Security augmentation grounded on Fresnel and Arnold transforms using hybrid chaotic structured phase mask

Correspondence Department of Applied Science, The NorthCap University, Sector 23-A, Gurugram, India Email: hukumsingh@ncuindia.edu Abstract A novel asymmetric double image encryption technique is introduced to protect the algorithm from the interlopers by using hybrid chaotic structured phase masks in Arnold and Fresnel transforms. Mostly random phase masks are used routinely in many algorithms which are not strong against many attacks. So, taking in consideration replacing of random phase masks with the new hybrid chaotic structured phase masks takes place in the algorithm. With the assistance of Fresnel zone plates, radial Hilbert mask and logistic map functions, the chaotic structured phase mask is accumulated and hybrid chaotic structured phase mask provides extra security to system. Arnold transform helps in pixel shuffling, and with the pixel reorganization, it also helps in image reaping and bordering. The propagation distance of the Fresnel transform and the constraints of the Arnold transform serve as keys for encryption and decryption. To check the sturdiness of hybrid chaotic structured phase mask, simulations have been done which are carried out by the support of MATLAB R2018a (9.4.0.813654).


INTRODUCTION
Optical technologies have become increasingly important for securing information and recognition. It has been widely explored to encrypt sensitive information because of their highspeed operation, parallel processing and multiple dimensional capabilities. The optical techniques possess many degrees of freedom such as amplitude, phase, polarization, large bandwidth, nonlinear transformations and multiplexing. All these can be combined in many ways to make information encryption more secure and more difficult to attack. Optical encryption techniques have played a vital role in the field of optical information processing over the past decade [1][2][3][4][5]. Different and reliable security in the transmission and storage of images is needed in several applications like medical or biometric images. Refregier and Javidi proposed an optical encryption method in Fourier domain, in which the encrypted image is obtained by double random phase encoding (DRPE) [6]. Many others could also be regarded as its variations which are based on Fourier transform (FT) and its different generalizations, such as fractional Fourier transform (FrFT) which was first introduced by Unnikrishnan et al. [7] in which a primary image is encoded to After that, the FrFT [8][9][10][11] based image encryption becomes a research focus in the community of image encryption. DRPEbased propagation distance and wavelength was first introduced by Situ and Zhang [12] in Fresnel transform (FrT) domain and, later on, it was extended as, by many other researchers [13][14][15][16][17][18][19], gyrator transform [20,21], fractional Mellin transform [22,23], fractional Hartly transform [24,25] and Arnold transform (ArT) [26,27]. These transforms are symmetric in nature, and symmetric cryptosystems are vulnerable and very sensitive to many attacks such as chosen plain image attacks, chosen cipher image attacks and known plain image attacks.

1.1
Main contributions 1. In the present work, the phase truncated part and phase reservation (PR) part are used as public and private keys for encryption. The proper usage of asymmetric cryptosystem in security applications demands the system to be robust to probable loss during both encryption and decryption. The proposed work uses asymmetric cryptosystem, i.e. different keys in encryption and decryption make the system stronger. 2. Arnold cat map is efficiently used in this procedure in which encoding of an image involves two procedures such as permutation and diffusion [35]. ArT is used along with FrT which helps in scrambling the input image, hence helping to add more security while encrypting and decrypting of images. 3. FrT is a lensless transform which has extra security parameters like propagation distances and wavelength, hence, increasing the security of system by means of extra parameters as compared to Fourier and fractional Fourier transforms. Structured phase mask (SPM) helps in performing image encoding [36][37][38][39][40] and the usage of SPM provides an additional benefit of having more keys at the time of enciphering an image which helps in increasing the security of the system. It consists of Fresnel zone plate (FZP) and radial Hilbert mask (RHM). Therefore, we proposed an asymmetric cryptosystem based on hybrid chaotic structured phase masks (HCSPMs) and it is attained through Fresnel and ArT. 4. The logistic map [41], FZPs [42] and RHM [43,44] are combined to make chaotic structured phase mask (CSPM) [45,46], and by taking hybrid of CSPM made the system much secure.

