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Abstract
As an important information medium, the digital image exists widely on the Internet. Quantum walks have the property of encrypting information. For the eneryption problem of optical digital images, an encryption scheme based on discrete cosine transform (DCT) and alternate quantum walks (AQW) is proposed in this paper. First, we use AQW and XOR operation to preprocess images in the spatial domain. Then, AQW are used to generate two random phase masks which can operate the preprocessed image and the DCT image, respectively. Finally, the encrypted image is obtained by using discrete cosine inverse exchange. The control parameters of AQW can replace the random phase mask as the key in the encryption and decryption process, so it is convenient for key management and transmission. The experimental simulation carried out the analysis of the image pixel histogram, the correlation of adjacent pixels, the robustness against noise and the sensitivity of secret keys, the results show that the image encryption method has strong security.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Digital image encryption technology originates from the early classical cryptography theory [1–3]. Classical cryptography usually considers binary streams, while digital images have large amounts of data, high real-time performance, and requirements for visual effects. Therefore, the security of images [4–6] is of great research significance. At present, the security protection technology of digital images mainly includes digital watermarking [7,8] and image encryption [9,10]. After the encryption operation, the image becomes the information similar to the random noise of the channel, to avoid eavesdropping during the network transmission, and then effectively protect the image data in transmission. The current image encryption technologies mainly include: image pixel scrambling technology [11,12],encryption technology based on secret segmentation and secret sharing [13], encryption technology based on modern cryptography system [14,15] and encryption technology based on chaotic systems [16,17]. At the same time, the image is transformed from the spatial domain to the frequency domain [18] the characteristics of the image signal can be analyzed from another perspective. With the development of computers and the network, image encryption technology has ushered in a broad prospect [19].
Together with relativity theory, quantum mechanics is regarded as the two basic pillars of modern physics. Quantum image processing [20,21] is a new interdisciplinary subject which integrates quantum information [22], quantum computing [23–26], image processing and mathematics. In the future, with the integration of computer technology and quantum cryptography [27–29], the computing platform with high concurrency and high throughput will bring new opportunities and challenges to cryptography. Compared to the classical random walk algorithm that simulates the random movement of particles in the underlying graph structure, the quantum walk simulates the quantum coherence evolution of particle movement on the graph [30–32]. As an important quantum computing model, the quantum walk was proposed in 1993 by Aharonov, Davidovich, and Zagur [33]. Continuous quantum walk was proposed by Farhi et al. [34] in 1998. In 2001, discrete quantum walk was formally proposed by Aharonov, Ambainis et al. [35]. Compared with the classical algorithm, the quantum walk algorithm has two main advantages: First, it can take less time to find the target node; Second, it can effectively avoid the eavesdropper’s attack with higher security.
Discrete quantum walks have the similar property of chaotic dynamics [36] in determining the irregular motion in the system, so it can be used to design the image encryption algorithm. In this paper, we propose an image encryption algorithm based on AQW [37] and two dimensional discrete cosine transform [38]. We study alternate quantum walks and generate a random probability distribution matrix by them. The probability distribution matrix is used to replace the random phase mask in traditional double random phase encoding (DRPE) [39]. In the spatial domain, AQW and XOR operation are used to preprocess the input image non-linearly to increase the security of the cryptographic system. In the DCT domain, the random phase mask is used to complete the image encryption. AQW parameters are used as encryption and decryption keys. To verify the robustness of the proposed method, a computer simulation is performed. The experimental simulation carried out the analysis of the image pixel histogram, the correlation of adjacent pixels, the robustness against noise and the sensitivity of secret keys. The simulation results show that the scheme has good concealment and security, and it is robust to noise .
2. Related work
2.1 Alternate quantum walks
The discrete quantum walk is proposed by referring to the concept of random walk in classical computation. A simple example of classical random walks is a random walk on an integer line: set a walker at the origin point and determine the direction of the walker’s step based on the state of the coin tossed. If the coin heads up, take a step to the left, otherwise, take a step to the right. Repeatation of this process creates a random walk, and each walk is independent of previous walks. This is called a special Markov process. And for a general one, the change of the states only depends on the current state of the system.
