Volume 17 Number 2
March 2020
Article Contents
Ya-Ru Liang and Zhi-Yong Xiao. Image Encryption Algorithm Based on Compressive Sensing and Fractional DCT via Polynomial Interpolation. International Journal of Automation and Computing, vol. 17, no. 2, pp. 292-304, 2020. doi: 10.1007/s11633-018-1159-2
Cite as: Ya-Ru Liang and Zhi-Yong Xiao. Image Encryption Algorithm Based on Compressive Sensing and Fractional DCT via Polynomial Interpolation. International Journal of Automation and Computing, vol. 17, no. 2, pp. 292-304, 2020. doi: 10.1007/s11633-018-1159-2

Image Encryption Algorithm Based on Compressive Sensing and Fractional DCT via Polynomial Interpolation

Author Biography:
  • Ya-Ru Liang received the B. Sc. degree in automation from Heilongjiang Institute of Technology, China in 2002, the M. Sc. degree in power electronics and power drives from Shenyang University of Technology, China in 2008, and the Ph. D. degree in mechanical engineering from Nanchang University, China in 2016. Currently, she is a lecturer in Department of Electronic Information Engineering, Jiangxi Agricultural University, China. Her research interests include image processing, image encryption and information security. E-mail: liangyaru@jxau.edu.cn ORCID iD: 0000-0002-2716-0704

    Zhi-Yong Xiao received the B. Sc. degree in electronic information engineering from Nanchang Hangkong University, China in 2001, and the M. Sc. degree in control theory and engineering from East China University of Technology, China in 2008. He is a Ph. D. degree candidate in mechanical engineering at School of Mechatronic Engineering, Nanchang University, China. His research interests include image processing, computer vision and pattern recognition. E-mail: zhyxiao@jxau.edu.cn (Corresponding author) ORCID iD: 0000-0003-0773-3461

  • Received: 2018-01-27
  • Accepted: 2018-09-03
  • Published Online: 2018-12-04
  • Based on compressive sensing and fractional discrete cosine transform (DCT) via polynomial interpolation (PI-FrDCT), an image encryption algorithm is proposed, in which the compression and encryption of an image are accomplished simultaneously. It can keep information secret more effectively with low data transmission. Three-dimensional piecewise and nonlinear chaotic maps are employed to obtain a generating sequence and the exclusive OR (XOR) matrix, which greatly enlarge the key space of the encryption system. Unlike many other fractional transforms, the output of PI-FrDCT is real, which facilitates the storage, transmission and display of the encrypted image. Due to the introduction of a plain-image-dependent disturbance factor, the initial values and system parameters of the encryption scheme are determined by cipher keys and plain-image. Thus, the proposed encryption scheme is very sensitive to the plain-image, which makes the encryption system more secure. Experimental results demonstrate the validity and the reliability of the proposed encryption algorithm.
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Image Encryption Algorithm Based on Compressive Sensing and Fractional DCT via Polynomial Interpolation

Abstract: Based on compressive sensing and fractional discrete cosine transform (DCT) via polynomial interpolation (PI-FrDCT), an image encryption algorithm is proposed, in which the compression and encryption of an image are accomplished simultaneously. It can keep information secret more effectively with low data transmission. Three-dimensional piecewise and nonlinear chaotic maps are employed to obtain a generating sequence and the exclusive OR (XOR) matrix, which greatly enlarge the key space of the encryption system. Unlike many other fractional transforms, the output of PI-FrDCT is real, which facilitates the storage, transmission and display of the encrypted image. Due to the introduction of a plain-image-dependent disturbance factor, the initial values and system parameters of the encryption scheme are determined by cipher keys and plain-image. Thus, the proposed encryption scheme is very sensitive to the plain-image, which makes the encryption system more secure. Experimental results demonstrate the validity and the reliability of the proposed encryption algorithm.

Ya-Ru Liang and Zhi-Yong Xiao. Image Encryption Algorithm Based on Compressive Sensing and Fractional DCT via Polynomial Interpolation. International Journal of Automation and Computing, vol. 17, no. 2, pp. 292-304, 2020. doi: 10.1007/s11633-018-1159-2
Citation: Ya-Ru Liang and Zhi-Yong Xiao. Image Encryption Algorithm Based on Compressive Sensing and Fractional DCT via Polynomial Interpolation. International Journal of Automation and Computing, vol. 17, no. 2, pp. 292-304, 2020. doi: 10.1007/s11633-018-1159-2
    • With the rapid development of network communication and multimedia technologies, information can be easy to obtain via the Internet. Correspondingly, however, the insecurity of information arises because of the possibilities of data tampering and theft. In this context the way information is expressed is through text, image, video or speech. An image is amongst the most important means of expressing information due to its comprehensiveness and intuitiveness. Thus, image encryption is necessary so as to resist illegal access to data. Encryption methods based on different transforms and scrambling techniques have been proposed in various literatures. These transforms include Fourier transform[1, 2], fractional Fourier transform (FrFT)[3, 4], gyrator transform[5], Hartley transform[6], Mellin transform[7], random transform[8], etc. For example, Unnikrishnan et al.[9] outlined an optical image encryption using FrFT, in which a primary image was encoded to stationary white noise with the fractional orders as the cipher keys. In order to enlarge the key space, an image encryption algorithm based on the multiple-parameter FrFT[10] was introduced and showed superior robustness to blind decryption compared with other methods that existed.

      The pixel scrambling and diffusion operation, which can change the positions and values of image pixels randomly, is another kind of encryption method[1113]. An encryption algorithm based on hyper-chaos was presented by Gao and Chen[14]. It employed a total image shuffling matrix to shuffle image pixels and combined the states of a hyper-chaotic system to change the grey values of the shuffled-image. Another example of an image encryption scheme using reverse 2-dimensional (2D) chaotic maps and dependent diffusion was found in [15], in which the scrambling and diffusion of pixels were achieved simultaneously. In the above mentioned encryption methods, the volume of encrypted image data was equal to or greater than that of the original image.

      In many applications, the real-valued data are necessary and can benefit the digital image processing. However, the output of the encrypted images via the conventional encryption methods are usually complex, which increases the burden of storage and transform. In 2004, a method of reality-preserving treatment for a fractional transform was introduced[16]. Subsequently, an image encryption based on the reality-preserving multiple-parameter fractional Fourier transform (MPFRFT) was proposed, in which the parameters of the reality-preserving MPFRFT enhanced the space of keys[17]. An image encryption algorithm based on the reality-preserving fractional Mellin transform (RpFrMT) was presented in [18], in which the output of the transform was real and the transform was nonlinear. However, the disadvantages were its floating-point output and data expansion with respect to the original image. In addition, the above mentioned reality preserving transform adopted the same way of transforming, i.e., a reality-preserving transform matrix must be constructed and it lost some properties compared with non-reality preserving fractional transform.