Related works
1. Fridrich proposed a chaos-based image cryptosystem composed of permutation and diffusion, in 1998, which has a high security since the transformed logistic map is adopted for initial permutation and nonlinear diffusion. Low-dimensional chaotic systems are always good to select for manipulating any encryption procedure, but on the another side, it also has some drawbacks such as simple dynamic characteristic, simple cipher code and less determined sequence parameters which may be the reason of serious security problems. 2. So, to overcome from low-dimensional chaotic systems, HCSPMs are introduced in the proposed work, which improves the strength of the scheme. 3. As compared to Ref. [46], the suggested scheme uses HCSPM in place of CSPM and asymmetric cryptosystem instead of symmetric cryptosystem. Usage of secondary images in HCSPM makes it more robust as compared with CSPM. Asymmetric cryptosystem uses different keys at the time of encrypting and decrypting of an image which makes the system robust as compared to symmetric cryptosystem which is vulnerable against many attacks. 4. Kumar et al. [47] proposed hybrid mask which are generated with the help of RPMs, whereas in our paper, hybrid masks are generated with the aid of CSPM. Hence, it increases the key length as compared to RPM. HCSPMs have more parameters than hybrid mask as it has initial value, bifurcation parameter, wavelength, focal length and radius.

Summary
Introducing HCSPMs in the cryptosystem, this novel mask helps in enlarging the key space and hence protects it from the hackers. HCSPMs are used first time in ArT and FrTs. Here, original image is converted into encrypted image with the support of haphazardness of HCSPMs. From literature point of view, HCSPM is not taken in consideration till now. Pixel scrambling technique, i.e. ArT, gives randomness in the image and with the help of HCSPMs makes the proposed system unique and new. In our proposed scheme, we have used HCSPM in both input and in frequency plane, which helps to achieve better performance as it increases the key domain. This scheme not only overcomes the problem of key space in an optical setup but also enhances the security. This paper is organized as follows. Theoretical background is discussed in Section 2. The proposed method and results are described in Sections 3 and 4. Section 5 presents the statistical study and, finally, the conclusions are made in Section 6.

THEORETICAL BACKGROUND
In this section, we briefly explain mathematical formulation of canonical transforms, namely ArT and FrT. In the present scheme, HCSPMs are used as phase masks.

Arnold transform (ArT)
The ArT, also known as 'cat mapping', was introduced by Arnold in the study of ergodic theory. This transform involves a process of clipping and splicing, which realigns the pixel matrix of a digital image. The ArT is unitary and energy saver. It aids in refining the security of the encrypted image and to eliminate the random phase masks (RPMs) of original DRPE. Given an N × N image I (x, y), the ArT of a pixel (x, y) in the image is calculated by Equation (1), where x and y are the coordinates of the pixel before applications of the ArT, and after that, the new position is denoted by the x 1 and y 1 coordinates. Using Equation (1), the ArT of an image I (x, y) can be expressed by In Equation (2), s denotes the pixel value of the original image I (x, y), and (x 1 , y 1 ) T is the transpose of (x 1 ,y 1 ) and, finally, the term on the right-hand side of '|' indicates the algorithm conditions. The original image will reappear after some iterations

Fresnel transform (FrT)
The FrT is a lensless transform and it has propagation distance s and wavelength λ which can be written in Equation (3) And h ,s denotes the kernel and is given by the expression as follows: where I (x, y) is the input image and Fr T ,s denotes the FrT parameters, where λ is the wavelength and s is the propagation distance. FrT has been used for adding more security with the idea of utilizing Propagation distance. The optical structure for the algorithm is depicted in Figure 2. There are three planes in the system, i.e. the input, the transform and the output planes. The first diffuser HCSPM 1 (x, y) is situated in the input plane. The second diffuser HCSPM 2 (x, y)is positioned in the transform plane. The ciphered image C(x, y) is attained in the output plane. The wave field I(x, y) × HCSPM 1 (x, y) propagates distance s 1 and is thus Fresnel transformed. The resultant signal is then multiplied by the second diffuser HCSPM 2 (u, v). The image field now propagates a distance s 2 being once again Fresnel transformed to form the ciphered image C(x, y).

