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Abstract
We present a compact optical encryption scheme by using a continuous-random-motion blurring model in an optical imaging system. Image encryption is performed by additive motion blur effects with continuous and random shifts of a camera. Real-time random phase modulation can be achieved without the use of random phase mask. Storage of the key is more convenient, which only requires parameters of motion. In addition, modulation characteristics are different from the traditional encryption schemes. On the premise of high security, modulation space is broadened, and flexibility of encryption is further improved. Simulations and experiments verify the validity of the motion blur-based crypto-system and demonstrate its security under several attacks. This novel method will be significant for the practical applications in the field of optical information security.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical image encryption has attracted growing attention due to its unique advantages, such as multidimensional, high parallelism and high-speed computing [1]. Since the optical encryption was pioneered by the proposal of the double random phase encoding (DRPE) [2], various schemes were designed as the supplements of random phase masks by using chaotic maps [3,4], modularization [5,6], amplitude encoding [7], and polarization modulation [8,9]. Several DRPE-patched optical cryptosystems were also proposed, for instance, DRPE in Fresnel domain [10,11], gyrator domains [12], and fractional Fourier domains [13]. Subsequently, optical encryption schemes based on interference [14], diffraction [15–17], and various other transformations of these systems [18–20] were further developed. In recent years, there are relatively novel training encryption combined with deep learning [21] or material-based optical encryption with metasurface [22]. In above schemes, flexibility of cryptosystems is limited by high-precision components of an actual optical path and most of them are demonstrated unsafe under attacks. Therefore, to explore compact optical crypto-systems with direct, succinct, and yet secure encryption engines is still in demand.
In imaging systems, the motion blur effects are generally unwanted for they cause degradations of the imaging quality. However, this is an opportunity of utilizing the motion blur effects for a random phase (and amplitude) modulation in an optical crypto-system. For imaging systems, the relative movement between the target and the sensor will destroy the image information and cause bad effects. So far, target recovery has always been an extremely difficult task in the field of computer vision. Previous blind image deconvolution (BID) methods are less suitable for larger blur kernels and require certain prior assumptions about convolution kernels or images. The maximum a posterior (MAP) method [23] abandons these constraints and achieves a truly practical BID. Due to the need to traverse entire solution space, speed of the image recovery algorithm [23] is slow to meet practical requirements. Subsequently, different modeling methods [24–31] based on the MAP framework appear successively. The frame-based prior method relies on home-made wavelet functions, which perform poorly in heterogeneous scenarios [24]. Sparse coding based priors, such as normalized sparsity prior [25], $L_0$ regularized prior [26], low-rank prior [27], dark channel prior [28], $L_0$-norm regularized gradient prior for text [29], bright channel prior [30] and hybrid dictionary prior [31], were considered for motion deblurring. Furthermore, another way of image blind deblurring [32] is to go directly end-to-end from blurred to clear images by deep learning of neural networks. The above deblurring methods all have a common problem, which exploit statistical priors (blur kernels, latent images or both) for clear retrievals. In the absence of such initial conditions, blind deblurring methods are ineffective. It can be a new paradigm for optical encryption, if path information in motion blurring is used as a key.
Thus, in this article, we propose an optical encryption scheme based on such motion-blur effect, simply by randomly shifting the imaging element. Firstly, we design a random trajectory-based motion blur data generation method. Random trajectories are generated by the Markov process, that is, the next position is randomly generated based on the previous point. A trajectory between two random points is generated by subpixel interpolation. Based on this, a long-exposure continuous random motion model is constructed, which can simultaneously modulate spatial amplitude and phase. It is suitable for continuous and uneven motion paths during camera instability shooting. Then, according to Abbe’s quadratic imaging principle, exploiting the obtained random trajectory kernel, inverse process of random motion can be implemented by phase filtering in Fourier domain. Random phase modulation is achieved in an environment without modulating elements because it is independent of optical imaging process. The results show that our optical cryptosystem can effectively hide natural images and accomplish decryption at macro microscale. In addition, it is able to provide robust security, including resistance to existing deblurring algorithms, breaking through the limitations of optical component modulation. We found that, unlike the previous encryption characteristics, the ciphertext has an reverse increasing value of entropy for higher security in the proposed optical cryptosystem. This discovery led to a more flexible and wider modulation range.