      Due to the high redundancy of digital images and the limited sensibility of the human eyes to high frequency image information, a certain degree of distortion of image information is allowed. Thus, much of the literature examines lossy image compression. A spectrum cutting is a kind of lossy data compression method. For example, an optical multi-image encryption based on Fourier transform and fractional Fourier transform was proposed in [19]. It performed the operation of Fourier transform to obtain the distributed spectrum, in which most energy in the frequency domain was centralized in the central part of the spectrum. Then, a spectrum cutting was carried out to allow multiple image to be encrypted into a single one. The popularly known Shannon′s sampling theory states that the sampling rate must be at least twice the signal bandwidth to avoid the loss of information while sampling a signal. To reduce the encryption and decryption processing time and the cost of storage and transmission of information, a compression method using high speed sampling, and then encryption is often adopted. This method wastes some sampling resources. In 2006, the theory of compressive sensing or compressive sampling (CS) was proposed in [2022], which achieved the sampling and compression simultaneously. Afterwards, many researchers have focused on CS-based image and signal processing[2328]. A compression-combined digital image encryption method based on compressive sensing was proposed by Huang and Sakurai[23], in which the block Arnold scrambling was used to permutate the positions of measurements, and encryption and compression were accomplished in one step. However, the security of this method was not so high due to the periodicity of Arnold transform. Although Rachlin and Baron[24] proved that the CS-based encryption scheme cannot achieve perfect security, it is still of interest owing to the high computational complexity of breaking through the security. Subsequently, Mayiami et al.[25] proposed a compressed sensing-based encryption which can achieve Shannon′s definition of perfect secrecy subject to a restriction on the number of measurements. An image encryption based on compressive sensing and a double random-phase encoding technique was provided in [26], where the image information was encrypted twice with low data transmission and smaller random phase masks so as to effectively improve information security. Generally, an image encryption based on compressive sensing treats the whole measurement matrix as the key. This renders the key too large to distribute and memorize. A novel image coding scheme was investigated by Hu et al.[27], in which the compression and encryption were achieved simultaneously under a parallel compressive sensing framework. And the measurement matrix was constructed by utilizing a Logistic-Tent system controlled by four keys. This decreased the burden of storage as compared with a key using the whole measurement matrix.

      In this paper, an image encryption algorithm based on CS and fractional discrete cosine transform (DCT) via polynomial interpolation (PI-FrDCT) will be proposed, which can keep information secret more effectively with low data transmission. Firstly, the original image is compressed and encrypted by CS. Then the PI-FrDCT is carried out. The PI-FrDCT is a unitary, real and orthogonal transform, in which a specific sequence, called a generating sequence (GS), has resolved the nonuniqueness of a fractional operator. Three-dimensional piecewise and nonlinear chaotic maps are employed to generate the random GS, in which the initial values and system parameters are treated as the cipher keys. Lastly, a simple bitwise exclusive OR (XOR) operation is performed to conceal the distribution property of the encrypted image after PI-FrDCT.

      The rest of this paper is organized as follows. Section 2 describes some related work. Section 3 introduces the image encryption and decryption process based on CS and PI-FrDCT. Section 4 gives the experimental results and provides analysis. And the conclusions are drawn in Section 5.

    • CS theory points out that if a signal is sparse in one basis then it can be recovered from a small number of projections onto a second basis that is incoherent with the first[2022]. Usually natural signals in the time domain are non-sparse, but may be sparse in some transform domains, such as discrete cosine transform (DCT), discrete Fourier transform (DFT), and discrete wavelet transform (DWT) domains. Consider a 1-D discrete signal x with length N, which may be represented in the ${{\varPsi }} $ domain:

      ${{\alpha }} = {{\varPsi x}}.$

      (1)

      If only $K$ of the N coefficients in ${{\alpha }} $ in (1) are non-zeros, $K << N$, ${{x}}$ is termed as $K$ sparse signal. The linear measurement process is expressed as

      ${{y}} = {{\varPhi x}} = {{\varPhi \varPsi }}'{{\alpha }} = {{\varTheta \alpha }}$

      (2)

      where ′ denotes transposition, ${{\varPhi }}$ is a measurement matrix with size $M \times N$ ($M < N$). And the sensor matrix ${{\varTheta }}$ is the product of ${{\varPsi }} $ and ${{\varPsi }} '$.

      In order to perfectly reconstruct the signal from ${{y}}$, the sensor matrix ${{\varTheta }}$ satisfies the restricted isometry property (RIP) for any K-sparse vector ${{v}}$ meeting (3)[29]:

      $1 - {\delta _K} \le \frac{{\left\| {{{\varTheta v}}} \right\|_2^2}}{{\left\| {{v}} \right\|_2^2}} \le 1 + {\delta _K}$

      (3)

      where ${\delta _K}$ is the smallest number of the isometry constant $\delta $ of ${{\varTheta }}$, $\delta \in (0,1)$.

      RIP ensures that ${{\varTheta }}$ will not map two different K-sparse signals into the same set, i.e., the formed matrix by each $M$ column vector extracted from ${{\varTheta }}$ is non-singular.

      Meanwhile, the measurement data length M is restricted by the inequation:

      $M \ge cK\log (\frac{N}{K})$

      (4)

      where $c$ is a small constant, $K$ is the sparsity degree.

      Since ${{x}}$ is K-sparse, ${{x}}$ can be rebuilt from ${{y}}$ by solving the optimal problem below:

      $\min {\left\| {{\alpha }} \right\|_0}\;\;{\rm{subject}}\;{\rm{to}}\;\;{{y}} = {{\varTheta \alpha }}.$

      (5)

      The above problem is optimal in theory. But, it is not feasible numerically because it is an NP-hard problem[30]. To overcome this difficulty, some relaxing reconstruction algorithms, such as basis pursuit (BP)[31], total variation (TV)[32], matching pursuit (MP)[33], orthogonal matching pursuit (OMP)[34], smooth l0 algorithm (SL0)[35], have been developed. Since SL0 not only solves the problem of an intractable computational load of the minimal l0 search, it also results in an algorithm which is much faster than those algorithms based on minimizing the l1 norm[35]. Thus, SL0 is adopted in the proposed algorithm.

