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.     

Image encryption based on the multiple-parameter discrete fractional Fourier transform and chaos function

In recent years, the chaos-based cryptographic algorithms have suggested some new and efficient ways to develop secure image encryption techniques. In this paper, we propose a new approach for image encryption based on the multiple-parameter discrete fractional Fourier transform and chaotic logistic maps in order to meet the requirements of the secure image transmission. In the proposed image encryption scheme, the image is encrypted by juxtaposition of sections of the image in the multiple-parameter discrete fractional Fourier domains and the alignment of sections is determined by chaotic logistic maps. This method does not require the use of phase keys. The new method has been compared with several existing methods and shows comparable or superior robustness to blind decryption.     

Image encryption based on the reality-preserving multiple-parameter fractional Fourier transform

In recent years, a number of methods have been proposed in the literature for the encryption of two-dimensional information by using the fractional Fourier transform, but most of their encryptions are complex values and need digital hologram technique to record information, which is inconvenient for digital transmission. In this paper, we propose a new approach for image encryption based on the real-valuedness and decorrelation property of the reality-preserving multiple-parameter fractional Fourier transform in order to meet the requirements of the secure image transmission. In the proposed scheme, the original and encrypted images are respectively in the spatial domain and the reality-preserving multiple-parameter fractional Fourier transformed domain determined by the encryption keys. Numerical simulations are performed to demonstrate that the proposed method is reliable and more robust to blind decryption than several existing methods.     

Novel image encryption algorithm based on multiple-parameter discrete fractional random transform

A new method of digital image encryption is presented by utilizing a new multiple-parameter discrete fractional random transform. Image encryption and decryption are performed based on the index additivity and multiple parameters of the multiple-parameter fractional random transform. The plaintext and ciphertext are respectively in the spatial domain and in the fractional domain determined by the encryption keys. The proposed algorithm can resist statistic analyses effectively. The computer simulation results show that the proposed encryption algorithm is sensitive to the multiple keys, and that it has considerable robustness, noise immunity and security.     

不对称离散分数傅里叶变换实现数字图像的加密变换

利用不对称分数傅里叶变换的特性,提出了一种图像加密变换的新方法。对图像的x,y方向分别实施不同级次的一维分数傅里叶变换,得到加密图像。解密方法就是对变换后的图像实施对应级次的分数傅里叶逆变换,只有当x,y方向的逆变换级次分别与原变换级次都相同或者满足周期条件时,才能恢复原图像。加密变换有效地提高了图像加密和防伪力度。数值计算验证了方法的正确性和可行性。     

基于变形分数傅里叶变换的六重密钥图像加密

提出一种利用变形分数傅里叶变换和双随机相位编码对图像加密的方法.对要加密的图像分别进行两次变形分数傅里叶变换和两次随机相位函数调制,使加密图像的密钥由原来两重增加到六重.利用全息元件,可以用光学系统实现这种加密和解密变换.计算机模拟结果表明,只有当六重密钥都完全正确时,才能准确地重建原图像,这种六重密钥加密方法提高了图像信息的安全保密性.     

基于分频域和菲涅耳域的光学图像加密方法

结合分数傅里叶变换及菲涅耳变换,在光学图像加密系统中分别具有多密钥性和无透镜性的优点,提出了基于分频域和菲涅耳域的光学图像加密方法。基于分数傅里叶变换的光学加密系统,引入菲涅耳变换及全息技术,使原有的加密系统在不增加光学元件的基础上提高了系统的安全性。理论分析和计算机仿真模拟证明了这种方法的可行性。     

Asymmetric multiple-image hiding using phase retrieval technique based on amplitude- and phase-truncation in fractional Fourier domain

