Devil’s Vortex Phase Structure as Frequency Plane Mask for Image Encryption Using the Fractional Mellin Transform

A frequency plane phase mask based on Devil’s vortex structure has been used for image encryption using the fractional Mellin transform. The phase key for decryption is obtained by an iterative phase retrieval algorithm. The proposed scheme has been validated for grayscale secret target images, by numerical simulation. The efficacy of the scheme has been evaluated by computing mean-squared-error between the secret target image and the decrypted image. Sensitivity analysis of the decryption process to variations in various encryption parameters has been carried out. The proposed encryption scheme has been seen to exhibit reasonable robustness against occlusion attack.


Introduction
Phase filters play an important role in designing an optical image encryption system.One of the most efficient zone plates is based on Devil's vortex Fresnel lens (DVFL) derived from Devil's staircase function called the Cantor function [1].This function, related to the standard Cantor set, finds applications in areas like wave-particle interactions [2], crystal growth [3], and mode locking of the 3D coherent states in high-Q laser cavities [4].Devil's staircase also shows up in the context [5,6] of dynamical systems and periodic structures with long spatial periods and so forth.Many such examples along with their physical interpretation are given in [5][6][7][8].Devil's lens (DL), also called blazed fractal zone plate, has a phase distribution much similar to a ramped stair-step pattern and has a quadratic fractal blazed profile [9,10].Devil's vortex lens (DVL) is phase-only Devil's lens modulated by a helical structure [11][12][13][14].When a spiral phase plate is introduced, the DVL generates multiple axial vortices.When a converging Fresnel lens is incorporated in the DVL, it results in a DVFL [15].Barrera et al. [16] have presented results on multiple image encryptions and multiplexing based on fractal encrypting masks, using joint transform correlator architecture.In the present study, we have used DVFL as a structured phase filter in the frequency domain of a fractional Mellin transform-based encryption system to attain enhanced security.
A number of image encryption studies carried out in the past are based on the double random phase encoding (DRPE) technique.However, DRPE using the Fourier transform (FT) suffers from the problems of security and its vulnerability to a variety of attacks [17].This has motivated researchers to design more secure systems by introducing additional parameters through phase filters and the use of a variety of integral transforms such as the fractional Fourier transform [18,19], gyrator transform [20,21], the Hartley transform [22,23], and the Arnold transform [24].It is now well known that optical and hybrid techniques for information security [25][26][27] have several advantages over the digital techniques.
The encryption scheme proposed in our paper is asymmetric and is based on the FrMT [28][29][30].Compared to most transform-based techniques which are linear in nature, the fractional Mellin transform (FrMT) is a nonlinear transform and could potentially provide security against most known attacks [17,31].Here, we have used FrMT on an arbitrary input image transformed to an annular domain.The transformed image is multiplied by a structured phase mask (DVFL) in the frequency domain, aimed at enhancing the   security by increasing the key space.This is followed by phase retrieval algorithm (PRA) [32,33] to generate the phase key for decryption.The paper is organised as follows.The construction of phase function based on DVFL is explained in Section 2. The details of the encryption-decryption processes and PRA are given in Section 3. In Section 4, we have presented the results of numerical simulation followed by conclusions in Section 5.

DVFL-Based Phase Function
A phase mask based on Devil's lens can be described by the one-dimensional Cantor function, a particular case of Devil's staircase.A triadic Cantor set in the interval [0, 1] can be defined as [9] where   (0) = 0,   (1) = 1,  is the order of the Cantor function, and  defines the number of horizontal sections of the function, having a value from 0 to 2  − 1.Here  and  are the start and end points of each segment of the Cantor set.For some basic values of , the values of  , and  ,+1 are provided in Table 1 [15].From (1), one can easily note that the function   () increases linearly from  , to  ,+1 .The phase function corresponding to Devil's lens is defined as where  = (/) 2 is the normalized quadratic radial coordinate and  is the lens radius.The phase function () increases linearly from 0 to 2.A DVL can be constructed by combining Devil's lens and a vortex lens (VL).A vortex lens is based on an azimuthally dependent vortex function, and its phase function can be given as   () = exp(), where  is a nonzero integer called topological charge and  is the azimuthal angle.Incidentally, the concept of topological charge has given rise to an important branch of optics called singular optics which plays an important role in various contexts (e.g., [34,35]).The phase function corresponding to DVL can be given as In the same manner, we can combine a Fresnel zone plate with DVL.The phase function of a Fresnel zone plate is defined as [15] where  0 is focal length and  is the incident wavelength.A combination of the Fresnel lens (FL) and DVL results in the formation of DVFL, having a phase function given by [15] A plot of DVFL, which is a combination of Devil's lens, a vortex lens, and a Fresnel lens, is shown in Figure 1.

