in networks, people from all over the world can send and receive information, perform projects, and communicate with friends by sending images and videos through the internet. Sending and receiving such information using images and videos via internet and other networks require some kind of secure routes. That is because images and videos may incorporate secret or sensitive information such as patients’ medical surveys, personal information, high expensive marketable designs, and secret manuscripts.

It has been reported in literature [1–4] that an encryption tool is an effective approach to protect such information when sending and receiving data through multiple ways of communications. This is because the only one who can decrypt and see such information is the only authorized entities that have security keys of decryption. In an encryption process, the security keys are the core of any encryption and decryption algorithm. They are used to convert the data from a readable state to an apparent nonsense and vice versa. The designer of an encryption scheme should share the security keys needed to recover the original information with intended recipients and consequently other unwanted individuals can be precluded [5–7].

Recently, the theory of mathematics and programming languages have been used intensively in modern cryptography; cryptographic algorithms are built based on computational and complex assumptions by which breaking such algorithms in practice by any adversary is very hard [8–12]. The high efficiency of any cryptographic algorithm is the most important criterion by which the robustness of encryption is measured. For instance, the data encryption standard (DES) as a traditional algorithm [13, 14] faces problems when used to encrypt large images and therefore its efficiency becomes low and weak. Other traditional encryption algorithms such as international data encryption algorithm (IDEA) require a large computational time and super computers when used in encrypting real time images [14]. Cryptographic algorithms that use less time are much more preferable for encrypting such real time images. In addition, some encryption schemes may be run very slowly and this increases the degree of security features yet they would be of little use when dealing with real time images.

Chaos theory has been raised in different fields of physics, engineering, biology, and economy in the past two decades. Since the 1990s, many researches and scientists have done extensive studies in the chaotic systems emanated from this theory. Due to such studies, researchers have come up with the fact that there is a close relationship between cryptography and chaos. Chaotic systems such as logistic and other systems have got much attention and have been applied in the process of encryption [7]. What makes the encryption algorithms based on such systems more robust and reliable than other algorithms is the complex properties of such chaotic systems. Such complex properties can be summarized as sensitivity to the initial conditions of the systems’ parameters, nonperiodicity of systems’ equilibrium states, the topological transitivity of the systems’ behaviors, and the pseudorandom property. Since the appearance of the first algorithm in images’ encryption that entirely depended on chaos by Matthews [1], many chaos-based encryption algorithms of images have been introduced in literature. Some of those cryptosystems have used one or several dimensional maps such as baker’s and cat’s maps serving the purpose of encryption of images [2–7]. Wang et al. [8] have used an extension of fractal Fourier transformation and a digital holographic scheme to propose a cryptographic algorithm. Due to the parameters used in this scheme, enhancement security of the encryption process has been provided. Furthermore, many cryptographic algorithms have adopted popular chaotic models that represent chaos by using mathematical models such as logistic map, Lorenz map, Henon map, and Rössler attractor. Lorenz map is characterized by its attractor having two nonlinear terms while in Rössler attractor there is only one nonlinear term and this makes the complexity of Lorenz attractor and its chaos higher than those in Rössler attractor. Other algorithms have divided the images into several blocks and tried to define a permutation for each block using a logistic map to encrypt the original image.

Other chaotic image encryption algorithms that incorporate several parameters and work under frequency domain are more powerful in the encryption procedures because of the strength of the security provided by such algorithms [9–14]. Kuo has introduced a novel image encryption method in [9]. The way this method works is by making a random change in the phase spectra of the original image. This can be done by using a pseudonoise image with binary phase spectra embedded in the phase spectra of the original image. This methodology of adding such noise is actually a security key system. With this methodology, a part of the encrypted image with such noise can be used to obtain a full recovery of the original image without any drawbacks. Therefore, encryption algorithms based on such noise are more suitable for secure transmission of data through different ways of communications. This is due to the ability of the algorithms to recover the original image to some extent with partial access to the encrypted image. In [15], a high-dimensional Lorenz chaotic system with perceptron model within a neural network has been introduced. Liu and Wang [16] have designed a stem-cipher algorithm using the piecewise linear chaotic map as a generator of a pseudorandom key stem sequence in order to robust the security and improve the dynamic degradation. Furthermore, in order to get a robust security, Liu and Wang [17] proposed a bit-level permutation with a high-dimensional chaotic map in order to encrypt color image. A novel color image encryption algorithm based on chaos has been proposed by Wang et al. [18]. In [19], the potential flaws of Zhu’s algorithm have been analyzed. Zhang and Wang have proposed an encryption algorithm based on a spatiotemporal chaos of the mixed linear-nonlinear coupled map lattices [20]. Based on spatiotemporal nonadjacent coupled map lattices, Zhang and Wang [21] have proposed an encryption algorithm.

