The core insight of the new technique is that color medical images carry redundant statistical structure that attackers can exploit. In a typical color image, the red, green, and blue channels are highly correlated with one another, and neighboring pixels within each channel tend to have nearly identical values. Classical encryption that treats an image as an ordinary file often leaves enough of this structure intact for statistical analysis to reconstruct recognizable content. The researchers attack this weakness head-on by first separating each image into its three constituent channels and then deliberately scrambling them with channels drawn from other images in the batch.

This inter-channel mixing serves two purposes at once. First, it drastically lowers the correlation between adjacent pixels, because a pixel in the encrypted red channel may now depend on data that originally lived in the green channel of an entirely different scan. Second, it binds the security of every image in the group to the security of the whole batch, so an adversary who somehow compromises one image still faces an incoherent jumble of fragments from the others. The authors describe the resulting scrambling process as introducing enough uncertainty and diffusion to defeat both statistical and differential attacks, the two workhorses of cryptanalysis against image ciphers.

Once the channels are mixed, the scheme applies two further layers of protection: pixel permutation and channel diffusion. Crucially, these operations are not driven by conventional pseudo-random number generators but by three distinct chaotic maps—the logistic map, the Henon map, and the Chebyshev map. Chaotic systems are prized in cryptography because they exhibit extreme sensitivity to initial conditions: a change of even one digit in the starting value produces a completely different sequence. By generating independent encryption keys from three different chaotic systems, the scheme multiplies the difficulty of predicting or reproducing its scrambling behavior.

The division of labor among the three maps is deliberate. The logistic map, a deceptively simple one-dimensional equation famous for producing complex behavior from a single parameter, contributes sequences for one stage of the process. The Henon map, a two-dimensional system introduced in 1976 and known for its strange attractor, adds a second source of pseudo-randomness with richer dynamics. The Chebyshev map rounds out the trio, and each map’s output feeds independent keys for permutation and diffusion stages. This multi-chaotic key architecture means an attacker would need to break three separate dynamical systems simultaneously rather than a single generator.

The numbers reported in the study are the currency in which image-encryption schemes are judged, and they are impressive. The encrypted images achieve an average information entropy of 7.9994 on a scale where the theoretical maximum is 8, meaning the pixel values are distributed almost perfectly uniformly and leak virtually no statistical information. The key space reaches 2 to the power of 512, a number so vast that brute-force search is computationally meaningless with any foreseeable hardware. For context, even a key space of 2 to the power of 100 is generally considered far beyond exhaustive attack.

Two further metrics measure resistance to differential attacks, which test whether changing a single pixel of the original image changes the encrypted output. The number of pixels change rate, or NPCR, came in at 99.61 percent, essentially the ideal value indicating that nearly every encrypted pixel changes when the plaintext is altered by even one bit. The unified average changing intensity, or UACI, measured 33.46 percent, sitting squarely in the optimal range around the theoretical expectation of 33.46 percent for a strong cipher. Together, these figures indicate that the scheme produces avalanche-like sensitivity: the smallest alteration in a medical image cascades into a completely different ciphertext.

The researchers validated their algorithm on publicly available medical datasets, including brain tumor MRI scans, a breast cancer histopathological database, dermatology images, and SARS-CoV-2 CT scans, giving the evaluation a realistic clinical flavor rather than relying on generic test images. Beyond raw security, the team emphasizes efficiency: the method is designed to support the secure transfer of real-time medical information while preserving the integrity and quality of the original images after decryption. That combination matters, because a cipher that garbles diagnostic detail or adds prohibitive latency is useless in an emergency room or a remote consultation.

The work arrives amid a broader surge of interest in medical image security, driven by the growth of the Internet of Medical Things and telemedicine. Recent literature has explored DNA-based operations, hyperchaotic systems, fuzzy logic substitution boxes, and even large language models for plausible-deniability schemes. Hosny and colleagues’ contribution stands out for its focus on encrypting multiple images simultaneously—a practical requirement when hospitals transmit entire imaging sessions—rather than one image at a time, and for the elegance of using the images’ own color channels as raw material for cross-contamination of information.

For clinicians and health IT administrators, the message is that color medical images can be protected with ciphers purpose-built for their statistical quirks rather than adapted from generic file encryption. For cryptographers, the study adds evidence that combining cross-image channel mixing with multiple independent chaotic key generators is a productive design pattern. As healthcare data breaches continue to make headlines, techniques like this one aim to ensure that a stolen stream of encrypted scans reveals nothing more than noise—uniformly distributed, information-theoretically opaque, and mathematically indifferent to the secrets it once carried.

Subject of Research: Chaotic-map-based encryption of multiple color medical images using cross-image channel mixing

Article Title: Efficient encryption of multi-color medical images using channel mixing and multi-chaotic keys

Article References: Efficient encryption of multi-color medical images using channel mixing and multi-chaotic keys. (n.d.). https://doi.org/10.1007/s12652-026-05131-1

Image Credits: AI Generated

DOI: 10.1007/s12652-026-05131-1

Keywords: medical image encryption, chaotic maps, channel mixing, logistic map, Henon map, Chebyshev map, information entropy, differential attacks, telemedicine security, color images, cryptography, Internet of Medical Things

The study, published in the journal Results in Engineering by Yu-Feng Sun, Jie-Sheng Wang and colleagues, describes an enhanced version of the Manta Ray Foraging Optimization algorithm, or MRFO. The original algorithm, first introduced a few years ago, mimics the cooperative hunting strategies that manta rays use to harvest plankton. Because plankton drifts unpredictably with seasons and weather, manta rays have evolved three distinct foraging behaviors: chain foraging, in which animals line up in arcs so that each one catches food missed by the one ahead; spiral foraging, where the group coils around a dense patch of food; and somersaulting, in which individuals flip back and forth around a promising spot. Translated into mathematics, these behaviors become rules that move a population of candidate solutions around a search space, gradually converging on the best answer.

