Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 7683687, 15 pages

http://dx.doi.org/10.1155/2016/7683687

## A Novel 1D Hybrid Chaotic Map-Based Image Compression and Encryption Using Compressed Sensing and Fibonacci-Lucas Transform

School of Information Science and Engineering, Lanzhou University, Lanzhou, Gansu 730000, China

Received 31 December 2015; Revised 17 April 2016; Accepted 18 April 2016

Academic Editor: Zhen-Lai Han

Copyright © 2016 Tongfeng Zhang 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

A one-dimensional (1D) hybrid chaotic system is constructed by three different 1D chaotic maps in parallel-then-cascade fashion. The proposed chaotic map has larger key space and exhibits better uniform distribution property in some parametric range compared with existing 1D chaotic map. Meanwhile, with the combination of compressive sensing (CS) and Fibonacci-Lucas transform (FLT), a novel image compression and encryption scheme is proposed with the advantages of the 1D hybrid chaotic map. The whole encryption procedure includes compression by compressed sensing (CS), scrambling with FLT, and diffusion after linear scaling. Bernoulli measurement matrix in CS is generated by the proposed 1D hybrid chaotic map due to its excellent uniform distribution. To enhance the security and complexity, transform kernel of FLT varies in each permutation round according to the generated chaotic sequences. Further, the key streams used in the diffusion process depend on the chaotic map as well as plain image, which could resist chosen plaintext attack (CPA). Experimental results and security analyses demonstrate the validity of our scheme in terms of high security and robustness against noise attack and cropping attack.

#### 1. Introduction

With the fascinating development of computer networks and multimedia communications during the past decades, a great demand for secure image transmission is increased. Image encryption is one of effective measures to prevent unauthorized access to images [1, 2]. Since the introduction of chaos theory to cryptography and the proposal of permutation and diffusion structure by Matthews [3] and Fridrich [4], respectively, Chaos-based encryption algorithms have received remarkable attentions [5–9] for the reason that some typical features of chaos, including ergodicity, sensitivity to initial condition, and random-like behavior, can be well connected with some conventional cryptographic properties such as confusion and diffusion [10]. Chen et al. [5] have proposed an image encryption algorithm which employs the three-dimensional cat map to shuffle the positions of the image pixels and uses another chaotic map to confuse the relationship between the encrypted and its original image. Huang [6] has designed pseudorandom chaotic sequence with the created secret keys depending on each other and used a two-dimensional Chebyshev map in diffusion process. In [7], Pareek et al. have proposed a new approach for image encryption based on chaotic Logistic maps. An external secret key of 80-bit length and two chaotic Logistic maps are employed. Gao and Chen [8] presented a new image encryption scheme which employs an image total shuffling matrix to shuffle the positions of image pixels and then uses a hyperchaotic system to confuse the relationship between the plain image and the cipher image. A symmetric block cipher based on the improved standard map was derived by Lian et al. [9]. Permutation and diffusion structure are usually adopted in the cryptosystems mentioned above, which suggests iterating the permutation and diffusion stage several rounds to earn good confusion and diffusion effect, whereas the permutation in those cryptosystem is almost the same in each round and independent of plain image.

The chaotic system utilized in the cryptosystem often could be divided into one-dimensional (1D) chaos and high dimensional (HD) chaos [11]. 1D chaos such as Logistic map [12] and Tent map [13, 14] has fast computational performance while HD chaotic systems like Lorenz system [15], hyperchaos [16], and chaotic standard map [17] have large parametric space and high security. 1D chaotic maps often have simple structures and are easy to implement. But their intrinsic weakness such as small key space and weak security enables image encryption algorithm feasible to be attacked. To overcome those limitations, some improvements on 1D chaotic system have been carried out. Mazloom and Eftekhari-Moghadam [18] introduce a kind of coupled nonlinear chaotic map to color image encryption. Zhou et al. used a combination of two existing 1D chaotic maps to generate two new 1D chaotic maps in series and parallel in the literature [19] and [20], respectively, where the former has two parameters but weak uniform distribution property and the latter shows good uniform distribution property but has only one parameter essentially.

Recently, some researchers proposed a kind of compression-combined encryption method based on compressive sensing (CS) [21–27]. CS includes sparse representation, linear measurement, and reconstruction processes. An image compression-encryption scheme is proposed in [21], which combines 2D compressive sensing with nonlinear fractional Mellin transform. The literature [23] proposes a double-image encryption compression scheme. The major core of the encryption system is that two circular matrices and the measurement matrix utilized in compressive sensing are designed by using a two-dimensional Sine Logistic modulation map. Liu et al. [25] present a combined compressive sensing and optical image encryption method using double random phase encoding-based block compressive sensing. These CS-based image encryption methods could achieve satisfactory security performance except for the resistance of differentia attack due to the absence of diffusion process. Zhang et al. [27] apply the encryption and compression scheme into the field of medical images. The novelties of their work lie in that Bernoulli measurement matrix in CS is constructed by using Chebyshev map and quantized measurements by max-Llyod are encrypted in diffusion phase.

Presently, we aim to design a novel 1D hybrid chaotic map which possesses large key space and satisfies uniform distribution property. Meanwhile, CS-combined with Fibonacci-Lucas transform image encryption is put forward based on the proposed 1D chaotic map. Bernoulli measurement matrix in CS is derived by the 1D chaotic map. Permutation phase is fulfilled by Fibonacci-Lucas transform, whose transform kernel could be controlled by the hybrid 1D chaotic map. Furthermore, a linear scaling on the measurement data is taken into consideration to accomplish diffusion operation. The contributions of this work are as follows: firstly, a 1D hybrid chaotic map is constructed; secondly, the proposed chaotic map shows good uniform distribution and could constitute the Bernoulli measurement matrix in CS, thus reducing the cost of transmission and improving image recovery quality; thirdly, Fibonacci-Lucas transform is utilized to accomplish block scrambling, whose transform kernel is different from each other in each permutation round; finally, the diffusion operation related to plaintext is carried out after a linear scaling of the measurements. The proposed method is described in detail in Section 2. Section 3 presents the simulated results and security analyses. The conclusion is drawn in Section 4.

#### 2. Proposed Method

##### 2.1. One-Dimensional (1D) Hybrid Chaotic Map

The proposed 1D chaotic map is fulfilled by hybrid structures including both series and parallel, as shown in Figure 1. It is a nonlinear combination of three different 1D chaotic maps which are considered as seed maps [18, 19]. The system is defined by the following equation:where , , and are three 1D chaotic maps (seed maps) with parameters , , and ; mod is modulo operation; and is the iteration number. The three chaotic maps , , and can be chosen from among the Tent map, Logistic map, and Sine map, who are defined by (2a)–(2c), respectively,where , , and represent the Tent map, Logistic map, and Sine map, respectively, , , and are their control parameters, and the three chaotic output sequences s all fall within . It is obvious that there are three different combinations for choice of , , and in (1). As an example, , , and take the forms of Logistic map, Tent map, and Sine map, respectively. Correspondingly, (1) can be rewritten as