# Protecting Compressive Ghost Imaging with Hyperchaotic System and DNA Encoding

Authors: Jingru Sun; Mu Peng; Fang Liu; Cong Tang
Journal: Complexity
 (2020)

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## Abstract

As computational ghost imaging is widely used in the military, radar, and other fields, its security and efficiency became more and more important. In this paper, we propose a compressive ghost imaging encryption scheme based on the hyper-chaotic system, DNA encoding, and KSVD algorithm for the first time. First, a 4-dimensional hyper-chaotic system is used to generate four long pseudorandom sequences and diffuse the sequences with DNA operation to get the phase mask sequence, and thenN phase mask matrixes are generated from the sequences. Second, in order to improve the reconstruction efficiency, KSVD algorithm is used to generate dictionary D to sparse the image. The transmission key of the proposed scheme includes the initial values of hyper-chaotic and dictionary D, which has plaintext correlation and big key space. Compared with the existing compressive ghost imaging encryption scheme, the proposed scheme is more sensitive to initial values and more complexity and has smaller transmission key, which makes the encryption scheme more secure, and the reconstruction efficiency is higher too. Simulation results and security analysis demonstrate the good performance of the proposed scheme.


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## Body

## 1. Introduction

In recent years, with the rapid development of computer network and communication technology, information security issues have become more and more important. As an emerging optical imaging technology [1–3], CGI (computational ghost imaging) has attracted the attention of researchers once it had appeared and has been widely used in military, encryption, radar, and other fields [4, 5]. Therefore, the security of the CGI is especially vital.CGI is developed based on ghost imaging technology [6], which can transmit image information through one optical path, with simple structure, strong anti-interference ability, and good imaging effect. In 2010, Clemente proposed an image encryption technology based on CGI [7]; as shown in Figure 1, this solution can encrypt plain-images into light intensity values and only requires a bucket detector without spatial resolution to receive the light intensity, which indicates a new research direction for optical information security [8]. To achieve high image reconstruction efficiency, Katz proposed a compressive ghost imaging (CSGI) scheme, which combines CGI with a compressive sensing (CS) algorithm to reduce the number of measurements required for image recovery by an order of magnitude [9–11].Figure 1
CGI encryption scheme.Then, Durfin et al. proposed a CSGI encryption scheme [12]. Zhao et al. further improved the security of optical encryption by utilizing the high fault tolerance of QR encoding, which reduce the size of the transmitted images and enhance the robustness [13]. Wu et al. proposed an optical multiple-image encryption scheme based on CGI, and this method can transmit multiple images at the same time; but with the distance as the keys, it is vulnerable to brute-force attacks [14]. Zhu et al. use fingerprint technology to produce a phase modulation matrix, the fingerprint has the uniqueness, but it is easy to be obtained, and as a transmitted key, fingerprint is too big [15]. Li et al. proposed a multiple-image CSGI encryption method based on the LWT and XOR operations [16]. Most works in the literature fail to associate the key with plaintext image and have big transmission keys. This motivates us to look for a novel CSGI encryption method with plaintext correlation, smaller transmission keys, larger key spaces, and high image reconstruction efficiency.Chaos has many excellent characteristics, such as pseudorandomness, ergodicity, and sensitivity to initial points and parameters [17–23] and has been widely used in image encryption and privacy protection [24–28], communication encryption [29], and other fields. Chaotic systems can be divided into low-dimensional chaotic system and high-dimensional chaotic system. Low-dimensional chaotic systems such as Henon chaotic system [30, 31] and Tent map [32, 33] et al. were first used in the encryption system [34] but have been proved not security enough [35]. High-dimensional chaotic systems such as three-dimensional Lorenz chaotic systems [36, 37], and Chen System [38] and Yu System [39, 40] et al. have more dimensional and higher complexity. Especially hyper-chaotic systems [41, 42] have two or more Lyapunov exponents greater than 0 and larger key space and higher complexity. Hyper-chaotic systems have been wildly used in chaotic image encryption scheme [28, 43–45]. The DNA encoding and decoding technology [46] is a kind of biological method to process information, which has the characteristics of large-scale parallelism, high-storage density, ultra-low-power consumption, unique molecular structure, and intermolecular recognition mechanism. DNA has great development prospects in the field of information encryption [47–52]. In this paper, a CSGI encryption scheme based on the hyper-chaotic system, DNA and KSVD technologies is proposed. First, given four transmission keys, input the four keys as initial values to the hyper-chaotic-system; second, 4 long chaotic sequences are generated by a hyper-chaotic system, then three of them are arranged into a phase sequence, and the other sequence is used to produce a DNA sequence. Third, diffuse the phase sequence with the DNA sequence by DNA operation and then get phase modulation matrixes, which are used as the input of the spatial light modulator (SLM). At the same time, get dictionary matrix D with the original image by KSVD and achieve original signal sparse representation through D. Finally, complete the encryption of the scheme. Compared with the existing CSGI encryption scheme, the proposed scheme has smaller double transmission key, larger key space, high key sensitivity, plaintext correlation, and unpredictability. The use of DNA further increases the complexity and randomness of the encryption scheme.The rest of this paper is organized as follows. In section2, the basic theories of the CGI, hyper-chaotic system, DNA technology, compressed sensing, and singular value decomposition are described. In Section 3, the system framework of our proposed scheme and the generation process of phase mask matrixes are described in detail. The simulation results and security analysis are performed in section 4. The paper is summarized in section 5.


