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20 pages, 6296 KiB  
Article
Privacy-Preserving Image Captioning with Partial Encryption and Deep Learning
by Antoinette Deborah Martin and Inkyu Moon
Mathematics 2025, 13(4), 554; https://doi.org/10.3390/math13040554 - 7 Feb 2025
Viewed by 404
Abstract
Although image captioning has gained remarkable interest, privacy concerns are raised because it relies heavily on images, and there is a risk of exposing sensitive information in the image data. In this study, a privacy-preserving image captioning framework that leverages partial encryption using [...] Read more.
Although image captioning has gained remarkable interest, privacy concerns are raised because it relies heavily on images, and there is a risk of exposing sensitive information in the image data. In this study, a privacy-preserving image captioning framework that leverages partial encryption using Double Random Phase Encoding (DRPE) and deep learning is proposed to address privacy concerns. Unlike previous methods that rely on full encryption or masking, our approach involves encrypting sensitive regions of the image while preserving the image’s overall structure and context. Partial encryption ensures that the sensitive regions’ information is preserved instead of lost by masking it with a black or gray box. It also allows the model to process both encrypted and unencrypted regions, which could be problematic for models with fully encrypted images. Our framework follows an encoder–decoder architecture where a dual-stream encoder based on ResNet50 extracts features from the partially encrypted images, and a transformer architecture is employed in the decoder to generate captions from these features. We utilize the Flickr8k dataset and encrypt the sensitive regions using DRPE. The partially encrypted images are then fed to the dual-stream encoder, which processes the real and imaginary parts of the encrypted regions separately for effective feature extraction. Our model is evaluated using standard metrics and compared with models trained on the original images. Our results demonstrate that our method achieves comparable performance to models trained on original and masked images and outperforms models trained on fully encrypted data, thus verifying the feasibility of partial encryption in privacy-preserving image captioning. Full article
(This article belongs to the Section E2: Control Theory and Mechanics)
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21 pages, 1171 KiB  
Article
Statistical Analysis of the Sum of Double Random Variables for Security Applications in RIS-Assisted NOMA Networks with a Direct Link
by Sang-Quang Nguyen, Phuong T. Tran, Bui Vu Minh, Tran Trung Duy, Anh-Tu Le, Lubos Rejfek and Lam-Thanh Tu
Electronics 2025, 14(2), 392; https://doi.org/10.3390/electronics14020392 - 20 Jan 2025
Viewed by 842
Abstract
Next- generation wireless communications are projected to integrate reconfigurable intelligent surfaces (RISs) to perpetrate enhanced spectral and energy efficiencies. To quantify the performance of RIS-aided wireless networks, the statistics of a single random variable plus the sum of double random variables becomes a [...] Read more.
Next- generation wireless communications are projected to integrate reconfigurable intelligent surfaces (RISs) to perpetrate enhanced spectral and energy efficiencies. To quantify the performance of RIS-aided wireless networks, the statistics of a single random variable plus the sum of double random variables becomes a core approach to reflect how communication links from RISs improve wireless-based systems versus direct ones. With this in mind, the work applies the statistics of a single random variable plus the sum of double random variables in the secure performance of RIS-based non-orthogonal multi-access (NOMA) systems with the presence of untrusted users. We propose a new communication strategy by jointly considering NOMA encoding and RIS’s phase shift design to enhance the communication of legitimate nodes while degrading the channel capacity of untrusted elements but with sufficient power resources for signal recovery. Following that, we analyze and derive the closed-form expressions of the secrecy effective capacity (SEC) and secrecy outage probability (SOP). All analyses are supported by extensive Monte Carlo simulation outcomes, which facilitate an understanding of system communication behavior, such as the transmit signal-to-noise ratio, the number of RIS elements, the power allocation coefficients, the target data rate of the communication channels, and secure data rate. Finally, the results demonstrate that our proposed communication can be improved significantly with an increase in the number of RIS elements, irrespective of the presence of untrusted proximate or distant users. Full article
(This article belongs to the Special Issue Wireless Sensor Network: Latest Advances and Prospects)
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10 pages, 6163 KiB  
Article
Diffractive Optical Encryption Systems Based on Multiple Wavelengths and Multiple Distances
by Yitong Wu, Haowei Duan, Yuze He and Yuanyuan Liu
Photonics 2024, 11(10), 922; https://doi.org/10.3390/photonics11100922 - 30 Sep 2024
Viewed by 974
Abstract
Coherent diffractive imaging is an optical methodology that encodes information about an object within the diffraction intensity. Here, we introduce a diffractive optical encryption system that utilizes multiple wavelengths and multiple distances, significantly expanding the size of the secret key space and enhancing [...] Read more.