2. Motion blur model
An optical setup of the motion-blur-based cryptosystem is shown in Fig. 1. The proposed cryptosystem is termed as motion-blur-based encryption (MBBE), because random motion is used for generating the keys here. We extend the concept of the microscopic Brownian motion to a macroscopic scale to realize continuous random motion with active modulation. The purpose is to borrow the unpredictability of Brownian motion to implement pseudorandom functions. Specifically, we propose a macroscopic Brownian motion-like optical model. The CCD camera has a random subjective dynamic ability to imitate the random trajectory of the “Brown particle”. As a “particle”, the camera is moved around the center point of the image. During the movement, the orientation and step size are arbitrary, to simulate trajectories of continuous-random motion while keeping the object stationary. The motion-blur function is updated by recording and storing a motion trajectory. This avoids the tedious process of intermediate images reading and pre-processing, and reduces the amount of data pre-processing of computers and highlights the simplicity of experimental operations. Information redistribution of the original image arises from the continuous random motions of the camera.
Fig. 1. Imaging configuration and process. (a) Imaging configuration. L1, L2: Lens. (b) Imaging process. The CCD camera as a continuous-random-motion particle, with random moving steps and rotation angles, is executed for continuous random motion. The motion in spatial domain was converted into a frequency domain phase shifting. (c) Random trajectories.
The spatial displacement corresponds to the phase shift in the frequency domain. Suppose that the object $f(x,y)$ is moved in an input plane, $x(t)$ and $y(t)$ are time-varying displacement components of a random path in orthogonal directions $\mathbf {x}$ and $\mathbf {y}$, respectively. For simplicity, we assume the movements continuous uniform motion with a constant speed. The exposure time of the CCD camera is set as $T$. The motion process is expressed as follows
where $g(x,y)$ is a recorded objective function after the displacement. Transforming Eq. (1) into Fourier domain, the expression will be converted into the following form3. Encryption and decryption
Under long-exposure shooting, continuous movements of the camera will produce phase modulation with motion blur effect on the target. This evolution can be described as
where $k=1, 2, \dots, m$ is the movement label, and $H_e$ is the synthesized optical transfer function. $M_k(u,v)$ is the motion-blur function which can be rectified by $P_{k-1}(u,v)$ is the transfer function of movement in the frequency domain, and can be modeled as where $P_{0}(u,v)=1$. It can be seen from Fig. 1(b) that the imaging process is a spatial frequency domain modulation. The phase random modulation function is realized through continuous random motion of the camera. The motion-blur function $M_k(u,v)$ generated by each step of displacement and the associated transfer function of movement $P_{k-1}(u,v)$ are combined to produce the final optical transfer function $H_e$. Finally, the ciphertext $g(x,y)$ is obtained at the output in the spatial domain.In Fig. 2(a), the prominent central crosses exist in both amplitude and phase of the plaintext. After random phase modulation of the MBBE system, the central cross bright line at the ciphertext amplitude is obviously weakened, while the one of phase disappears. The decryption process shown in Fig. 2(b) is composed of two steps: the generation of the inverse optical blur transfer function and deconvolution in frequency domain. The step size and angle of the key correspond to the horizontal and vertical components of Cartesian space. According to the rules of the motion blur transfer function and the temporal sequence of motion, the frequency domain deconvolution operation is performed by synthesizing the inverse function in the frequency domain. The ciphertext $g(x,y)$ is used as input. The plaintext information is recovered through the inverse modulation of space-frequency domain with $H_e^{-1}(u,v)$. Finally, the decrypted image $f^{\prime }(x,y)$ is output in spatial domain. Figure 2(c) exhibits results of encryption and decryption using the MBBE system more intuitively. When encrypting the plaintext that contains more complex information, the ciphertext can completely hide original information.