What makes the CEED problem so stubborn is its mathematical character. The relationship between a generator’s output and its fuel consumption follows a cubic curve, and the emissions of carbon dioxide, nitrogen oxides and sulfur dioxide each follow their own cubic functions. The combined objective is nonlinear, nonconvex and discontinuous, riddled with local minima that can trap classical techniques such as lambda iteration or gradient-based methods. To make matters harder, the problem is genuinely multi-objective, so the researchers converted it into a single objective by applying price penalty factors that fold the cost of each pollutant into one total figure in dollars per hour.

Constraint handling is where many algorithms stumble. Traditional approaches add a fixed penalty whenever a solution violates the power balance requirement, the equation demanding that total generation minus transmission losses equals demand. But a fixed penalty is a blunt instrument. Set it too low and the algorithm happily returns infeasible schedules; set it too high and the search becomes so restricted that it cannot explore properly. Finding the right value through trial and simulation is time-consuming and inefficient. The team’s answer was to make the penalty itself dynamic, drawing on five different mathematical functions: sinusoidal, hyperbolic tangent, arctangent, linear and hyperbolic secant. Each function is raised to an order that changes during the run, so the penalty strength adapts to both the size of the violation and the stage of the optimization. Early on, mild penalties preserve exploration; later, penalties tighten to push solutions toward feasibility.

The second major innovation targets the heart of the MRFO algorithm. In the original design, a parameter called C governs the switch between exploration, casting the net wide across the search space, and exploitation, homing in on the best solution found so far. This parameter rises monotonically from zero to one, a predictable march that can leave the algorithm poorly balanced at certain stages. The researchers replaced it with a chaotic switching factor, drawn from nine well-known chaotic maps including Chebyshev, logistic, tent, sine, Gauss and singer maps, combined with an exponential adjustment function. The result is a parameter that still trends upward but oscillates along the way, periodically reopening the search space and preventing the population from settling prematurely into a local optimum.

To find out which chaotic map worked best, the team ran all nine enhanced variants, named C1MRFO through C9MRFO, through the CEC-2022 benchmark suite of twelve test functions, using a population of thirty and five hundred iterations averaged over thirty independent runs. The Chebyshev-based variant, C1MRFO, emerged as the overall winner, ranking first among the nine and beating the original MRFO, which finished last. Wilcoxon rank-sum tests confirmed that most of the improvements were statistically significant, with p-values below the conventional 0.05 threshold. When C1MRFO was pitted against a roster of state-of-the-art metaheuristics, including the dhole optimization algorithm, evolutionary mating algorithm, dung beetle optimizer, snow geese algorithm, particle swarm optimization, grey wolf optimizer, sparrow search algorithm and marine predators algorithm, it achieved the best final rank, taking first place on many of the test functions while keeping computational times competitive.

The real test, however, came when the algorithm was applied to actual dispatch problems. The researchers simulated a six-unit power system under four load demands of 150, 175, 200 and 225 megawatts. Across all four scenarios, C1MRFO produced the lowest comprehensive total cost, which combines fuel cost with the weighted penalties for carbon dioxide, nitrogen oxide and sulfur dioxide emissions. At 150 megawatts, for example, it reached a total cost of about 10,137 dollars per hour, edging out the grey wolf optimizer and clearly beating algorithms such as the arithmetic optimization algorithm, Harris hawks optimization, whale optimization algorithm, sine cosine algorithm, particle swarm optimization and the lambda iteration method. The pattern held at every load level, with the margin over the weakest classical method growing to thousands of dollars per hour at 225 megawatts.

The dynamic penalty functions delivered a further, striking gain. When the sinusoidal penalty function was applied with the parameter order set to dy5, the total cost at 150 megawatts dropped from roughly 10,137 to 9,065.71 dollars per hour, and similar reductions appeared at the higher loads, reaching 14,621.45 dollars per hour at 225 megawatts. Comparing the five penalty shapes, the sinusoidal function consistently produced the lowest minimum costs, beating the hyperbolic tangent, arctangent, linear and hyperbolic secant alternatives by margins ranging from hundreds to nearly two thousand dollars per hour depending on the load. Notably, when the penalty order was fixed rather than dynamic, performance was worst across the board, confirming that the adaptivity of the penalty, not merely its presence, is what matters.

To verify that each improvement contributes on its own, the team conducted an ablation study on a much larger twenty-unit system with a total demand of 2,500 megawatts. The results formed a clear hierarchy: the original MRFO achieved 61,835.16 dollars per hour, the chaos-enhanced C1MRFO reached 61,826.11, and the fully improved version combining chaos with the optimal sinusoidal dynamic penalty, sin(dy5)MRFO, achieved the best figure of 61,807.36 dollars per hour. All three outperformed the phasor particle swarm optimization benchmark, which came in at 61,895.57 dollars per hour. The stepwise improvement demonstrates that the two enhancements are not redundant but superimpose their benefits, and that the framework scales to systems far larger than the six-unit case.

The authors are careful to note where the work goes next. The current approach converts the multi-objective problem into a single objective through price penalty factors, but future versions could adopt Pareto-based multi-objective optimization to generate full sets of non-dominated trade-off solutions. The test cases also assume deterministic demand, so extending the algorithm to stochastic dispatch with wind and solar integration is another goal, as is combining the dynamic penalty with feasibility-first constraint-handling strategies. For now, the study offers a compelling demonstration that borrowing the foraging logic of a filter-feeding fish, perturbing it with chaos, and teaching it to calibrate its own punishments can squeeze meaningful savings and emission reductions out of one of power engineering’s hardest scheduling problems.

Subject of Research: An improved manta ray foraging optimization algorithm with chaotic switching factors and dynamic penalty functions for solving the combined economic emission dispatch problem in power systems.