Chaotic spiral phase mask (CSPM)
At the time of image encoding, instead of using RPMs, we have established CSPMs. Chaotic map produces vast number of random iterative values. These are non-correlation, pseudorandomness and ergodicity. The logistic map is termed as one-dimensional (1D) non-linear chaos function and is defined as where r denotes the bifurcation parameter, i.e. 0<r<4. ; and x 0 is the original value [41]. Equation (5) comes into play when Equation (4) is iterated n times and it is a recursive function which requires r and x 0 as needed values. The sequence x n becomes chaotic if there is a minor variation in x 0 and r belongs to [3.5699456,4]. The chaotic random phase mask (CRPM) is given as where (x, y) denotes the coordinates of CRPM, which is achieved by one-dimensional CRPM, given by Equation (7).
where x i ∈ (0, 1) is transformed into two-dimensional matrix and regarded as Y and represented by Equation (8), CRPM consists of three private keys x 0 , r and n. Figure 3(a) shows the CRPM. It generates many hitches in aligning optical setup. To overcome this, SPM has been created, which is a combination of FZPs and RHM. The radial transform is beneficial in image processing as it controls upto what degree the enhancement of the chosen edges occurs. Using RHM makes an image edge-enhanced in contrast to input image. The radial Hilbert functions (P, ) are the log polar coordinates and are given by Equation (9), H (P, ) = e iP (9) where P = 8 infers the topological charge, i.e. the order of RHM, and is depicted in Figure 3(b). The complex part of Fresnel wave front is given in Equation (10) where f = 400 mm is the focal length, λ = 632.8 nm being the wavelength and pixel spacing = 0.023; here r is the radius and the FZP is shown in Figure 3(c).
The SPM produced by combination of Equations (9) and (10) is depicted in Figure 3(d) and the expression is given by Equation (11) SPM (P, , r ) = H (P, ) × U (r ) (11) Finally, the chaotic spiral phase mask (CSPM) is shown in Figure 3(e) and it is generated in Equation (12) by using Equations (6) and (11)   Here arg{⋅}, which is the argument operator, indicates phase operation. Precise use of all the constraints benefits in sustaining the safety of the scheme. Suggested cryptosystem is harmless and enough to overwhelm any kind of outbreaks.

Hybrid chaotic spiral phase masks (HCSPMs)
In the proposed algorithm, we have used CSPM which is a fusion of logistic map, FZP and RHM, and taking hybrid of it makes it stronger. To increase the security, we introduce HCSPM which helps in enlarging the key space. HCSPMs are used which enables to add more protection in the system. The HCSPM has several advantages over RPM as it provides extra key length and many parameters which hold the system security. The hybrid CSPM is the angle of Fourier transform of the secondary image P 1 (x, y) and P 2 (x, y) and is given by Equation (13) and (14).
where Arg is the argument; FT is Fourier transform; and P 1 and P 2 are the secondary images used at the time of making hybrid of CSPM. The HCSPM 1 is treated as primary mask and the HCSPM 2 as secondary mask which helps in enhancing of an image, and the private keys produced by them are used at the time of decoding of an image. The secondary images P 1 and P 2 that help in making HCSPM are shown in Figure 4

PROPOSED METHOD
This proposed scheme uses HCSPMs as the primary and secondary keys in encryption method. Practice of converting the input image into stationary white noise is done by encryption process. Let I(x, y)∕I 1 (x, y) be the original images of Lion/check that has to be encoded, and the steps of converting it into encrypted image are as follows.

Encryption
Step 1: ArT is applied on the original image and the product is multiplied with HCSPM 1 . In the input plane, the resulting complex image is subjected to FrT having propagation distances s 1 . Thus, two forms are carried out from it, one is absolute form (N 1 ) and the another is angular form (M 1 ). Hence, an intermediate image T(u,v) is obtained and is given by Equation (15).
Step 2: The intermediate image T(u,v) in the frequency plane is multiplied with the second hybrid CSPM, i.e. HCSPM 2 and the resultant bonded with FrT having propagation distance s 2 . Now two parts are again obtained: one is phase truncation (N 2 ) and another is PR (M 2 ). Hence, ciphered images C (x, y)/C 1 (x, y) are attained by Equation (16).
M 1 and M 2 are preserved as private keys which are used at the time of decoding the encoded image in decryption procedure.

Decryption
The decryption process is simply the inverse of the encryption process, and follow the same steps but in reverse order. The flowchart of decryption method is shown in Figure 5(b).
Step 1: The propagation distances and the two private keys that are generated at the time of encryption help in decryption. The ciphered images C(x, y)∕C 1 (x,y) are multiplied with private key M 2 , and after that, FrT with propagation distance s 2 is applied on it and the intermediate image T is obtained which is given by Equation (17), Step 2: Now, Equation (17) is multiplied by private key M 1 , and by applying FrT with propagation distance s 1 and inverse ArT, the original images I(x,y)/ I 1 (x,y) are recovered and represented by Equation (18).
Thus, the key space of the encryption and decryption procedure consists of the distances s 1 and s 2 , the wavelength λ and the two diffusers HCSPM 1 and HCSPM 2 . The HCSPMs offer the procedure of extra parameters which help in maintaining the security of the scheme.

