Discrete Logarithmic Factorial Problem and Einstein Crystal Model Based Public-Key Cryptosystem for Digital Content Confidentiality

Public-key encryption is extensively used to provide digital data confidentiality and deliver the security features, such as nonrepudiation (digital signature) and secure key exchange. Conventional public-key schemes are based on mathematical problems with inflexible constraints, and the security of digital contents relies on computational complexity. In the era of emerging technologies, most public-key image encryption schemes are susceptible to various threats. We propose a novel public-key encryption in this article with near-ring criteria and provide confidentiality to private data with the microstates of the Einstein crystal model. The virtual oscillator generated by microstates of initial oscillators for the common secrets with the public-key scheme produces unique states to encrypt digital data. The privacy-preserved structure, that mimics the data stream of digital content with the behavior of the improved Einstein crystal model, describes a system in terms of microstates to generate diffusion in the plain data with unique states of a virtual oscillator. The performance and digital forensic evaluations, such as randomness, histogram uniformity, pixels’ correlation, pixels’ similarity, visual strength, pixels’ incongruity, key sensitivity, linear and differential attacks, noise, and occlusion attacks analyses, certify the resistivity of the proposed algorithm against potential threats and provide superior capacity in comparison to existing methodologies to hostile certain attacks.

not uniformly distributed, hence this scheme may also be 66 vulnerable to statistical attacks. 67 Nowadays, most image encryption schemes are developed 68 on the notion of confusion [13], [14] and diffusion com- in the image data [16]. By aiming at all the above problems, 76 we developed a new encryption scheme that provides the 77 ultimate performance and security. is insecure against CCA if the message is a small integer. For 89 instance, if the message is 200-bit integer and the public expo-90 nent is 3, then the available public message will be a 600-bit 91 integer. This means the message can be recovered using a 92 non-modular cube root, which is simple and easy to compute 93 [18]. Furthermore, for chosen-ciphertext (CCA) security, 94 a variation of the Cramer-Shoup (CS) method [19] makes use 95 of the computational Diffie-Hellman (CDH) assumption. The 96 high-security cost of this cipher is that the size of ciphertexts 97 is much larger than with the CS scheme (which is based on 98 the decisional Diffie-Hellman assumption) [20]. 99 The proposed discrete logarithmic factorial problem 100 (DLFP) based public-key establishment scheme in this article 101 provides a secure way to generate common keys. The adver-102 sary can't bypass the protocol even with one of the secrets 103 from pairs, and the developed scheme resists the CPA and 104 CCA attacks. To share the digital content securely on the 105 shared key, we developed a privacy-preserved structure that 106 mimics the data stream with the behavior of the improved 107 Einstein crystal model, which reflects wave-particle dual-108 ity [21], [22]. This model describes a system in terms of 109 microstates, whereas each microstate acts as a harmonic 110 oscillator in a three-dimensional potential. We generated 111 diffusion in the plain data with unique states of a virtual 112 oscillator followed by a random walk on inimitable points 113 without including the full dynamics (Monte Carlo). In the 114 first approximation, we introduced a simplified interaction 115 between the oscillators by allowing data transfer randomly 116 from one oscillator to another with no overhead. The stochas-117 tic model of uncorrelated states has similar behavior to gener-118 ate sequence as in quantum chaos and provides the chaos tran-119 sition from a Poisson to a Gaussian distribution. The security 120 and performance measures for the proposed model validate 121 the effectiveness in comparison with existing techniques.

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This article is organized into six sections. The preliminar-123 ies of the Einstein crystal model, Monte Carlo modification, 124 and DLFP key establishment are explained in Section II. The 125 anticipated algorithm and its execution on standard images 126 are deliberated in Section III, with performance and security 127 evaluations assessed in Section IV. Digital forensic analysis 128 of the outcomes from the proposed strategy is presented in 129 Section V, and concluding notes with upcoming prospects are 130 given in Section VI. • an ordered pair (N , •) is a semigroup, and for each 148 element n 1 , n 2 , n 3 ∈ N , (n 1 + n 2 ) · n 3 = n 1 · n 3 +n 2 · n 3 149 Assumptions: For a factorial problem, the component ω of 150 non-abelian group, near-ring N and sub near-rings N 1 , N 2 ∈ 151 N , determine the elements a 1 ∈ N 1 and a 2 ∈ N 2 that satisfy 152 ω = a 1 a 2 .

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Given the prime p for DLFP, when the generator of the 154 cyclic group Z p is α and element β ∈ Z p , find an integer For a DLFP, N is a near-ring non-abelian identity con- .

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• The generated common secret between Alice and Bob is Let the communicating parties, Alice and Bob, agree on 176 some prime numbers, p , in order to generate the common 177 secret. Fig. 1 demonstrates the establishment of the common 178 key using the secret credentials. each of them. After generating and sharing the public keys, 185 they are able to develop a common secret between them using 186 their private key pairs with each other's shared public key. The 187 evaluation of the common secret between Alice and Bob from 188 their private keys is demonstrated in Table 1.