Chaotic economic systems such as monopoly and duopoly are sophisticated systems on which the chaos that occurs in them is more difficult than those found in Lorenz, logistic, and Rössler. In [22], the author has introduced a new Cournot duopoly model on which an unknown demand function without inflection points has been studied. This model has shown complex dynamical properties such as hard bifurcation and bad chaos. In comparison with logistic and other models, the Cournot model can be used extensively in the encryption scheme and thus strength cryptographic algorithm can be presented. To the reader’s knowledge, such chaotic economic models have not been used in literature before. Other chaotic economic systems can be found elsewhere [23–28] that are suitable in the encryption process for many reasons as they share some characteristics with cryptography. Of these reasons is that they have much more security keys, sensitive dependence to the initial conditions, hard bifurcation, and bad chaos. Nonlinear dynamical systems are characterized by their complex behavior such as bifurcation and chaos. The real-life applications originated by those systems have been extensively investigated. These applications may be classified into two parts: man-made applications and other applications simulated from nature. Due to the complex behavior of those applications which results because of chaos, several cryptographic techniques have been proposed and discussed in literature in the last two decades [29–33]. In [34], a detailed survey on chaotic cryptographic techniques has been elaborately reported. A hyperchaotic map has been used in [29] in order to encrypt and decrypt images in such a manner that the security of breaking the encryption is very difficult. What makes the encryption process difficult in this algorithm is that in the algorithm a permutation of the image that needs to be encrypted is done by an ergodic matrix of a hyperchaotic sequence. In [30], new novel image encryption and decryption techniques have been introduced. The Poker shuffle approach has been used to control the process of encryption. Multi-chaotic systems have been used to encrypt color images in [31]. In this study, four chaotic maps have been incorporated in the encryption scheme. The authors in [31] have used the so-called Henon map in encrypting a gray image. In this algorithm, the Arnold cat map is combined with Henon map in order to shuffle the pixels’ position of an image and hence an encrypted image is yielded. More papers on different types of cryptographic techniques that use chaotic systems in the encryption process can be found in [35–41].

The proposed algorithm covers many challenges which can be addressed as follows. (1) Robust images encryption and decryption techniques: an encryption algorithm can be used to convert the data into a strong encrypted file and therefore secure transmission of them via different sort of nowadays communications. This will save the data from unauthorized people and intentionally reduce the quality of perception. (2) Algorithmic code: the algorithmic steps of the encryption and decryption scheme should be prepared in a way that facilitates handling compressed format of multimedia. Based on that the encrypted and decrypted files will be modified without any crashing or damages. (3) Time complexity: many algorithms face problems when dealing with large multimedia data. It is important for cryptographic algorithms to overcome this disadvantage and speed up the behavior of the algorithm. One way to do that is to try to encrypt important parts of the multimedia data in such a way that makes the inverse process of encryption very quick without any crashing. (4) Chaotic economic systems: introducing new chaotic economic systems with hard bifurcation and bad chaos is obligatory in order to get a robust encryption algorithm. This is the motivation of our proposed paper. It is shown in literature that no one has used such systems in the encryption scheme; however, they can be used to introduce strong encryption and decryption techniques. Some of those systems can be found elsewhere [42–49].

The paper is organized as follows. In Section 2, the chaotic economic map is presented. In Section 3, the algorithmic steps of our proposed algorithm are outlined. Some experimental results are obtained in Section 4 and then some conclusion is given in Section 5.

#### 2. Chaotic Economic Map (CEM)

In literature, it is known that the logistic map , where and , is a one-dimensional discrete chaotic map. It has been used recently as a scheme for the process of images encryption. Its parameter represents the key security in the encryption process. This parameter has a great impact on the complex behavior of this map. It is reported that when with certain initial value of , the equilibrium point of the map becomes asymptotically stable and hence cannot be used in encryption. In , the map will behave periodically and therefore very weak encryption can be raised. Chaotic behavior of the map can be found in with periodicity disappearing. In the latter case however the chaos exists but the encryption scheme based on this map with is still weak. This weakness is due to the small key security this map is based on. To overcome this limitation, we suggest the following proposed map:

Equation (1) is a nonlinear chaotic economic map that incudes six important parameters. The parameter captures the size of the market demand while represents the slope of the market price. The parameter is a fixed marginal cost and is called the speed of adjustment parameter. is a constant. The chaotic behavior of the map is shown in Figure 1 at , respectively, and , , and . It is clear that the proposed map includes periodic windows in the third case which in turn make the map unsuitable for the encryption scheme. These windows can be eliminated using different values for the map’s parameters as we will see in the correlation analysis.