Fig. 2. Flow chart of encryption and decryption. (a) Encryption flowchart. When the plaintext $f(x,y)$ is transmitted in space-frequency domain, it will be modulated by the fuzzy transfer function in the Fourier domain. The final output in spatial domain is a motion-blurred image (i.e., ciphertext $g(x,y)$). (b) Decryption flowchart. The ciphertext $g(x,y)$ is used as input to finally obtain the output decrypted image $f^{\prime }(x,y)$ in space domain through the inverse operations of space domain to frequency domain. (c) Results of encryption and decryption.
4. Comparison with DRPE
We take the typical optical encryption scheme, DRPE, as a comparison to illustrate the encryption particularity of MBBE.
4.1 Characteristics analysis
Physically, both DRPE and MBBE are optical imaging processes. The DRPE is a phase filtering imaging system. However, the MBBE is an imaging system with motion blurring that usually unwanted. Two methods are extremely different in the role of optical image encryption. First of all, during modulation processes of DRPE in both spatial and frequency domains, pixel-to-pixel alignment is needed for discrete case, which requires the experiment to be extremely precise. Otherwise a small deviation will cause the experiment to fail. The random number preparation of masks are extremely difficult because of precision lack for encryption systems in visible light. In the MBBE system, it only needs to control mechanical movement without quantitative analysis, which may greatly reduce costs of preparation and difficulties of operation. We use the concept of phase modulation with motion for the optical information security technology. Considering that the uncomplicated motion blur can play encryption effect, a complex motion blur can be analyzed by Fourier transform and Abbe’s theory. The MBBE is easier to implement random phase modulation.
The most obvious difference between DRPE and MBBE is that DRPE conserves nearly all information of the object image, except the high-frequencies lost by diffraction. On the contrary, The MBBE loses part of information in motion blurring processes due to the convolution smoothing. In Fig. 3 we compare information entropy and gray level histograms of DRPE and MBBE. The DRPE encrypts the plaintext into a white-noise-like ciphertext with chaotic redistribution of pxiels (Figs. 3(a1)~(a4)), and hence increase information entropy. The DRPE can change gray value of the plaintext, but strong correlation is not alleviated. From the gray distribution histogram in Fig. 3(a4), it can be seen that gray value distribution of the ciphertext of DRPE is still very high, nearly 1000 pixels. In contrast, the MBBE in Fig. 3(b) represents a convolution process which can smooth gray value distributions to make it uniform. The synthesized optical transfer function $H_e$ contains both amplitude and phase information. This indicates that the motion-blurred image belongs to a coarse gray level image, showing randomness in large scale. Motion blur process greatly destroys the statistical characteristics of plaintext, as shown in Fig. 3(b4). Furthermore, the motion-blurred image in MBBE is low-entropy. This low-entropy feature is selective, which will be explained in details in the next section. When the number of motion reaches limit, information entropy can approach a minimum. Therefore, the MBBE cryptosystem has a large modulation space.
Fig. 3. Comparison of information entropy and gray level histograms of MBBE and DRPE. (a1)~(a4) The plaintext, ciphertext and their corresponding histograms of the DRPE. (b1)~(b4) The plaintext, ciphertext and their corresponding histograms of the MBBE. $E$ is information entropy.
4.2 Entropy selection adjustability
In previous optical cryptosystems [4,6], encryptions are processes of increasing entropy, that is, entropy of the corresponding ciphertexts must be greater than that of the plaintexts. However, the entropy modulation is varied in the MBBE system. Figure 4(a) displays four sets of plaintexts with different sizes. Figure 4(b) reveals the change curves of the entropy value of ciphertexts corresponding to two kinds of images with Fig. 4(a). There are mainly four characteristics displayed. First, entropies of two images with the same size (the pixel number along one direction $N$ for an $N \times N$ image) have similar fluctuations when the number of movements is greater than thirty. Second, the entropy of “car” increases and the one of “watermelon” decreases during encryptions. Third, “watermelon” is sensitive to even size , and “car” is the opposite. Fourth, when size is even, entropies decrease as the number of movements increases until it converges. When it is odd, the fluctuation range of entropy gradually becomes smaller and tends to be stable. The high-frequency information of images is not easily scrambled, but it can be completely done after thirty movements. Therefore, in subsequent discussion, the default global information has been scrambled.
