An Efficient Public Key Cryptosystem Based on Dihedral Group and Quantum Spin States

The enciphering schemes based on medium transformations by following the strict guidelines are almost used everywhere. We have developed the structure to simulate the digital data with quantum spin states rather than following or creating the strict guidelines. We simulate the pixels of an image with the dihedral group and spin states for a defined phase to create confusion in it. The scope of this article is concerned in the development and deployment of public key cryptosystem, its performance and security analyses.


I. INTRODUCTION
The ability to process the digital contents competently and transfer securely is more dynamic, and has affected our veracity in the progressive growth of internet of things (IoTs). The computer or digital devices concerning crime, in which the computer can either be a tool, target or evidence, referred as cybercrime. As the information processing depends on information technology, the inhibition of activities and the control are dominant to the attainment of organization or individuals. To manage the computer frameworks from assaults or unauthorized access, cybersecurity bargain with the software or hardware equipment. The release of the intimate information, alteration or loss in digital contents may perhaps affected by the prohibited access [1]- [2]. Secure data transmission/reception across the communal complexes has incredible consequences and gradually imperative due to manipulation and larceny in digital contents. In everyday existence and at economic forums necessitating the digital contents security structure to enhance their privacy level and become completely predominant [3]- [4].
The advancement of quantum machines with the sufficient computational power could have distracted consequences for the advanced digital security. For instance, factorization and discrete log-based algorithms, such as RSA, DSA etc., which The associate editor coordinating the review of this manuscript and approving it for publication was Sedat Akleylek . assumes to be secured can be illuminated effectually, if the universal quantum framework developed [5]- [8].
The technology dependent organizations and individuals give birth to rising wave of cybercrime. The consistent advancements in the frameworks are of high significance and the quantum expansions bring us near to such a variation, where we can explore the field lies at the intersection of quantum novelties and cybersecurity [9]- [12].
The digital contents privacy in quantum era reviews all the perspectives affecting the security of computation and communication by the development of quantum progressions [13]- [19].
This field can be comprehensively categorized into three classifications reliant on the access and design of quantum challenges. The first classification sustains the existing tasks remains protected, whereas the other two classifications emphasize the conceivable outcomes in quantum developments [20]- [24].
In the presented article, we have developed a public key digital contents confidentiality scheme which simulates the data/pixels of the content with the dihedral and spin states systems. Dihedral is the noncommutative cryptographic group, which is an attractive area to provide security to high-end applications. The working principle of this group is based on the random polynomials chosen by the communicating parties to secure key exchange, enciphering-deciphering, and authentication challenges. The order of this group is more challenging to length based assaults and brute-force attacks [25]. The combination of spin states with the noncommutative cryptographic group provides legitimate security to high-end applications and also satisfy the quantum challenges.
This article is organized in 6 segments. The basic terminologies of public key protocol, dihedral group representation and quantum spin states sorted out in segment 2. The proposed security structure, algorithm and implementation on standard digital contents canvassed in segment 3, whereas the performance and security analyses evaluated in segment 4. Digital forensics analyses for the proposed structure presented in segment 5, and the concluding remarks with future projections canvassed in segment 6.

II. PRELIMINARIES
The basic terminologies and implementations of Rabin cryptosystem, Dihedral groups representations, and the entanglement of quantum spin states discussed in subsections. The proposed cryptosystem based on these terminologies (see Fig. 4), its implication on standard digital images and performance/ security investigations discussed in rest of the article.

A. RABIN CRYPTOSYSTEM
The insolvability based Rabin public key cryptosystem resolves the square root modulo problematic of composite integers. It can be deliberated as a variant of RSA cryptosystem and has computational advantage over the RSA by consuming the public exponent e = 2 [26]- [27]. Its decryption practice faintly faster than the RSA, as it requires computations of two modular exponentiations and the Chinese Remainder Theorem (CRT).
Description of Rabin criteria (see Fig. 1) in the proposed algorithm: Let Alice send the enciphered digital data to Bob at a phase 332 • , Bob will share his public key with Alice after generating public and private keys. At this instant, Alice will encode the phase value at the public key of Bob and transmit over the public channel. To recover the phase value, Bob will utilize his private key and then apply the phase over the enciphered digital contents to decipher it.

