Double-image encryption scheme combining DWT-based compressive sensing with discrete fractional random transform

doi:10.1016/j.optcom.2015.05.043

Optics Communications

Volume 354, 1 November 2015, Pages 112-121

https://doi.org/10.1016/j.optcom.2015.05.043 Get rights and content

Highlights

•
A fractional RT is constructed by two RCMs controlled by the 2D sine Logistic modulation map.
•
The upper half of a random circular matrix is chosen as the measurement matrix.
•
A double-image encryption scheme combining DWT-based CS with DFrRT is proposed.
•
The proposed scheme can achieve encryption and compression simultaneously.

Abstract

A new discrete fractional random transform based on two circular matrices is designed and a novel double-image encryption–compression scheme is proposed by combining compressive sensing with discrete fractional random transform. The two random circular matrices and the measurement matrix utilized in compressive sensing are constructed by using a two-dimensional sine Logistic modulation map. Two original images can be compressed, encrypted with compressive sensing and connected into one image. The resulting image is re-encrypted by Arnold transform and the discrete fractional random transform. Simulation results and security analysis demonstrate the validity and security of the scheme.

Introduction

With the increasing computing ability of classical computers, the security of classical cryptography, including image encryption algorithms, will face a growing threat. Therefore, to protect the secret image from leaking during the process of transmission and storage, various image encryption schemes have been presented recently. Due to the excellent properties of unpredictability, ergodicity and sensitivity to their parameters and initial values, chaotic maps have been applied to design image encryption schemes widely [1], [2], [3], [4], [5]. Moreover, some optical transforms have been developed as pixel diffusion tools for the security of images, for instance, fractional Fourier transform (FrRT) [6], Gyrator transform (GT) [7], Fresnel transform (FST) [8], and fractional random transform [9]. However, the two techniques used in image encryption have their own weakness and the schemes using simple chaotic map have been found insecure [10], [11], [12], [13]. Most of the above-mentioned transforms are linear. As is well known, the linear encryption system is relatively vulnerable to chosen and known plaintext attacks [14], [15]. Considering their own strength and weakness, combining transform operation with chaotic system together can further make up for their respective defects [16], [17].

Recently, Candes and Donoho proposed a new signal processing theory called compressive sensing (CS) [21], [22], which can sample signals in space domain and fulfill the compressive sampling simultaneously at a significantly lower rate than that of the Shannon–Nyquist sampling theorem. Subsequently, by combining CS, many image encryption schemes have been studied. By utilizing the properties of the measurement matrix, an image encryption scheme based on CS was proposed [23]. To resist consecutive packet loss and malicious cropping attack, a digital image encryption method [24] was constructed by combining CS with bitwise XOR. Nevertheless, the CS-based encryption schemes above-mentioned cannot achieve perfect security. To enhance the security of the encryption system, Liu et al combined CS with double random phase encoding optical encryption technique, which realized digital and optical double encryption [25]. Moreover, an image encryption scheme combining fractional Fourier transform with iterative kernel steering regression in double random phase encoding was provided to achieve de-noising before encryption and compression [26]. Unfortunately, most of the above-mentioned encryption schemes based on CS consume the long keys or render the keys too long to distribute and memorize, since the whole measurement matrix was considered as the key. To overcome these shortcomings, image compression–encryption hybrid schemes [27], [28] were devised where the measurement matrix is controlled by a short key and the compression and encryption could be fulfilled simultaneously. Lately, Liu et al.designed an image encryption scheme by utilizing CS and chaos theory [29] and Zhou et al. employed CS together with the nonlinear fractional Mellin transform (FrMT) to encrypt the image, which are all succeeded in resisting common attacks, shortening the length of the key and enhancing the security [30].

