Protecting Compressive Ghost Imaging with Hyperchaotic System and DNA Encoding

. 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 then N 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.


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][2][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]. erefore, 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][10][11].
en, 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. is motivates us to look for a novel CSGI encryption method with plaintext correlation, smaller transmission keys, larger key spaces, and high image reconstruction efficiency.
e DNA encoding and decoding technology [46] is a kind of biological method to process information, which has the characteristics of largescale 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][48][49][50][51][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. ird, 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. e use of DNA further increases the complexity and randomness of the encryption scheme. e rest of this paper is organized as follows. In section 2, 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. e simulation results and security analysis are performed in section 4. e paper is summarized in section 5.

CGI.
In CGI, as shown in Figure 2, a spatial laser beam transmits through a spatial light modulator (SLM), which introduces an arbitrary phase mask matrix φ(x, y), generating a spatially incoherent beam. Knowing the random phase and the distribution of laser light field U in (x, y), one can evaluate the distribution of the light intensity U i (x, y) right after the SLM: rough 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 is z distance away from the SLM, and the speckle filed I i (x, y) can be calculated: where h z (x, y) is the transfer function in the spatial domain at a distance z, ⊗ represents the convolution operation, and I i (x, y) is defines as the reference light. e signal light intensities detected by a bucket detector placed behind the object, which can be represented by a transmission function of the object T(x, y) and written as To construct the object's transmission function T(x, y), the reference light speckle filed I i (x, y) cross-correlated with the signal light intensities B i : where G(x, y) denotes the recovered object information, 〈·〉 � (1/N) i · is an ensemble average over N measurements, I i (x, y) is calculated by the receiver according to equation (2), and B is the average value for the measured components B i [53].

Hyper-Chaotic
System. In our proposed CSGI encryption scheme, the phase mask matrix required on the SLM is generated by the hyper-chaotic system:   Complexity By setting parameters a � 35, b � 3/8, c � 55, and d � 1.3, we obtain four Lyapunov exponents, including two positive Lyapunov exponents, λ 1 � 1.4164 and λ 2 � 0.5318, a zero Lyapunov exponent λ 3 � 0, and a negative Lyapunov exponent λ 4 � −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.

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.

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 on L1 norm and other algorithms.
Suppose a signal is x ∈ R N×1 , before sampling the signal x, select a suitable and orthogonal sparse base Ψ ∈ R N×N to sparsely represent the signal x as where s is the sparse representation of x on a sparse basis Ψ. s has K nonzero elements, and other N − K(N ≫ 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 measuring x, 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 ∈ R M×N (M < N). Bernoulli matrix, Gaussian distribution matrix, Hadamar matrix, Toeplitz matrix, etc. are often used in compressed sensing technology. e measurement of signal x can be expressed as where ΦΨ � Θ is a sensor matrix and y ∈ R M×1 is the measurement result.
In the end, use the compressed sensing reconstruction algorithm to reconstruct s from y: e approximate solution vector can be obtained by applying inverse transformation for s:

Singular Value Decomposition (SVD)
. Suppose a real matrix E K ′ ∈ R m×n can be decomposed can be decomposed into where E ∈ R m×n is a singular value matrix whose nonzero elements are only located on the diagonal. U ∈ R m×m and V ∈ R n×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 Decomposing E K ′ by equation (10), then we can get Complexity Table 2: DNA encode rule. Table 3: DNA encode rule.  With the eigen-decomposition of E K ′ E ′T K and E ′T K E K ′ , the left singular matrix U and the right singular matrix V can be obtained.

Proposed Encryption Scheme of CSGI
e proposed encryption scheme of CSGI includes three main parts: the generated of the phase mask φ, original image sparse presentation, and CSGI encryption. Next, the implementation processes will be introduced in detail.

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 as T, whose size is M × M, the initial values are x 0 , y 0 , z 0 , and w 0 and the N phase mask matrices φ(x, y) can be realized as follows: (1) Use the initial values x 0 , y 0 , z 0 , and w 0 to produce four pseudorandom sequences C x, C y, C z, and C w by iterating equation (5) the new sequence P st can be obtained.
, and w 0 � C w(L − 1), and repeat step 1 to step 6 N − 1 times, and then we can get N phase mask matrices φ(x, y).

Original Image Sparse Presentation with KSVD.
Suppose the original signal (image) is a matrix X ∈ R m×n . D ∈ R m×K is dictionary matrix, and each column of the dictionary is called atomic vector d k . S is the sparse matrix. Ideally, there is X � DS, with the original signal X sparse representation through D. erefore, solving the dictionary matrix and sparse matrix can be converted into an optimization problem as follows: where s i (i � 1, 2, . . . , K) is the row vector of the sparse matrix S and ‖s i ‖ 0 ≤ T 0 is the limitation, namely, each row of the sparse matrix has nonzero elements as few as possible. is problem can be converted to a nonconstrained optimization problem by using Lagrangian multiplier method, which is In order to simplify the optimization problem, ‖s i ‖ 0 is replaced by ‖s i ‖ 1 .