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Security analysis of the proposed key establishment 190 algorithm: An adversary can bypass the protocol by obtain-191 ing Alice's or Bob's private key in the following attacks. The Einstein crystal model describes a system in terms of 207 microstates wherein the atoms in the crystal do not interact 208 directly. Each atom acts as a three-dimensional harmonic 209 oscillator with a central potential, and the total system con-210 sists of N atoms [24], [25]. The atoms in the system share the 211 total energy in the system. Each atom in a three-dimensional 212 structure consists of three independent oscillators in the 213 x, y, and z directions. In other words, a system with N oscil-214 lators consists of N 3 atoms. Each oscillator has potential 215 v (x) = 1 2 k x 2 , where k is the spring constant and x is the 216 derivation of the equilibrium position. The energies of such 217 oscillators are quantized with several possible values (∈ = 218 h v n) where n = 0, 1, 2, . . . can only be integers. The 219 measurement of energy is h v = ∈, and for the dimensionless 220 energy states, we choose the symbol q. Let us consider a 221 simplified four-oscillator system with N = 4 and quantized 222 level q = 2. In Fig. 2, we describe the states by using a simple 223 illustration with possible energy levels for N = 4 oscillators. 224 There are generally two possibilities.
We assumed a small modification to the Einstein crystal 261 model to include some of the facts below. The system consists 262 of two parts, A and B, so that each oscillator belongs to 263 either A or B. In the first approximation, we introduced a 264 simplified interaction between the oscillators by allowing 265 energy to move randomly from one oscillator to another by 266 conserving the total energy. This means the system will move 267 from one microstate, S 1 , to another microstate, S 2 , with the 268 same energy. Hence, both microstates are equally probable, 269 since all microstates are equally probable.

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To simulate the random transmission of energy in the 271 system, the process is as follows.

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• Select an oscillator (particle) at random, i 1 , and transport 273 the energy from this oscillator.

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• Select another oscillator (particle) at random, i 2 , and 275 receive the energy from n 1 .

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We generate a virtual state oscillator from oscillators A and 277 B with the same energy. The resulting dynamics of the oscil-278 lators are expressed in Fig. 3.

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Monte Carlo simulation 280 In the stochastic model, we generate a sequence of uncor-281 related microstates. For each sample, we randomly generate 282 a new microstate, ensuring no correlation with the previous 283 state [28]. We sample many such states in a long sequence 284 before making statistical predictions about the probability 285 of a microstate. This motion is observed to be in a zigzag 286 path and is referred to as a Monte Carlo model for a micro-287 canonical system. There is also a strong connection between 288 the random walk and the Schrödinger equation by replacing 289 time with imaginary time. Quantum chaos is also linked to a 290 Monte Carlo diffusion process for the quantum energy-level 291 spacing sequences. These levels acts as a function to execute 292 a random walk, which infers uncorrelated random increments 293 in the level spacing, and transforms the chaos transition from 294 a Poisson to a Gaussian distribution. The random walk of 295 Monte Carlo microstates is shown in Fig. 4.

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Digital content that navigates through the flowchart described 298 in Fig. 5 produces encrypted streams with high randomness. 299 To demonstrate our idea clearly, we consider here a simplified 300 22-oscillator system with N = 22 atoms (DLFP, Section II-A) 301 and quantized level q = 256 (for eight-bit images).    Carlo random walk sequence with the common secret, 315 as explained in Section II-B.

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• By performing a bitwise XOR operation between the 317 microstates of the two generated oscillators, we develop 318 a virtual oscillator.

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• For Alice, we operate layer-wise plain image pixels 320 with the microstates of the virtual oscillator to generate 321 diffusion in their values.

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• To diffuse the pixel values completely, we pass each 323 layer of the generated cipher image from the virtual 324 VOLUME 10, 2022  where p (x n ) is the probability of source x n and is expressed 351 in bits. For dissimilarity in eight-bit digital content, the opti-352 mum Shannon entropy is 8. Table 2 depicts entropy analysis 353 for plain and encrypted content, as well as assessment with 354 existing techniques.

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The outcomes of the proposed method in Table 2 are 356 reasonably close to the perfect estimation of Shannon 357 entropy and outperform the previous methods. These 358 results demonstrate that there is a negligible data loss 359 and the structure in Fig. 5 is resistant to entropy 360 attacks.

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We processed original and encrypted image content hav-363 ing 256 dark-dimension intensities in order to estimate 364 the consistency of histograms for the proposed method-365 ology [34], [35]. An evaluation of the original and 366 encrypted content for the Airplane and Baboon images 367 from the proposed methodology are demonstrated in Fig. 7 368 and Fig. 8. 369 We evaluated the plain and encrypted content in 370 Figs. 7 and 8 to ensure consistency in encrypted content to 371 make the factual assaults hard.  horizontal, diagonal, and vertical directions [36], [37].