Coherent diffractive imaging is an optical methodology that encodes information about an object within the diffraction intensity. Here, we introduce a diffractive optical encryption system that utilizes multiple wavelengths and multiple distances, significantly expanding the size of the secret key space and enhancing the overall security of the system by incorporating these parameters as keys. The system adopts single optical path design, compact structure and is easy to implement, overcoming the disadvantage of single key space of traditional encryption system. This system can encrypt images into a series of diffraction intensity maps (i.e., ciphertexts), and exhibits a high sensitivity to minor variations in wavelength or distance during the process of decryption, showing excellent anti-cracking ability. Furthermore, the system also has considerable robustness, ensuring that the information still can be effectively recovered even in instances of partial loss. Numerical simulation results are presented to demonstrate feasibility and effectiveness of the proposed method. Our study provides novel concepts and methodologies to the advancement of optical encryption technology, while also offering significant technical assistance to the domain of information security. Full article
(This article belongs to the Section Optical Interaction Science)
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24 pages, 10448 KiB  
Article
Optical Color Image Encryption Algorithm Based on Two-Dimensional Quantum Walking
by Guohao Cui, Xiaoyi Zhou, Hao Wang, Wentao Hao, Anshun Zhou and Jianqiang Ma
Electronics 2024, 13(11), 2026; https://doi.org/10.3390/electronics13112026 - 22 May 2024
Cited by 3 | Viewed by 1206
Abstract
The double random phase encoding (DRPE) image encryption method has garnered significant attention in color image processing and optical encryption thanks to its R, G, and B parallel encryption. However, DRPE-based color image encryption faces two challenges. Firstly, it disregards the correlation of [...] Read more.
The double random phase encoding (DRPE) image encryption method has garnered significant attention in color image processing and optical encryption thanks to its R, G, and B parallel encryption. However, DRPE-based color image encryption faces two challenges. Firstly, it disregards the correlation of R, G, and B, compromising the encrypted image’s robustness. Secondly, DRPE schemes relying on Discrete Fourier Transform (DFT) and Discrete Fractional Fourier Transform (DFRFT) are vulnerable to linear attacks, such as Known Plaintext Attack (KPA) and Chosen Plaintext Attack (CPA). Quantum walk is a powerful tool for modern cryptography, offering robust resistance to classical and quantum attacks. Therefore, this study presents an optical color image encryption algorithm that combines two-dimensional quantum walking (TDQW) with 24-bit plane permutation, dubbed OCT. This approach employs pseudo-random numbers generated by TDQW for phase modulation in DRPE and scrambles the encrypted image’s real and imaginary parts using the generalized Arnold transform. The 24-bit plane permutation helps reduce the R, G, and B correlation, while the generalized Arnold transform bolsters DRPE’s resistance to linear attacks. By incorporating TDQW, the key space is significantly expanded. The experimental results validate the effectiveness and security of the proposed method. Full article
(This article belongs to the Section Computer Science & Engineering)
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8 pages, 2556 KiB  
Proceeding Paper
Phase-Image-Encryption-Based Elliptic Curve and Double-Random-Phase Encoding
by Arabind Kumar, Sanjay Yadav and Tarul Garg
Eng. Proc. 2023, 59(1), 155; https://doi.org/10.3390/engproc2023059155 - 11 Jan 2024
Viewed by 823
Abstract
In this paper, we proposed an enhanced asymmetric cryptosystem scheme for image encryption using a combination of Elliptic Curve and Fourier transformations. Our proposed encryption and decryption process is highly secure with a smaller key size compared to other schemes due to the [...] Read more.
