Mathematical Problems in Engineering

Volume 2018, Article ID 6764052, 10 pages

https://doi.org/10.1155/2018/6764052

## Image Encryption Technique Combining Compressive Sensing with Double Random-Phase Encoding

^{1}Department of Mathematics, Shantou University, Shantou, Guangdong 515063, China^{2}School of Mathematics, Jiaying University, Meizhou, Guangdong 514015, China

Correspondence should be addressed to Shouzhi Yang; nc.ude.uts@gnayzs

Received 8 June 2017; Revised 9 September 2017; Accepted 22 January 2018; Published 3 April 2018

Academic Editor: Tomasz Kapitaniak

Copyright © 2018 Huiqing Huang and Shouzhi Yang. 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 new image compression-encryption method based on compressive sensing and double random-phase encoding is proposed, which can complete image compression and encryption simultaneously. We utilize the hyperchaotic system to generate a measurement matrix and two random-phase masks first. Then the original image is measured by measurement matrix to accomplish encryption and compression at the same time, next the compressed image is reencrypted by the double random-phase encoding technique with the two random-phase masks, and lastly the resulting image is confused and diffused by using hyperchaotic system simultaneously. Some numerical simulations verify the validity and the reliability of the proposed algorithm.

#### 1. Introduction

With the development of network and communication technology, encryption technique becomes more and more important for the information security and network security. It is one of the most important methods in protecting network security, which can prevent the illegal easy stealing, distorts, duplicates, and spreads of sensitive information. In 1989, a typical image encryption method with advantages of good performance and high security was based on chaos theory which was proposed by Matthews [1]. Subsequently, all kinds of chaos-based image encryption techniques have been reported [2–7]. Guan et al. employed Arnold cat map and Chen’s chaotic system to shuffle the positions and change the gray values of image pixels [3]. In [2], Chen et al. proposed a symmetric image encryption scheme based on 3D chaotic cat maps. Gao and Chen proposed a novel image encryption algorithm based on hyperchaos, which uses a new image total shuffling matrix to shuffle the pixel positions of the plain image and then the states combination of hyperchaos is used to change the gray values of the shuffled-image [6]. A chaos-based image encryption algorithm with variable control parameters is proposed [7].

The double random-phase encoding (DRPE) was first proposed in 1995 [8]; since then, many researchers have proposed and analyzed a lot of encryption algorithms based on DRPE [9–14]. In [9], Zhang and Karim proposed a new encryption technique to encrypt color images using existing optical encryption systems for gray-scale images. Subsequently, Unnikrishnan et al. proposed an optical architecture that encodes a primary image to stationary white noise by using two statistically independent random-phase codes, the encoding is done in the fractional Fourier domain, and the optical distribution in any two planes of a quadratic phase system is related by fractional Fourier transform of the appropriately scaled distribution in the two input planes [10]. After that, a lensless optical security system based on double random-phase encoding in the Fresnel domain was designed [11]. To enhance security further, a novel image encryption method is proposed by utilizing random-phase encoding in the fractional Fourier domain to encrypt two images into one encrypted image with stationary white distribution [12].

Recently, encryption methods [15–18] based on compressive sensing (CS) [19, 20] have been widely studied. Zhang et al. proposed a new color image encryption algorithm combining compressive sensing with Arnold transform, which can encrypt the color image into a gray image [15]. In [16], an image information encryption method based on compressive sensing and double random-phase encoding is proposed. Lately, Zhou et al. proposed a novel image compression-encryption scheme by combining 2D compressive sensing with nonlinear fractional Mellin transform [17]. To overcome the low-security and reduce the possible transmission burden, an efficient image compression-encryption scheme based on hyperchaotic system and 2D compressive sensing is proposed [18].

In this paper, a new image encryption method based on CS and DRPE technique is proposed which can accomplish encryption and compression at the same time. In this scheme, the original image is measured by the measurement matrix first, where the measurement matrix is controlled by hyperchaotic system with initial conditions. And then the two random-phase masks generated by the hyperchaotic system performs DRPE with the compressed image. Lastly, the resulting image is confused and diffused by using hyperchaotic system simultaneously.

#### 2. Preliminaries for Proposed Technique

In this section, some preliminaries about the CS theory and DRPE technique used in image encryption algorithm are introduced.

##### 2.1. Compressive Sensing

CS is a new sample theory, which can reconstruct original signal by directly sampling a sparse or compressible signal at a rate much lower than the Nyquist rate. For a 1D signal in with length could be represented aswhere is the matrix with as columns and is the coefficient sequence of signal . Suppose that measurements of are taken through the following linear measurement:where is an measurement matrix incoherent with basis matrix . In fact, the magic of CS is that can be designed such that can be recovered approximately form the measurements when the matrix satisfies the Restricted Isometry Property (RIP) [22].

To recover the signal from , it is required to solve the optimal problem below:

The problem above can be solved by greed iterative algorithm, one of the most commonly used algorithms is the orthogonal matching pursuit (OMP) method [23].

##### 2.2. Double Random-Phase Encoding

In 1995, Refregier and Javidi proposed the DRPE technique [8]. The encoded image is obtained by random-phase encoding in both the input and the Fourier planes. If two random-phase masks are used to encrypt the image in the input and Fourier planes, respectively, the input image is transformed into a complex-amplitude stationary white noise.

Let denote the image to be encoded and denote the encoded image. and stand for the key function in the spatial and frequency domain, respectively, and the values of which are distributed from 0 to 1 with uniform probability.

The encoding and decoding procedures are shown as follows:where and represent the 2D fast Fourier transformation and inverse 2D fast Fourier transformation, respectively.

##### 2.3. Hyperchaotic System

In the proposed encryption scheme, a new hyperchaotic system generated from Gao et al.’s chaotic system is used in key scheming, which is defined by [24]where , , , , and are the control parameters of the hyperchaotic system. When , , , , and , the system is in a hyperchaotic state. The hyperchaos attractors are shown in Figure 1. With parameters , , , , and , its Lyapunov exponents are , , , and , respectively. Since the hyperchaotic system has two positive Lyapunov exponents, the prediction time of the hyperchaotic system is shorter than the original chaotic system [25]; as a result, it is safer than chaos in security algorithm. Because of this advantage of the hyperchaotic system, we will use it to generate the keys in the compression and encryption stage of our algorithm.