Article Title: Manta ray foraging optimization algorithm with dynamic penalty function and chaos switching factor to solve combined economic emission dispatch problem

Article References: Sun, Y.-F., Wang, J.-S., Zhang, X.-Y., Li, Y.-X., Guan, X.-Y., & Liu, X. (2026). Manta ray foraging optimization algorithm with dynamic penalty function and chaos switching factor to solve combined economic emission dispatch problem. Results in Engineering, 32, Article 113383. https://doi.org/10.1016/j.rineng.2026.113383

Image Credits: AI Generated

DOI: Not provided

Keywords: manta ray foraging optimization, combined economic emission dispatch, metaheuristic algorithms, chaotic maps, dynamic penalty function, power system optimization, emission reduction, fuel cost minimization, CEC-2022 benchmarks, swarm intelligence, constraint handling, Results in Engineering

The scheme, developed by El Mahdi Mouloua and colleagues, rests on two novel operators that work in tandem. The first is a chaotic polycyclic shift, denoted by the Greek letter phi with a chaotic vector index, which generalizes the familiar left and right bit rotations used in classical cryptography. Instead of shifting data by a fixed amount, the operator displaces pixels and their bits by amounts dictated by a chaotic sequence, so that the pattern of movement never repeats and cannot be predicted without the secret key. This operator drives the confusion mechanism of the cipher, scrambling the relationship between the original image and its encrypted form.

The second operator addresses the other pillar of strong encryption: diffusion. In cryptographic terms, confusion hides the connection between the key and the ciphertext, while diffusion ensures that a change to a single element of the plaintext spreads throughout the entire encrypted output. The researchers call their diffusion engine the chaotic avalanche effect, symbolized by xi. It creates a strong binding between each encrypted pixel and the following original pixel, so that information from one part of the image cascades into the encryption of subsequent parts. A single flipped bit in the input therefore triggers an avalanche of changes across the ciphertext, denying attackers the ability to modify localized regions of an image undetected.

The intellectual roots of the work reach into an unexpected corner of mathematics: the theory of polycyclic codes. One of the co-authors, Oussama Kabbouch, had previously explored the algebraic structure of polycyclic codes with collaborators, building on earlier generalizations of cyclic codes introduced by Lopez-Permouth, Parra-Avila and Szabo in 2009. By importing the cyclic shift idea from coding theory into the chaotic domain, the team transformed a static algebraic operation into a dynamic, key-dependent one. The chaotic vector that steers the shift is generated by chaotic maps, whose extreme sensitivity to initial conditions means that even a minute difference in the key produces a completely different scrambling pattern.

Chaos theory has long fascinated cryptographers for precisely this reason. Chaotic systems, such as the logistic map and its more elaborate relatives like the intertwining logistic map, exhibit deterministic but seemingly random behavior. Their trajectories never settle into a pattern, yet they can be reproduced exactly by anyone holding the same initial values. This combination of reproducibility and unpredictability makes them natural candidates for generating the keystreams and permutation sequences that encryption demands. The new scheme taps this property to control both the polycyclic shifts and the avalanche diffusion, ensuring that every stage of the cipher is driven by the same chaotic engine and the same secret key material.

The experimental evaluation reported in the paper reads like a checklist of modern cryptographic benchmarks, and the scheme passes each one. The key space, which measures how many possible secret keys an attacker would need to brute-force, is described as large, a critical property given that many published chaotic ciphers have been broken because their key spaces proved too small. Key sensitivity is similarly high: encrypting the same image with keys that differ by a single digit yields entirely different ciphertexts, which prevents an attacker from gleaning information through near-miss keys.

Statistical tests on the encrypted images tell a similar story. The histograms of the encrypted outputs are uniform, meaning that every pixel value appears with equal frequency and no statistical fingerprint of the original image survives. The correlation between adjacent pixels, which in natural images is typically very high because neighboring pixels tend to look alike, drops to low values in the ciphertext, destroying the structural redundancy that attackers exploit in known-plaintext and chosen-plaintext attacks. Information entropy, a measure of randomness, reaches high values close to the theoretical maximum of eight bits per pixel for eight-bit images, indicating that the encrypted data is essentially indistinguishable from noise.

Two additional metrics, standard in the image encryption literature, quantify the scheme’s resistance to differential attacks. The number of pixels change rate, or NPCR, measures the fraction of pixels that change when the plaintext is altered by a single bit, while the unified average changing intensity, or UACI, measures the average strength of those changes. Ideal values are close to 99.6 percent and 33.4 percent respectively, and the reported strong encryption performance of the scheme, together with low peak signal-to-noise ratio values between original and encrypted images, indicates that the chaotic avalanche effect delivers the full diffusion strength needed to defeat these attacks. Low PSNR here is a virtue: it confirms that the encrypted image leaks essentially nothing about the original.

The broader context makes the contribution timely. The literature is littered with image encryption schemes that looked impressive on paper but collapsed under cryptanalysis, including DNA-based ciphers and designs built on the two-dimensional logistic-adjusted-sine map, both of which were broken by dedicated attacks. Meanwhile the demand for image security keeps growing, driven by telemedicine platforms transmitting diagnostic scans, the Industrial Internet of Things streaming visual sensor data, and cloud storage services holding vast archives of personal photographs. Recent research has explored hyperchaotic maps, memristive neural networks, quantum walks and DNA encoding as ingredients for next-generation ciphers, and the Moroccan-Romanian team positions its polycyclic approach within this competitive landscape, arguing that a mathematically grounded shift operator offers a cleaner and more principled confusion mechanism than ad hoc permutation tricks.

The researchers, who contributed equally to the work alongside co-authors from institutions in Morocco and Romania, suggest that the combination of a generalized cyclic structure with chaotic dynamics could extend beyond still images, potentially informing the protection of video streams and other multimedia content where efficiency and security must coexist. For now, the scheme stands as a demonstration that classical algebra and modern chaos theory can be fused into a cipher that is more than the sum of its parts: a system in which the ancient idea of the cyclic shift is reborn, steered by randomness that only the legitimate key holder can reproduce, and in which every pixel of an image guards the secrets of every other.