RESULTS AND DISCUSSIONS
The recommended cryptosystem is supported by the MATLAB R2018a

Numerical analysis
The demonstration of the proposed procedure is checked by various analysis that has been carried out in terms of mean square error (MSE) and peak signal to noise ratio (PSNR). To check the quality of the decrypted image, MSE and PSNR methods are calculated between the original and decrypted images. MSE is calculated using the original I 0 (x, y) and the decrypted image I d (x, y). The MSE expression is given by Equation (19), The calculated value of MSE for lion and check images are 3.26 × 10 −23 and 5.92 × 10 −23 , respectively. Less value of MSE demonstrates good similarity with input image. MSE is also examined by fluctuating Fresnel distances as s 1 = The precise use of constraints will lead to accurate plotting which certifies about the robust arrangement of the scheme.
Degree of transparency of noise in the noised image is amounted by estimating PSNR and is evaluated between the input and the decoded image, and the PSNR expression is shown by Equation (20).
The PSNR values for the grayscale lion and binary check images are figured between original and decrypted images as 628.69 and 622.71 dB, respectively. High value of PSNR confirms less error in the decrypted image. Figure 9(a) and (b) demonstrates the plot of propagation distances (s 1 , s 2 ) with PSNR, respectively.
The calculated values of MSE and PSNR for the recommended scheme is described in Table 1 and these obtained values are equated with Ref. [46].

STATISTICAL STUDY
To investigate the potential and effectiveness of the suggested system, statistical analysis has been done.

Histogram analysis
Histogram is one of the statistical analyses among others and is carried out generally for cipher images. For image verification, histograms play a very vital part. In the encryption procedure, a plain image is changed into encrypted image.  (d) it is conclude that histograms of encrypted grayscale and binary images are similar to each other which ensures that the desired algorithm is robust against attacks and the attacker cannot be able to recover any information from this. Histograms of HCSPM 1 and HCSPM 2 are also shown in Figure  10(e) and (f), respectively.

Noise attack analysis
The strength of the projected effort has been verified by including Gaussian noise. The Gaussian noise consists of mean zero and unit standard deviation. Let C be the ciphered image which is free from noise and C ′ be the noise attained by ciphered image. Efficacy of the proposed algorithm has been carried [48] out by the noise equation and it is given by Equation (21). where K signifies the noise strength parameter. The quality of the encrypted images is checked in Figure 11(a)-(f) by doing noise attack analysis on grayscale and binary images with K = 0.2, 0.5 and 1.0, respectively. Figure 12 demonstrates the plot of both the images among MSE and the noise factor (K). The graphical representation of MSE and K illustrates that the scheme is strong because as the noise factor increases, the MSE also increases, which, in turn, affects the quality of image.

Occlusion attack analysis
Many inspections have been done only for encrypted images, particularly for occlusion analysis. Occlusion on encrypted images shows the robustness of the scheme. By occluding some part of encrypted image, the corresponding changes reflect the decrypted image. Figure 13 shows the occlusion analysis on  Figure 13(a)-(c) represents the 25%, 50% and 75% occluded area on ciphered image, respectively, and Finally, from the occlusion analysis, it is well proven that our cryptosystem is highly robust.

CONCLUSION
We proposed a novel asymmetric cryptosystem scheme for grayscale and binary images using HCSPMs in Fresnel and Arnold domain. Our method uses two HCSPMs (HCSPM 1 , HCSPM 2 ) which is consist of logistic map, radial Hilbert transform and FZPs. HCSPMs not only increase the key space but also augment the security of the system because of presence of too many parameters in it. As compared to previous work, it is a new and unique approach towards the safety. The proposed algorithm has been authenticated in the FrT domain. Numerical study has been done using various methods and to check the sturdiness of the proposed system. Statistical analysis validates the sustainability and competence of the system. Hence, the proposed technique is said to be a well alternate to traditional approaches and it offers tremendously good tactic for fortifying images.