Fractional Fourier transform based image multiplexing and encryption technique for four-color images using input images as keys

A digital technique for multiplexing and encryption of four RGB images has been proposed using the fractional Fourier transform (FRT). The four input RGB images are first converted into their indexed image formats and subsequently multiplexed into a single image through elementary mathematical steps prior to the encryption. The encryption algorithm uses two random phase masks in the input- and the FRT domain, respectively. These random phase masks are especially designed using the input images. As the encryption is carried out through a single channel, the technique is more compact and faster as compared to the multichannel techniques. Different fractional orders, the random masks in input-, and FRT domain are the keys for decryption as well as de-multiplexing. The algorithms to implement the proposed multiplexing-, and encryption scheme are discussed, and results of digital simulation are presented. Simulation results show that the technique is free from cross-talk. The performance of the proposed technique has also been analyzed against occlusion, noise, and attacks using partial windows of the correct random phase keys. The robustness of the technique against known-, and chosen plain-text attacks has also been explained.     

Image encryption and decryption using fractional Fourier transform and radial Hilbert transform

A technique for image encryption using fractional Fourier transform (FRT) and radial Hilbert transform (RHT) is proposed. The spatial frequency spectrum of the image to be encrypted is first segregated into two parts/channels using RHT, and image subtraction technique. Each of these channels is encrypted independently using double random phase encoding in the FRT domain. The different fractional orders and random phase masks used during the process of encryption and decryption are the keys to enhance the security of the proposed system. The algorithms to implement the proposed encryption and decryption scheme are discussed, and results of digital simulation are presented.     

Optical image encryption using fractional Fourier transform and chaos

We propose a new method for image encryption using fractional Fourier transform and chaos theory. Random phase masks are generated using iterative chaos functions. The input image is combined with the first random phase mask at the object plane and is then transformed using the fractional Fourier transform. After the first fractional Fourier transform, the second random phase mask, again generated by using the chaos functions, is used at the fractional plane. The second fractional Fourier transform operation is then carried out to obtain the encrypted image. Three types of chaos functions have been used: the logistic map, the tent map and the Kaplan–Yorke map. The mean square error and the signal-to-noise ratio between the decrypted image and the input image for the correct order and the incorrect order of the fractional Fourier transform have been calculated. The computer simulations are presented to verify the validity of the proposed technique.     

Color image encryption and decryption using fractional Fourier transform

We propose the encryption of color images using fractional Fourier transform (FRT). The image to be encrypted is first segregated into three color channels: red, green, and blue. Each of these channels is encrypted independently using double random phase encoding in the FRT domain. The different fractional orders and random phase masks used during the process of encryption and decryption are the keys to enhance the security of the proposed system. The algorithms to implement the proposed encryption and decryption scheme are discussed, and results of digital simulation are presented. The technique is shown to be a powerful one for colored text encryption. We also outline the implementation of the algorithm and examine its sensitiveness to changes in the fractional order during decryption.     

Asymmetric image encryption using phase-truncated discrete multiple-parameter fractional Fourier transform

An asymmetric image encryption scheme is proposed using a phase-truncated discrete multiple-parameter fractional Fourier transform (DMPFRFT). After applying a pixel-scrambling operation and random-phase mask, an asymmetric ciphertext with stationary white noise can be obtained using phase truncation in the DMPFRFT domain. Using the phase key, an inverse pixel-scrambling operation, and the parameters of the DMPFRFT, the original image can be successfully retrieved. Numerical simulations were conducted to demonstrate the validity and the security of the proposed method, and electro-optical hybrid setups are suggested for encryption and decryption.     

Modified computational integral imaging-based double image encryption using fractional Fourier transform

In this paper, we propose an image encryption technique to simultaneously encrypt double or multiple images into one encrypted image using computational integral imaging (CII) and fractional Fourier transform (FrFT). In the encryption, each of the input plane images are located at different positions along a pickup plane, and simultaneously recorded in the form of an elemental image array (EIA) through a lenslet array. The recorded EIA to be encrypted is multiplied by FrFT with two different fractional orders. In order to mitigate the drawbacks of occlusion noise in computational integral imaging reconstruction (CIIR), the plane images can be reconstructed using a modified CIIR technique. To further improve the solution of the reconstructed plane images, a block matching algorithm is also introduced. Numerical simulation results verify the feasibility and effectiveness of the proposed method.     