The quantum walk is the quantum analog of the classcial random walk, which quantizes the classical Markov process under the theory of quantum mechanics. Specifically, we use a wave function $\left |\psi _{p}\right \rangle$ to describe the position of the particle:
where $\left |x\right \rangle$ represents the position of the particle at $x$, and $c_{x}$ is its corresponding complex amplitude satisfying the normalizating condition. The probability that we find a particle at $x$ is: In addition, we use $\left |\psi _{q}\right \rangle$ to represent the qubits describing Markov state variables, which is also called the coin state: $\left |\psi _q\right \rangle =\cos \alpha \left |0\right \rangle +\sin \alpha \left |1\right \rangle$. Therefore, the initial state of the whole system can be written as: To consider the random walk of a particle, we need to introduce two unitary operators, the coin operator $\hat {C}$ and the operator $\hat {S}$. The process of quantum walks can be seen as:2.2 Discrete cosine transform
The discrete cosine transform (DCT) is a real even function transformation in the real number field. The transformation kernel of DCT is a cosine function. For images with high correlation, DCT can reduce image correlation very well. The basic idea of DCT is to extend a real function symmetrically into a real even function, and the Fourier transform of a real even function must also be a real even function. Cosine transform of continuous function and discrete function are all based on this principle. The forward transformation kernel of two-dimensional DCT is defined as follows:
The inverse transformation of two-dimensional DCT transform is defined as:
3. Image encryption scheme
The quantum walk is a quantum computing model used to construct modern cryptographic mechanism within the concepts of quantum mechanics. Discrete quantum walks can be used to design an image encryption algorithm on account of the similarity of chaotic dynamics in determining the irregular motion in the system. We propose an image encryption scheme in the DCT transform domain based on quantum walks. In our scheme, the AQW is used to generate a probability distribution matrix which is set to replace the two random phase masks $CRPM1$ and $CRPM2$ in traditional double random phase encoding. The encryption and decryption processes of the scheme are shown in Fig. 1.
3.1 Alternate quantum walks to obtain the key
In this scheme, two random phase masks are generated by alternate quantum walks and the AQW is controlled by different parameters. First, parameters $\left (N_{1}, \alpha _{1}, \beta _{1}\right )$, $\alpha _{1}, \beta _{1} \in [0, \pi / 2]$, are set and the AQW is performed to produce a probability distribution matrix $P_{1}$ of size $N_{1} \times N_{1}$. Similarly, we set parameters $\left (N_{2}, \alpha _{2}, \beta _{2}\right ), \alpha _{2}, \beta _{2} \in [0, \pi / 2]$, to obtain probability distribution matrix $P_{2}$ of size $N_{2} \times N_{2}$.
Assuming that the size the original image is $m \times n$, then the dimensions of the two random phase masks, $CRPM1$ and $CRPM2$, generated by the AQW, must be $m \times n$ as well. Therefore, we need to adjust $P_{1}$ and $P_{2}$ to the normal image size according to the actual situation.
3.2 Encryption process
Step1: Converts $\operatorname {Re} P_{1}$ to an integer value in the interval [0,255].
Step2: Perform bit-by-bit (XOR) operation to the gray original image $I$ and the generated key $K$ to obtain the preprocessed image $I_{1}$. Step3: After the modulation of the first random phase encoding $CRPM_{1}$ and the two-dimensional DCT, the preprocessed image $I_{1}$ is expressed as follows:Step5: Finally, the encrypted image $ReI$ is obtained by the inverse transform of the discrete cosine transform (IDCT).
3.3 Decryption process
The image decryption process is the inversion of the image encryption process.
Step1: The two-dimensional DCT is performed on the encrypted image $ReI$. And the complex conjugate template $CRPM_{2}^{*}$ of the second random phase mask $CRPM_{2}$ is used to modulate.
4. Experiments and discussion
To verify the effectiveness and security of the encryption method, encryption and decryption experiments on three gray images with the size of $300 \times 300$ pixels are carried out, and the analyzation of the image pixel histogram, the correlation of adjacent pixels, the robustness against noise, and the sensitivity of secret keys are proposed as well.
4.1 Simulation results
In the simulation experiment, we encrypted and decrypted three gray images with the size of $300 \times 300$ pixels. For the initial parameter values of alternate quantum walks used in this scheme, $\left (N, \alpha , \beta \right )$, we set as $\left (N_{1}=500, \alpha _{1}=\pi /2, \beta _{1}= \pi / 3\right )$ and $\left (N_{2}=500, \alpha _{2}=0, \beta _{2}= \pi / 6\right )$. Through different parameters, we carried out multi-group simulation experiments to form a comparison. The encryption result of this scheme is shown in Fig. 2(b), and the image of decryption using the correct key is shown in Fig. 2(c). For further quantitative analysis, the correlation coefficient $C$ is used to measure the difference between the decrypted image and the original image. The correlation coefficient between two images is defined as follows:
4.2 Histogram analysis
Histogram analysis provides a useful visualization of the intensity of each pixel in the image relative to the pixel in the image. A good optical image cipher system should show the same distribution for different encrypted images. Figure 3, Fig. 4 and Fig. 5 show gray pixel histograms of the original image and the encrypted image, respectively.