    • For a signal of length N, its fractional DCT (FrDCT) matrix ${{{C}}_\alpha }$ is defined as[3639]

      $\begin{split} {{{C}}_\alpha } \!\!= & 2{\rm Re}\left[ {\sum\limits_{n = 1}^{\frac{N}{2}} {{{{U}}_n}\lambda _n^\alpha } } \right]\!\! =\!2{\rm Re}\left[ {\sum\limits_{n = 1}^{\frac{N}{2}} {{U_n}{{\rm e}^{j({\varphi _n} + 2\pi {q_n})\alpha }}} } \right]\!\!=\\ & \sum\limits_{n = 1}^{\frac{N}{2}} {({{{A}}_n}\cos {\omega _n}\alpha + {{{B}}_n}\sin {\omega _n}\alpha )} \end{split}$

      (6)

      where $\alpha $ is the fractional order, $N$ is an integer multiple of 4, ${{{U}}_n} = {{{u}}_n}{{u}}_n^ * $ is a unitary matrix, ${{{u}}_n}$ is the n-th eigenvector of the $N \times N$ DCT matrix, ${{{A}}_n} = 2{\mathop{\rm Re}\nolimits} \left[ {{U_n}} \right]$, ${{{B}}_n} = 2{\mathop{\rm Im}\nolimits} \left[ {{U_n}} \right]$, ${\omega _n} = {\varphi _n}\; + 2\pi {q_n}$, $0 < {\varphi _n} < \pi $, $n = 1,$$ 2,\cdots,$ ${\displaystyle\frac{N}{2}}$, $N$ is an integer multiple of 4, ${q_n}$ is an arbitrary integer. ${{q}} = ( {q_1},{q_{2,}}\cdots,$${q_{\frac{N}{2}}})$ resolves the nonuniqueness of a fractional operator and is called “the generating sequence” of the FrDCT. The detailed derivation of ${{{C}}_\alpha }$ can be found in [36, 38]. The FrDCT inherits all features of DCT, such as having a unique orthonormal basis, the reality preservation, index additivity, and non-periodicity.

      To compute ${{{C}}_\alpha }$, one can write an interpolation formula for (6) that allows the construction of ${{{C}}_\alpha }$. The FrDCT matrix via polynomial interpolation is expressed as

      ${{{C}}_\alpha } = \sum\limits_{r = 0}^{N - 1} {{{{C}}_{{\alpha _r}}}{l_r}(\alpha )} \qquad \qquad \qquad $

      (7)

      where ${\alpha _r}$ is considered as the r-th “fraction”, and $ {l_r}{\rm{(}}\alpha {\rm{) = }}$ $\displaystyle\sum\nolimits_{n = 1}^{\frac{N}{2}} {({d_{2n - 1,r}}\cos {\omega _n}\alpha + {d_{2n,r}}\sin {\omega _n}\alpha )} $, ${{D}} \!=\! \left\| {{d_n}_r} \right\| \!=\! $${{{H}}^{ - 1}}$, $ {{H}} =\left\| {{h_{rn}}} \right\|$ with ${h_{r,2n - 1}} = \cos {\omega _n}{\alpha _r}$, ${h_{r,2n}} = \sin {\omega _n}{\alpha _r}$, ${\rm{ }}r = 0,1,\cdots,N - 1$, $n = 1,2,\cdots,\displaystyle\frac{N}{2}$ . The detailed derivation of the interpolation formula of ${{{C}}_\alpha }$ is demonstrated in [36, 39].

      Equation (7) is the general interpolation formula, which gives the FrDCT matrix as a weighted combination of its values at the $N$ different fractions ${\alpha _r}$.

      From (7), the FrDCT via polynomial interpolation (PI-FrDCT) ${{{S}}_\alpha }$ of a given sequence ${{s}}$ can be obtained by

      ${{{S}}_\alpha }{\rm{ = }}\sum\limits_{r = 0}^{N - 1} {{{{S}}_{{\alpha _r}}}{l_r}(\alpha )}. $

      (8)

      Thus, ${{{S}}_\alpha }$ can be obtained by merely evaluating the FrDCTs ${{{S}}_{{\alpha _r}}}$ of ${{s}}$ for $N$ different fractions ${\alpha _r}$. The inverse PI-FrDCT of ${{{S}}_\alpha }$ can be expressed as

      ${{s}} = {{{C}}_{ - \alpha }}{{{S}}_\alpha }{\rm{ = }}\sum\limits_{r = 0}^{N - 1} {{{{C}}_{{\alpha _r}}}{{{S}}_\alpha }{l_r}( - \alpha )}. $

      (9)

      For simplicity, ${\alpha _r} = nr$, $r = 0,\;1,\cdots,\;N - 1$ and $n = 1,\;2,\;3,\cdots$ is chosen.

      In this paper, the condition number of matrix ${{H}}$ is calculated for different n when ${\alpha _r} = nr$. When ${\alpha _r} = 5r$, $r = 0,\;1,\cdots,\;N - 1$, the coefficient matrix ${{H}}$ is found not singular and the equation set (7) is well-conditioned. Of course, n can be set to any other values satisfying (7). Choosing ${\alpha _r} = 5r $, $r = 0,\;1,\cdots,\;N - 1$, yields

      ${{{C}}_\alpha }{\rm{ = }}\sum\limits_{r = 0}^{N - 1} {{{{C}}_{^{5r}}}{l_r}(\alpha )}. $

      (10)

      At last, The PI-FrDCT of a given sequence ${{s}}$ can be expressed as

      ${{{S}}_\alpha } = {l_0}(\alpha ){{s}} + {l_1}(\alpha ){{{s}}^{_5}} + {l_2}(\alpha ){{{s}}^{_{10}}} + \cdots + {l_{N - 1}}(\alpha ){{{s}}^{_{5(N - 1)}}}$

      (11)

      where ${{s}}$ is the original sequence, ${{{s}}^{_5}}$ is the 5th DCT of ${{s}}$, ${{{s}}^{_{10}}}$ is the 5th DCT of ${{{s}}^{_5}}$, etc.

      From (11), the PI-FrDCT can be completed by merely calculating the multiple DCT of the signal and ${l_r}$ with the arguments ${\varphi _n}$ of the eigenvalues and the generating sequence ${{q}} = \left( {{q_1},{q_2},\cdots,{q_\frac{N}{2}}} \right)$ due to ${\omega _n} = {\varphi _n} + 2\pi {q_n}$. It does not require the calculation of the orthonormal set ${{{u}}_{ \pm n}}$ of eigenvectors compared with (7), and allows fast computation.