We propose a multiple-image hiding scheme based on the amplitude- and phase-truncation approach, and phase retrieval iterative algorithm in the fractional Fourier domain. The proposed scheme offers multiple levels of security with asymmetric keys. Multiple input images multiplied with random phase masks are independently fractional Fourier transformed with different orders. The individual keys and common keys are generated by using phase and amplitude truncation of fractional spectrum. After using two fractional Fourier transform, the resultant encrypted image is hided in a host image with phase retrieval iterative algorithm. Using the correct universal keys, individual keys, and fractional orders, one can recover the original image successfully. Computer simulation results with four gray-scale images support the proposed method. To measure the validity of the scheme, we calculated the mean square error between the original and the decrypted images. In this scheme, the encryption process and generation of decryption keys are complicated and should be realized using computer. For decryption, an optoelectronic setup has been suggested.     

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 multifractional Fourier transforms

We propose a new image encryption algorithm based on a generalized fractional Fourier transform, to which we refer as a multifractional Fourier transform. We encrypt the input image simply by performing the multifractional Fourier transform with two keys. Numerical simulation results are given to verify the algorithm, and an optical implementation setup is also suggested.     

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.     

Optical image encryption by random shifting in fractional Fourier domains

A number of methods have recently been proposed in the literature for the encryption of two-dimensional information by use of optical systems based on the fractional Fourier transform. Typically, these methods require random phase screen keys for decrypting the data, which must be stored at the receiver and must be carefully aligned with the received encrypted data. A new technique based on a random shifting, or jigsaw, algorithm is proposed. This method does not require the use of phase keys. The image is encrypted by juxtaposition of sections of the image in fractional Fourier domains. The new method has been compared with existing methods and shows comparable or superior robustness to blind decryption. Optical implementation is discussed, and the sensitivity of the various encryption keys to blind decryption is examined.     

光学图像加密与多参数加权类分数傅里叶变换

为了提高图像加密的安全性, 提出了一种多参数加权类分数傅里叶变换。此类多参数加权类分数傅里叶变换是C.C.Shih提出的四项加权类分数傅里叶变换的一种扩展, 除了分数阶数, 还有四个在四项加权系数之中的自由参数, 称其为向量参数。同时给出此多参数加权类分数傅里叶变换的离散形式, 并把这种算法应用到光学图像加密中。此算法在应用一次二维分数傅里叶变换可以有十个密键：一类为阶数参数; 另一类为向量参数, 因此这种加密算法在增加了安全性的同时, 加密过程的复杂度降低。数值仿真验证了此算法的有效性和可靠性。     

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.     

Image encryption algorithm based on the multi-order discrete fractional Mellin transform

A multi-order discrete fractional Mellin transform (MODFrMT) is constructed and directly used to encrypt the private images. The MODFrMT is a generalization of the fractional Mellin transform (FrMT) and is derived by transforming the image with multi-order discrete fractional Fourier transform (MODFrFT) in log-polar coordinates, where the MODFrFT is generalized from the closed-form expression of the discrete fractional Fourier transform (DFrFT) and can be calculated by fast Fourier transform (FFT) to reduce the computation burden. The fractional order vectors of the MODFrMT are sensitive enough to be the keys, and consequently key space of the encryption system is enlarged. The proposed image encryption algorithm has significant ability to resist some common attacks like known-plaintext attack, chosen-plaintext attack, etc. due to the nonlinear property of the MODFrMT. Additionally, Kaplan-Yorke map is employed in coordinate transformation process of the MODFrMT to further enhance the security of the encryption system. The computer simulation results show that the proposed encryption algorithm is feasible, secure and robust to noise attack and occlusion.     