Subject of Research: Chaotic image encryption using polycyclic shift operators

Article Title: Design of a secure image encryption scheme using chaotic polycyclic shift

Article References: Mouloua, E. M., Ben-Hdech, A., Bejan, C. A., Es-Sabry, M., Najmeddine, M., Kabbouch, O., Bucerzan, D., & Essaid, M. (2026). Design of a secure image encryption scheme using chaotic polycyclic shift. Cluster Computing, 29(13), Article 771. https://doi.org/10.1007/s10586-026-06588-7

Image Credits: AI Generated

DOI: 10.1007/s10586-026-06588-7

Keywords: image encryption, chaos theory, polycyclic codes, cryptography, confusion and diffusion, chaotic maps, multimedia security, NPCR, UACI, information entropy, Cluster Computing, data security

The motivation for the work begins with a fundamental mismatch. Standard encryption algorithms such as AES, DES and RSA were designed to protect streams of text or raw data, and they perform reasonably well on such inputs. Digital images, however, are different beasts. They are large, they contain highly redundant pixel information, and neighboring pixels tend to be strongly correlated with one another, often sharing values that differ by only a few bits. When conventional ciphers are applied directly to image data, the processing becomes slow and the underlying structure of the picture can leak through. Steganography, the art of hiding one file inside another, offers concealment but not true cryptographic resistance to cryptanalysis. For applications like medical imaging and military reconnaissance, where confidentiality is non-negotiable, researchers have therefore turned to specialized image encryption methods built around two classical operations: confusion, which rearranges the positions of pixels, and diffusion, which alters the values of pixels.

The first pillar of the new scheme is chaos theory. Chaotic systems are nonlinear dynamical processes in which a minuscule change in the starting conditions produces wildly divergent outcomes over time, a phenomenon popularly known as the butterfly effect. This extreme sensitivity, combined with apparent randomness, makes chaotic maps attractive for generating pseudo-random keystreams that depend on secret initial values. The particular tool chosen here is the Arnold Cat Map, a deceptively simple transformation that stretches and folds a two-dimensional space, named for the cartoon cat whose face becomes an unrecognizable blur when the map is applied repeatedly. In the encryption pipeline, the Cat Map acts as the confusion layer: it permutes the coordinates of every pixel according to a matrix equation involving two secret positive integers, m and n, applied modulo the image dimension N. In the reported experiments the team selected m and n equal to 7 and iterated the transformation four times. Because the map preserves pixel values while shuffling their locations, the picture retains its histogram but loses all recognizable geometry, appearing as pure noise.

The second pillar borrows from molecular biology. DNA cryptography exploits the four-letter alphabet of nucleotides, adenine, cytosine, guanine and thymine, along with the complementary base-pairing rules that govern how these letters bind together. In a computational setting, binary data can be encoded into quaternary DNA sequences, and operations such as substitution and exclusive OR can be performed according to the pairing rules A with T and C with G. The researchers begin with a chosen DNA sequence representing each nucleotide as an eight-bit binary string. From each string they extract pairs of bits, starting with the least significant two bits, then the next two, and so on, assembling these fragments into seed values. The resulting binary strings are concatenated, a designated bit is flipped through a significant-bit-changing strategy, and permutation selection produces four key strings. These are arranged into an eight-by-eight key matrix, which serves as the secret keystream for the diffusion stage. The DNA layer thus introduces an additional dimension of randomness and an enlarged key space beyond what the chaotic map alone provides.

The full encryption procedure weaves these components together in a carefully ordered sequence. First, the original image is passed through the Arnold Cat Map, which scatters pixel positions across the frame. Next, the shuffled image is divided into blocks of eight-by-eight pixels. Each block is then combined with the DNA-derived key matrix using the XOR operation, a bitwise comparison that outputs a one whenever the two input bits differ. This XOR step constitutes the diffusion layer, transforming pixel values rather than merely relocating them. Finally, all processed blocks are concatenated to form the encrypted image. Decryption simply reverses the pipeline: the cipher image is broken into blocks, XORed with the same key matrix to recover the shuffled picture, and the inverse Cat Map transformation restores the original pixel coordinates. Crucially, an adversary would need both the DNA sequence parameters and the Cat Map keys to reconstruct anything meaningful, a requirement that substantially raises the bar against brute-force attempts.

To evaluate the scheme, the team ran MATLAB 2019a simulations on an Intel Core i3-1005G1 processor with 6 gigabytes of RAM, testing the algorithm on a hand X-ray, a chest X-ray and the classic Baboon test image. The encrypted outputs were visually indistinguishable from random noise, and decryption with the correct keys restored the originals faithfully. The researchers measured encryption and decryption times across images of varying dimensions, finding that processing time scales with image size, an expected but practically important result for real-time or resource-constrained systems. They also note that changing the image dimension alters the period of the Cat Map, which in turn affects key size and the strength of the confusion effect, trading improved security against increased computational and memory costs.

Quantitative security metrics told a consistent story. The mean square error between original and encrypted images was high, while the peak signal-to-noise ratio, or PSNR, was correspondingly low. In image encryption, a low PSNR is a virtue rather than a flaw: it signals that the cipher image bears almost no resemblance to the source, which is precisely what a defender wants. Histogram analysis reinforced this conclusion. The histograms of the encrypted images differed dramatically from those of the originals and appeared uniformly distributed, denying an attacker the statistical fingerprints that leak information about the underlying content. Correlation coefficients between adjacent pixels, measured horizontally, vertically and diagonally, dropped to nearly zero after encryption, whereas natural images typically exhibit correlations close to one in all three directions. This near-total destruction of neighboring-pixel relationships is a hallmark of effective confusion.

