A novel encryption scheme for high-contrast image data in the Fresnelet domain

In this paper, a unique and more distinctive encryption algorithm is proposed. This is based on the complexity of highly nonlinear S box in Flesnelet domain. The nonlinear pattern is transformed further to enhance the confusion in the dummy data using Fresnelet technique. The security level of the encrypted image boosts using the algebra of Galois field in Fresnelet domain. At first level, the Fresnelet transform is used to propagate the given information with desired wavelength at specified distance. It decomposes given secret data into four complex subbands. These complex sub-bands are separated into two components of real subband data and imaginary subband data. At second level, the net subband data, produced at the first level, is deteriorated to non-linear diffused pattern using the unique S-box defined on the Galois field F28. In the diffusion process, the permuted image is substituted via dynamic algebraic S-box substitution. We prove through various analysis techniques that the proposed scheme enhances the cipher security level, extensively.


Introduction
Data encryption is an integral part of multimedia technologies. Usually, internet users require to deliver, obtain, or store secret information [1]. Extensive information exchange over internet demands safety from unwanted surveillance, thieving, and bogus publicity [2]. In this frame of reference, images are exposed to ciphers which are unrecognizable by the human eye [3]. The highest familiar way to escape from any trouble is to convert the confidential data into dummy formation [4]. The information appeared can only be restored by those who know the reverse scheme used to attain the initial pattern with exclusive key specification [5][6][7][8][9][10][11][12][13]. Data encryption enables data integrity, confidentiality, and data authentication [14]. This is achieved using a digital data encryption which transforms the meaningful information into dummy data. This strategy has been evolved to preserve the information and is heavily based on cryptography [4].
Cryptography is used for data confidentiality. Claud Shannon [15], introduced the substitution-permutation network (SPN), each layer of which uses substitution followed by permutation. Cryptographic procedures are helpful to preserve the secret information by scrambling it a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 into an uncorrelated data [4]. Improved security and confidentiality can be served if the two schemes, confusion and diffusion, are joined together into one hybrid system [16][17][18]. The gradual development of cryptographic systems witnesses the worth of the substitution box (S − box). It plays the role of a standout in symmetric key cryptography and a predominant source to produce nonlinearity [1,19]. The indispensable involvement of the S-box demands new construction algorithms and the recent literature introduces some safer and more reliable Sboxes [1,2,[20][21][22][23][24]. Further applications of S boxes in digital image ciphering, water marking as well as in steganography have become quite popular as well [3,[25][26][27][28][29][30]. In this work, we propose an efficient strategy that uses the algebraic properties of the Galois field to structure an Sbox which exhibits extra-ordinary features as compared to some prevailing designs at a very low computational labour.
In recent literature, many innovative and advanced techniques of image encryption have become pivot of attention. Liu and Wang introduced image encryption algorithms based on one-time keys [31] and spatial bit-level permutations along with chaotic systems [32]. In [33], a chaotic encryption scheme based on perceptron model is proposed. [34] describes an encryption strategy using DNA complementary rule with chaos. Some of the most recent image encryption techniques are detailed in [35][36][37][38][39][40][41][42]. However, according to our learning there is no previous work on the applications of the Fresnelet transform in conjunction with Sbox cryptography. In this paper, we use the cryptography to manipulate the significant information of the secret data.
Later part of this paper is ordered as follows. Section Fresnelet Transform provides a theoretical explanation of the Fresnelet transform. Section Algorithm for algebraic S-box describes the detailed algorithm used to develop an algebraic S-box. The cryptographic forte of the Sbox is examined in Performance analysis of S-box, through some highly significant parameters. The remaining study present a comprehensive model for encryption and decryption, including simulations and evaluations. Lastly, Conclusion Section concludes the paper.