B. DIHEDRAL GROUP
When an object remains unchanged under some transformations or functions, then it follows the rotational or reflexive symmetry [28]. Fig. 2 (a-d) follows the mirror symmetry due to its reflexive property, and if an object is rotated around its center at some angles (like 90 • to 270 • ), then it will come back to its original position or shape [29]. The order of the symmetry is the total number of rotations or reflections of an object in which it comes back to its original shape [30]. The order of symmetry of the butter fly is 4 at an angle of 90 • (see Fig. 2 (e-h)).
Dihedral group D n is the subset of the symmetry group S n . It is the regular polygon with n rotations and reflections and having 2n elements, while the symmetry group S n have n! elements [31]. Dihedral group in matrices form represented as: where r k is the counterclockwise rotation matrix through an angle of 2π /n and s k is the reflection matrix across the line that makes an angle of π/n with the x-axis. Let us select the dihedral group D 4 which has 8 elements, 4 elements represents the rotation and the other 4 about the reflection of the square [32]. These elements are given as: The rotation about the angles of 0 • , 90 • , 180 • , 270 • denoted as a 0 , a 1 , a 2 , a 3 , and the reflection about the two perpendicular lines and diagonal segments of the square denoted as b 1 , b 2 , c 1 , c 2 . The Cayley table and general notation of the group D 4 gives as follows: C. QUANTUM SPIN SYSTEM The Spin system specifies the rotation of particles around axes. The particles with half-integer spin follows the Fermi-Dirac measurement (Fermions) and Pauli's principle, whereas the whole integer spin follows the Bose-Einstein (Bosons) criteria to share the quantum states [33].
We have followed the half integer spin criteria in the anticipated mechanism, which have been formulated in view of quantum spin systems in literature [34]- [36]. The formulated spin system matrices for the rotation in x, y and z directions are: where θ is the angle of rotational symmetry about x, y and z axes, and demonstrate the position at which the spin system act. Also, the phase of rotational symmetry satisfies the spin half algebra and Kramer's arbitrary spin system. Let entangle these 2 × 2 matrices by introducing an identity matrix to create a set E of 4 × 4 matrices, i.e., The spin states for the rotation about R x , R y , R z and their entanglements specified in Fig. 3.

III. PROPOSED ALGORITHM
The spin system produces infinite states between -720 0 and 720 0 phases. The data, when followed by the defined entangled state or states produce ciphers of high randomness. The proposed enciphering-deciphering structure for the digital contents confidentiality demonstrated in Fig. 4. Over the insecure line of communication, let us consider the establishment of secure channel between Alice and Bob. After the distribution of secrets using Rabin cryptosystem, first they apply the secret at Dihedral group matrices directly, then they evaluate the spin states by taking mod 720 of the shared secret, and finally they will choose the entangled state or states by taking mod 24 of the secrets. Now both Alice and Bob can transfer the data to each other securely over insecure line of communication by smearing the plain data with spin states at finite state automation.

A. EXPERIMENTATION OF ANTICIPATED STRUCTURE
The pixels of standard Pepper and House images operated with the combination of dihedral matrices and entangled state E 20 (by taking mod24 of 332) at phase 332 • for the proposed structure followed in Fig. 5.

IV. PERFORMANCE ANALYSES AND SECURITY EVALUATIONS
To sustain the security assessments and performance analyses for the anticipated algorithm, we have performed the standard evaluations in the subsections of 4 on standard images of size 512 × 512 taken from SIPI (signal and image processing institute) image database [37]. These evaluations consist of uncertainty assessments, factual examinations and sensibility analyses for the enciphered information to perceive the affectability of the anticipated structure.