      For any fraction $\alpha $, one can iterate the DCT algorithm by $5(N - 1)$ times. Since fast algorithms are available for the DCT, with a computational complexity of $N{\log _2}N$ operations[33], the calculation of K distinct PI-FrDCTs using (11) has complexity of $5(N - 1){\log _2}N + $KN operations, instead of the $K{N^2}$ operations required for direct calculation. Hence, with the approach of (11), an improvement can be obtained for $K > 5{\log _2}N$.

      The PI-FrDCT of an image ${{F}}$ can be calculated as follows:

      Let ${{T}} \in {{\bf R}^{M \times N}}$ be an intermediate matrix, and be denoted by ${({t_{ij}})_{M \times N}}$. The matrices G and F are denoted by ${({g_{ij}})_{M \times N}}$ and ${({f_{ij}})_{M \times N}}$, respectively. If they are supposed to be images transformed and to be transformed, respectively, then

      $\left\{ \begin{aligned} & {{{t}}_{i,:}} \!=\! {l_0}(\alpha ){{{f}}_{i,:}} \!+\! {l_1}(\alpha ){{f}}_{i,:}^5 \!+\! {l_2}(\alpha ){{f}}_{i,:}^{10} \!+\! \cdots \!+\!{l_{N - 1}}(\alpha ){{f}}_{i,:}^{5(N \!- \!1)},\\ & \quad\qquad\qquad i = 0,1,\cdots\!,M - 1\\ & {{{g}}_{:,j}}\!= \!{l_0}(\beta ){{{t}}_{:,j}} \!+\! {l_1}(\beta ){{t}}_{:,j}^5 \!+\! {l_2}(\beta ){{t}}_{:,j}^{10} \!+\! \cdots \!+\!{l_{M - 1}}(\beta ){{t}}_{:,j}^{5(M - 1)},\\ & \quad\qquad\qquad j = 0,1,\cdots\!,N - 1 \end{aligned} \right.$

      (12)

      where $\alpha $ and $\beta $ are the fractional orders of row and column, respectively, $M$ and $N$ are integer multiples of 4, ${{{t}}_{i,:}}$ and ${{{g}}_{:,j}}$ are the i-th row of the matrix T and the j-th column of the matrix G, respectively, and

      $\begin{aligned} & {l_r}(\alpha ) \!=\!\!\sum\limits_{n = 1}^\frac{N}{2} {({d_{2n - 1,r}}\cos ({\varphi _n} \!\!+\! 2\pi {p_n})\alpha \!+\! {d_{2n,r}}\sin ({\varphi _n} \!+\! 2\pi {p_n})\alpha )} \\ & {l_r}(\beta ) \!=\! \!\sum\limits_{n = 1}^\frac{M}{2} {({d_{2n - 1,r}}\cos ({\varphi _n} \!\!+\! 2\pi {q_n})\beta \!+\! {d_{2n,r}}\sin ({\varphi _n} \!+\! 2\pi {q_n})\beta )} \end{aligned}$

      (13)

      where ${{p}} = \left( {{p_1},{p_2},\cdots\!,{p_\frac{N}{2}}} \right)$ and ${{q}} = \left( {{q_1},{q_2},\cdots\!,{q_\frac{M}{2}}} \right)$ are respectively generating sequences of the rows and the columns of PI-FrDCT.

      The inverse PI-FrDCT is simply given by

      $\left\{ \begin{aligned} & \begin{aligned}{{{t}}_{:,i}} = & {l_0}( - \beta ){{{g}}_{:,i}} + {l_1}( - \beta ){{g}}_{:,i}^5 + {l_2}( - \beta ){{g}}_{:,i}^{10} + \cdots +\\ & {l_{M - 1}}( - \beta ){{g}}_{:,i}^{5(M - 1)},\quad i = 0,1,\cdots,N - 1\end{aligned} \\ & \begin{aligned}{{{f}}_{j,:}} = & {l_0}( - \alpha ){{{t}}_{j,:}} + {l_1}( - \alpha ){{t}}_{j,:}^5 + {l_2}( - \alpha ){{t}}_{j,:}^{10} + \cdots + \\ & {l_{N - 1}}( - \alpha ){{t}}_{j,:}^{5(N - 1)},\quad j = 0,1,\cdots,M - 1.\end{aligned} \end{aligned} \right.$

      (14)
    • Three-dimensional (3D) piecewise and nonlinear chaotic maps[40] are employed to generate the GS, since they have more initial and system parameters than in one-dimensional case:

      $\begin{split} & \left\{ \begin{aligned} & {x_{n + 1}} = \frac{{4{b_1}{b_3}{x_n}(1 - {x_n})}}{{1 + 4({b_1}{b_3} - 1){x_n}(1 - {x_n})}}\\ & {y_{n + 1}} = \frac{{4{b_1}^2{y_n}(1 - {y_n})}}{{1 + 4({b_1}^2 - 1){y_n}(1 - {y_n})}}\\ & {z_{n + 1}} = \frac{{4{b_3}^2{z_n}(1 - {z_n})}}{{1 + 4({b_3}^2 - 1){z_n}(1 - {z_n})}}\\ &\qquad\qquad\qquad{x_n} \in \left[ {0,\;\frac{1}{2}} \right] \end{aligned} \right.\\ & \left\{ \begin{aligned} & {x_{n + 1}} = \frac{{4{b_2}{b_4}{x_n}(1 - {x_n})}}{{1 + 4({b_2}{b_4} - 1){x_n}(1 - {x_n})}}\\ & {y_{n + 1}} = \frac{{4{b_2}^2{y_n}(1 - {y_n})}}{{1 + 4({b_2}^2 - 1){y_n}(1 - {y_n})}}\\ & {z_{n + 1}} = \frac{{4{b_4}^2{z_n}(1 - {z_n})}}{{1 + 4({b_4}^2 - 1){z_n}(1 - {z_n})}}\\ &\qquad\qquad\qquad{x_n} \in \left( {\frac{1}{2},\;1} \right] \end{aligned} \right. \end{split}$

      (15)

      where ${x_n},\;\;{y_n} \in \left( {0,\;1} \right)$, $n = 0,\;1,\;2,\cdots$, and ${x_0}$, ${y_0}$ are the initial values. If $0.52 < {b_1}{b_3},\;{b_2}{b_4},\;{b_1},{b_2},{b_3},{b_4} \le 6$, the 3D piecewise and nonlinear chaotic maps will exhibit chaotic behavior. Fig. 1 exhibits the chaotic nature of $x,\;y,\;z$ in (15) and Fig. 2 shows the bifurcation behavior of $x$.