Fresnelet transform
The propagation phenomena of wave are structured via diffraction principle using the Fresnel transform [43]. The Fresnel transform in wavelet domain produces the bases of Fresnelet transform. These bases are used to reconstruct the digital off axis hologram with certain composition of parameters. The parameters can be tuned to desired value of the resolution scale, adjusted with particular wavelength and the specific distance between observing plane to the propagating objects. Fresnelet transform is demonstrated to simulate the approximation the monochromatic waves propagation. Propagation of monochromatic waves is shown as a function L 2 O 2 ðRÞ, with the Fresnel transform model in terms of convolution integral as follows: The kernel k τ (p) is the one-dimensional prorogation model with given parameter τ > 0 in (2), where, τ depends on the propagating distance d, based on the wavelength value λ as shown: A novel encryption scheme for high-contrast image data Moreover, the kernel k τ (p) can be extended to two-dimensional wave propagation using the tensor product for L 2 O 2 ðR 2 Þ, as follows: The K τ (p, q) is also known as the separable kernel to cover the two-dimensional Fresnel transform's family [14]. To obtain the precise reconstruction of decomposed data, the Fresnel transform unitary property facilitates, prominently. Furthermore, the extension of the separable nature of one-dimensional wavelet transform into two-dimensional wavelet transform can be achieved using the Riesz basis [43]. The Riesz basis for O 2 ðRÞ of a two-parameter fc j;l g j; l2Z can be defined on O 2 ðRÞ as family of the wavelet transform can be defined in terms of convolution integrals, such that where, the Haar wavelet is used as an orthonormal basis generation for O 2 ðRÞ. This is the simplest form of a multi-resolution decomposition by the wavelet transform, and can be used to get the desired reconstruction of the input original data as well [44]. This composition leads to the Fresnelet basis using the the Haar wavelet in combination of the Fresnel transform as follows: For constant value of τ, an orthonormal basis in terms of Fresnelet transform can be obtained using: b j;l ðpÞ ¼ ðc j;l Þ $ t ðpÞ, as follows: The coefficients representation using Fresnelet is done by ν j, l in (9). Moreover, the separable nature is extended the Fresnelet transform's from one-dimensional data to two-dimensional data. Following this, the four combinations are obtained using the tensor product m ðllÞ t ; m ðlhÞ t ; m ðhlÞ t ; and m ðhhÞ t ; for producing the approximation (lower-lower subband) and three high frequency details (lower-high subband, high-lower subband, and high-high subband) as follow: In (10)- (13), the representation of scaling functions is done using x and representation of the wavelet functions is done using y. These functions are established a low-low filter in (10) and high details filter in (11)- (13). Following this, the four Fresnelet coefficients are generated using the basis function μ to data Λ, as follows: The low-low filter data is demonstrated using the coefficient measure L ðllÞ t;d and the high-details data are represented using the coefficients measure L ðlhÞ t;d , L ðhlÞ t;d , and L ðhhÞ t;d , respectively. Theses Fresnelet coefficients are used to propagate the input image data. It transforms the input image data from meaningful information to dummy image using the Fresnelet forward transform in the form four complex subbands data [14].
The Fresnelet transform unitary property is used in reconstruction of secret data by applying the conjugate transpose to encrypted data as shown in The communication of encrypted data in digital form is distributed into real parts only. Therefore, the complex data is partition into two parts: real part a and magnitude of imaginary part b (in second and third image of Fig 2). Moreover, on reconstruction stage the imaginary part b (magnitude value) with i is added up into a (real value) to obtain the complex data using inverse process.
To transfer the given data with huge isolation and upgraded discreetness, the presented method implies the Fresnelet transform that holds distance parameters and the wavelength as keys, that are necessary for reconstruction of the correct information. Furthermore, the initial pattern of the information data is obtained in the reconstruction stage with the accurate parameters use during the inverse Fresnelet procedures.

Algorithm for algebraic S-box
In this section, we describe the algorithm used for construction of our S-box. Generally, Sbox is considered as the most influential component to produce the nonlinearity effect that relates to the confusion creating capability in ciphering. In order to grasp the structural properties of an S-box, we first need to go through some fundamental facts.
An n × n S-box can be defined as a vector Boolean function F : F n 2 ! F n 2 . The algebraic structure of the background Galois field used to construct an S-box, plays very basic role. For the underlying construction of 8 × 8 S-box, we use It is worth mentioning that in advanced encryption standard, AES S-box algorithm uses F 8 2 based on X 8 + X 4 + X 3 + X + 1. In fact, any degree 8 irreducible generating polynomial could be used for the field F 8 2 but this choice obviously affects the calculations. We define the general linear group GLðn; FÞ as a group formed by all n × n invertible matrices over the field F. A projective general linear group of degree n over a field F is defined to be the quotient of GLðn; FÞ by its center. For this paper, we form the 8 × 8 S-box by considering the action of the aforementioned Galois field F 8 2 on PGLð2; F 8 2 Þ, i.e. we take a function f : In above expression, f is known as a linear fractional transformation (LFT) with a 1 , a 2 , a 3 and a 4 2 F 8 2 satisfying the condition a 1 a 4 − a 2 a 3 6 ¼ 0. The algebraic complexity and nonlinearity of LFT gives incentive to deploy this for the process of byte substitution. For our S-box, in  Table 1. This highly nonlinear S-box attains the nonlinearity measure 112 that is quite similar to the recent state-of-the-art AES S-box but it utilizes a straightforward and comparatively simple approach as compared to the AES S-box. In the following section we use some highly significant analysis techniques to figure out the cryptographic strength of our S-box.