Information entropy provided further evidence of robustness. For an eight-bit image, maximum theoretical entropy is eight bits, achieved when all 256 gray levels occur with equal probability. The encrypted images produced entropy values approaching 7.9980, edging out comparable schemes in the literature that reported values of 7.9966 and 7.9968. The team also subjected the method to differential attack analysis using two standard indices: the number of pixels change rate, NPCR, and the unified average changing intensity, UACI. The scheme achieved NPCR values of approximately 99 percent and UACI values near 33 percent, the theoretical benchmarks for resisting differential attacks. These figures mean that flipping a single pixel in the plaintext produces a completely different ciphertext, making chosen-plaintext and known-plaintext attacks extraordinarily difficult to mount.

The study positions itself within a broader research landscape in which one-dimensional chaotic maps have given way to richer two- and three-dimensional systems, and in which DNA operations are increasingly paired with chaos to compensate for their individual weaknesses. Earlier efforts combined chaotic maps with SHA-256 hashing, fractal structures, substitution boxes and hyperchaotic dynamics, each adding layers of complexity. The new contribution distills these lessons into a lightweight framework that pairs a well-understood permutation mechanism with biologically inspired key generation, achieving strong security without the computational overhead of multi-dimensional chaotic systems. The authors acknowledge that their future work will probe the scheme’s resistance to chosen-plaintext and known-plaintext attacks under varying ACM parameters and DNA sequences. For now, the results suggest that the humble tools of molecular biology, when married to the mathematics of chaos, can offer a surprisingly powerful shield for the images the world increasingly depends on keeping secret.

Subject of Research: Hybrid image encryption combining DNA cryptography and the Arnold Cat Map chaotic transformation

Article Title: Enhanced image encryption using a hybrid technique based on dna cryptography and arnold cat map

Article References: Bano, M., Iqbal, J., Habib, U., Hajjej, F., & Ullah, I. (2026). Enhanced image encryption using a hybrid technique based on dna cryptography and arnold cat map. Cluster Computing, 29(13), Article 770. https://doi.org/10.1007/s10586-026-06499-7

Image Credits: AI Generated

DOI: 10.1007/s10586-026-06499-7

Keywords: image encryption, DNA cryptography, Arnold Cat Map, chaotic maps, XOR operation, cryptography, information entropy, PSNR, histogram analysis, NPCR, UACI, data security

The stakes are far from academic. Maintenance routinely consumes the largest share of a software project’s lifetime budget, often dwarfing the original cost of development. If engineers could reliably flag, early in a project’s life, which modules are likely to become maintenance nightmares, they could refactor them before the costs spiral. The traditional approach relies on static code metrics—counts of lines, cyclomatic complexity, coupling between classes, depth of inheritance trees, and similar structural measurements—fed into statistical or machine learning models that classify modules as maintainable or problematic. Over the past two decades, researchers have tried everything from regression analysis and decision trees to support vector machines and ensembles, and more recently deep learning architectures such as long short-term memory networks, which are better at capturing nonlinear relationships among dozens of correlated metrics.

Yet deep learning brings its own difficulties, and the new framework is essentially a systematic campaign against them. The first obstacle is class imbalance. In real software repositories, the vast majority of modules are reasonably maintainable, while the genuinely troublesome ones form a small minority. A naive classifier can achieve deceptively high accuracy simply by predicting that everything is fine, while completely failing at its actual job of identifying the rare problem modules. Goel and Kaur address this with an improved variant of SMOTE, the Synthetic Minority Over-sampling Technique, which generates synthetic examples of the underrepresented class by interpolating between existing minority samples. Their improved version expands the sample generation space, producing more diverse synthetic instances so that the decision boundary the network learns is not distorted by the lopsided original distribution.

The second obstacle is feature selection. Software metrics datasets contain a bewildering array of overlapping measurements—average lines of code, average cyclomatic complexity in its several variants, counts of methods, comments, semicolons, coupling measures, and cohesion statistics—many of which carry redundant or misleading information. Feeding all of them into a neural network invites overfitting and slows training. The framework employs a binary version of a relatively new metaheuristic called the Multi-Strategy Sand Cat Optimization Algorithm, or BMSCSOA. Inspired by the hunting behavior of sand cats, which can detect prey underground with remarkable sensitivity, the algorithm treats each candidate solution as a binary vector in which a one means the corresponding feature is selected and a zero means it is discarded. Its multi-strategy design lets the search alternate between exploration of the vast feature space and exploitation of promising regions, gradually converging on the most salient subset of metrics.

The third and perhaps most technically inventive component concerns how the LSTM network itself is configured. Hyperparameters—the learning rate, dropout rate, batch size, number of layers, neurons per layer, epochs, and patience settings—profoundly affect performance, but tuning them by hand or by brute-force grid search is prohibitively expensive. Worse, standard optimization runs risk getting trapped in local optima, settling for a mediocre configuration simply because the search started in an unlucky region. The authors tackle this with a two-pronged strategy. First, they initialize the search population using chaotic maps combined with fast random opposition-based learning, or FROBL. Chaotic maps inject deterministic but ergonomically scattered sequences into the initial population, ensuring broad, even coverage of the search space, while opposition-based learning evaluates both a candidate solution and its mirror image, roughly doubling the chance of starting near a good region of the landscape.

Once the population is seeded intelligently, the framework turns to hyperparameter optimization proper, deploying a hybrid of two nature-inspired algorithms: the Tunicate Swarm Optimization Algorithm, which mimics the jet-propulsion and swarming behavior of marine tunicates, and the Walrus Optimization Algorithm, modeled on the social and foraging behavior of walruses. By running these optimizers and hybrids of them against the LSTM’s hyperparameter space, the framework searches for the configuration that yields the best classification performance on the balanced, feature-selected data. The result is a sequential pipeline—improved SMOTE, then BMSCSOA feature selection, then chaotic FROBL initialization, then tunicate-walrus hyperparameter tuning—that the authors call the Metaheuristic-ChaoticFROBL-LSTM framework, wrapped in a HybridMetaLSTM architecture.