Complexity
So, the main problem is converted to D 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]. e optimization of D can be described as follows: Suppose the sparse matrix S is known, we can implement the columnwise update of dictionary matrix D. s T k is the k-th row vector of S, and E k denotes residual, so where E k � X − j≠k d j s T j . Aforementioned optimization question can be converted into where d k and s T k 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 E k and then rebuild new matrix E k ′ . Hence, the optimization turns into rough SVD, we can get Replace d k with the first column vector u 1 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 s ,T k . Replace s T k 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 matrix D and sparse matrix S from original signal. Figure 4 shows the process of CSGI encryption. e detailed steps are as follows:

CSGI Encryption.
(1) e phase masks matrices φ(x, y) generated in Section 3.1 are uploaded to SLM, and the laser beam is phase modulated by SLM according to equation (1). (2) e sparse matrix S and dictionary matrix D are gained by sparse representation of the original image according to Section 3.2. (3) e sparse matrix S 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 I i (x, y) can be further obtained according to equation (2). (4) e total light intensity B i (x, y) can be calculated by a bucket detector behind the image according to equation (3). (5) e initial values of the chaotic system and dictionary matrix D transmit through the private channel as the transmission key. And, B i (x, y) transmits through the public channel. Figure 5 shows the decryption process, and the detailed steps are as follows:

Decryption Process.
(1) e transmission key is received through the private channel x 0 , y 0 , z 0 , w 0 , and D, and the random phase mask matrices are calculated using the received transmission key according to the method in Section 3.1.

Simulation Results and Security Analysis
In this part, the proposed scheme is simulated with MATLAB R2016a to verify the feasibility. As shown in Figure 6(a), a grayscale image of 128 × 128 size is used as the original image. e initial values of the hyper-chaotic system are set as (x 0 � 1, y 0 � 0.949, z 0 � 1, and w 0 � 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). e following is the brief descriptions of the computational complexity of our algorithm and comparison with other algorithms.
In generation of phase mask matrix, the size of gray image is m × m, the main operations are "addition", "multiplication," and "mod", and then the operand is N(40m 2 + 19M 0 ), 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 5m 3 where the number of iterations is 10, and for projection and image reconstruction, the operand is 4 Nm 2 and 4m 3 , respectively. In order to guarantee the quality of results, we set N ≫ m in our experiments. e total operand is N(44m 2 + 19M 0 ) + 4m 3 , because the generation of phase mask matrix and sparse representation can be performed simultaneously. erefore, the computational complexity of CSGI can be expressed as Θ(Nm 2 ). Compared with other algorithms, the computational complexity of using QR code and compressed sensing to encrypt the image (QR-CGI-OE) [13] is Θ(Nm 2 ). 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 Θ(Nm 2 ). ese 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 0.532 μm. e image is placed at a distance z � 200 mm from the SLM, and the transmitted light is collected into a bucket detector. en, the image can be reconstructed according to Section 3.4 and Figure 6(c).

Key Space Analysis.
If the cryptographic scheme has enough large key space, it can resist brute-force attacks. Here, the transmission keys are x 0 , y 0 , z 0 , w 0 , and D. D is smaller and can be ignored. e operational precision of the computer is 10 16 ≈ 2 52 , and the key space of our proposed scheme is 2 52 × 2 52 × 2 52 × 2 52 ≈ 2 208 , which is much larger than the security requirement of the key space 2 100 . us, the key space of our proposed scheme is strong enough and can effectively resist brute-force attacks.

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. e private keys are set as (x 0 � 1, y 0 � 0.949, z 0 � 1, and x 0 � 1). In decryption process, we change the value of the private keys to (x 0 ′ � 1 + 10 − 15 , y 0 ′ � 0.949, z 0 ′ � 1, and w 0 ′ � 1) and then use to reconstruct the image. e 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.

Correlation Analysis.
To evaluate the quality of the reconstructed image, the correlation coefficient between the reconstructed image G and the original image T can be calculated by 8 Complexity where  and XOR-LWT-OE is conducted in this experiment. Figure 9 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. e 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.
Make measurement 3000 times on Lena, and the reconstructed images are shown in Figures 10(b)-10(f ). e r TG 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. en, we compare maximum r TG 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.

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. e test result is expressed as p. According to NIST test rules, to pass the test, p has to be greater than 0.01. e pseudorandom sequences generated by the hyper-chaotic map are successfully passed the NIST SP 800-22 statistical test. e test results are listed in Table 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 Figure 11, Figure 11  10 Complexity phase mask matrices are added; for Gaussian noise, mean value is zero and variance is 0.005 and r TG � 0.9549; Figure 11(b) is added Salt and pepper noise, density is 0.005, and r TG � 0.9306; Figure 11(c) is added speckle noise, mean is zero, variance is 0.01, and the r TG � 0.9119. Obviously, the proposed scheme can resist noise attacks well.

Conclusion
In this paper, a CSGI encryption scheme based on hyperchaotic-system and DNA and KSVD technology is proposed for the first time. e 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. e original image is sparse by dictionary D generated by KSVD. e 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.

Data Availability
e data used to support the findings of the study are available from the corresponding author upon request.

Conflicts of Interest
All authors declare that they have no conflicts of interest.