      Figure 1.  From left to right are chaotic behaviors of $x,\;y,\;z$ (${x_0}$=0.823 2, ${y_0}$=0.235, ${z_0}$=0.865 4, ${b_1}$=1.5, ${b_2}$=0.892, ${b_3}$=3.256, ${b_4}$=1.238 9)

      Figure 2.  Bifurcation behavior of $x$ (${x_0}$=0.011)

    • In this paper, a disturbance factor $\Delta $ associated with the plain-image is introduced and generated by[38]

      $\Delta = \frac{{\displaystyle\sum\limits_{i,j} {f(i,j)} }}{{255 \times M \times N}}$

      (16)

      where $f(i,j)$ indicates the pixel value of plain-image at $(i,j)$ and $M \times N$ is the size of image.

      With the disturbance factor, the actual initial values and system parameters can be modified such that:

      $z' = z + \Delta $

      (17)

      where $z$ represents given initial values and system parameters. $z'$ represents the actual ones. If there is even a one bit difference between two plain images, the actual initial values and system parameters will be completely different due to the introduction of $\Delta $.

    • The encryption and decryption process of the proposed method is shown in Fig. 3 and the image encryption steps are as follows.

      Figure 3.  Flowchart of the encryption and decryption process

      Step 1. Construct measurement matrix ${{\varPhi }}$ with size $M \times N$[41]:

      Generate a sequence with length $N + {m_1}$ by a logistic map, i.e., ${x'_{n + 1}} = \mu {x'_n}(1 - {x'_n})$ with an initial value ${x'_0}$ and a system parameter $\mu $, discard previous ${m_1}$ entries for enhancing randomness and confusion to obtain the sequence ${{\eta }} = [{\eta _1},{\eta _2},\cdots,{\eta _N}]$ and mark $\varepsilon = [{\varepsilon _1},{\varepsilon _2},\cdots,{\varepsilon _N}]$ as the result of index sequence of sorting ${{\eta }} $ in descending order.

      Sort the natural sequence ${{n}} = [1,2,\cdots,N]$ according to the index sequence $\varepsilon $, mark $pp = [p{p_1},p{p_2},\cdots,p{p_N}]$ as the result of sorting ${{n}}$.

      Generate the Hadamard matrix ${{J}}$ of order N, choose the row vectors, ${{J}}(p{p_1},\;\;:)$, ${{J}}(p{p_2},\;\;:)$, $\cdots,\;\;{{J}}(p{p_M},\;\;:)$ and then form the measurement matrix ${{\varPhi }}$, i.e.,

      ${{\varPhi }} = \left[ \begin{aligned} & {{J}}(p{p_1},\;:)\\ & {{J}}(p{p_2},\;:)\\ & \quad\quad\vdots \\ & {{J}}(p{p_M},\;:) \end{aligned} \right].$

      Step 2. Obtain ${{{I}}_2}$ with size $M \times M$ by one-time measuring or two-time measuring ${{{I}}_1}$ using ${{\varPhi }}$.

      a) For one-time measuring, apply CS to ${{{I}}_1}$ and mark ${{{I}}_{1'}}$ with size $M \times N$ as the result of the measurement, and then ${{{I}}_2}$ is taken column-wise from ${{{I}}_{1'}}$.

      b) For two-time measuring, obtain ${{{I}}_2}$ in accordance with

      ${{{I}}_2} = {{\varPhi \varPsi }}({{\varPhi \varPsi }}{{{I}}_1})'$

      (18)

      where ${{\varPsi}} $ is the discrete wavelet transform matrix and ${{{I}}_1}$ is the original image.

      Step 3. Perform the PI-FrDCT for ${{{I}}_2}$ in accordance with (12), and mark ${{{I}}_3}$ as the result of PI-FrDCT.

      For an image ${{{I}}_2}$ of size $M \times M$, two generating sequences of length $\displaystyle\frac{M}{2}$ are necessary for PI-FrDCT. First, a chaotic sequence of length $m + M + {M^2}$ is generated by iterating (15); and the previous $m$ entries are discarded to obtain the sequences ${{x}} \!=\! [{x_0},{x_1},\cdots\!,{x_{{\frac{M}{2}}{ - 1}}}]$, ${{y}}\!=\![{y_0},{y_1},\cdots\!,$${y_{{\frac{M}{2}}{ - 1}}}]$ and ${{z}} = [{z_0},{z_1},\cdots,{z_{{M^2} - 1}}]$. $m$ entries are discarded to further confirm the randomness of the sequences. Since the GS takes integer values, ${{p}}$ and ${{q}}$ are uniformly quantized to map intervals (0, 0.25], (0.25, 0.5], (0.5, 0.75], (0.75, 1) into integers 0, 1, 2, 3, respectively. The matrix ${{Q}}$ of size $M \times M$, which serves as a diffusion matrix for the XOR operation, is taken column-wise from ${{zz}} = [z{z_0}\;z{z_1}\cdots z{z_{{M^2} - 1}}]$ with

      $z{z_i} = ({z_i} \times {10^9})od 256,\;i = 0,1, \cdots ,{M^2} - 1 .$

      (19)

      Step 4. Non-uniformly quantize ${{{I}}_3}$ with 8 bits in accordance with (20) to obtain ${{{I}}_4}$:

      ${{{I}}_4} = {\rm{round}}(128 \times \tan (\frac{{{{I}}_3}}{1000}) + 128).$

      (20)

      Step 5. Carry out bitwise XOR to conceal the distribution property of ${{{I}}_4}$ and make the energy uniformly distributed over the entire image ${{E}}$.

      ${{E}} = {{{I}}_4} \oplus {{Q}}$

      (21)

      where $ \oplus $ denotes the bitwise XOR operation.

      The decryption process is the XOR followed by the dequantization, the inverse operation of PI-FrDCT and the reconstruction operation with the SL0 algorithm. All the cipher keys do not change except for the fractional orders, which are $ - \alpha $ and $ - \beta $, respectively.

      Instead of minimizing the l1 norm using linear programming (LP), the SL0 algorithm directly minimizes the l0 norm and is about two to three orders of magnitude faster than the interior-point LP solvers, while keeping the same accuracy. The detailed description of the SL0 algorithm can be found in [35].