The empirical evaluation is unusually broad. The team tested the framework on five datasets spanning very different eras and styles of software: UIMS and QUES, classic object-oriented systems from the early 1990s originally studied by Li and Henry; JM1, a large procedural dataset from the PROMISE repository of empirical software engineering data; and two newly collected datasets, AN and AK, drawn from Android NET versions 13 and 14 and Apache Kafka versions 3.6 through 3.7 respectively, which the authors themselves curated and published in 2025. The features span the full alphabet soup of software measurement, from Chidamber-Kemerer metrics such as weighted methods per class and depth of inheritance tree to dozens of line-count and complexity statistics. Performance was assessed with three standard classification metrics: accuracy, the area under the ROC curve, and the F1-score, which balances precision against recall.

The reported results are striking. Across all five datasets, the framework achieved accuracy ranging from 97.89 percent to 99.01 percent, AUC values between 97.94 and 99.13 percent, and F1-scores from 96.15 to 97.51 percent. According to the authors, these figures outperform a battery of existing state-of-the-art techniques, including gradient boosting, random forests, optimized extreme learning machines, naive Bayes, support vector machines, and earlier metaheuristic-tuned neural approaches. The consistency of the gains across datasets as different as a 1990s user interface management system and a modern streaming platform suggests the pipeline is not merely overfitting to one data distribution, although independent replication on additional industrial codebases would strengthen the claim considerably.

The broader significance of the work lies less in any single algorithm than in its demonstration that the messy, practical problems of applying machine learning to software engineering data can be engineered away layer by layer. Imbalance, redundancy, and hyperparameter sensitivity are not exotic difficulties; they afflict nearly every predictive model built on real-world software repositories, from defect prediction to effort estimation. By showing that a carefully orchestrated combination of resampling, binary swarm-based feature selection, chaos-assisted population initialization, and hybrid metaheuristic tuning can lift LSTM performance to near-ceiling levels, the study offers a template that other researchers in predictive software analytics are likely to adapt. It also joins a rapidly growing literature in which metaheuristics—sand cats, tunicates, walruses, chimp optimizers, and their many relatives—are routinely hybridized with deep networks for tasks ranging from renewable energy forecasting to medical diagnosis.

Caveats remain, as they always do. The datasets, while diverse, are still modest in size compared with the massive code corpora of large technology companies, and the computational cost of running multiple metaheuristics in sequence is nontrivial, which may matter for teams hoping to integrate such predictions into continuous integration pipelines. The authors report no external funding and declare no conflicts of interest, and they have made their datasets available for other researchers to scrutinize and extend. Whether chaotic maps and walrus-inspired search will become standard equipment in the software quality toolbox is an open question, but the message of the study is clear: the machinery for predicting which code will hurt tomorrow is getting sharper, and the economics of software maintenance may never look quite the same.

Subject of Research: A metaheuristic and chaos-based LSTM framework for predicting software maintainability

Article Title: A novel Metaheuristic-ChaoticFROBL-LSTM framework for enhancing software maintainability prediction

Article References: Goel, V., & Kaur, A. (2026). A novel Metaheuristic-ChaoticFROBL-LSTM framework for enhancing software maintainability prediction. Cluster Computing, 29(14), Article 829. https://doi.org/10.1007/s10586-026-06597-6

Image Credits: AI Generated

DOI: 10.1007/s10586-026-06597-6

Keywords: software maintainability prediction, LSTM, metaheuristic optimization, chaotic maps, opposition-based learning, SMOTE, feature selection, sand cat optimization, tunicate swarm algorithm, walrus optimization, software metrics, machine learning

The mathematical heart of the construction is a Lorenz system whose parameters are not fixed. In the classical Lorenz equations, the parameters sigma, rho, and beta are constants that shape the famous butterfly-shaped attractor. Kareem and Pirdawood instead allow these parameters to be dynamically modified based on the system’s own state trajectory and on information derived from the plaintext image itself. This produces what the authors describe as non-stationary chaos: a chaotic process whose statistical properties shift over time rather than settling into a predictable pattern. The practical consequence is a significant improvement in resistance to phase-space reconstruction attacks, a family of techniques in which an eavesdropper attempts to rebuild the underlying attractor from observed outputs and thereby reconstruct the pseudo-random keystream that protects the image.

Chaotic maps have long been attractive to cryptographers because of their extreme sensitivity to initial conditions and their high nonlinearity. A tiny change in the starting state of a chaotic system produces an exponentially diverging trajectory, which is precisely the property one wants in a keystream generator. However, as the authors note, many existing image encryption schemes suffer from degraded chaotic properties, particularly when implemented on finite-precision digital hardware where chaotic orbits can fall into short, repeating cycles. Others rely too heavily on a single protection mechanism, leaving them vulnerable once that mechanism is compromised. The new paradigm is designed to avoid both pitfalls by layering multiple, independent transformations whose parameters are continuously refreshed by the hyperchaotic system.

The encryption pipeline unfolds in four sequential stages. In the first, the bits of the image undergo permutation driven by a hyperchaotic sequence, shuffling the individual bits of each pixel rather than merely rearranging whole pixels. Bit-level permutation is far more destructive to an image’s internal structure than pixel-level shuffling, because it scrambles the binary representation itself, destroying correlations at the finest granularity. In the second stage, the scrambled bits are reinterpreted as DNA sequences. Under standard DNA encoding rules, groups of two binary bits are mapped to one of the four nucleotide bases, A, C, G, or T, and the specific mapping rule is selected dynamically by a second chaotic sequence, so that the encoding scheme itself varies across the image and from one encryption run to the next.