## 2. Basic Theories

### 2.1. CGI

In CGI, as shown in Figure2, a spatial laser beam transmits through a spatial light modulator (SLM), which introduces an arbitrary phase mask matrix φx,y, generating a spatially incoherent beam. Knowing the random phase and the distribution of laser light field Uinx,y, one can evaluate the distribution of the light intensity Uix,y right after the SLM:(1)Uix,y=Uinx,yeiφix,y.Figure 2
The basic theory of CGI.Through Fresnel diffraction, the light field distribution of signal light in front of object plane is the same as reference light, the light travels to the object plane which isz distance away from the SLM, and the speckle filed Iix,y can be calculated:(2)Iix,y=Uix,y⊗hzx,y2,where hzx,y is the transfer function in the spatial domain at a distance z, ⊗ represents the convolution operation, and Iix,y is defines as the reference light. The signal light intensities detected by a bucket detector placed behind the object, which can be represented by a transmission function of the object Tx,y and written as(3)Bi=∫dxdyIix,yTx,y.To construct the object’s transmission functionTx,y, the reference light speckle filed Iix,y cross-correlated with the signal light intensities Bi:(4)Gx,y=1N∑i=1NBi−BIix,y,where Gx,y denotes the recovered object information, ⋅=1/N∑i⋅ is an ensemble average over N measurements, Iix,y is calculated by the receiver according to equation (2), and B is the average value for the measured components Bi [53].


### 2.2. Hyper-Chaotic System

In our proposed CSGI encryption scheme, the phase mask matrix required on the SLM is generated by the hyper-chaotic system:(5)x˙=ay−x+yz,y˙=bx−y−xz+w,z˙=xy−cz,w˙=dw−xz.By setting parametersa=35, b=3/8, c=55, and d=1.3, we obtain four Lyapunov exponents, including two positive Lyapunov exponents, λ1=1.4164 and λ2=0.5318, a zero Lyapunov exponent λ3=0, and a negative Lyapunov exponent λ4=−39.1015 [54]. By this means, the system exhibits hyper-chaotic behavior. Figure 3 depicts the phase portraits of the hyper-chaotic system. Here, we take the fourth-order Runge-Kutta method to solve (5) and obtain the four hyper-chaotic sequences.Figure 3
Phase portraits. (a)x−y plane. (b) x−z plane. (c) x−w plane. (d) y−z plane.
(a)(b)(c)(d)


### 2.3. DNA

DNA is a long-chain polymer, and the basic elements are four nucleic acid bases, namely,A (adenine), C (cytosine), G (guanine), and T (thymine), where A and T, C and G are complementary, respectively. In a binary system, 0 and 1 are complementary. It can be concluded that 00 and 11 are complementary, and 10 and 01 are complementary. Encoding the four bases A, C, G, and T with 0 and 1, eight encoding methods can be obtained, as shown in Table 1. Each of the DNA coding rules corresponds to an operation rule, and the following algorithm is based on the encoding rule 1 and rule 2. According to the binary calculation rule, we can get the corresponding rules of DNA addition, subtraction, and complement rule listed in Tables 2 and 3.Table 1
DNA encode rule.
Rule12345678A0000010110101111T1111101001010000C0110001100110110G1001110011001001Table 2
DNA encode rule.
AdditionACGTSubtractionACGTAACGTAATGCCCGTACCATGGGTACGGCATTTACGTTGCATable 3
DNA encode rule.
Complement123456ATTCCGGTCGGACACGATGATGACATTC


### 2.4. Compressed Sensing

Compressed sensing technology uses sparse basis such as DCT or DFT to represent the signal sparsely, measures the signal based on Gaussian random matrix, and then reconstructs the signal based onL1 norm and other algorithms.Suppose a signal isx∈ℝN×1, before sampling the signal x, select a suitable and orthogonal sparse base Ψ∈ℝN×N to sparsely represent the signal x as(6)x=Ψs,where s is the sparse representation of x on a sparse basis Ψ. s has K nonzero elements, and other N−KN≫K elements’ values are 0.Sparse operation and the measurement matrix must satisfy restricted isometry property (RIP). Discrete cosine transform, fast-Fourier transform, etc. are common sparse operations.During measuringx, in order to reduce the number of the measurements and ensure that the measuring result contain as much information of x as possible, we need an appropriate measurement matrix Φ∈ℝM×NM<N. Bernoulli matrix, Gaussian distribution matrix, Hadamar matrix, Toeplitz matrix, etc. are often used in compressed sensing technology. The measurement of signal x can be expressed as(7)y=Φx=ΦΨs=Θs,where ΦΨ=Θ is a sensor matrix and y∈ℝM×1 is the measurement result.In the end, use the compressed sensing reconstruction algorithm to reconstructs˜ from y:(8)s˜=mins˜L1=minΨTxL1s.t.y=Φx=Θs.The approximate solution vector can be obtained by applying inverse transformation fors˜:(9)x˜=Ψs˜.


### 2.5. Singular Value Decomposition (SVD)

Suppose a real matrixEK′∈ℝm×n can be decomposed can be decomposed into(10)EK′=UEVT,where E∈ℝm×n is a singular value matrix whose nonzero elements are only located on the diagonal. U∈ℝm×m and V∈ℝn×n are both-unit orthogonal matrices, and U means the left singular matrix and V means the right singular matrix, respectively. Generally, E is represented as(11)E=σ100000σ100000⋯00000⋯0m×n.DecomposingEK′ by equation (10), then we can get(12)EK′EK′T=UEVTVETUT=UEETUT,EK′TEK′=VEUTUETVT=VEETVT.With the eigen-decomposition of EK′EK′T and EK′TEK′, the left singular matrix U and the right singular matrix V can be obtained.