基于不对称非标准傅里叶变换的光学图像加密

针对以往光学图像加密系统中输入面和频谱面对称性的缺点，在不增加系统元件的基础上，利用球面波照射不对称非标准傅里叶变换系统进行图像加密。通过把相位掩模置于该系统的傅里叶变换平面，利用不对称非标准傅里叶变换系统的输入面和频谱面的不对称性以及频谱面对于点光源相关参数的依赖性，克服了以前光学加密系统中输入面和频谱面的对称性所带来的安全隐患，并且获得了除相位掩模以外的另外四重密匙。理论分析和模拟实验表明：该方法不仅可行，而且多增加了几重密匙，增强了系统的安全性能。     

Double image encryption based on iterative fractional Fourier transform

We present an image encryption algorithm to simultaneously encrypt two images into a single one as the amplitudes of fractional Fourier transform with different orders. From the encrypted image we can get two original images independently by fractional Fourier transforms with two different fractional orders. This algorithm can be independent of additional random phases as the encryption/decryption keys. Numerical results are given to analyze the capability of this proposed method. A possible extension to multi-image encryption with a fractional order multiplexing scheme has also been given.     

Triple image encryption scheme in fractional Fourier transform domains

We proposed a triple image encryption scheme by use of fractional Fourier transform. In this algorithm, an original image is encoded in amplitude part and other two images are encoded into phase information. The key of encryption algorithm is obtained from the difference between the third image and the output phase of transform. In general case, random phase encoding technology is not required in the proposed algorithm. Moreover, all information of images is preserved in theory when image are decrypted with correct key. The optical implementation of the algorithm is presented with an electro-optical hybrid structure. Numerical simulations have demonstrated the efficiency and the security of this algorithm. Based on this scheme a multiple image algorithm is expanded and designed.     

基于分数阶Fourier变换的数字图像实值加密方法

构造了一种新的保实化的分数阶Fourier变换,提出了一种基于该变换的数字图像实值加密方法。利用保实分数阶Fourier变换的保实特性和阶数可加性等完成了数字图像的加密与解密,明文和密文分别位于空域和由密钥决定的保实分数阶Fourier变换域中,具有较强的抗统计破译能力。密图是一个实值图像,便于显示和存储。仿真实验结果表明,该加密方法密钥简单,无数据膨胀,对参数敏感度高,具有一定的鲁棒性和安全性。在信息安全领域具有良好的研究前景和实用价值。     

Optical image encryption based on multichannel fractional Fourier transform and double random phase encoding technique

The optical image encryption based on multichannel fractional Fourier transform (FRT) and double random phase encoding technique is proposed. Optical principles of encoding and decoding are analyzed in detail. With this method, one can encrypt different parts of input image, respectively. The system security can be improved to some extent, not only because fractional orders and random phase masks in every channel can be set with freedom, but also because the system parameters among all channels are independent. Numerical simulation results of optical image encryption based on four channel FRT and double random phase encoding are given to verify the feasibility of the method.     

Optical image encryption based on the multiple-parameter fractional Fourier transform

A novel image encryption algorithm is proposed based on the multiple-parameter fractional Fourier transform, which is a generalized fractional Fourier transform, without the use of phase keys. The image is encrypted simply by performing a multiple-parameter fractional Fourier transform with four keys. Optical implementation is suggested. The method has been compared with existing methods and shows superior robustness to blind decryption.     

A 3D image encryption technique using computer-generated integral imaging and cellular automata transform

A novel three-dimensional (3D) image encryption approach by using the computer-generated integral imaging and cellular automata transform (CAT) is proposed, in which, the two-dimensional (2D) elemental image array (EIA) digitally recorded by light rays coming from the 3D image is mapped inversely through the virtual pinhole array according to the ray-tracing theory. Next, the encrypted image is generated by using the 2D CAT scrambling transform for the 2D EIA. The reconstructed process is carried out by using the modified computational integral-imaging reconstruction (CIIR) technique; the depth-dependent plane images are reconstructed on the output plane. The reconstructed 3D image quality of the proposed scheme can be greatly improved, because the proposed encryption scheme carries out in a computer which can avoid the light diffraction caused by optical device CIIR, and solves blur problem caused by CIIR by using the pixel-averaging algorithm. Furthermore, the CAT-based encryption algorithm is an error-free encryption method; CAT as an orthogonal transformation offers considerable simplicity in the calculation of the transform coefficient, that is, it can improve the quality of the reconstructed image by reducing energy loss compared with the traditional complicated transform process. To show the effectiveness of the proposed scheme, we perform computational experiments. Experimental results show that the proposed scheme outperforms conventional encryption methods.     