The third stage introduces a confusion layer built on a DNA-based XOR operation. Here, the DNA-encoded image data is combined with a nucleotide stream derived from the secret key, performing the familiar exclusive-or logic in the four-letter DNA alphabet rather than on raw binary. Because the encoding rule and the keystream both evolve chaotically, an attacker who recovers part of the nucleotide stream gains little leverage over the rest. Finally, a fourth stage applies chaotic bit masking, a last diffusion pass in which chaotic sequences once again modify the bit values directly. The authors formally prove the perfect reconstructability of the scheme, meaning the entire chain of transformations can be exactly inverted with the correct key, so decryption recovers the original image without any loss.

Security analysis of the scheme yields numbers that approach the theoretical ideals used to benchmark image ciphers. When a single pixel of a plaintext image is changed, the encrypted versions differ at a Number of Changing Pixel Rate exceeding 99.61 percent, and the Unified Average Changing Intensity reaches approximately 33.46 percent, both near the optimal values expected of a cryptographically strong system. These metrics, NPCR and UACI, measure whether an attacker can probe the cipher by making tiny changes to the input and observing the output; near-ideal results indicate that even a one-bit change in the plaintext cascades into a completely different ciphertext, a property known as plaintext sensitivity that defeats differential attacks.

Empirical validation on standard test images reinforces the formal results. The encrypted images exhibit uniformly distributed histograms, meaning that no gray level occurs more often than any other and no statistical fingerprint of the original image survives. Information entropy, a measure of randomness, reaches approximately 7.997 on an eight-bit scale where the theoretical maximum is 8, indicating ciphertext that is nearly indistinguishable from pure noise. Correlation coefficients between adjacent pixels, which in natural images are typically close to 1 because neighboring pixels look alike, fall below 0.01 in absolute value in the encrypted output. In visual terms, the encrypted images appear as featureless static, revealing nothing about faces, text, or structures hidden within.

The choice of DNA operations is more than an aesthetic flourish. DNA computing and DNA cryptography have emerged as a fertile research area because the four-symbol alphabet and the combinatorial explosion of possible encoding rules add layers of complexity that are awkward to replicate in pure binary arithmetic. Recent literature has explored eight-base DNA encodings, DNA-based linear feedback shift registers, and diffusive DNA coding operations combined with hyperchaotic systems with cross-feedback structures. Kareem and Pirdawood’s contribution is to bind these DNA operations tightly to a non-degenerate hyperchaotic map, ensuring that the rule selection, the nucleotide keystream, and the bit masks all derive from the same continuously evolving chaotic state, so the layers reinforce rather than duplicate one another.

The work arrives amid a broader surge of research into chaos-based image security, driven in part by the explosion of visual data transmitted across the Internet of Things, in medical telemetry, and in cloud storage. Related efforts in recent years have produced memristive hyperchaotic maps for pseudorandom number generation, fractional-order Hopfield neural networks applied to medical image privacy, lightweight chaotic ciphers for resource-constrained IoT devices, and hybrid schemes combining elliptic curve cryptography with genetic algorithms. The field has also learned hard lessons from cryptanalysis: several published chaotic ciphers have been broken when their chaos degenerated under finite precision or when their permutation and diffusion stages were insufficiently coupled. The new scheme’s designers explicitly target these weaknesses, grounding the design in formal proofs of key and plaintext sensitivity rather than relying solely on empirical testing.

For practitioners, the significance lies in the combination of provable properties and measured performance. A key space above 2^456 dwarfs the roughly 2^128 offered by many mainstream block ciphers in raw key size terms, and the formal sensitivity proofs provide assurance that the scheme does not harbor structural shortcuts. The authors, who contributed equally to the research, report no external funding and declare no competing interests. As data breaches grow in severity and images increasingly serve as high-capacity carriers of private information, encryption paradigms that combine nonlinear dynamics with unconventional alphabets may become essential infrastructure. This study suggests that the marriage of hyperchaos and DNA-style computation offers a cryptographically strong framework for keeping the visual world’s most sensitive frames out of unauthorized hands.

Subject of Research: Hyperchaotic image encryption using DNA sequence operations and multi-stage dynamic bit-level scrambling

Article Title: A hyperchaotic image encryption paradigm fusing DNA sequence operations with bit-level multi-stage dynamic scrambling

Article References: Kareem, S. R., & Pirdawood, M. A. (2026). A hyperchaotic image encryption paradigm fusing DNA sequence operations with bit-level multi-stage dynamic scrambling. Cluster Computing, 29(13), Article 772. https://doi.org/10.1007/s10586-026-06605-9

Image Credits: AI Generated

DOI: 10.1007/s10586-026-06605-9

Keywords: image encryption, hyperchaotic system, Lorenz system, DNA cryptography, bit-level permutation, chaotic maps, NPCR, UACI, information entropy, phase-space reconstruction attacks, Cluster Computing, cybersecurity

The research team, led by Ahmed Kamal of Alexandria University together with H. A. El-Kamchochi, Adel El-Fahar, and Esam A. A. Hagras of Delta University for Science and Technology, calls the framework CCEC-SIE, short for Chaotic Complex Elliptic Curve-based Secure Image Encryption. The scheme’s central innovation is deceptively simple: instead of defining elliptic curves over ordinary prime fields, the researchers define them over Gaussian primes, complex numbers of the form c plus di whose norm equals an ordinary prime. Because each coordinate in such a field is itself a complex number, the number of available points on the curve grows quadratically, expanding the cryptographic space without requiring larger primes or heavier computation.

The mathematics behind this expansion follows directly from Hasse’s theorem, which bounds the number of points on an elliptic curve over a finite field. For a rational prime p, the field contains roughly p elements, but a Gaussian prime built from the same p yields a field of order p squared. In a worked example using the curve y squared equals x cubed plus x plus 1 over the prime 7, the rational version produces a modest set of points, while the Gaussian extension yields 55 points, comfortably within the Hasse bound of 36 to 64 for a field of order 49. Scaled to cryptographic sizes, this means a 256-bit parameter effectively doubles the key space, a gain achieved with the same curve equation and no increase in prime size.