In this paper, we proposed an enhanced asymmetric cryptosystem scheme for image encryption using a combination of Elliptic Curve and Fourier transformations. Our proposed encryption and decryption process is highly secure with a smaller key size compared to other schemes due to the use of Elliptic Curve Cryptography. The experimental results prove that the image-encryption scheme proposed in this research is effective and has strong anti-attack and key sensitivity. Computer-based simulations have been performed for this scheme to complete the measurable examination utilizing histograms plots, and correlation distribution of adjacent pixels. Moreover, the security of this encryption scheme relies on Elliptic Curve Cryptography, which has high security. The validation of the scheme is shown using a grayscale image and all the computations are performed in MATLAB (R2021a). The security against several attacks like noise is also shown. Full article
(This article belongs to the Proceedings of Eng. Proc., 2023, RAiSE-2023)
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14 pages, 6209 KiB  
Article
Nonlinear Encryption for Multiple Images Based on a Joint Transform Correlator and the Gyrator Transform
by Ronal A. Perez, Juan M. Vilardy, Elisabet Pérez-Cabré, María S. Millán and Cesar O. Torres
Sensors 2023, 23(3), 1679; https://doi.org/10.3390/s23031679 - 3 Feb 2023
Cited by 10 | Viewed by 2141
Abstract
A novel nonlinear encryption–decryption system based on a joint transform correlator (JTC) and the Gyrator transform (GT) for the simultaneous encryption and decryption of multiple images in grayscale is proposed. This security system features a high level of security for the single real-valued [...] Read more.
A novel nonlinear encryption–decryption system based on a joint transform correlator (JTC) and the Gyrator transform (GT) for the simultaneous encryption and decryption of multiple images in grayscale is proposed. This security system features a high level of security for the single real-valued encrypted image and a high image quality for the multiple decrypted images. The multispectral or color images are considered as a special case, taking each color component as a grayscale image. All multiple grayscale images (original images) to encrypt are encoded in phase and placed in the input plane of the JTC at the same time without overlapping. We introduce two random-phase masks (RPMs) keys for each image to encrypt at the input plane of the JTC-based encryption system. The total number of the RPM keys is given by the double of the total number of the grayscale images to be encrypted. The use of several RPMs as keys improves the security of the encrypted image. The joint Gyrator power distribution (JGPD) is the intensity of the GT of the input plane of the JTC. We obtain only a single real-valued encrypted image with a high level of security for all the multiple grayscale images to encrypt by introducing two new suitable nonlinear modifications on the JGPD. The security keys are given by the RPMs and the rotation angle of the GT. The decryption system is implemented by two successive GTs applied to the encrypted image and the security keys given by the RPMs and considering the rotation angle of the GT. We can simultaneously retrieve the various information of the original images at the output plane of the decryption system when all the security keys are correct. Another result due to the appropriate definition of the two nonlinear operations applied on the JGPD is the retrieval of the multiple decrypted images with a high image quality. The numerical simulations are computed with the purpose of demonstrating the validity and performance of the novel encryption–decryption system. Full article
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12 pages, 2016 KiB  
Article
Double Image Encryption System Using a Nonlinear Joint Transform Correlator in the Fourier Domain
by Ronal A. Perez, Elisabet Pérez-Cabré, Juan M. Vilardy, María S. Millán and Cesar O. Torres
Sensors 2023, 23(3), 1641; https://doi.org/10.3390/s23031641 - 2 Feb 2023
Cited by 3 | Viewed by 2324
Abstract
In this work, we present a new nonlinear joint transform correlator (JTC) architecture in the Fourier domain (FD) for the encryption and decryption of two simultaneous images. The main features of the proposed system are its increased level of security, the obtention of [...] Read more.
In this work, we present a new nonlinear joint transform correlator (JTC) architecture in the Fourier domain (FD) for the encryption and decryption of two simultaneous images. The main features of the proposed system are its increased level of security, the obtention of a single real-valued encrypted signal that contains the ciphered information of the two primary images and, additionally, a high image quality for the two final decrypted signals. The two images to be encrypted can be either related to each other, or independent signals. The encryption system is based on the double random phase encoding (DRPE), which is implemented by using a nonlinear JTC in the FD. The input plane of the JTC has four non-overlapping data distributions placed side-by-side with no blank spaces between them. The four data distributions are phase-only functions defined by the two images to encrypt and four random phase masks (RPMs). The joint power spectrum (JPS) is produced by the intensity of the Fourier transform (FT) of the input plane of the JTC. One of the main novelties of the proposal consists of the determination of the appropriate two nonlinear operations that modify the JPS distribution with a twofold purpose: to obtain a single real-valued encrypted image with a high level of security and to improve the quality of the decrypted images. The security keys of the encryption system are represented by the four RPMs, which are all necessary for a satisfactory decryption. The decryption system is implemented using a 4f-processor where the encrypted image and the security keys given by the four RPMs are introduced in the proper plane of the processor. The double image encryption system based on a nonlinear JTC in the FD increases the security of the system because there is a larger key space, and we can simultaneously validate two independent information signals (original images to encrypt) in comparison to previous similar proposals. The feasibility and performance of the proposed double image encryption and decryption system based on a nonlinear JTC are validated through computational simulations. Finally, we additionally comment on the proposed security system resistance against different attacks based on brute force, plaintext and deep learning. Full article
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14 pages, 3939 KiB  
Article
Privacy-Preserving Image Captioning with Deep Learning and Double Random Phase Encoding
by Antoinette Deborah Martin, Ezat Ahmadzadeh and Inkyu Moon
Mathematics 2022, 10(16), 2859; https://doi.org/10.3390/math10162859 - 11 Aug 2022
Cited by 5 | Viewed by 2373
Abstract
Cloud storage has become eminent, with an increasing amount of data being produced daily; this has led to substantial concerns related to privacy and unauthorized access. To secure privacy, users can protect their private data by uploading encrypted data to the cloud. Data [...] Read more.