Optical image encryption using Hartley transform and logistic map

We propose a new method for image encryption using Hartley transform with jigsaw transform and logistic map. Logistic map has been used to generate the random intensity mask which is known as chaotic random intensity mask. The problem of bare decryption with Hartley transform has been solved by using the jigsaw transform. In the proposed technique, the image is encrypted using two methods in which the second method is the extension of the first method. In the first method, the image is encrypted using Hartley transform and jigsaw transform. In the second method, the image is encrypted using Hartley transform, jigsaw transform and logistic map. The mean square errors and the signal to noise ratio have been calculated. Robustness of the technique in terms of blind decryption and the algorithmic complexity has been evaluated. The optical implementation has been proposed. The computer simulations are presented to verify the validity of the proposed technique.     

Image encryption based on extended fractional Fourier transform and digital holography technique

We present a new optical image encryption algorithm that is based on extended fractional Fourier transform (FRT) and digital holography technique. We can perform the encryption and decryption with more parameters compared with earlier similar methods in FRT domain. In the extended FRT encryption system, the input data to be encrypted is extended fractional Fourier transformed two times and random phase mask is placed at the output plane of the first extended FRT. By use of an interference with a wave from another random phase mask, the encrypted data is stored as a digital hologram. The data retrieval is operated by all-digital means. Computer simulations are presented to verify its validity and efficiency.     

Double image encryption by using Arnold transform and discrete fractional angular transform

Based on Arnold transform and discrete fractional angular transform, a double image encryption algorithm is designed. Two original images are regarded as the amplitude and phase of a complex function. Arnold transform is introduced for scrambling the pixels at a local area of the complex function. Subsequently the changed complex function is converted by discrete fractional angular transform. The operations mentioned will be performed many times. The amplitude of final output complex function is the encrypted image and its phase is regarded as the key of encryption algorithm. The parameters of the two transforms serve as the additional keys for enhancing the security. Some numerical simulations have been done to validate the performance of this encryption scheme.     

Fractional Fourier transform and geometric quantization

Generalized Fourier transformation between the position and the momentum representation of a quantum state is constructed in a coordinate independent way. The only ingredient of this construction is the symplectic (canonical) geometry of the phase-space: no linear structure is necessary. It is shown that the “fractional Fourier transform” provides a simple example of this construction. As an application of this technique we show that for any linear Hamiltonian system, its quantum dynamics can be obtained exactly as the lift of the corresponding classical dynamics by means of the above transformation. Moreover, it can be deduced from the free quantum evolution. This way new, unknown symmetries of the Schrödinger equation can be constructed. It is also argued that the above construction defines in a natural way a connection in the bundle of quantum states, with the base space describing all their possible representations. The non-flatness of this connection would be responsible for the non-existence of a quantum representation of the complete algebra of classical observables.     

Novel single-channel color image encryption algorithm based on chaos and fractional Fourier transform

A new color image encryption algorithm based on fractional Fourier transform (FrFT) and chaos is proposed. The colors of the original color image are converted to HSI (hue-saturation-intensity), and the S component is transformed by the random-phase encoding based on FrFT to obtain a new random phase. The I component is transformed by double random-phase encoding based on FrFT using the H component and the new random phase as two phase plates. Then chaos scrambling technology is used to encrypt the image, which makes the resulting image nonlinear and disorder both in spatial domain and frequency domain. Additionally, the ciphertext is not a color image but a combination of a gray image and a phase matrix, so the ciphertext has camouflage property to some extent. The results of numerical simulations demonstrate the effectiveness and the security of this algorithm.     

Double image encryption based on discrete multiple-parameter fractional Fourier transform and chaotic maps

A double image encryption method is proposed by utilizing discrete multiple-parameter fractional Fourier transform and chaotic maps. One of the two original images scrambled by one chaotic map is encoded into the amplitude of a complex signal with the other original image as its phase. The complex signal multiplied by another chaotic random phase mask is then encrypted by discrete multiple-parameter fractional Fourier transform. The parameters in chaotic map and discrete multiple-parameter fractional Fourier transform serve as the keys of this encryption scheme. Numerical simulations have been done to demonstrate the performance of this algorithm.     

Fractional Fourier-domain random encoding and pixel scrambling technique for double image encryption

A double image encryption method is proposed using fractional Fourier-domain random encoding and pixel scrambling technique. One of the two original images is encoded into the phase function of a synthesized input signal after being scrambled, and the other original image encoded into its amplitude. The phase function serves as phase mask in the input domain, and the synthesized input signal is then encrypted into stationary white noise by utilizing random phase encoding in fractional Fourier domain. The two original images can be retrieved without cross-talk by using the correct keys with fractional orders, the random phase mask and the pixel scrambling operator. Numerical simulations and security analysis have been done to prove the validity and the security of the proposed encryption method.