Performance analysis of S-box
In this section, we analyze the proposed S-box through some widely accepted parameters. The detailed analysis is presented in the following subsections. We compare the results with the famous S-boxes, as named earlier.

Strict avalanche criterion
This criterion is used to gauge the confusion creating capability of an S-box. A function F : F n 2 ! F n 2 would be regarded to satisfy SAC if a single input-bit change assures change in 50% output-bits. The results presented in Table 2, shows that our S-box fulfils the requirements of SAC.

Linear and differential approximation probabilities
This is a measure of the unevenness of an event, mathematically defined by: where x represents all possible inputs to the S-box and Γ x and Γ y give the parity of the input and output bits respectively. We further use the differential approximation probability, which determines the differential uniformity of an S-box. Its mathematical expression is given by; Here Δx and Δy represent the input and out put differentials respectively. The smaller LP and DP measures guarantee the stronger S-box.

Bit independence criterion
In bit independence criterion, input bits are altered exclusively, and then output bits are scrutinized for their independence. Bit independence has great worth in cryptographic structures. The goal of reaching the maximum complexity and perplexity in a system can be achieved through this property of increasing independence between the bits. In cryptographic systems, the increased independence between bits is an essential requirement as it makes harder to understand and forecast the design of the system. The results of BIC are compared in Table 2. It is evident that, in BIC, our S-box has similarity with the Xyi S-box.
Based on the performance evaluation of the newly designed S-box, it is evident that the Sbox shows extra-ordinary results and can be considered for further multimedia applications.

Encryption method
The secret image is encrypted to keep the confidential information, the Fresnelet transform is employed based on the wavelet family, known as Haar. At first, the secret image Λ is propagated using the Fresnelet transform Δ τ at the distance d 1 = 1m, as follow: In the second phase, a scrambled data ψ is generated from the components of decomposed image Λ using the IΔ τ (inverse Fresnelet transform) at a distance, d 2 = 10 −2 m, as follow: : The encrypted data from the secrete image are obtained in Fig 2 in form of complex properties of the Fresnelet transform. Moreover, complex data is separated into the imaginary part ψ im and the real part ψ re for getting unique pattern of non-linear pattern using S-box.

Single unit dummy data
To manage the data the encrypted dummy data, we employ the inverse wavelet transform (WT) for reconstructing the two parts (real and imaginary) into single unit dummy data. The inverse wavelet with the Haar transform is employed.
The subband data C 0 is the low-passed and the diagonal detail zeros valued data. The maximum intensity values of image M 2 R 2 of horizontal detail and the vertical detail are used α 1 . D re data and α 2 .D im to derive the α 1 and α 2 as encryption normalization values. A scale parameter α is presented as a strength factor which regulates the participation of scrambled dummy data for getting single unit dummy data as follow: The above reconstruction process with the inverse wavelet transform (IWT) gives an information dummy data E after uniting the real and imaginary parts of D in reconstruction phase.

S-Box encryption
A typical 512 × 512 image of Signature, 256 × 256 image of DDNT, and 256 × 256 image of USAF are encrypted using the proposed method. We use the inverse of S-box to decrypt the ciphered image. Fig 3(A), 3(B) and 3(C) show the true, encrypted and the decrypted image of Signature. Figs 4 and 5 show the encrypted and the decrypted image of DDNT and USAF information data using S-box and the proposed method. One can see that our algorithm is capable to recover the true secret image with high accuracy as measured out in terms of coefficient correlation.

The decryption process
The opposite of an encryption process is decryption process. The wavelet transform (WT) is considered to decompose the encrypted data image E into four subband data sets E ðllÞ jÀ 1 , E ðhlÞ jÀ 1 ,  The final data are split by α so that the scrambled information data of same values are obtained. The retrieved imaginary and real parts of the scrambled data are rejoined in complex data pattern. Lastly the obtained complex scrambled data is refined by the inverse Fresnelet transforms applying the exact keys specification of the Fresnelet transform for the sake of obtaining the private information image on employing the inverse cipher S-box.