## 2.1. CGI

In CGI, as shown in Figure2, a spatial laser beam transmits through a spatial light modulator (SLM), which introduces an arbitrary phase mask matrix φx,y, generating a spatially incoherent beam. Knowing the random phase and the distribution of laser light field Uinx,y, one can evaluate the distribution of the light intensity Uix,y right after the SLM:(1)Uix,y=Uinx,yeiφix,y.Figure 2
The basic theory of CGI.Through Fresnel diffraction, the light field distribution of signal light in front of object plane is the same as reference light, the light travels to the object plane which isz distance away from the SLM, and the speckle filed Iix,y can be calculated:(2)Iix,y=Uix,y⊗hzx,y2,where hzx,y is the transfer function in the spatial domain at a distance z, ⊗ represents the convolution operation, and Iix,y is defines as the reference light. The signal light intensities detected by a bucket detector placed behind the object, which can be represented by a transmission function of the object Tx,y and written as(3)Bi=∫dxdyIix,yTx,y.To construct the object’s transmission functionTx,y, the reference light speckle filed Iix,y cross-correlated with the signal light intensities Bi:(4)Gx,y=1N∑i=1NBi−BIix,y,where Gx,y denotes the recovered object information, ⋅=1/N∑i⋅ is an ensemble average over N measurements, Iix,y is calculated by the receiver according to equation (2), and B is the average value for the measured components Bi [53].


## 2.2. Hyper-Chaotic System

In our proposed CSGI encryption scheme, the phase mask matrix required on the SLM is generated by the hyper-chaotic system:(5)x˙=ay−x+yz,y˙=bx−y−xz+w,z˙=xy−cz,w˙=dw−xz.By setting parametersa=35, b=3/8, c=55, and d=1.3, we obtain four Lyapunov exponents, including two positive Lyapunov exponents, λ1=1.4164 and λ2=0.5318, a zero Lyapunov exponent λ3=0, and a negative Lyapunov exponent λ4=−39.1015 [54]. By this means, the system exhibits hyper-chaotic behavior. Figure 3 depicts the phase portraits of the hyper-chaotic system. Here, we take the fourth-order Runge-Kutta method to solve (5) and obtain the four hyper-chaotic sequences.Figure 3
Phase portraits. (a)x−y plane. (b) x−z plane. (c) x−w plane. (d) y−z plane.
(a)(b)(c)(d)


## 2.3. DNA

DNA is a long-chain polymer, and the basic elements are four nucleic acid bases, namely,A (adenine), C (cytosine), G (guanine), and T (thymine), where A and T, C and G are complementary, respectively. In a binary system, 0 and 1 are complementary. It can be concluded that 00 and 11 are complementary, and 10 and 01 are complementary. Encoding the four bases A, C, G, and T with 0 and 1, eight encoding methods can be obtained, as shown in Table 1. Each of the DNA coding rules corresponds to an operation rule, and the following algorithm is based on the encoding rule 1 and rule 2. According to the binary calculation rule, we can get the corresponding rules of DNA addition, subtraction, and complement rule listed in Tables 2 and 3.Table 1
DNA encode rule.
Rule12345678A0000010110101111T1111101001010000C0110001100110110G1001110011001001Table 2
DNA encode rule.
AdditionACGTSubtractionACGTAACGTAATGCCCGTACCATGGGTACGGCATTTACGTTGCATable 3
DNA encode rule.
Complement123456ATTCCGGTCGGACACGATGATGACATTC


## 2.4. Compressed Sensing

Compressed sensing technology uses sparse basis such as DCT or DFT to represent the signal sparsely, measures the signal based on Gaussian random matrix, and then reconstructs the signal based onL1 norm and other algorithms.Suppose a signal isx∈ℝN×1, before sampling the signal x, select a suitable and orthogonal sparse base Ψ∈ℝN×N to sparsely represent the signal x as(6)x=Ψs,where s is the sparse representation of x on a sparse basis Ψ. s has K nonzero elements, and other N−KN≫K elements’ values are 0.Sparse operation and the measurement matrix must satisfy restricted isometry property (RIP). Discrete cosine transform, fast-Fourier transform, etc. are common sparse operations.During measuringx, in order to reduce the number of the measurements and ensure that the measuring result contain as much information of x as possible, we need an appropriate measurement matrix Φ∈ℝM×NM<N. Bernoulli matrix, Gaussian distribution matrix, Hadamar matrix, Toeplitz matrix, etc. are often used in compressed sensing technology. The measurement of signal x can be expressed as(7)y=Φx=ΦΨs=Θs,where ΦΨ=Θ is a sensor matrix and y∈ℝM×1 is the measurement result.In the end, use the compressed sensing reconstruction algorithm to reconstructs˜ from y:(8)s˜=mins˜L1=minΨTxL1s.t.y=Φx=Θs.The approximate solution vector can be obtained by applying inverse transformation fors˜:(9)x˜=Ψs˜.


## 2.5. Singular Value Decomposition (SVD)

Suppose a real matrixEK′∈ℝm×n can be decomposed can be decomposed into(10)EK′=UEVT,where E∈ℝm×n is a singular value matrix whose nonzero elements are only located on the diagonal. U∈ℝm×m and V∈ℝn×n are both-unit orthogonal matrices, and U means the left singular matrix and V means the right singular matrix, respectively. Generally, E is represented as(11)E=σ100000σ100000⋯00000⋯0m×n.DecomposingEK′ by equation (10), then we can get(12)EK′EK′T=UEVTVETUT=UEETUT,EK′TEK′=VEUTUETVT=VEETVT.With the eigen-decomposition of EK′EK′T and EK′TEK′, the left singular matrix U and the right singular matrix V can be obtained.


## 3. Proposed Encryption Scheme of CSGI

The proposed encryption scheme of CSGI includes three main parts: the generated of the phase maskφ, original image sparse presentation, and CSGI encryption. Next, the implementation processes will be introduced in detail.