Key exchange in the scheme adapts the classical ElGamal protocol to this Gaussian domain. The sender picks a random ephemeral key, computes two cipher points on the curve, and transmits them to the receiver, who recovers the shared secret using a private scalar. Because every point coordinate carries both real and imaginary components, the resulting shared key is itself a complex number, which then seeds the entire encryption pipeline. The authors demonstrate the protocol with complete numerical examples, showing that the receiver’s recovered point matches the sender’s original plaintext point exactly, confirming the algebraic correctness of the construction.

Once the shared key is established, the scheme turns to chaos theory for bulk encryption. Classical one-dimensional chaotic maps such as the Logistic and Tent maps are workhorses of image encryption, but they suffer from well-documented weaknesses: restricted chaotic ranges, non-uniform output distributions, periodic windows, and vulnerability to parameter estimation attacks. The researchers address these flaws by splitting each map’s control parameter into a primary value and a dynamically selected coefficient derived from the secret key, then applying a modulo-1 operation to keep iterations bounded. The resulting Iterative Logistic Map and Iterative Tent Map sustain chaotic behavior across a parameter interval roughly ten times wider than their classical counterparts, with Lyapunov exponents remaining positive throughout the operating range. The enhanced maps also passed the full NIST SP800-22 randomness test suite over a thousand independent iterations on million-bit sequences.

These chaotic sequences feed a second major component: a Complex Invertible Key Matrix, or CIKM. The construction generalizes the classical Hill cipher, a linear encryption technique long considered too fragile for modern use, into the complex domain. A chaotic matrix whose entries are complex numbers generated independently by the two enhanced maps is embedded in a block structure alongside identity matrices, producing a larger key matrix that is its own inverse. This self-invertibility means the same matrix encrypts and decrypts, eliminating the computationally expensive inversion step entirely. Pixel-level scrambling is handled separately by Arnold’s Cat Map, whose control parameters are derived from the shared secret key, ensuring that both the permutation and the diffusion stages are keyed and image-dependent.

Authentication is woven into the design rather than bolted on. Before transmission, the sender computes a hash of the encrypted image, combines it with a hash derived from a curve point tied to the plaintext, and produces a digital signature using the sender’s private key. Crucially, the signature is verified before decryption begins, so any tampered or corrupted ciphertext is rejected without wasting computational resources on a meaningless reconstruction. The verification requires only a single complex point multiplication, which the authors note is more efficient than existing authenticated image encryption schemes. This pre-decryption check is particularly valuable in distributed or resource-constrained environments where bandwidth and processing cycles are at a premium.

The security evaluation is extensive. Encrypted versions of standard test images including Lena, Cameraman, Baboon, and Peppers at resolutions from 256 by 256 up to 1024 by 1024 showed uniformly flat histograms, with chi-square tests confirming statistical uniformity that outperformed a recent Sine-map-based competitor. Pixel correlation coefficients, which approach one in natural images, dropped to near zero in all horizontal, vertical, and diagonal directions. Information entropy of the ciphertexts approached the theoretical maximum of 8 bits per pixel for 8-bit grayscale images. Against differential attacks, the scheme achieved a Number of Pixels Change Rate of 99.62 percent and a Unified Average Changing Intensity of 33.46 percent, both matching or exceeding the ideal theoretical values. The total key space, computed from the secret key components, image hashes, private keys, chaotic map parameters, and scrambling controls, reaches an estimated 2 to the power of 896, far beyond the 2 to the 128 threshold generally considered safe against brute-force enumeration.

Perhaps the most surprising results concern computational efficiency, an area where complex arithmetic might be expected to hurt rather than help. Because multiplication of Gaussian integers splits into independent computations of real and imaginary parts, hardware multipliers handle inputs on the order of p rather than p squared, roughly halving the digit count per operand. Under schoolbook long multiplication, this reduces time complexity from order n squared to order n squared divided by 4, a fourfold speedup that persists even when two parallel multipliers compute the real and imaginary parts simultaneously. Elliptic curve operations are deliberately restricted to a constant number per session, independent of image size, so the dominant cost remains efficient pixel-level operations and complex matrix diffusion. The authors report low computational overhead suitable for near real-time secure image transmission at standard resolutions.

The scheme does make a deliberate trade-off. Because the complex self-invertible matrix applies a global transformation, any modification to the ciphertext, whether from noise injection, data loss, or deliberate cropping, propagates across the entire decrypted image. Peak signal-to-noise ratio measurements under Gaussian noise, salt-and-pepper noise, and occlusion attacks at 12.5, 25, and 50 percent data loss all showed severe degradation, rendering decrypted outputs unusable. The authors frame this not as a vulnerability but as a feature aligned with high-security requirements: the digital signature catches any corruption before decryption, and the extreme sensitivity guarantees that tampering cannot go undetected. Future work, they note, will extend the framework to video and hyperspectral imagery and explore adaptive key management strategies, but for now the Gaussian domain offers image encryption a broader, faster, and more defensible mathematical home.

Subject of Research: Gaussian elliptic curve cryptography and chaotic complex matrix methods for secure image encryption and authentication

Article Title: Secure image encryption and authentication using Gaussian elliptic curve and complex invertible matrix

Article References: Kamal, A., El-Kamchochi, H. A., El-Fahar, A., & Hagras, E. A. A. (2026). Secure image encryption and authentication using Gaussian elliptic curve and complex invertible matrix. Cluster Computing, 29(14), Article 798. https://doi.org/10.1007/s10586-026-06373-6

Image Credits: AI Generated

DOI: 10.1007/s10586-026-06373-6

Keywords: image encryption, elliptic curve cryptography, Gaussian integers, chaotic maps, ElGamal key exchange, digital signature, Hill cipher, Arnold Cat Map, cryptography, information security, complex numbers, differential attacks