Statistical analysis
In this section we gauge the security strength of the proposed algorithm by using some most significant statistical analysis, s described below.
• Entropy: Entropy analysis measures the randomness of a system. Table 3, represents the information entropy for different images (both original and the encrypted). It is obvious that the results favor the proposed methodology.  • Contrast: The contrast analysis is used to identify objects in an image. The numerical results for contrast of t images true and encrypted images are arranged in Table 3, which show that the proposed scheme is efficient.
• Correlation: The correlation coefficient, between the pixels at the same indices in both the plain and the encrypted image, measures the similarity between the pixels pattern of both images. Results arranged in Table 3 witness effectiveness of the proposed method.
• Homogeneity: The homogeneity analysis determines the closeness of the elements distribution in the gray level co-occurrence matrix (GLCM) to GLCM diagonal.
• Differential analysis: A desirable feature of an encryption algorithm is to show high sensitivity to single-bit change in the plain image. For this purpose two measures, NPCR and UACI, are commonly used. NPCR stands for the number of pixels change rate of image as a result of one pixel change in the plain image. However UACI means unified average intensity of differences between the plain and the encrypted images. In our case, the NPCR is over 99% and the UACI is over 33%. These results prove that our algorithm is highly sensitive and robust against the differential attacks and even single-bit difference in two plain image results in absolutely different encrypted images.

Simulation and evaluation
In the given process, we deal with a sampling interval size Δ = 10 nm of a hypothetical CCD plane, a wavelength λ = 632.8 nm (nm = nanometer), and distances d 1 = 1m (m = meter) and d 2 = 10 −4 m [14]. These parameters are employed in the Fresnelet transform operations for the encryption as well as decryption stages and are considered as the key parameters. The obtained information images are predicted through the evaluation of correlation coefficients (CC) with the original information image data. The index of CC followed with in the range of 0 to 1. 0 value show almost no correlation between original and decrypted image while 1 reflect almost same out compare to original image. To analyze the presented algorithm, our main focus on encryption analysis. The decrypted out put is almost clear with average value in terms of coefficient correlation value (0.9995). Notice from Table 3 that the given scheme provides good encryption of secret information data as compared to ordinary S-box encryption methods [2].

Processing speed
It is an important feature of an efficient encryption algorithm to elapse lesser processing time. We tested our algorithm on MATLAB R-2016a, with i5 − 2520M CPU 2.50 GHZ and 8GB memory and the results are presented in Table 4. it is evident that our scheme is quite efficient.

Classical attacks
By Kerckhoff's principle, a sscure cryptosystem is one which can't be broken even if everything is known, except for the secret key. Keeping this fact in view, classically, there are four main attacks [36], that are worth-studying.
• Ciphertext only attack: When a string of ciphertext is known to the attacker.
• Known Plaintext attack: When a string of plaintext along with the corresponding ciphertext is known to the attacker.
• Chosen plaintext attack: When attacker has temporary access to the encryption mechanism such that he can choose a plaintext string and obtain its ciphered version.
• Chosen ciphertext attack: When attacker has temporary access to the decryption mechanism such that he can choose a ciphertext string and can obtain its corresponding plaintext string. Among the above-mentioned attacks, Chosen ciphertext attack is of highest importance. If a cryptosystem can resist this attack, it would surely resist the other attacks as well [36,40].
Obviously the proposed scheme is highly sensitive to the subband-distribution, the irreducible monic polynomial used to construct the background Galois field and the S-box parameters a 1 , a 2 , a 3 and a 4 . Minor change in these will cause serious effects on the outcome and therefore our scheme is quite safe against the aforementioned attacks.

Conclusion
To overcome the drawback of general S-box encryption in highly contrast data, this paper presents a new encryption approach with improved cipher security of the encrypted data. The proposed method is considered better approach for two significant reasons. First, the Fresnelet transform is used for encrypting the data of information image with various distance parameters as key. Second, the algebraic s-box provide highly nonlinearity in an encrypted data. Experimental studies validate that an encryption of digital image using the proposed scheme shown great superiority over the recently existing encryption methods. Although the proposed method in this study intents at the image encryption, however, its limitation does not restrict it to only this area and can be significantly implied in various fields of security information.

Supporting information
Data Availability Statement: Signature image is taken from flicker image database: https:// www.flickr.com/photos/wyncliffe/14353455608/in/photostream/. The person who associated a work with this deed has dedicated the work to the public domain by waiving all of his or her rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission. Rest of the data used in the proposed method are available within the paper.