### 3.1. Generation of the Phase Mask Matrix

Chaotic systems have some significant features, such as deterministic, pseudorandomness, and ergodicity, and they are sensitive to initial points and parameters. Supposing that the original image is denoted asT, whose size is M×M, the initial values are x0, y0, z0, and w0 and the N phase mask matrices φx,y can be realized as follows:(1)
Use the initial valuesx0, y0, z0, and w0 to produce four pseudorandom sequences C_x, C_y, C_z, and C_w by iterating equation (5) for M0+L times, where L=M×M. To get rid of transient effect, we discard the first M0 numbers of each sequence.(2)
Define twoL length sequences, named Phase_s and DNA_s:(13)Phase_si=C_xi+C_yi,C_zi3×1016mod256,DNA_si=C_wi×1016mod256.(3)
Convert sequencesPhase_s and DNA_s to binary sequence, and then convert Phase_s and DNA_s into DNA sequence L×4P_sb and DNA_sb, respectively according to DNA encoding rules 1 and 3.(4)
According to the DNA complement rule and(14)P_sti=DNA_C1p_sbiif DNA_sbi=Aandimod2=0,DNA_C2p_sbiif DNA_sbi=Candimod2=0,DNA_C3p_sbiif DNA_sbi=Gandimod2=0,DNA_C4p_sbiif DNA_sbi=Tandimod2=0,DNA_C5p_sbiif DNA_sbi=Aandimod2=1,DNA_C6p_sbiif DNA_sbi=Candimod2=1,DNA_C3p_sbiif DNA_sbi=Gandimod2=1,DNA_C4p_sbiif DNA_sbi=Tandimod2=1,the new sequenceP_st can be obtained.(5)
Convert the sequenceP_sti to decimal, and normalize the phase values so that they are uniformly distributed in the range 0,2π:(15)P_sti=modP_sti,2×π.(6)
Transform the sequenceP_sti into M×M pixels, and obtain phase mask matrix:(16)φx,y=P_sti.(7)
Setx0=C_xL−1, y0=C_yL−1, z0=C_zL−1, and w0=C_wL−1, and repeat step 1 to step 6 N−1 times, and then we can get N phase mask matrices φx,y.


### 3.2. Original Image Sparse Presentation with KSVD

Suppose the original signal (image) is a matrixX∈ℝm×n. D∈ℝm×K is dictionary matrix, and each column of the dictionary is called atomic vector dk. S is the sparse matrix. Ideally, there is X=DS, with the original signal X sparse representation through D. Therefore, solving the dictionary matrix and sparse matrix can be converted into an optimization problem as follows:(17)minD,SX−DSF2s.t∀i,si0≤T0,minD,S∑isi0,or s.t.minD,SX−DSF2≤ε,where sii=1,2,…,K is the row vector of the sparse matrix S and si0≤T0 is the limitation, namely, each row of the sparse matrix has nonzero elements as few as possible. This problem can be converted to a nonconstrained optimization problem by using Lagrangian multiplier method, which is(18)minD,SX−DSF2+λsi1.In order to simplify the optimization problem,si0 is replaced by si1.So, the main problem is converted toD and S two objective optimization problems, and the common method of S optimizing is orthogonal matching pursuit (OMP) algorithm, which has been discussed in [55]. The optimization of D can be described as follows:Suppose the sparse matrixS is known, we can implement the columnwise update of dictionary matrix D. skT is the k-th row vector of S, and Ek denotes residual, so(19)X−DSF2=X−∑j=1KdjsjTF2=X−∑j≠kdjsjT−dkskTF2=Ek−dkskTF2,where Ek=X−∑j≠kdjsjT. Aforementioned optimization question can be converted into(20)mindk,skTEk−dkskTF2,where dk and skT become the variables to optimize, and equation (20) can be described as a least squares problem which can be solved by using SVD. Extract all nonzero terms of Ek and then rebuild new matrix Ek′. Hence, the optimization turns into(21)mindk,skTEk′−dkskTF2.Through SVD, we can get(22)Ek′=UEVT.Replacedk with the first column vector u1 of left singular matrix, and then get one column of D. Multiply the first row of right singular matrix and the largest singular value; afterwards, we can obtain sk,T. Replace skT of sparse matrix S with the new result. Singular values of E should be ordered from largest to smallest.Repeat the above steps to update each column of dictionary, and then we can gain the final dictionary matrixD and sparse matrix S from original signal.


### 3.3. CSGI Encryption

Figure4 shows the process of CSGI encryption. The detailed steps are as follows:(1)
The phase masks matricesφx,y generated in Section 3.1 are uploaded to SLM, and the laser beam is phase modulated by SLM according to equation (1).(2)
The sparse matrixS and dictionary matrix D are gained by sparse representation of the original image according to Section 3.2.(3)
The sparse matrixS is placed at z distance from the SLM. According to Fresnel diffraction, we can get the light field distribution that can be obtained in the front of the image, and the light field intensity Iix,y can be further obtained according to equation (2).(4)
The total light intensityBix,y can be calculated by a bucket detector behind the image according to equation (3).(5)
The initial values of the chaotic system and dictionary matrixD transmit through the private channel as the transmission key. And, Bix,y transmits through the public channel.Figure 4
CSGI encryption.


### 3.4. Decryption Process

Figure5 shows the decryption process, and the detailed steps are as follows:(1)
The transmission key is received through the private channelx0, y0, z0, w0, and D, and the random phase mask matrices are calculated using the received transmission key according to the method in Section 3.1.(2)
The same computed intensity patternsIix,y are obtained as step 3 of Section 3.3.(3)
The computed intensity patternsIix,y are correlated with the total intensities of the light field Bix,y received from the public mode, and the estimated matrix S˜ of S is reconstructed, according to equation (3).(4)
According toX˜=DS˜, use dictionary matrix D to reconstruct the original image from S˜.Figure 5
CSGI encryption.