    • The proposed algorithm is tested using two images, Lena and Peppers, both with size 512×512 (Figs. 4(a) and 4(d)). An average value should be subtracted to remove the direct current (DC) component, so that the quantization error can be as small as possible in the quantization process after PI-FrDCT. 128 is selected as an approximation of the DC value for convenience. $\alpha $, $\beta $, ${x'_0}$, $\mu $, ${m_1}$, ${b_1}$, ${b_2}$, ${b_3}$, ${b_4}$, ${x_0}$, ${y_0}$, ${z_0}$, $m$ and $\Delta $ serve as the cipher keys. Since the sensitivity of the fractional orders $\alpha $ and $\beta $ to the decrypted image quality is not high, they serve as auxiliary keys. According to Section 2.3, ${x'_0}$, $\mu $, ${m_1}$, ${b_1}$, ${b_2}$, ${b_3}$, ${b_4}$, ${x_0}$, ${y_0}$, ${z_0}$ and $m$ are randomly chosen within the specified intervals. In our experiment, $\alpha $ and $\beta $ are set to be 0.768 9 and 0.457 8, a disturbance factor $\Delta $ associated with the plain-image is produced in the encryption process according to (16). The common initial value and parameter ${x'_0}$, $\mu $, ${m_1}$, ${b_1}$, ${b_2}$, ${b_3}$, ${b_4}$, ${x_0}$, ${y_0}$, ${z_0}$ and $m$ are set to be 0.110 5, 3.992 4, 1 000, 3.256 9, 0.892 54, 2.567 2, 1.862, 0.234 5, 0.568 97, 0.987 5, 2 000, respectively. These parameters can of course be set to be any other values satisfying the conditions. The actual cipher keys of ${x'_0}$, $\mu $, ${m_1}$, ${b_1}$, ${b_2}$, ${b_3}$, ${b_4}$, ${x_0}$, ${y_0}$ and ${z_0}$ are obtained by (17). The encryption and decryption results are shown in Fig. 4. Here, the compression ratio is 1.78, i.e., the size of the encrypted image is 384×384. As can be seen, the decrypted images contain the main information in spite of some distortion. To evaluate the security and effectiveness of the proposed algorithm, statistical analysis, key space analysis and robustness analysis are carried out.

      Figure 4.  Test image and results: (a) Original Lena, (b) Encrypted image of (a), (c) Decrypted image of (b), (d) Original Peppers, (e) Encrypted image of (d), and (f) Decrypted image of (e)

      In addition, the encryption algorithm performing one-time measuring in the CS stage is called Algorithm 1, and the encryption algorithm using the two-time measuring in the process of the encryption is called Algorithm 2. The experiments are performed in order to analyze the difference between Algorithms 1 and 2 with the same compression ratio.

    • Histograms of both original test and encrypted images are shown in Fig. 5. From Figs. 5 (b) and 5 (d), it can be found that histograms of the encrypted images exhibit a uniform distribution and are different from those of the original images. None of the useful information is leaked to the adversary. The introduction of PI-FrDCT and XOR operations makes the encrypted scheme more secure. Hence, the proposed algorithm is secure enough to resist statistical analysis attacks by histogram analysis.

      Figure 5.  Histograms: (a) Original Lena, (b) Encrypted Lena, (c) Original Peppers, and (d) Encrypted Peppers

    • The information entropy is expressed as

      $H{\rm{(}}m{\rm{)}} = - \sum\limits_{i = 0}^{L - 1} {P({m_i})} {\log _2}P({m_i})$

      (22)

      where ${m_i}$ is the i-th gray value for an L-gray level image and $P({m_i})$ is its normalized occurrence frequency. A larger entropy usually means a more uncertain distribution. The maximum entropy for an image with 256 gray levels is 8 when all gray levels are equally distributed, indicating the largest randomness. As shown in Table 1, the values of information entropy for different encrypted images using Algorithms 1 and 2 are larger than those for different original images, and are very close to 8. It indicates that the distribution of the image grey value is completely random and possesses an ability to resist entropy based attacks. Thus, it can be concluded that the proposed algorithm can effectively resist entropy based attacks.

      Original imageAlgorithm 1Algorithm 2
      Lena7.445 57.998 47.998 5
      Peppers7.571 57.998 67.998 4
      Barbara7.466 47.998 77.998 7
      Boat7.123 87.998 67.998 8
      Plane6.705 47.998 67.998 4
      Camera7.048 07.998 37.998 4

      Table 1.  Information entropy of original and encrypted images

    • Correlation analysis is considered here. Pairs of all adjacent pixels in the horizontal, vertical or diagonal directions from both the original image and encrypted image are selected and then the correlation coefficients of adjacent pixels for each direction are calculated as

      ${r_{xy}} = \frac{{\displaystyle\sum\limits_{i = 1}^N {({x_i} - \bar x)({y_i} - \bar y)} }}{{\sqrt {\displaystyle\sum\limits_{i = 1}^N {{{({x_i} - \bar x)}^2}}\displaystyle\sum\limits_{i = 1}^N {{{({y_i} - \bar y)}^2}} } }}$

      (23)

      where $\bar x = \displaystyle\frac{1}{N}\displaystyle\sum\limits_{i = 1}^N {{x_i}} $ and $\bar y = \displaystyle\frac{1}{N}\displaystyle\sum\limits_{i = 1}^N {{y_i}} $, ${x_i}$ and ${y_i}$ are gray values of two adjacent pixels in an image, $N$ represents the number of pairs of adjacent pixels in the same image.

      The values of the correlation coefficients of adjacent pixels in the encrypted image are reduced compared with those in the corresponding encrypted image and the original image, as shown in Table 2. The correlation coefficients of the proposed Algorithm 2 are relatively smaller than those of Algorithm 1 in Table 2. The results show the coefficients of the proposed Algorithm 2 are all sufficiently low, which indicates a satisfactory confusion effect.