Fig. 4. Entropy selectivity. (a) Four sets of sizes for each of the two plaintexts “car” and “watermelon” are tested. The initial entropy is marked in the upper left corner of each image. (b) In the purple box is the change curve of entropy of ciphertexts with number of movements. The dotted line represents the “car”, and the solid line represents the “watermelon”. Colors are used to distinguish image sizes. Spectra distributions of the “car” (c1) and the “watermelon” (c2).
Traditional encryption is a process of pixel scrambling to strengthen the difference between adjacent pixels, showing an increase in entropy. Ciphertexts are close to full entropy (For instance, the full entropy of an 8-bit image is 8). Compared with traditional encryption, MBBE system can modulate inverse entropy. The ciphertext is close to the gray distribution characteristics of a uniform gray level image, which is a process of crippling the difference between adjacent pixels. The reason for inverse entropy is that motion blur is essentially the process of convolution smoothing. With the increase in motion steps, the high-frequency part of information gets lost, only the low-frequency components of the image remain.
In the MBBE scheme, we can not allow entropy of the blurred image, or ciphertext, getting extremely low, because in such a case, most of information is lost. The drop of entropy is usually slow with the increase of the number of motion steps. Moreover, the changes of entropy with the number of movements can be flexibly selected by initial entropy and image size, regardless of the image contents. In Fig. 4(a), the gray image “watermelon” is a natural image with more details and higher initial entropy. Comparatively, the initial entropy of a simple gray image “car” is low. Figures 4(c1) and (c2) visually compare the image distribution characteristics of the two. Compared with spectrum diagrams, they indicate that the image with low initial entropy carries more zero-gray values, while the image features with high initial entropy are relatively soft. Consequently, for images with low initial entropy, the expansion of zero outliers is equivalent to information scrambling, which is why it still exhibits the behavior of increasing entropy.
Due to the parity selectivity of the two-dimensional Fourier transform with data size, the simple gray image “car” will have a more sensitive modulation response with odd sizes. The complex image “watermelon” is the other way around. Therefore, this novel flexible choice of encryption can break the modulation limitation of entropy, which is no longer limited to the value close to 8 of traditional ciphertext. For complex images, the modulation space is doubled. In order to obtain a wider range of entropy modulation and the best encryption effect, odd sizes are selected by the simple images, and even sizes are used for the complex images. The camera movements of the MBBE system are selected as one hundred times, when the decryption effect is still clear. The purpose of the hundred-step displacement system is to implicitly emphasize that the adjustable space as the number of movements is flexible and ample. The boundary value of the camouflage effect can be seen from the relationship curve in Fig. 4(b). Motion times with $30$ is the movement boundary value of image camouflage, when the entropy fluctuation is feeble.
5. Test results and discussions
5.1 Potential security analysis
In the absence of prior information, blind deblurring methods are ineffective to crack the MBBE system. We take two classic deblurring methods, as examples, to verify the security and effectiveness of the MBBE system. Firstly, without knowing the key (random paths), the blind-deconvolution algorithm cannot effectively attack the MBBE to get a clear image. This proves that the MBBE is safe against the attack of deblurred algorithms. Secondly, for the Wiener algorithm, it is only effective when accurate point spread function is known, which also verifies the effectiveness of the MBBE system.
Figure 5(a) shows restoration results of the three algorithms, which are also selective for image size. We evaluate reconstructed images using mean square error (MSE). The MSE of decrypted images by MBBE system are between $9.43\times 10^{-16}$ and $7.27 \times 10^{-14}$. When the key is completely known, the ciphertext with an even size can be clearly reproduced by Wiener algorithm. Nevertheless, deblurring methods are invalid even if the completely correct key is used when image size is odd. Hence, leveraging the advantage of two-dimensional Fourier transform, if we want to ensure absolute security of MBBE system, whether the key is known or not, we only need to input an odd-sized plaintext. In addition, power spectral density (PSD) is used to further demonstrate safety and efficacy of MBBE system. Figures 5(b) and (c) show the PSD plots of “car” and “watermelon”, respectively. The horizontal axis represents frequency, and the vertical axis is PSD. Two plaintext and decrypted images have similar PSD curves. Since motion is equivalent to smooth filtering, PSD of decrypted images will drop. The PSD curves of two ciphertext prove that MBBE system can change the energy distribution of plaintext.