## 3.1. Generation of the Phase Mask Matrix

Chaotic systems have some significant features, such as deterministic, pseudorandomness, and ergodicity, and they are sensitive to initial points and parameters. Supposing that the original image is denoted asT, whose size is M×M, the initial values are x0, y0, z0, and w0 and the N phase mask matrices φx,y can be realized as follows:(1)
Use the initial valuesx0, y0, z0, and w0 to produce four pseudorandom sequences C_x, C_y, C_z, and C_w by iterating equation (5) for M0+L times, where L=M×M. To get rid of transient effect, we discard the first M0 numbers of each sequence.(2)
Define twoL length sequences, named Phase_s and DNA_s:(13)Phase_si=C_xi+C_yi,C_zi3×1016mod256,DNA_si=C_wi×1016mod256.(3)
Convert sequencesPhase_s and DNA_s to binary sequence, and then convert Phase_s and DNA_s into DNA sequence L×4P_sb and DNA_sb, respectively according to DNA encoding rules 1 and 3.(4)
According to the DNA complement rule and(14)P_sti=DNA_C1p_sbiif DNA_sbi=Aandimod2=0,DNA_C2p_sbiif DNA_sbi=Candimod2=0,DNA_C3p_sbiif DNA_sbi=Gandimod2=0,DNA_C4p_sbiif DNA_sbi=Tandimod2=0,DNA_C5p_sbiif DNA_sbi=Aandimod2=1,DNA_C6p_sbiif DNA_sbi=Candimod2=1,DNA_C3p_sbiif DNA_sbi=Gandimod2=1,DNA_C4p_sbiif DNA_sbi=Tandimod2=1,the new sequenceP_st can be obtained.(5)
Convert the sequenceP_sti to decimal, and normalize the phase values so that they are uniformly distributed in the range 0,2π:(15)P_sti=modP_sti,2×π.(6)
Transform the sequenceP_sti into M×M pixels, and obtain phase mask matrix:(16)φx,y=P_sti.(7)
Setx0=C_xL−1, y0=C_yL−1, z0=C_zL−1, and w0=C_wL−1, and repeat step 1 to step 6 N−1 times, and then we can get N phase mask matrices φx,y.


## 3.2. Original Image Sparse Presentation with KSVD

Suppose the original signal (image) is a matrixX∈ℝm×n. D∈ℝm×K is dictionary matrix, and each column of the dictionary is called atomic vector dk. S is the sparse matrix. Ideally, there is X=DS, with the original signal X sparse representation through D. Therefore, solving the dictionary matrix and sparse matrix can be converted into an optimization problem as follows:(17)minD,SX−DSF2s.t∀i,si0≤T0,minD,S∑isi0,or s.t.minD,SX−DSF2≤ε,where sii=1,2,…,K is the row vector of the sparse matrix S and si0≤T0 is the limitation, namely, each row of the sparse matrix has nonzero elements as few as possible. This problem can be converted to a nonconstrained optimization problem by using Lagrangian multiplier method, which is(18)minD,SX−DSF2+λsi1.In order to simplify the optimization problem,si0 is replaced by si1.So, the main problem is converted toD and S two objective optimization problems, and the common method of S optimizing is orthogonal matching pursuit (OMP) algorithm, which has been discussed in [55]. The optimization of D can be described as follows:Suppose the sparse matrixS is known, we can implement the columnwise update of dictionary matrix D. skT is the k-th row vector of S, and Ek denotes residual, so(19)X−DSF2=X−∑j=1KdjsjTF2=X−∑j≠kdjsjT−dkskTF2=Ek−dkskTF2,where Ek=X−∑j≠kdjsjT. Aforementioned optimization question can be converted into(20)mindk,skTEk−dkskTF2,where dk and skT become the variables to optimize, and equation (20) can be described as a least squares problem which can be solved by using SVD. Extract all nonzero terms of Ek and then rebuild new matrix Ek′. Hence, the optimization turns into(21)mindk,skTEk′−dkskTF2.Through SVD, we can get(22)Ek′=UEVT.Replacedk with the first column vector u1 of left singular matrix, and then get one column of D. Multiply the first row of right singular matrix and the largest singular value; afterwards, we can obtain sk,T. Replace skT of sparse matrix S with the new result. Singular values of E should be ordered from largest to smallest.Repeat the above steps to update each column of dictionary, and then we can gain the final dictionary matrixD and sparse matrix S from original signal.


## 3.3. CSGI Encryption

Figure4 shows the process of CSGI encryption. The detailed steps are as follows:(1)
The phase masks matricesφx,y generated in Section 3.1 are uploaded to SLM, and the laser beam is phase modulated by SLM according to equation (1).(2)
The sparse matrixS and dictionary matrix D are gained by sparse representation of the original image according to Section 3.2.(3)
The sparse matrixS is placed at z distance from the SLM. According to Fresnel diffraction, we can get the light field distribution that can be obtained in the front of the image, and the light field intensity Iix,y can be further obtained according to equation (2).(4)
The total light intensityBix,y can be calculated by a bucket detector behind the image according to equation (3).(5)
The initial values of the chaotic system and dictionary matrixD transmit through the private channel as the transmission key. And, Bix,y transmits through the public channel.Figure 4
CSGI encryption.