Most image encryption schemes can only deal with a single image. However, in some cases, a double-image encryption scheme or a multi-image encryption scheme is necessary. Consequently, double-image encryption schemes have been drawn widespread attention [18], [19], [20]. Combining Chirikov standard map with chaos-based fractional random transform, Zhang and Xiao [18] presented a double-image encryption scheme in which some classical types of attacks such as known plaintext attack and chosen plaintext attack are invalid. A double image encryption scheme based on logistic maps and discrete fractional random transform was proposed [19], which strengthens the nonlinearity in spatial domain and discrete fractional random transform (DFrRT) domain and could resist chosen plaintext and ciphertext-only attacks. Furthermore, to enrich the double image encryption system, a novel double image encryption scheme using chaos-based local pixel scrambling technique and gyrator transform [20] was provided, which improved the efficiency in encryption, storage or transmission. Generally, two plain images are encrypted into two cipher images in most of double-image encryption schemes. These schemes can achieve effective encryption without compression. To achieve fast and efficient double-image encryption–compression, in this paper, we will propose a novel double-image encryption scheme by combining CS with the discrete fractional random transform (DFrRT). Two original images are encrypted and compressed simultaneously by employing CS. The two compressed images are combined into an enlarged one in the way of connecting them sequentially. Finally, Arnold transform and the discrete fractional random transform are introduced to re-encrypt the resulting image to enhance the reliability and confidentiality of the cryptographic system. In the process, the DFrRT and the measurement matrix in CS are constructed by random circular matrices. The random circular matrices are produced by using the 2D sine Logistic modulation map (2D-SLMM). Experimental results and security analysis show that different types of digital images can be encrypted with high security, small storage space and acceptable encryption speed.

The rest of the paper is arranged as follows: In Section 2, some basic fundamental knowledge is reviewed. In Section 3, the proposed image encryption scheme is described. In Section 4, some numerical simulations are given. Concluding remarks are summarized in the final section.

Section snippets

Compressive sensing

Compressive sensing can sample analog band limited signal at low rate and compress the signal in a high ratio, which breaks through the Nyquist sampling theorem. CS theory can effectively reduce the data storage, promote the processing capacity and reduce the amount of processing data together with hardware burden. In CS theory, the polynomial s is considered as an arbitrary one-dimensional signal, which is an $N \times 1$ column in R^N. It can be represented by using the orthogonal wavelet basis matrix $Ψ$

Double-images encryption scheme based on CS and DFrRT

The proposed image encryption scheme based on CS and DFrRT is illustrated in Fig. 1(a). Without loss of generality, the size of the two original images is $N \times N$ and the detailed procedures are as follows:

Step 1. Image sparse representation: Most images are sparse in discrete wavelet transform (DWT) domain. Images $I_{1} (x, y)$ and $I_{2} (x, y)$ can be represented by using the discrete wavelet basis: $G_{1} = DWT [I_{1} (x, y)] G_{2} = DWT [I_{2} (x, y)]$ Most coefficients of the matrices $G_{1}$ and $G_{2}$ are close to zero, i.e., $G_{1}$ and $G_{2}$

Simulation results and performance analyses

To verify the validity and reliability of the proposed scheme, numerical simulations have been performed on a matlab 7.0 platform. The two original images “Lena” and “Pepper” are all with resolution 256×256 as shown in Fig. 2(a) and (b), respectively, and the discrete symlet8 wavelet is adopted as the sparse basis matrix Ψ. The parameters are taken as $x_{0} = 0.3$ , $x_{1} = 0.3547$ , $y_{0} = 0.4$ , $y_{1} = 0.7842$ , $μ = 1$ , $β = 3$ in Eq. (10), $λ_{1} = 2$ , $λ_{2} = 3$ in Eq. (9), $η = 0.333$ in Eq. (8) and $α = 0.2$ in Eq. (15), the encryption and

Conclusion

An efficient double-image encryption scheme combining compressive sensing with the discrete fractional random transform is designed. The measurement matrix in compressive sensing and the discrete fractional random transform are constructed with the random circular matrix controlled by the 2D sine Logistic modulation map. The images to be encrypted are represented in the domain of the discrete wavelet transform. Two original images are compressed by compressive sensing simultaneously and

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant nos. 61262084 and 61462061), the Foundation for Young Scientists of Jiangxi Province (Jinggang Star) (Grant no. 20122BCB23002), the Natural Science Foundation of Jiangxi Province (Grant no. 20132BAB201019), the Opening Project of Shanghai Key Laboratory of Integrate Administration Technologies for Information Security (Grant no. AGK2014004), and the Innovation Project of Jiangxi Graduate Education (Grant no.

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Double-image encryption scheme combining DWT-based compressive sensing with discrete fractional random transform

Highlights

Abstract

Introduction

Section snippets

Compressive sensing

Double-images encryption scheme based on CS and DFrRT

Simulation results and performance analyses

Conclusion

Acknowledgments

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Inf. Sci.