Fig. 5. Tests of safety and effectiveness. (a) The effect of image size on two deblurring methods. Evaluate with MSE in the upper left corner of each reconstructed image. (b) PSD of the “car” and (c) PSD of the “watermelon”.
5.2 Key characteristic analysis
Security performance of an encryption system must consider the possibility of being attacked, while ensuring sensitivity of key itself and capacity of key space. Figure 6(a) shows sensitivity of the key of MBBE. Each set of key pairs (i.e., single-step displacement) contains two elements, namely step length and angle, corresponding to displacement of horizontal axis X and vertical axis Y in space coordinate system. After MBBE performs random phase modulation with a hundred-step shift on a plaintext, we use disturbed keys to recover the ciphertext. Taking displacement in X direction as an example, we apply perturbation to any fifty subkeys. When the perturbation $\Delta$ is 0.5 pixels, key sensitivity of the MBBE can reach $99.92 \%$ according to the following relative error expression
where $m_\Delta$ is the number of subkeys with perturbation. $l_{max}$ is maximum pixel distance allowed for each step of displacement, which range is $[-160, 160]$. $M$ is the number of key pairs.Fig. 6. Key characteristics and security analysis. (a) Key sensitivity. The influence of path disturbance on decryption under a hundred-step displacement, (b) and (c) are orderliness of the keys. Blue, yellow, and red represent motion-blur transfer functions corresponding to paths 1, 2, and 3, respectively. (c) Step-by-step decryption of images in different path sequences. For a three-step random path of MBBE system, there are 6 forms of permutation and combination. (c1) is when the path sequence and each time displacement is accurately known, that is, when the sequence is blue, yellow, and red. (c2)–(c6) respectively represent when there is an error in the path order prediction and the path displacement is accurately known. (d) Potential safety analysis. (e) Scatterplot of cross-correlation functions between a plaintext and its ciphertext.
In Fig. 6(a), we test key sensitivity. It is the default completely correct path order. For convenience of explanation, in Figs. 6(b) and (c), orderliness of paths is explained by taking three-step displacement as an example. In Fig. 6(c), each blurred image contains different blur functions. By decrypting the ciphertext generated by three-step displacement step by step, the difference of decryption in different path sequences is highlighted. Figure 6(c1) is to use correct path sequence. The final recovery result in the third column indicates that decryption can be successful in this case. Figures 6(c2)-(c6) are cases where the other five path sequences are deviated, and none of them can recover the plaintext information. Hence, orderliness of paths plays a decisive role in the success of decryption. It can be concluded that the key space is $M!$ for our method. When $M \geq 30$, key space far exceeds $2^{100}$, which is the normal standard for measuring key space.
Figure 6(d) depicts the ability of MBBE to resist potential attacks. Most of existing encryption methods are “one time one key” encryption mechanism. That is, keys are one-time. However, due to limitations of actual optical structure at present, it greatly hinders realization of “one time one key” in experimental aspect. It is conducive to an attacker using a text-based attack to steal plaintext information that needs to be hidden. In contrast, the key generation method of MBBE is completely independent of encrypted images. Plaintexts and ciphertexts can be “several to several” or “one to several”. Multiple plaintexts use the same key, or the same plaintext uses different keys to generate completely unrelated corresponding ciphertexts. The scatter plot in Fig. 6(e) proves that plaintexts $f(x,y)$ and ciphertexts $g(x,y)$ are completely unrelated. Therefore, MBBE has a strong anti-attack ability, even a plain text attack is invalid for the MBBE system.
While ensuring relatively high sensitivity of the key, the MBBE also has a huge key space, which is sufficient to resist brute force attacks. This increases difficulty of guessing accurate point spread functions for deblurring algorithms. In addition, MBBE has a high degree of flexibility in key replacement. Plaintexts, ciphertexts and keys are not related to each other, liberating the shackles of “one time one key” in optical practice. Security of optical encryption has been reasonably improved.