## 3.4. Decryption Process

Figure5 shows the decryption process, and the detailed steps are as follows:(1)
The transmission key is received through the private channelx0, y0, z0, w0, and D, and the random phase mask matrices are calculated using the received transmission key according to the method in Section 3.1.(2)
The same computed intensity patternsIix,y are obtained as step 3 of Section 3.3.(3)
The computed intensity patternsIix,y are correlated with the total intensities of the light field Bix,y received from the public mode, and the estimated matrix S˜ of S is reconstructed, according to equation (3).(4)
According toX˜=DS˜, use dictionary matrix D to reconstruct the original image from S˜.Figure 5
CSGI encryption.


## 4. Simulation Results and Security Analysis

In this part, the proposed scheme is simulated with MATLAB R2016a to verify the feasibility.As shown in Figure6(a), a grayscale image of 128×128 size is used as the original image. The initial values of the hyper-chaotic system are set as x0=1,y0=0.949,z0=1,andw0=1, and then referring to the mentioned point in Section 3, we obtain N different random phase mask matrices and the sparse representation of original image which is shown in Figures 6(b) and 6(c). The following is the brief descriptions of the computational complexity of our algorithm and comparison with other algorithms.Figure 6
(a) The original image. (b) The random phase mask matrix. (c) The spares matrix. (d) The reconstructed image.
(a)(b)(c)(d)In generation of phase mask matrix, the size of gray image ism×m, the main operations are “addition”, “multiplication,” and “mod”, and then the operand is N40m2+19M0, where N is the number of measurements. In the step of sparse representation, the main operation is the computation of sparse matrix S and dictionary matrix D, the operand is 5m3 where the number of iterations is 10, and for projection and image reconstruction, the operand is 4Nm2 and 4m3, respectively. In order to guarantee the quality of results, we set N≫m in our experiments. The total operand is N44m2+19M0+4m3, because the generation of phase mask matrix and sparse representation can be performed simultaneously. Therefore, the computational complexity of CSGI can be expressed as ΘNm2. Compared with other algorithms, the computational complexity of using QR code and compressed sensing to encrypt the image (QR-CGI-OE) [13] is ΘNm2. And, the method based on the LWT, XOR operations (XOR-LWT-OE) [16], spends most time on the process of measurement, whose computational complexity can be expressed as ΘNm2. These illustrate that the computational complexity of the method used in this paper is identical to some relatively new algorithms’, whereas our algorithm have better performance and less times of the measurement, which is stated below.During the encryption of the CSGI, the wavelength of the plane wave is selected 0.532μm. The image is placed at a distance z=200mm from the SLM, and the transmitted light is collected into a bucket detector. Then, the image can be reconstructed according to Section 3.4 and Figure 6(c).


### 4.1. Key Space Analysis

If the cryptographic scheme has enough large key space, it can resist brute-force attacks. Here, the transmission keys arex0, y0, z0, w0, and D. D is smaller and can be ignored. The operational precision of the computer is 1016≈252, and the key space of our proposed scheme is 252×252×252×252≈2208, which is much larger than the security requirement of the key space 2100. Thus, the key space of our proposed scheme is strong enough and can effectively resist brute-force attacks.


### 4.2. Key Sensitivity Analysis

A highly secure computation ghost imaging system must be sensitive to the key. To verify the security performance of our proposed scheme, a security test is carried out. The private keys are set asx0=1,y0=0.949,z0=1,andx0=1. In decryption process, we change the value of the private keys to x0′=1+10−15,y0′=0.949,z0′=1,andw0′=1 and then use to reconstruct the image. The sampled object is shown in Figure 7. Obviously, the information related to the plaintext image cannot be obtained when the private key is changed slightly.Figure 7
Reconstructed images for the sample object. (a) The original image. The reconstructed image with (b) the correct keys and (c) the incorrect keys.
(a)(b)(c)


### 4.3. Correlation Analysis

To evaluate the quality of the reconstructed image, the correlation coefficient between the reconstructed imageG and the original image T can be calculated by(23)rTG=ET−ETG−EGDTDG,where DT and DG are the square deviations of the reconstructed image and the original image, respectively, Dx=1/N∑i=1Nxi−Ex2 and Ex=1/N∑i=1Nxi. The larger the correlation coefficient, the better the imaging effect. Ideally, the correlation coefficient rTG=1.In order to get a good reconstruction result, we performed a lot of experiments by changing the measurementN. The rTG value of Figures 8(a)–8(c) is 0.2418, 0.5494, and 0.9533 Obviously, from left to right, the reconstructed image is getting better.Figure 8
Reconstructed images with different measurements. The measurement andrTG of is (a) 1000, 0.2418, (b) 2000, 0.5494, and (c) 2800, 0.9533.
(a)(b)(c)The comparison experiments for compressed sensing using different sparse basis such as DCT, DFT, and KSVD are conducted on Lena. Besides, simulation of QR-CGI-OE and XOR-LWT-OE is conducted in this experiment. Figure9 shows the curve of the correlation coefficient change with the measurement based on several different means, the abscissa represents the number of measurements, the ordinate represents the correlation coefficient between the decrypted image and the original image. As shown in Figure 9, the correlation coefficient increases with the increase of measurement. The reconstructed image with high quality can be obtained with the increase of the measurement number. In addition, the result suggests that based on KSVD sparse representation, a high-quality image can be reconstructed in less measurements than other methods.Figure 9
The relation curve of correlation coefficientrTG with the measurement based on DCT, DFT, KSVD, QR-CGI-OE, and XOR-LWT-OE.Make measurement 3000 times on Lena, and the reconstructed images are shown in Figures10(b)–10(f). The rTG value of DCT, DFT, QR-CGI-OE, XOR-LWT-OE, and KSVD is 0.7353, 0.7821, 0.4540, 0.6248, and 0.9729, respectively. Then, we compare maximum rTG and measurements of KSVD with other sparse basics and algorithms, as shown in Table 4. By the way, after measuring 7100 times, the decrypted QR code can just be recognized and original image can be restored in QR-CGI-OE.Figure 10
The reconstructed image based different sparse transformation. (a) The original image of Lena. (b–f) The reconstructed image based on DCT, DFT, and KSVD sparse representation, respectively, and QR-CGI-OE AND XOR-LWT-OE algorithms.
(a)(b)(c)(d)(e)(f)Table 4
The best reconstructedrTG and measurement of different sparse basis and other algorithms.
Sparse basis or algorithmsMaxrTGMeasurementNo sparse representation0.824318000DCT0.84786400DFT0.84395300QR-CGI-OE0.82107100XOR-LWT-OE0.93436200KSVD0.97293000