Cloud storage has become eminent, with an increasing amount of data being produced daily; this has led to substantial concerns related to privacy and unauthorized access. To secure privacy, users can protect their private data by uploading encrypted data to the cloud. Data encryption allows computations to be performed on encrypted data without the data being decrypted in the cloud, which requires enormous computation resources and prevents unauthorized access to private data. Data analysis such as classification, and image query and retrieval can preserve data privacy if the analysis is performed using encrypted data. This paper proposes an image-captioning method that generates captions over encrypted images using an encoder–decoder framework with attention and a double random phase encoding (DRPE) encryption scheme. The images are encrypted with DRPE to protect them and then fed to an encoder that adopts the ResNet architectures to generate a fixed-length vector of representations or features. The decoder is designed with long short-term memory to process the features and embeddings to generate descriptive captions for the images. We evaluate the predicted captions with BLEU, METEOR, ROUGE, and CIDEr metrics. The experimental results demonstrate the feasibility of our privacy-preserving image captioning on the popular benchmark Flickr8k dataset. Full article
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24 pages, 7341 KiB  
Article
A Novel Virtual Optical Image Encryption Scheme Created by Combining Chaotic S-Box with Double Random Phase Encoding
by Peiran Tian and Runzhou Su
Sensors 2022, 22(14), 5325; https://doi.org/10.3390/s22145325 - 16 Jul 2022
Cited by 26 | Viewed by 2198
Abstract
The double random phase encoding (DRPE) system plays a significant role in encrypted systems. However, it is a linear system that leads to security holes in encrypted systems. To tackle this issue, this paper proposes a novel optical image encryption scheme that combines [...] Read more.
The double random phase encoding (DRPE) system plays a significant role in encrypted systems. However, it is a linear system that leads to security holes in encrypted systems. To tackle this issue, this paper proposes a novel optical image encryption scheme that combines a chaotic S-box, DRPE, and an improved Arnold transformation (IAT). In particular, the encryption scheme designs a chaotic S-box to substitute an image. The chaotic S-box has the characteristics of high nonlinearity and low differential uniformity and is then introduced to enhance the security of the DRPE system. Chaotic S-boxes are resistant to algebraic attacks. An IAT is used to scramble an image encoded by the DRPE system. Meanwhile, three chaotic sequences are obtained by a nonlinear chaotic map in the proposed encryption scheme. One of them is used for XOR operation, and the other two chaotic sequences are explored to generate two random masks in the DRPE system. Simulation results and performance analysis show that the proposed encryption scheme is efficient and secure. Full article
(This article belongs to the Section Optical Sensors)
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22 pages, 13333 KiB  
Article
Multi-Image Compression–Encryption Algorithm Based on Compressed Sensing and Optical Encryption
by Jingjin Wei, Miao Zhang and Xiaojun Tong
Entropy 2022, 24(6), 784; https://doi.org/10.3390/e24060784 - 2 Jun 2022
Cited by 19 | Viewed by 2862
Abstract
In order to achieve large-capacity, fast and secure image transmission, a multi-image compression–encryption algorithm based on two-dimensional compressed sensing (2D CS) and optical encryption is proposed in this paper. Firstly, the paper uses compressed sensing to compress and encrypt multiple images simultaneously, and [...] Read more.