5.3 Demonstrations and discussions
In order to verify that MBBE can effectively correspond to practical operation, we carry out an experimental verification, illustrated in Fig. 7. The experiments are simply done by shooting a movie on a computer screen displayed by Microsoft PowerPoint, with a cell phone camera. To eliminate the problem of recording fixed points, we start the movie firstly and then initiate the imaging. Since the moment when the camera initiates recording and the starting point of the actual displacement cannot accurately correspond, we need to further proofread the motion model utilizing the stored key. The motion process is approximately continuous along the horizontal direction. Figure 7(a) is a “heart” image recorded in a real scene. Figure 7(b) is the blurred image of a moving heart taken with a long exposure, or the ciphertext. The camera’s exposure time is one second, and the process of heart movement is completely recorded during the exposure time. Figure 7(c) is a motion blur image simulation based on the path parameters of the actual motions. Compared with Fig. 7(b), it can be seen that MBBE is consistent with the recording process in actual long exposure. We can not guarantee that the movie moves with a uniform speed. It is reasonable that we assume the movie moves slowly at first, then smoothly moves with a uniform speed, and finally slows down. Therefore, we compare the decryption effects under four multi-step displacement models, i.e., with 13, 9, 7, and 5 segments of uniform motion, respectively. Figures 7(d1),(e1),(f1), and (g1) display the corresponding restoration results of Fig. 7(b) using the motion-blur transfer functions. Comparing Figs. 7(b) and (d1), using the motion-blur transfer function generated by path parameters can restore a reasonable clear image.
Fig. 7. Recovery of long-exposure pictures taken in real scenes. (a) The image of a heart symbol. (b) The motion blurred image of the heart, the ciphertext. (c) Blurred image under long exposure simulated by the MBBE with 13 steps. (d1) Decrypted image with the motion-blur transfer function generated according to (c). (d1)~(g1) Decryption under different multi-step displacement of MBBE systems (13-step, 9-step, 7-step, and 5-step, respectively). (d2)~(g2) The motion velocity curve simulated by the constructed discretized data (13-step, 9-step, 7-step, and 5-step, respectively).
The feature similarity index measure (FSIM) is used as the quality evaluation standard [33]. Phase congruency (PC) and gradient magnitude (GM) complement each other to elicit FSIM. FSIM can be mathematically expressed as
Our encryption experiments are primary. We do not include more complicate cases when the camera takes zig-zag-like random shifts in two-dimensional or three dimensional spaces (with defocusing effects) and moves with different speeds. These situations may be extremely difficult to be described in mathematics. In a complex random shift, we believe that, the acceleration and deceleration in each shifting segment (maybe a complex curve) should be taken into account. The effect of fixed points (at which the camera or the target stops) on the motion blur should be included in the mathematical model. The nonlinear responses of the camera during long exposure should also be considered.
Our theoretical model of the optical encryption with motion blur is also quite terse. Even though we only consider that the camera moves continuously with uniform speeds, the encryption scheme is still demonstrated safe. Therefore we indicate that the trajectory information is extremely crucial. We cannot attack it without knowing that prior information. From this point of view, our work arises a natural challenge to scientists who devote themselves to solving the motion deblurring problem, it may be possible that we can reconsider the task as a cryptoanalysis.
6. Conclusion
We design a generation method of motion blur patterns based on random trajectories and propose a succinct optical cryptosystem based on the random motion of the camera in an optical imaging system. The CCD camera is used to record continuous-random motion trajectories without paying attention to the intermediate level images during motion. Random phase and amplitude modulations can be implemented in the environments without additional optical elements. Our optical cryptosystem can effectively hide natural images and completely retrieve them at macro-microscale, with characteristics of high security, more flexibility and free from the limitation of optical components. Simulations and experiments demonstrate the validity of our proposal. Our proposal may provide an alternative optical encryption technique, and also a challenge to the motion deblurring problem as a cryptoanalysis task.
Funding
National Natural Science Foundation of China (11874132, 12074094, 61975044); Interdisciplinary Research Foundation of HIT (IR2021237).
Disclosures
The authors declare no conflicts of interest.
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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