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where P and C signify the plain and encrypted content,  The correlation coefficients in Table 3 are quite close to 404 zero, which implies the variables are either hardly related or 405 dissimilar, and show results superior to the existing approach. 406

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Similarity analyses measure the resemblance of pixels 408 from among different digital content. To observe the 409   Table 4.
434 Table 4 shows that there is no pixel resemblance between 435 plain and encrypted image content. Moreover, the approxima-436 tions of SSIM, SC, and NCC have better consequences than 437 the existing approach.

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Visual strength analysis is a statistical approach for mea-440 suring the chromatic quality and texture of an image by 441 reflecting the spatial association of pixels in the gray level 442 co-occurrence matrix (GLCM) [42], [43]. The classification 443 VOLUME 10, 2022 TABLE 4. Pixel-based similitude analyses for plain-encoded content and assessments with the most recent existing approach.    The following parameters are used to analyze the evaluations 497 of these investigations. where P k, l and C k, l are the pixel positions for the plain 505 and encrypted information in the k th row and l th column, 506 respectively, and I MAX is an estimate of the digital content's 507 maximum possible pixel. A higher MSE esteem and a more 508 consistent PSNR can increase the quality of digital content 509 encryption, or vice versa [46]. Table 6 depicts the analysis of 510 standard digital content for the attainability of the anticipated 511 structure.

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To determine what happened to the digital content, we per-514 formed a systematic data evaluation while keeping an 515 archived sequence of evidence. To sustain the resistivity of 516 VOLUME 10, 2022  the foreseen structures, we performed linear, key sensitivity, 517 noise, and differential assault analyses as follows.

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The key employed by the cryptanalyst conducts a linear 520 assault to identify the logic used in encryption and decryp-521 tion to perceive immediate information for the association 522 between particular bits of plain and encrypted data [49].

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The analyst will attempt to decode the information using 524 all available keys to find the similarities in the ciphers. The The sensitivity of the key is determined by how much of key-533 space is available to withstand a brute force attack [50]. The 534 total number of keys needed to encrypt or decrypt the algo-535 rithm is specified. We analyzed brute force, computational, 536 and ciphertext attacks on the proposed algorithm.  Ciphertext analysis: To analyze ciphertext attacks, an ana-552 lyst requires the key matrix to bitwise XOR the cipher image 553 [52]. For an eight-bit image having dimensions of 64 × 64, 554 the analyst requires (64 × 64)! combinations of eight-bit 555 values to decode the image. To crack the cipher images 556 in Section III-B of this article, analysts need to compute 557 (512 × 512)! combinations of eight-bit values, which is much 558 harder than cracking the key with a brute force attack.

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During transmission or reception, it is possible that the infor-561 mation may be affected by noises and can be tempered 562 over the insecure channel. A cryptosystem should be capable 563 to resist the attacks and recover the data up to a certain 564 level even after tempering in data [53], [54]. To validate 565 the robustness of the algorithm, we estimated the MSE 566 and PSNR by introducing Gaussian noise and occlusion 567 to transmitted content. The Gaussian noise analysis, hav-568 ing normalized power 0.000001, 0.000003, 0.000005, and 569 0.000007, for the gray level transmitted image is depicted in 570 Table 7, and the occlusion analysis for the encrypted images 571 occluded by 1/8, 1/4, and 1/2, 3/5 are given in Fig. 12 and 572 Table 8.

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By varying the noise strength from 0.000001 to 0.000007, 574 there seems to be a minute variation in the noise ratio and 575 the error estimation, which proves the robust efficiency of the 576 proposed structure against noise assaults.

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The effects of MSE and PSNR after introducing occlusion 578 attack to encrypted images in Table 8 and the decrypted 579 images in Fig. 12 are indicating that the proposed algo-580 rithm can withstand up to 50% occlusion attack. Hence, the 581 proposed method provides better security against noise and 582 resists the occlusion attack.     Tables 9 and 10.

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The outcome in Table 9 is legitimately near a seamless 607 estimation of 1, which indicates that the foreseen structure has 608 better results over the existing approach to opposing attacks, 609 whereas the UACI results in Table 10 show that the proposed

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In the next-generation frameworks, when adversaries will be 616 fully equipped with AI technologies, it is predicted that most 617 public-key image encryption schemes will be susceptible to from key pairs is known. The virtual oscillator generated by 623 microstates of initial oscillators for the common secrets, on 624 the proposed public-key scheme, produces unique states to 625 generate diffusion in the plain data followed by a random 626 walk, comparable to quantum chaos, on inimitable points. 627 The performance and digital forensic assessments certified 628 the superior resistivity of the proposed method in compar-629 ison to existing schemes to hostile attacks. The developed 630 structure in this article can be extended to applications of 631 already developed models, such as secure transfer of satellite 632 and drone imageries, low-profile mobile applications, audio-633 video encryptions, etc., and we strongly believe that there is 634 room for further improvements to envisioned structures with 635 even better cryptographic properties.