### 4.4. NIST Statistical Test

In this paper, the NIST SP 800-22 test suite [56] is used to analyse randomness and discover potential defects in the structure of the pseudorandom sequence generator. During the test, we used the default values that came with the NIST test. The test result is expressed as p. According to NIST test rules, to pass the test, p has to be greater than 0.01. The pseudorandom sequences generated by the hyper-chaotic map are successfully passed the NIST SP 800-22 statistical test. The test results are listed in Table 5.Table 5
NIST statistical test result.
Statistical testXYZWp valueResultp valueResultp valueResultp valueResultFrequency0.105232Passed0.150434Passed0.200545Passed0.331051PassedBlock frequency0.553902Passed0.426452Passed0.626177Passed0.788040PassedRuns0.873186Passed0.072802Passed0.893688Passed0.216107PassedLongest run0.713956Passed0.935258Passed0.497594Passed0.985966PassedRank0.357115Passed0.437155Passed0.771378Passed0.193581PassedFFT0.291282Passed0.840006Passed0.186356Passed0.354010PassedNon-overlapping template0.263903Passed0.640982Passed0.886167Passed0.433739PassedOverlapping template0.121652Passed0.468763Passed0.969480Passed0.660406PassedUniversal0.522018Passed0.532899Passed0.257854Passed0.881329PassedLinear complexity0.985256Passed0.847794Passed0.913437Passed0.824185PassedSerial test-10.271726Passed0.082269Passed0.368673Passed0.229263PassedSerial test-20.437868Passed0.130755Passed0.308462Passed0.182765PassedApproximate entropy0.992196Passed0.939959Passed0.214085Passed0.105079PassedCumulative sums0.116982Passed0.083702Passed0.182438Passed0.182438PassedRandom excursions0.030601Passed0.058385Passed0.054319Passed0.200294PassedRandom excursions variant0.040671Passed0.470229Passed0.032714Passed0.042780Passed


### 4.5. Noise Addition

As the phase mask matrices may be attacked by noise, to test the robustness of this scheme, we add Gaussian noise, salt and pepper noise, and speckle noise on the phase mask matrices, respectively. As shown in Figure11, Figure 11(a) is the decryption image when the phase mask matrices are added; for Gaussian noise, mean value is zero and variance is 0.005 and rTG=0.9549; Figure 11(b) is added Salt and pepper noise, density is 0.005, and rTG=0.9306; Figure 11(c) is added speckle noise, mean is zero, variance is 0.01, and the rTG=0.9119. Obviously, the proposed scheme can resist noise attacks well.Figure 11
The decrypted images when the phase mask matrices are attacked by noise, and the measurement is 3000. (a) Gaussian noise: mean value is zero, variance is 0.005 andrTG=0.9549. (b) Salt and pepper noise: density is 0.005 and rTG=0.9306. (c) Speckle noise: mean is zero, variance is 0.01, and rTG=0.9119.
(a)(b)(c)


## 4.1. Key Space Analysis

If the cryptographic scheme has enough large key space, it can resist brute-force attacks. Here, the transmission keys arex0, y0, z0, w0, and D. D is smaller and can be ignored. The operational precision of the computer is 1016≈252, and the key space of our proposed scheme is 252×252×252×252≈2208, which is much larger than the security requirement of the key space 2100. Thus, the key space of our proposed scheme is strong enough and can effectively resist brute-force attacks.


## 4.2. Key Sensitivity Analysis

A highly secure computation ghost imaging system must be sensitive to the key. To verify the security performance of our proposed scheme, a security test is carried out. The private keys are set asx0=1,y0=0.949,z0=1,andx0=1. In decryption process, we change the value of the private keys to x0′=1+10−15,y0′=0.949,z0′=1,andw0′=1 and then use to reconstruct the image. The sampled object is shown in Figure 7. Obviously, the information related to the plaintext image cannot be obtained when the private key is changed slightly.Figure 7
Reconstructed images for the sample object. (a) The original image. The reconstructed image with (b) the correct keys and (c) the incorrect keys.
(a)(b)(c)