In order to achieve large-capacity, fast and secure image transmission, a multi-image compression–encryption algorithm based on two-dimensional compressed sensing (2D CS) and optical encryption is proposed in this paper. Firstly, the paper uses compressed sensing to compress and encrypt multiple images simultaneously, and design a new structured measurement matrix. Subsequently, double random phase encoding based on the multi-parameter fractional quaternion Fourier transform is used to encrypt the multiple images for secondary encryption, which improves the security performance of the images. Moreover, a fractional-order chaotic system with more complex chaotic behavior is constructed for image compression and encryption. Experimental results show that the algorithm has strong robustness and security. Full article
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15 pages, 4170 KiB  
Review
Single-Shot Compressed Imaging via Random Phase Modulation
by Cheng Zhang, Ru Zhang, Yuanyuan Zhu, Hairong Yang, Chuan Shen and Sui Wei
Appl. Sci. 2022, 12(9), 4536; https://doi.org/10.3390/app12094536 - 29 Apr 2022
Cited by 3 | Viewed by 1669
Abstract
Compressed sensing (CS) provides an innovative framework for signal sampling, which enables accurate recovery of the sparse or compressible signal from a small set of linear measurements far fewer than the Nyquist rate in traditional signal processing. In compressed sensing, random modulation plays [...] Read more.
Compressed sensing (CS) provides an innovative framework for signal sampling, which enables accurate recovery of the sparse or compressible signal from a small set of linear measurements far fewer than the Nyquist rate in traditional signal processing. In compressed sensing, random modulation plays a key role, which can spread out the signal information more or less evenly across all locations. There are many modulation techniques, such as amplitude modulation, frequency modulation, phase modulation, spectrum modulation, and so on. Among these modulation techniques, phase modulation is vital due to the efficiency and convenience of modulation. In this paper, we review both the theoretical and application of compressed sensing and several compressed imaging systems using random phase modulation. First, we review the fundamentals of compressed sensing, dividing it into three parts: sparse representation, incoherent measurement, and nonlinear reconstruction algorithm. We then show how phase modulation can be applied to compressed sensing and compressed imaging, where the presentation can be divided into six main parts, corresponding to different aspects of phase modulation applied in compressed sensing or compressed imaging: (1) Fundamentals of compressed sensing. (2) Principles of phase modulation. (3) Single-shot compressed imaging with spatial-domain single random phase mask (CI-SSRPM). (4) Single-shot compressed imaging with a random convolution using a double random phase mask (CI-DRPM). (5) Single-shot compressed imaging with Fourier-domain single random phase mask (CI-FSRPM). (6) Single-shot compressed imaging with double random phase encoding (CI-DRPE). Full article
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23 pages, 7936 KiB  
Article
Double Image Encryption Scheme Based on Compressive Sensing and Double Random Phase Encoding
by Rui Zhang and Di Xiao
Mathematics 2022, 10(8), 1242; https://doi.org/10.3390/math10081242 - 10 Apr 2022
Cited by 14 | Viewed by 2189
Abstract
In order to overcome the shortcomings of the standard compressive sensing (CS) encryption framework, a novel fusion application scheme of CS and optical transformation technology is proposed. The proposed scheme, making full use of the feature of CS to achieve compression and encryption [...] Read more.
In order to overcome the shortcomings of the standard compressive sensing (CS) encryption framework, a novel fusion application scheme of CS and optical transformation technology is proposed. The proposed scheme, making full use of the feature of CS to achieve compression and encryption simultaneously, compresses and encrypts two images into one image, which not only reduces storage space and transmission bandwidth, but also improves the security performance of encryption. In the proposed scheme, the two original images are first sampled with CS, and then double random phase coding is performed to obtain two small-sized images. Meanwhile, the two original images are directly encrypted with double random phase coding to obtain the authentication information. Next, we combine two small-sized images and authentication information into one image, and finally perform double random phase coding again to obtain the final encrypted image. It should be emphasized that the proposed scheme has the function of image authentication. Experiment results validate the effectiveness and advancement of the proposed fusion application scheme. Full article
(This article belongs to the Special Issue Mathematical Methods for Computer Science)
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18 pages, 37056 KiB  
Article
An Optical Image Encryption Method Using Hopfield Neural Network
by Xitong Xu and Shengbo Chen
Entropy 2022, 24(4), 521; https://doi.org/10.3390/e24040521 - 7 Apr 2022
Cited by 12 | Viewed by 2414
Abstract
In this paper, aiming to solve the problem of vital information security as well as neural network application in optical encryption system, we propose an optical image encryption method by using the Hopfield neural network. The algorithm uses a fuzzy single neuronal dynamic [...] Read more.