The Encryption Scheme
The scheme proposed in our paper is based on the FrMT, a nonlinear integral transform [28][29][30].According to this, an arbitrary image is first transformed to log-polar coordinates and is then subjected to the fractional Fourier transform (FrFT), which is a generalization of the Fourier transform in fractional order [18] and provides additional degree of freedom for encryption.The FrMT is inspired by the FrFT.A two-dimensional FrMT of order ( 1 ,  2 ) is the FrFT of the same order ( 1,  2 ) of a function in its log-polar transformation.
In a Cartesian coordinate system, the two-dimensional FrMT of order ( 1 ,  2 ) of an image (, ) is given by [28] where When the image is transformed to an annular domain by log-polar transformation, its FrMT of order ( 1 ,  2 ) can be written [28] as where  is a constant.Since FrMT involves log-polar transformation of the input image prior to its transformation by FrFT, it requires the setting up of parameters for transforming the input image to an annular domain.Hence, a few parameters are set in advance such as centre position of annular domain,   ,   , the radii of the innermost ( in ) and outermost ( out ) rings of annular domain, and the number of sampling points along distance axis   and along angle axis   .Some recent image encryption studies have preferred structured phase masks over the commonly used random phase masks (RPM) because of some advantages.Unlike in the conventional DRPE scheme, here we have used a structured phase mask based on Devil's vortex Fresnel lens (DVFL) in the frequency plane [15] by multiplying the FrMT transformed image with phase function based on DVFL.This is followed by an iterative phase retrieval algorithm (PRA).
In the iterative process, the initial amplitude is taken from the FrMT transformed image and the initial phase is randomly generated with the uniform distribution in [0, 2] as in a recent study [30].To initiate PRA, we take FrFT of order  of the initial input.The amplitude of the resulting complex function is replaced by a secret (target) amplitude image.Next, we perform the inverse FrFT to obtain complex distribution in the image plane.The phase of the transformed result substitutes the phase of input in the first step and the next iteration starts.The above steps are repeated until the algorithm converges or reaches a preset number of iterations.
We can summarize the steps of encryption scheme as follows.
(i) Start with an arbitrary input image for generating ciphertext.
(ii) Transform the arbitrary image to log-polar coordinates and then subject it to a fractional Fourier transformation, resulting in a fractional Mellin transformed image.
(iii) Multiply the transformed image by a structured phase mask based on (DVFL).
(iv) Apply the PRA based on FrFT.
Flow chart of the encryption scheme including the PRA is shown in Figure 2. In our scheme, we consider an arbitrary real valued image (, ).The selected annular domain image of (, ) is   (, ) with inner and outer radii, respectively, as  in and  out .The image in the annular domain is then subjected to FrFT of order  (taken same in both directions, i.e.,  1 =  2 = ).This transformed image is multiplied by a DVFL-based structured phase function.Thus, the transformed result (, V), a complex-valued image, can be expressed as where  1 ,  2 , and  3 are the phases of DL, VL, and FL, respectively, and   (, V) and  0 are, respectively, the amplitude and phase of the FrMT transformed image.The next step of the encryption process is the use of an iterative PRA to determine the phase key corresponding to the secret image [30].The input amplitude of the complexvalued image for PRA can be written as The next step of PRA can be expressed mathematically as Initially (, V) is taken as uniform random distribution in the interval [0, 2] and subsequently it is generated from the PRA iteratively.  on the right hand side of (10) represents the FrFT of order  (taken same in both directions, i.e.,  1 =  2 = ).Here, (, ) is the phase distribution.The final step of the PRA can be written by taking the inverse FrFT as

International Journal of Optics
where  4 (, V) is the phase distribution obtained by iterative method and used in the first step.The phase key can be made complicated by modulating the phase functions, thereby enhancing the security of the system.The amplitude   (, V) of the transformed image is used to generate a phase distribution given by where  is a positive number equal to 1 + max(  ).
The final phase key (, V) is obtained from the following expression: The keys of the proposed scheme include the order  of FrFT in the iterative process, the final phase key (, V), and the order  of the FrMT along with parameters of the structured phase mask, DVFL.
Decryption Process.Generally, the decryption process is reverse of the encryption.The transformed image (, V) is multiplied by the final phase key obtained from (13).
The resulting expression undergoes the fractional Fourier transformation of the order  to give complex distribution of the decrypted image.The amplitude distribution of the decrypted image is given by