      AlgorithmImageHorizontalVerticalDiagonal
      Original Lena0.952 9760.984 7950.948 743
      Algorithm 1Encrypted Lena0.005 3580.019 3800.003 486
      Algorithm 2Encrypted Lena0.000 536–0.000 2200.000 684
      Original Peppers0.977 3720.980 0180.970 839
      Algorithm 1Encrypted Peppers0.031 067–0.004 8430.005 990
      Algorithm 2Encrypted Peppers–0.002 496–0.002 3260.000 855
      Original Barbara0.919 6310.968 7760.914 610
      Algorithm 1EncryptedBarbara–0.001 703–0.005 070–0.017 998
      Algorithm 2Encrypted Barbara0.006 552–0.004 922–0.001 693
      Original Boat0.957 8120.957 6370.941 071
      Algorithm 1Encrypted Boat0.006 925–0.027 532–0.006 066
      Algorithm 2Encrypted Boat0.001 0100.001 783–0.001 020
      Original Plane0.962 6520.960 9470.925 657
      Algorithm 1Encrypted Plane0.068 6140.085 2470.004 917
      Algorithm 2Encrypted Plane–0.014 8700.014 3700.006 899
      Original Camera0.986 5670.990 6480.985 069
      Algorithm 1Encrypted Camera0.112 1010.164 1020.010 177
      Algorithm 2Encrypted Camera–0.001 7030.002 3530.001 361

      Table 2.  Correlation coefficients of adjacent pixels

      As an example, the experimental results of the Lena image are exhibited in Fig. 6. From Figs. 6 (a)-6 (c), it can be seen that the joint distribution of gray values of adjacent pixels of the original image in three directions distributes near the diagonal of the coordinate plane. This demonstrates a strong correlation in the original image. And from Figs. 6 (d)-6 (f), the joint distribution of gray values of adjacent pixels of the encrypted images in three directions exhibit relatively uniform. Therefore, we can conclude that the statistical analysis attack on our scheme is infeasible.

      Figure 6.  Joint distribution of gray values of adjacent pixels of the original Lena and encrypted Lena: (a), (b) and (c) are the horizontal, vertical and diagonal correlations of the original Lena, respectively. (d), (e) and (f) are the horizontal, vertical and diagonal correlations of the encrypted image, respectively.

      In conclusion, the results in Fig. 6 and Table 2 demonstrate that the correlation between adjacent pixels in the encrypted image is greatly reduced.

    • To evaluate the ability to resist differential attacks, two criteria are defined: the number of pixels change rate (NPCR) and the unified average changing intensity (UACI). The NPCR and UACI are defined as[38, 42]

      ${{\rm NPCR}} = \frac{{\displaystyle\sum\limits_{i,j} {{{C}}(i,j)} }}{{M \times N}} \times 100{\text%} $

      (24)

      ${{C}}(i,j) = \left\{ \begin{aligned} & 0,\;{{{T}}_1}(i,j) = {{{T}}_2}(i,j)\\ & 1,\;{{{T}}_1}(i,j) \ne {{{T}}_2}(i,j) \end{aligned} \right.$

      (25)

      where $M \times N$ is the size of images ${{{T}}_1}$ and ${{{T}}_2}$. ${{{T}}_1}(i,j)$ and ${{{T}}_2}(i,j)$ indicate the pixel values of two cipher-images at $(i,j)$, corresponding to two plain-images that are different by one pixel.

      ${{\rm UACI}} = \frac{{\sum\limits_{i,j} {\left| {{{{T}}_1}(i,j) - {{{T}}_2}(i,j)} \right|} }}{{255 \times M \times N}} \times 100\% .$

      (26)

      For an image with 8 bits for each pixel, the expected values of NPCR and UACI are respectively 99.61% and 33.46%[43]. As shown in Table 3, the NPCRs are all above 99.60% and the UACIs of the encrypted image are all over 33.20%, which indicates that the values of each pixel pair at the same location in two encrypted images are substantially different. Thus, the encryption scheme is very sensitive with respect to a small change in the plain-images even by as little as one pixel difference. The scheme is even more secure due to combining CS, PI-FrDCT and XOR operations. In conclusion, the results in Table 3 demonstrate that the proposed encrypted algorithm has an ability to resist a differential attack.

      AlgorithmNPCR (%)UACI (%)
      LenaAlgorithm 199.618 933.333 5
      Algorithm 299.621 633.300 9
      PeppersAlgorithm 199.629 733.201 7
      Algorithm 299.634 533.219 3
      BarbaraAlgorithm 199.610 733.217 8
      Algorithm 299.605 333.241 6
      BoatAlgorithm 199.611 433.224 1
      Algorithm 299.613 433.222 6
      PlaneAlgorithm 199.610 133.333 0
      Algorithm 299.605 333.342 7
      CameraAlgorithm 199.625 033.284 1
      Algorithm 299.625 033.259 8

      Table 3.  NPCRs and UACIs for different encrypted images

    • In the proposed algorithm, the cipher keys are $\alpha $, $\beta $, ${x'_0}$, $\mu $, ${m_1}$, ${b_1}$, ${b_2}$, ${b_3}$, ${b_4}$, ${x_0}$, ${y_0}$, ${z_0}$, $m$ and $\Delta $. The fractional orders $\alpha $ and $\beta $ serve as auxiliary keys. Fig. 7 illustrates the decrypted images of Lena with just one wrong key during decryption. Intuitively, the decrypted images are completely disordered, as shown in Fig. 7. The results show that any slight mismatch in the cipher key will lead to a incorrect decoding. If the error is up to ${10^{ - 15}}$, the key space size is ${10^{135}}$ if ${x'_0}$, $\mu $, ${b_1}$, ${b_2}$, ${b_3}$, ${b_4}$, ${x_0}$, ${y_0}$ and ${z_0}$ are used as cipher keys. In addition, the choice of $\alpha $, $\beta $, ${m_1}$ and $m$ makes the actual key space even larger. The introduction of PI-FrDCT and 3D piecewise and nonlinear chaotic maps enlarges the key space and enhances the security of the encryption scheme. Therefore, the key space is large enough to resist brute-force attacks.

      Figure 7.  Decrypted images with only one wrong key. The deviations are respectively: (a) $\Delta {x'_0} = {10^{ - 15}}$, (b) $\Delta \mu = {10^{ - 15}}$, (c) $\Delta {m_1} = 1$, (d) $\Delta {b_1} = {10^{ - 15}}$, (e) $\Delta {b_2} = {10^{ - 15}}$, (f) $\Delta {b_3} = {10^{ - 15}}$, (g) $\Delta {b_4} = {10^{ - 15}}$, (h) $\Delta {x_0} = {10^{ - 15}}$, (i) $\Delta {y_0} = {10^{ - 15}}$, (j) $\Delta {z_0} = {10^{ - 15}}$, (k) $\Delta m = 1$, (l) use $\Delta $ of Peppers.