In this paper, aiming to solve the problem of vital information security as well as neural network application in optical encryption system, we propose an optical image encryption method by using the Hopfield neural network. The algorithm uses a fuzzy single neuronal dynamic system and a chaotic Hopfield neural network for chaotic sequence generation and then obtains chaotic random phase masks. Initially, the original images are decomposed into sub-signals through wavelet packet transform, and the sub-signals are divided into two layers by adaptive classification after scrambling. The double random-phase encoding in 4f system and Fresnel domain is implemented on two layers, respectively. The sub-signals are performed with different conversions according to their standard deviation to assure that the local information’s security is guaranteed. Meanwhile, the parameters such as wavelength and diffraction distance are considered as additional keys, which can enhance the overall security. Then, inverse wavelet packet transform is applied to reconstruct the image, and a second scrambling is implemented. In order to handle and manage the parameters used in the scheme, the public key cryptosystem is applied. Finally, experiments and security analysis are presented to demonstrate the feasibility and robustness of the proposed scheme. Full article
(This article belongs to the Special Issue Computational Imaging and Image Encryption with Entropy)
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12 pages, 1389 KiB  
Article
Improving Continuous Variable Quantum Secret Sharing with Weak Coherent States
by Yijun Wang, Bing Jia, Yun Mao, Xuelin Wu and Ying Guo
Appl. Sci. 2020, 10(7), 2411; https://doi.org/10.3390/app10072411 - 1 Apr 2020
Cited by 2 | Viewed by 2227
Abstract
Quantum secret sharing (QSS) can usually realize unconditional security with entanglement of quantum systems. While the usual security proof has been established in theoretics, how to defend against the tolerable channel loss in practices is still a challenge. The traditional ( [...] Read more.
Quantum secret sharing (QSS) can usually realize unconditional security with entanglement of quantum systems. While the usual security proof has been established in theoretics, how to defend against the tolerable channel loss in practices is still a challenge. The traditional ( t,n ) threshold schemes are equipped in situation where all participants have equal ability to handle the secret. Here we propose an improved ( t,n ) threshold continuous variable (CV) QSS scheme using weak coherent states transmitting in a chaining channel. In this scheme, one participant prepares for a Gaussian-modulated coherent state (GMCS) transmitted to other participants subsequently. The remaining participants insert independent GMCS prepared locally into the circulating optical modes. The dealer measures the phase and the amplitude quadratures by using double homodyne detectors, and distributes the secret to all participants respectively. Special t out of n participants could recover the original secret using the Lagrange interpolation and their encoded random numbers. Security analysis shows that it could satisfy the secret sharing constraint which requires the legal participants to recover message in a large group. This scheme is more robust against background noise due to the employment of double homodyne detection, which relies on standard apparatuses, such as amplitude and phase modulators, in favor of its potential practical implementations. Full article
(This article belongs to the Special Issue Quantum Communications and Quantum Networks)
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7 pages, 1140 KiB  
Article
Optical Image Encryption System Using Several Tilted Planes
by Juan M. Vilardy O., Carlos J. Jimenez and Cesar O. Torres M.
Photonics 2019, 6(4), 116; https://doi.org/10.3390/photonics6040116 - 7 Nov 2019
Cited by 6 | Viewed by 3377
Abstract
A well-known technique for optical image encryption is the double random phase encoding (DRPE) technique, which uses two random phase masks (RPMs), one RPM at the input plane of the encryption system and the other RPM at the Fourier plane of the optical [...] Read more.
A well-known technique for optical image encryption is the double random phase encoding (DRPE) technique, which uses two random phase masks (RPMs), one RPM at the input plane of the encryption system and the other RPM at the Fourier plane of the optical system, in order to obtain the encrypted image. In this work, we propose to use tilted planes for the Fourier and the output planes of the optical DRPE encryption system with the purpose of adding two new security keys, which are the angles of the tilted planes. The optical diffraction on a tilted plane is computed using the angular spectrum of plane waves and the coordinate rotation in the Fourier domain. The tilted distributions at the intermediate and output planes of the optical DRPE encryption system are the second RPM and the encrypted image, respectively. The angles of the tilted planes allow improvement to the security of the encrypted image. We perform several numerical simulations with the purpose of demonstrating the validity and feasibility of the proposed image encryption system. Full article
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