A. UNCERTAINTY ANALYSES
To specify the randomness in the transmitted message, the average of information characterizes as entropy. To specify the arbitrary trials from set of dissimilar events . . , d n } with associated likelihoods, the average consequences of the source referred as Shannon entropy [38]. It can be determined for the digital contents as: where 2 N is the average information and d i is the source image. The ideal Shannon entropy for perfectly indiscrimination in 8-bit digital contents is 8 [39]. The evaluations of entropies for the plain and enciphered layer wise digital contents, and their assessments with the most recent existing methodologies depicted in Table 2.
We have certified the outcomes in 8-bit digital enciphered contents for the anticipated algorithm of Fig. 4 exceptionally close to the ideal Shannon entropy, and have superior outcomes when associated with existing methodologies. The mechanism of the foreseen plan is sheltered upon entropy attacks, as the secretion of information is inappropriate.

B. HISTOGRAMS CONSISTENCY ANALYSES
To evaluate the algorithm of the anticipated plan, we have measured the consistency in enciphered contents histograms [41]- [43]. The processed plain and enciphered contents of Pepper and House images having 256 dark facet levels with distinctive constituent demonstrated in Fig. 6. Fig. 6 demonstrate the consistency in enciphered contents which classifies the factual assaults hard, and the plain contents have sharp drops after sharp upsurges.

C. PIXELS' CORRELATION ANALYSES
To observe the adjacent pairs of pixels associated in horizontal, vertical and diagonal orders, we have performed the correlation analyses. Let us pick the 10000 adjacent pixels' pairs from each content initially and endure the association among the adjacent pixels of plain and enciphered contents. In sense of assessable investigation, the enciphered information assistances the relation of pixels to improve the barrier [44]- [46]. We have computed the two-dimensional correlation coefficients for the numerous pairs of plain and enciphered contents with the following expression.
where M and N are the height and width of the contents, plain and enciphered contents signified as P and C, and the mean approximation of P and C are P and C. VOLUME 8, 2020 The plain and enciphered contents correlation coefficients demonstration and assessments with the existing methodology followed in Table 3.
The enciphered contents correlation coefficients in Table 2 are exceptionally close to zero, which is the requisite for the competent enciphering plan and has inferior values on comparison with existing approach. The demonstration of horizontal, vertical and diagonal analyses on standard Pepper and House images in picture form (see Fig. 7) evaluated by the following expression: The adjacent pixels approximation signified by x and y, the variances signified by σ 2 x and σ 2 y , and the covariance of x and y random variables is σ x,y .

D. PIXELS' SIMILITUDE ANALYSES
The similitude analyses signified the structure resemblance between the plain and enciphered contents. We have evaluated the normalized cross-correlation (NCC), structural content (SC) and structure similarity index measure (SSIM) to observe the variation in structure between the plain and enciphered contents. The resemblance and traces of correlation measured by NCC, whereas SC regulates the noise level and sharpness, and SSIM relates the luminance, divergence  and assembly between the plain and enciphered contents. The similitude analyses evaluated here by the following expressions: where P i, j and C i, j are the plain and enciphered contents, the mean values are µ p and µ c , and the standard deviation is σ p c . The NCC, SC and SSIM esteems are fairly close to unity, if there is any structure similarity or correlation traces between the plain and enciphered contents [44]. Higher the SC estimation derives the eminence of image.
The similitude analyses and their assessments with existing approaches conceded in Table 4 for the plain-enciphered contents. VOLUME 8, 2020 In the light of Table 4, NCC, SC and SSIM esteems are fairly close to zero, which represents the structure dissimilarity and correlation invalidation between the plain and enciphered contents.