## 4.3. Correlation Analysis

To evaluate the quality of the reconstructed image, the correlation coefficient between the reconstructed imageG and the original image T can be calculated by(23)rTG=ET−ETG−EGDTDG,where DT and DG are the square deviations of the reconstructed image and the original image, respectively, Dx=1/N∑i=1Nxi−Ex2 and Ex=1/N∑i=1Nxi. The larger the correlation coefficient, the better the imaging effect. Ideally, the correlation coefficient rTG=1.In order to get a good reconstruction result, we performed a lot of experiments by changing the measurementN. The rTG value of Figures 8(a)–8(c) is 0.2418, 0.5494, and 0.9533 Obviously, from left to right, the reconstructed image is getting better.Figure 8
Reconstructed images with different measurements. The measurement andrTG of is (a) 1000, 0.2418, (b) 2000, 0.5494, and (c) 2800, 0.9533.
(a)(b)(c)The comparison experiments for compressed sensing using different sparse basis such as DCT, DFT, and KSVD are conducted on Lena. Besides, simulation of QR-CGI-OE and XOR-LWT-OE is conducted in this experiment. Figure9 shows the curve of the correlation coefficient change with the measurement based on several different means, the abscissa represents the number of measurements, the ordinate represents the correlation coefficient between the decrypted image and the original image. As shown in Figure 9, the correlation coefficient increases with the increase of measurement. The reconstructed image with high quality can be obtained with the increase of the measurement number. In addition, the result suggests that based on KSVD sparse representation, a high-quality image can be reconstructed in less measurements than other methods.Figure 9
The relation curve of correlation coefficientrTG with the measurement based on DCT, DFT, KSVD, QR-CGI-OE, and XOR-LWT-OE.Make measurement 3000 times on Lena, and the reconstructed images are shown in Figures10(b)–10(f). The rTG value of DCT, DFT, QR-CGI-OE, XOR-LWT-OE, and KSVD is 0.7353, 0.7821, 0.4540, 0.6248, and 0.9729, respectively. Then, we compare maximum rTG and measurements of KSVD with other sparse basics and algorithms, as shown in Table 4. By the way, after measuring 7100 times, the decrypted QR code can just be recognized and original image can be restored in QR-CGI-OE.Figure 10
The reconstructed image based different sparse transformation. (a) The original image of Lena. (b–f) The reconstructed image based on DCT, DFT, and KSVD sparse representation, respectively, and QR-CGI-OE AND XOR-LWT-OE algorithms.
(a)(b)(c)(d)(e)(f)Table 4
The best reconstructedrTG and measurement of different sparse basis and other algorithms.
Sparse basis or algorithmsMaxrTGMeasurementNo sparse representation0.824318000DCT0.84786400DFT0.84395300QR-CGI-OE0.82107100XOR-LWT-OE0.93436200KSVD0.97293000


## 4.4. NIST Statistical Test

In this paper, the NIST SP 800-22 test suite [56] is used to analyse randomness and discover potential defects in the structure of the pseudorandom sequence generator. During the test, we used the default values that came with the NIST test. The test result is expressed as p. According to NIST test rules, to pass the test, p has to be greater than 0.01. The pseudorandom sequences generated by the hyper-chaotic map are successfully passed the NIST SP 800-22 statistical test. The test results are listed in Table 5.Table 5
NIST statistical test result.
Statistical testXYZWp valueResultp valueResultp valueResultp valueResultFrequency0.105232Passed0.150434Passed0.200545Passed0.331051PassedBlock frequency0.553902Passed0.426452Passed0.626177Passed0.788040PassedRuns0.873186Passed0.072802Passed0.893688Passed0.216107PassedLongest run0.713956Passed0.935258Passed0.497594Passed0.985966PassedRank0.357115Passed0.437155Passed0.771378Passed0.193581PassedFFT0.291282Passed0.840006Passed0.186356Passed0.354010PassedNon-overlapping template0.263903Passed0.640982Passed0.886167Passed0.433739PassedOverlapping template0.121652Passed0.468763Passed0.969480Passed0.660406PassedUniversal0.522018Passed0.532899Passed0.257854Passed0.881329PassedLinear complexity0.985256Passed0.847794Passed0.913437Passed0.824185PassedSerial test-10.271726Passed0.082269Passed0.368673Passed0.229263PassedSerial test-20.437868Passed0.130755Passed0.308462Passed0.182765PassedApproximate entropy0.992196Passed0.939959Passed0.214085Passed0.105079PassedCumulative sums0.116982Passed0.083702Passed0.182438Passed0.182438PassedRandom excursions0.030601Passed0.058385Passed0.054319Passed0.200294PassedRandom excursions variant0.040671Passed0.470229Passed0.032714Passed0.042780Passed


## 4.5. Noise Addition

As the phase mask matrices may be attacked by noise, to test the robustness of this scheme, we add Gaussian noise, salt and pepper noise, and speckle noise on the phase mask matrices, respectively. As shown in Figure11, Figure 11(a) is the decryption image when the phase mask matrices are added; for Gaussian noise, mean value is zero and variance is 0.005 and rTG=0.9549; Figure 11(b) is added Salt and pepper noise, density is 0.005, and rTG=0.9306; Figure 11(c) is added speckle noise, mean is zero, variance is 0.01, and the rTG=0.9119. Obviously, the proposed scheme can resist noise attacks well.Figure 11
The decrypted images when the phase mask matrices are attacked by noise, and the measurement is 3000. (a) Gaussian noise: mean value is zero, variance is 0.005 andrTG=0.9549. (b) Salt and pepper noise: density is 0.005 and rTG=0.9306. (c) Speckle noise: mean is zero, variance is 0.01, and rTG=0.9119.
(a)(b)(c)


## 5. Conclusion

In this paper, a CSGI encryption scheme based on hyper-chaotic-system and DNA and KSVD technology is proposed for the first time. The hyper-chaotic system is used to generate four long pseudorandom sequences, the sequences are diffused with DNA operation, and then the phase mask matrices for encryption can be obtained. The original image is sparse by dictionaryD generated by KSVD. The transmission key of the scheme is composed with the initial values of the hyper-chaotic system and the dictionary D. Compared with the existing scheme, the proposed scheme has a small transmission key which ensures the security of the private key, big key space, highly sensitive to the key, high complexity, and strong plaintext correlation and ensures the security of the scheme. Simulation results and security analysis show that the proposed scheme can resist most of the known attacks well and has high security and great performance.



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*Source: 8815315-2020-10-05.xml*