Numerical Simulations and Discussion
The validity of the proposed scheme has been verified by performing some numerical simulations using MATLAB 7.6.An arbitrary input image Lena is chosen, which has 256 × 256 pixels and gray level is in the range of 0-255 (Figure 3(a)).
To start the encryption process, some parameters have to be set since a selected annular domain of the ciphertext is transformed by FrMT.The FrFT in the scheme has been computed using an algorithm proposed by Garcia et al. [36].
For simplicity, the FrMT orders in both the directions are assumed to be the same and equal to 0.3.The center position of the annular domain is taken as (  ,   ) = (96, 108), with the inner and outer radii being, respectively,  in = 20,  out = 80 [30].The annular domain with these parameters has been shown in Figure 3(b), whereas the FrMT transformed amplitude image   (, V) is shown in Figure 3(c).The orders of the FrFT and IFrFT in the PRA are all taken equal to 0.7.
The iterative process ends when the correlation coefficient (CC) value exceeds the predefined value 0.996.Through PRA, we are able to obtain the corresponding decryption phase key which is the main key in the proposed scheme.When even one incorrect parameter of DVFL is used, one can hardly retrieve the original image with other correct parameters.
Using this scheme we have carried out numerical simulations using Lena as the arbitrary image and a secret image for encryption.As shown in Figure 4(a), the secret image chosen to be encrypted is a grayscale image of Barbara having a size of 256 × 256 pixels.The corresponding phase key generated from the PRA has been shown in Figure 4(b), and the decrypted image using the correct keys, that is, ( = 3,  = 632.8nm,  = 3,  = 4 cm,  = 0.3,  = 0.7), is shown in Figure 4(c).Figures 5(a  In order to assess the efficiency and security of the proposed scheme, the mean-squared-error (MSE) between the original image and the decrypted image has been calculated.If   (, ) and   (, ) denote, respectively, the pixel values of the original image and the decrypted image, the MSE function can be written as Besides the decryption phase key obtained from the PRA, the proposed scheme also depends on some other factors such as the DVFL parameters, as well as orders of the FrMT and FrFT.In order to study the sensitivity of the encryption scheme to orders of the FrMT and FrFT, the MSE values have been plotted against the orders of the FrMT and FrFT in Figures 6(a) and 6(b), respectively.These results imply that the decryption of the original image is very sensitive to the orders of FrMT and FrFT.Based on the above numerical   International Journal of Optics simulations, one can see that all the correct keys are required in the decryption process.If any key of encryption process is incorrect, it becomes increasingly difficult to retrieve the secret image.The robustness of the encryption scheme is tested against occlusion attacks.As the decryption key contains phase key generated by PRA, it has the phase information of the secret image.Thus, in this scheme, the phase key plays a vital role as it bears the signature of the secret image.We have subjected the decryption phase key to occlusion attacks as shown in Figure 7.The corresponding retrieved images shown alongside indicate reasonable robustness of the scheme up to 50% occlusion.It can be concluded that apparently the main information of the secret image can be recognized visually from the decrypted images.

Conclusions
An encryption scheme for grayscale image has been proposed, which uses a nonlinear fractional Mellin transform, a structured phase mask (Devil's vortex Fresnel lens) in the frequency plane, employing a phase retrieval technique to generate the decryption phase key corresponding to each secret input image.The use of FrMT with DVFL phase mask enlarges the key space, thereby making the proposed scheme more secure.The proposed scheme has been validated for a few grayscale images using MATLAB 7.6.The sensitivity of the scheme has also been studied for various parameters such as phase keys and orders of FrMT and FrFT.The scheme is considered secure against most of the known attacks by virtue of its design being nonlinear and asymmetric and having a larger key space.However, with the rapid increase in computational capability and emergence of new techniques, one can expect to break security of the system in the future.In this context, it may be worthwhile to carry out a cryptanalysis of the proposed scheme in the future with a view to investigating the endurance of the scheme against various other attacks.
)-5(d) show the decrypted images with only one incorrect parameter (with all other parameters being correct).It is seen from Figure5(d) that the decryption is not very sensitive to a change in the parameter .

Figure 5 (
e) shows the decrypted image with two incorrect parameters  and .The decrypted image is unrecognizable (except in Figure5(d)), thus highlighting the scheme's sensitivity to the encryption parameters.