4.3 Correlation analysis
The correlation coefficient of adjacent pixels is used for the correlation analysis of the encryption key. The adjacent pixels of the original image should be highly correlated with values closer to 1, while the pixels of the encrypted image should be closer to 0. Figure 6, Fig. 7 and Fig. 8 show the horizontal, vertical and diagonal analysis of the correlation coefficients of adjacent pixels of the original image and the encrypted image.
Table 1. This table shows the pixel correlation coefficients of three simulated images before and after encryption
4.4 Information entropy analysis
Information entropy is a measure of the degree of randomness of the signal source. The information entropy calculation formula is as follows:
The information entropy of the original image and the encrypted image is shown in Table 2.Table 2. Information entropy analysis
4.5 Noise analysis
In the process of transmission and storage of encrypted image, it may be polluted by noise, so it is necessary to test the anti-noise attack of encrypted image. Gaussian noise refers to a type of noise whose probability density function obeys Gaussian distribution. The probability density function is expressed as follows:
where $z$ represents the gray value, $\mu$ represents the mean or expected value of $z$, and $\delta$ represents the standard deviation of $z$. Salt and pepper noise is a common noise in digital image. Salt and pepper noise is a kind of noise caused by signal pulse intensity, and the algorithm to generate the noise is relatively simple. Figure 9 shows the decrypted image affected by salt and pepper noise and Gaussian noise under different parameters. The results show that the proposed encryption method has the ability to resist noise.
Fig. 9. Figures (a) and (b) show decrypted images affected by salt-and-pepper noise with intensity of 0.01 and 0.1. Figures (c) and (d) represent decrypted images affected by Gaussian noise with a mean of 0 and variances of 0.01 and 0.1.
4.6 Keys security analysis
An efficient cryptographic system should provide a large enough key space. Theoretically, the technique proposed by our AQW provides an infinite key space allowance. The main secret keys include: $\left (N, \alpha , \beta \right )$ of AQW for XOR operation; $\left (N_{1}, \alpha _{1}, \beta _{1}\right )$ of AQW for $CRPM1$; $\left (N_{2}, \alpha _{2}, \beta _{2}\right )$ of AQW for $CRPM2$. However, assuming that the computational accuracy is on the order of $10^{-16}$, then our system has a key space of dimension $\left (10^{16}\right )^{3} \times \left (10^{16}\right )^{3} \times \left (10^{16}\right )^{3}=10^{144}$, which is sufficient for an optical image cipher system [40,41].
Similarly, key sensitivity analysis is used to verify the performance of many information security protocols.Here, the ability of the technology to withstand small deviations from its key parameters is assessed. An effective cryptographic system should recognize changes in the composition of its key parameters. We adjust the parameters of the alternate quantum walk as the key, and the simulation experiment is shown in Fig. 10. All attempts to compromise the integrity of the encrypted image have failed for different substitutions of key parameters.
The evaluation of sensitivity uses two parameters: number of pixel change rate (NPCR) and unified average changing intensity (UACI) [42]. Their definitions are as follows:
where $M*N$ is the size of the image, and $D(i, j)$ is defined as follows:Table 3. Information entropy analysis
4.7 Comparison with existing works
An encryption scheme based on discrete cosine transform (DCT) and alternate quantum walks (AQW) is proposed in this paper. In this subsection, we compare the image encryption scheme based on AQW with the existing related schemes. We randomly selected 3000 pairs of pixel points from Fig2 and analyzed their correlation coefficients from vertical, horizontal and diagonal directions respectively. The comparison results with other schemes are shown in Table 4. The experimental analysis proves that our algorithm can guarantee image security well.
Table 4. Performance comparison between the AQW image encryption scheme and existing schemes
5. Conclusion
This paper explores the integration of quantum walks into an optical image encryption framework. The discrete quantum walk is studied, and the random probability distribution sequence is generated by it, and the encryption matrix is generated by the sequence operation. Different encryption matrices are used to encrypt images in the spatial domain and DCT domain. Experimental results based on simulation demonstrate the effectiveness of the proposed optical method. This shows the potential applications of quantum technology in optical information security. On this basis, it is possible to realize quantum image encryption based on quantum walk in a quantum system.
Funding
Project of Shandong Province Higher Educational Science and Technology Program (J18KZ012); Natural Science Foundation of Shandong Province (ZR2019YQ01); National Natural Science Foundation of China (11975132, 61772295).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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