    • For a noise attack, the Gaussian noises are added into the encrypted image[38]:

      ${{{E}}_N} = {{E}}(1 + k{{G}})$

      (27)

      where ${{{E}}_N}$ and ${{E}}$ are the noise-affected and noiseless encrypted images, respectively. $k$ represents the intensity level of the added noise. ${{G}}$ is the Gaussian noise with zero-mean and unit standard deviation. When $k$ is respectively set to 0.01, 0.05, 0.1 and 0.5, the corresponding decrypted images of Lena and Peppers are exhibited in Fig. 8. As shown in Figs. 8 (a)-8 (h), the quality of the decrypted images decreases with the increase of added noise intensity. The content of decrypted image can still be identified despite the interference of noise to some extent. Figs. 9 (a)-9 (c) respectively show the encrypted images with an occlusion of ${\displaystyle\frac{1}{64}}$, ${\displaystyle\frac{1}{16}}$ and ${\displaystyle\frac{1}{4}}$. Figs. 9 (d)-9 (i) show the corresponding decrypted images for Lena and Peppers. The quality of the decrypted images decreases as the occlusion size increases. Although the encrypted image is partly occluded, the main content of the original images can be still recognized from the decrypted image, as shown in Figs. 9 (d)- 9 (i). Hence, the proposed encryption scheme has a certain robustness against noise and occlusion attacks.

      Figure 8.  Decrypted images with Gaussian noise intensity levels of: (a), (e) k=0.01; (b), (f) k=0.05, (c), (g) k=0.1 and (d), (h) k=0.5

      Figure 9.  Results of anti-occlusion attacks: the encrypted images for Lena with an occlusion of (a) ${\displaystyle\frac{1}{64}}$, (b) ${\displaystyle\frac{1}{16}}$, (c)${\displaystyle\frac{1}{4}}$; (d), (e) and (f) are decrypted images from (a), (b), (c), respectively; (g), (h), (i) are decrypted images for Peppers under the same occlusions as in (a), (b), (c), respectively.

    • To measure the similarity of original and decrypted images for different compression ratios, the peak signal-to-noise ratio (PSNR)[31] is adopted.

      ${\rm{PSNR}} = 10{\log _{10}}\frac{{{{255}^2} \times M \times N}}{{\displaystyle\sum\limits_{i = 1}^M {\displaystyle\sum\limits_{j = 1}^N {\left[ {{{O}}(i,j) - {{I}}(i,j)} \right]} } }}\;({\rm{dB}})$

      (28)

      where $M \times N$ is the size of image, ${{I}}(i,j)$ and ${{O}}(i,j)$ are the pixel values of the original and reconstructed images at $(i,j)$, respectively.

      In order to obtain the same compression ratio, the number of rows of the sensor matrix in Algorithm 1 is smaller than that in Algorithm 2. As shown in Tables 4 and 5, the quality of the reconstructed images decreases as the compression ratio increases. From Table 4, the quality of the reconstructed image using Algorithm 1 is visually inferior to that using Algorithm 2 when the compression ratio is 4:1. For instance, Lena′s mouth and eyebrows become blurred. In particular, from the reconstructed images in Tables 4 and 5 and PSNR in Tables 6 and 7, more information is lost when adopting Algorithm 1 than adopting Algorithm 2, when the compression ratio is 7.11:1. Even though the PSNR of the proposed algorithm is relatively small when the compression ratio is up to 7.11:1, the main content of the reconstructed images can be obtained. Tables 6 and 7 list the PSNR of the proposed algorithm and the algorithm in [19]. Compressive sensing is a signal processing tool to efficiently acquire and reconstruct a signal. Its principle is based on optimization and sparsity. The signal can be sampled at a frequency smaller than that required by the Nyquist theory. Although the PSNRs of proposed algorithm are not superior to those in [19], both of [19] and ours are of significance and we need not focus on their PSNRs too much.

      Compression ratioDecrypted imagesSize of
      encrypted
      image
      Algorithm 1Algorithm 2
      1.306:1448×448
      1.78:1384×384
      2.56:1320×320
      4:1256×256
      7.11:1192×192

      Table 4.  Reconstructed image for different compression ratios of Lena

      Compression ratioDecrypted imagesSize of
      encrypted
      image
      Algorithm 1Algorithm 2
      1.306:1448×448
      1.78:1384×384
      2.56:1320×320
      4:1256×256
      7.11:1192×192

      Table 5.  Reconstructed image for different compression ratios of Peppers

      Compression ratioPSNR (dB)
      Reference [19]Algorithm 1Algorithm 2
      1.306:143.716 239.542 036.351 6
      1.78:140.329 334.940 032.278 5
      2.56:137.812 430.687 328.547 9
      4:135.350 826.336 926.100 6
      7.11:132.677 219.827 423.456 7

      Table 6.  PSNR for different compression ratios of Lena

      Compression ratioPSNR (dB)
      Reference [19]Algorithm 1Algorithm 2
      1.306:139.396 338.241 835.960 5
      1.78:135.930 034.311 932.770 8
      2.56:134.088 429.559 628.998 6
      4:132.414 024.090 225.788 3
      7.11:130.738 916.953 722.448 8

      Table 7.  PSNR for different compression ratios of Peppers

    • It is clear that in the proposed scheme, the initial values and system parameters of the encryption scheme are determined by cipher keys and plain-image due to introduction of a disturbance factor. The actual initial values and system parameters are different for different plain-images. A disturbance factor makes the encryption system more effective in resisting the known-plaintext and chose-plaintext attacks. In addition, GSs used in the PI-FrDCT and the matrix Q used in the XOR operation are generated by 3D piecewise and nonlinear chaotic maps and are also different for different plain-images. Hence, the correct decrypted results cannot be obtained when a disturbance factor for another image is adopted, as shown in Fig. 7 (l). Thus, the proposed algorithm can resist the known-plaintext and chosen-plaintext attacks.

    • In this paper, we proposed an encryption scheme based on CS and PI-FrDCT, in which the compression and encryption are achieved simultaneously. In the encryption phase, we utilized the CS theory to encrypt and compress the original image. To enhance the security of the encrypted image, the resulting image after CS is re-encrypted by PI-FrDCT. The real-valued output is quantized with 8 bits for a coefficient, which makes the encrypted results convenient for storage, display and transmission. Meanwhile, GSs and the XOR matrix are generated by 3D piecewise and nonlinear chaotic maps, which further strengthen the security of the encryption scheme. The sensitivity of cipher keys is high due to employing the 3D piecewise and nonlinear chaotic maps. Experimental results show the effectiveness and high security of the proposed scheme. For example, it has a high sensitivity, a sufficiently large key space, a robustness against noise and occlusion attacks and an ability to resist common attacks.

    • This work was supported by National Natural Science Foundation of China (Nos. 61662047 and 61462061).

Reference (43)

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