E. PIXELS' INCONSISTENCY ANALYSES
We have analyzed the divergence of image pixels' by estimating the mean absolute error (MAE), mean square error (MSE) and peak signal to noise ratio (PSNR) [45]. The precision of continuous variables for the disparity in enciphered contents regarding the plain image estimated here by MAE. The enciphered contents eminence can be specified by MSE and PSNR, whereas the average absolute difference between the plain and enciphered contents estimated here by MAE. The aggregate square error between the plain and enciphered content estimated by MSE, and the peal error by PSNR. The following expressions are examined here to estimate the inconsistency between the plain and enciphered  data.
The pixels' position for the plain and enciphered contents at i th row and j th column specified by P i, j and C i, j , and the extreme pixel position specified by I MAX .
The quality of enciphered contents can be enhanced by superior the MSE esteems and inferior the PSNR estimations, or vice-versa [49]. The evaluation of the proposed structure for inconsistency analyses followed in Table 5.

F. VISUAL STRENGTH ANALYSES
The visual strength analyses measure the image chromatic quality with respect to gray level co-occurrence matrix (GLCM). We have perceived the visual quality of images by observing the homogeneity, energy and contrast analyses of GLCM. The acquaintance of distribution in GLCM to GLCM diagonally performed by homogeneity, the aggregate of squared components of the content measured by energy, and the object in image's texture recognized by contrast [50]. The evaluation of GLCM analyses analyzed here with the following expressions.
The row and column position of pixels are i and j, homogeneity and energy of the image lies in the range of 0 and 1, and the range of contrast between 0 and (size (image) − 1) 2 . The contrast esteem must be superior for the great number of variations in the image's pixels, whereas the consistent image has 1 energy and 0 contrast value. Table 6 exhibits the analyses of GLCM for the projected enciphering plan and their assessments with most recent existing methodologies.

V. DIGITAL DATA FORENSIC ANALYSES
Digital data forensic specifically investigate the structure of enciphering/deciphering by specifying a documented chain of evidence to inspect what happened to digital data or medium. There are many tools available to investigate the systems/structures resistivity in sense of their encoded or damaged information [51]- [52]. We have performed the noise and differential assaults at the anticipated structure to affirm their resistivity.

A. NOISE ASSAULTS ANALYSE
It is possible in the digital communication either the channel or information affected by noises during the transmission or reception [53]. We investigate the effect of Gaussian noise having normalized power 0.000001 and 0.000005 in the VOLUME 8, 2020   encoded information to determine the strength of the anticipated structure. The consequences of the introduced noise in the encoded information and assessments with the existing methodology exhibited in Table 7.
The PSNR estimation has very minute variation when the Gaussian noise strength varies from 0.000001 to 0.000005, and has superior results over existing approach, which affirms the improbable strength of anticipated structure against Gaussian noise assaults.

B. DIFFERENTIAL ASSAULTS ANALYSES
The strength of the projected structure by variation of a plain image solitary pixel modifies the encoded information allied with a probability of half-pixel variation demonstrated by differential assault analysis. The number of pixels changing rate (NPCR) bound together to find the unified average change intensity (UACI) for a precise objective to assess the consequence of miniature variation in the plain content on its encoded one [54]- [57]. The NPCR and UACI for the two enciphered contents, in which one of them is vacillated by a single pixel estimated by the following expressions: The width and height of the digital content represented by W and H , and K is the two dimensional set similar to the enciphered image size. If C 1 (i, j) = C 2 (i, j), then K (i, j) =1; otherwise K (i, j) =0.
The consequences of the NPCR and UACI for the enciphered contents and their assessments with the most recent existing methodologies exhibited in Table 8 -9. In the light of Tables 8-9, the NPCR and UACI consequences have superior capacity to hostile attacks as NPCR esteems are fairly close to the perfect estimation of 1 and the proposed strategy has superior outcomes and great degree  touchy to a miniature change in the plain content over existing methodology [58]- [59].

VI. CONCLUDING REMARKS AND FUTURE PROJECTIONS
The upcoming frameworks will comprise of quantum and the association of traditional and quantum gadgets, and we will deal with these challengers when it marks our society. The proposed structure is appropriate for the traditional and quantum gadgets associations, and real time enciphering solicitations due to small execution time and capable of attacks hostility. The implication of the projected structure of Fig. 4 can be enhanced to a variety of digital mediums, such as audio/video calls, satellite imaginaries, etc.