Asymmetric multiple-image encryption based on coupled logistic maps in fractional Fourier transform domain

A multiple-image encryption scheme is proposed based on the asymmetric technique, in which the encryption keys are not identical to the decryption ones. First, each plain image is scrambled based on a sequence of chaotic pairs generated with a system of two symmetrically coupled identical logistic maps. Then, the phase-only function of each scrambled image is retrieved with an iterative phase retrieval process in the fractional Fourier transform domain. Second, all phase-only functions are modulated into an interim, which is encrypted into the ciphertext with stationary white noise distribution by using the fractional Fourier transform and chaotic diffusion. In the encryption process, three random phase functions are used as encryption keys to retrieve the phase-only functions of plain images. Simultaneously, three decryption keys are generated in the encryption process, which make the proposed encryption scheme has high security against various attacks, such as chosen plaintext attack. The peak signal-to-noise is used to evaluate the quality of the decrypted image, which shows that the encryption capacity of the proposed scheme is enhanced considerably. Numerical simulations demonstrate the validity and efficiency of the proposed method.     

Novel optical image encryption scheme based on fractional Mellin transform

A novel nonlinear image encryption scheme is proposed by introducing the fractional Mellin transform (FrMT) into the field of image security. As a nonlinear transform, FrMT is employed to get rid of the potential insecurity of the optical image encryption system caused by the intrinsic object-image relationship between the plaintext and the ciphertext. Different annular domains of the original image are transformed by FrMTs of different orders, and then the outputs are further encrypted by comprehensively using fractional Fourier transform (FrFT), amplitude encoding and phase encoding. The keys of the encryption algorithm include the orders of the FrMTs, the radii of the FrMT domains, the order of the FrFT and the phases generated in the further encryption process, thus the key space is extremely large. An optoelectronic hybrid structure for the proposed scheme is also introduced. Numerical simulations demonstrate that the proposed algorithm is robust with noise immunity, sensitive to the keys, and outperforms the conventional linear encryption methods to counteract some attacks.     

The multiple-parameter discrete fractional Hadamard transform

Discrete fractional Hadamard transform (DFrHaT) is a generalization of the Hadamard transform, which has been widely used in signal processing. In this paper, we present the multiple-parameter discrete fractional Hadamard transform (MPDFrHaT), which has multiple order parameters instead of only one in DFrHaT. The proposed MPDFrHaT is shown to possess all of the desired properties of DFrHaT. In fact, it will reduce to DFrHaT when all of its order parameters are the same. We also propose a novel encryption technique, double random amplitude (DRA) encoding scheme, by cascading twofold random amplitude filtering. As a primary application, we exploit the multiple-parameter feature of MPDFrHaT and double random amplitude encoding scheme for digital image encryption in the MPDFrHaT domain. Results show that this method can enhance data security.     

基于级联分数傅里叶变换系统的数字水印技术

提出一种基于分数傅里叶变换和随机相位编码的光学加密数字水印技术,可成为一种信息隐藏及保护的有效方案.该数字水印技术对于噪音叠加和常见的图像处理操作具有较强的稳健性.该技术根据光学级联分数傅里叶变换系统,利用两个随机相位分布函数对水印信息编码并经过迭代分数傅里叶变换嵌入到变换域的载体图像中.在水印检测和提取过程中,两个相位分布函数作为密钥.随机相位编码技术的引入,进一步提高了数字水印系统的密钥空间.增强了系统的安全性.该数字水印技术基于光学分数傅里叶变换原理,可以利用光学变换系统方便地实现.     

Fractional Fourier plane image encryption technique using radial hilbert-, and Jigsaw transform

A new method for image encryption using integral order radial Hilbert transform (RHT) filter in the fractional Fourier transform (FRT) domain has been proposed. The technique is implemented using the popular double random phase encoding method in the fractional Fourier domain. The random phase masks (RPMs), integral orders of the RHT, fractional orders of FRT, and indices of the Jigsaw transform (JT) have been used as keys for encryption and decryption. Simulation results have been presented and the schematic representation for optical implementation has been proposed. The mean-square-error and signal-to-noise ratio between the decrypted image and the input image have been calculated for the correct as well as incorrect orders of the RHT. Effect of occlusion and noise on the performance of the proposed scheme has also been studied. The robustness of the technique has been verified against attack using partial windows of the correct random phase masks. Similar investigations have also been carried out for the chosen-, and the known-plain-text attacks.