A novel 1D powered Chebyshev quadratic map-based image encryption using dynamic permutation-diffusion.

Skip to main content An official website of the United States government Here's how you know Here's how you know Official websites use .gov A .gov website belongs to an official government organization in the United States. Secure .gov websites use HTTPS A lock ( Lock Locked padlock icon ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites. Search Log in Dashboard Publications Account settings Log out Search… Search NCBI Primary site navigation Search Logged in as: Dashboard Publications Account settings Log in Search PMC Full-Text Archive Search in PMC Journal List User Guide PERMALINK Copy As a library, NLM provides access to scientific literature. Inclusion in an NLM database does not imply endorsement of, or agreement with, the contents by NLM or the National Institutes of Health. Learn more: PMC Disclaimer | PMC Copyright Notice Sci Rep . 2026 Feb 17;16:9469. doi: 10.1038/s41598-026-38483-x Search in PMC Search in PubMed View in NLM Catalog Add to search A novel 1D powered Chebyshev quadratic map-based image encryption using dynamic permutation-diffusion Boukansous Sarra Boukansous Sarra 1 School of Computer Science and Technology, Zhejiang Gongshang University, Hangzhou, 310018 P.R. China Find articles by Boukansous Sarra 1, ✉ , Hao Sun Hao Sun 2 School of Information and Electronic Engineering, Zhejiang Gongshang University, Hangzhou, 310018 P.R. China Find articles by Hao Sun 2, ✉ , Mohit Dua Mohit Dua 3 Departement of Computer Engineering, National Institute of Technology, Kurukshetra, 136119 India Find articles by Mohit Dua 3 , Shelza Dua Shelza Dua 4 Departement of Electronic and Communication Engineering, Kurukshetra, 136119 India Find articles by Shelza Dua 4 , Deepti Dhingra Deepti Dhingra 5 Departement of Computer Science & Engineering, Panipat Institute of Engineering & Technology, Samalkha, 132102 India Find articles by Deepti Dhingra 5 Author information Article notes Copyright and License information 1 School of Computer Science and Technology, Zhejiang Gongshang University, Hangzhou, 310018 P.R. China 2 School of Information and Electronic Engineering, Zhejiang Gongshang University, Hangzhou, 310018 P.R. China 3 Departement of Computer Engineering, National Institute of Technology, Kurukshetra, 136119 India 4 Departement of Electronic and Communication Engineering, Kurukshetra, 136119 India 5 Departement of Computer Science & Engineering, Panipat Institute of Engineering & Technology, Samalkha, 132102 India ✉ Corresponding author. Received 2025 Sep 17; Accepted 2026 Jan 29; Collection date 2026. © The Author(s) 2026 Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/ . PMC Copyright notice PMCID: PMC13005032  PMID: 41702983 Abstract Chaotic maps have gained significant attention in image encryption systems due to their intrinsic features, including sensitive dependence on initial conditions, non-linear dynamics, high entropy, and ergodicity. These properties enable the generation of complex, pseudo-random sequences that enhance unpredictability and randomness over time. However, many existing chaotic maps suffer from limited chaotic ranges and low adaptability, resulting in reduced encryption strength and vulnerability to differential or statistical attacks. This limitation motivates the development of a new chaotic system with stronger non-linearity, wider chaotic behavior, and improved dynamic flexibility. Based on this, the 1D-Powered Chebyshev Quadratic Map (1D-PCQM) is integrated into a cryptographic framework that aims to improve both security and efficiency. The proposed image encryption scheme referred as Dynamic Triple Chaotic Map Image Encryption Scheme (D3CM-IES) is based on the use of three chaotic maps: the Sine-cosine map, the Sine-Tangent-Sine (STS) map, and the novel 1D-PCQM map. These maps are employed to generate chaotic sequences, and a dynamic selection mechanism is then used to select one of these sequences during encryption, enhancing attack resistance. This scheme is performed through four main steps: (i) generating three chaotic sequences built from the aforementioned chaotic maps; (ii) performing a dynamic selection procedure that selects a single chaotic sequence for encryption; (iii) dynamic switching, where spatial locations (the spatial position of pixels) within the image are rearranged; and (iv) dynamic diffusion, where pixel values are modified to further conceal the image content. A series of experiments and tests were conducted to measure the performance of the proposed scheme and evaluate its ability to withstand known cryptographic attacks. Statistical and security results demonstrated that the proposed scheme exhibits a high degree of robustness. In particular, the method achieves Number of Pixels Change Rate (NPCR) of 99.66% and a Unified Average Changing Intensity (UACI) of 49.94%. Which reflect a high resistance to differential attacks small changes in the plain-text lead to significantly altered cipher-text. These results confirm the method’s effectiveness in providing strong confusion and diffusion, essential for modern image encryption systems. Keywords: Quadratic map, Chebyshev map, Chaotic map, Image encryption, Dynamic permutation and diffusion, Security analysis Subject terms: Engineering, Mathematics and computing, Physics Introduction Due to the prevalence of contemporary communication methods, sharing digital photographs between individuals has become rapid and simple in today’s society. But still, this link isn’t always trustworthy because supplied photos can be modified or altered during transmission. In this regard, manipulation of sensitive images, such as military, medical, and satellite images, by unauthorized parties can lead to major problems for both individuals and organizations. That’s why researchers are working to develop information security strategies like cryptography 1 , 2 and steganography 3 , 4 in order to combat this risk. To achieve this goal, two techniques are used: spatial domain encryption and frequency domain encryption, which transforms an image from its spatial domain to the frequency domain using complex mathematical transformations such as Meixner Moment Transform 5 , Wavelet Transform 6 , Hahn Moment Transform 7 , Fourier Transform, fast FFT 8 . Consequently, transform-based coding systems aren’t as well suited for real-time applications as spatial coding systems because they typically involve more complicated computations and take more resources to set up. The strong security and low complexity of spatial domain cryptosystems make them a popular choice for designing solutions that overcome this limitation, where encryption is implemented on the pixel level using confusion/diffusion in order to alter their true values 9 . One of the most prominent techniques that have been studied Block-Based Encryption, Pixel Shuffling, Image Slicing, Bit-Plane Encryption, DNA-Based Encryption, Rubik’s Cube Method, Chinese Remainder Theorem (CRT) 10 , Fractal-Based Encryption, Quantum Walks 2 , 11 – 13 and, chaotic systems or non-linear dynamics or the so-called “butterfly effect” in which explained by to complex systems that are highly responsive to beginning conditions. Even minor alterations to these conditions can lead to unexpected outcomes 10 . Many researchers have contributed to the study of dynamic systems, where one-dimensional or multidimensional chaotic models have been developed and integrated with various technologies for encryption applications. For example, Li et al. 14 proposed a two-dimensional enhanced logistic modular map and employed dynamic vector-level operations for image encryption. Feng et al. 15 introduced a novel four-dimensional fractional-order Hopfield neural-network–based medical image encryption scheme characterized by a large key space and strong security. Yu et al. 16 presented a six-dimensional memristive chaotic system exhibiting hyperchaotic behavior, which was subsequently utilized to design an image encryption algorithm. In 17 a memristive tri-neuron Hopfield neural network was constructed to implement an image encryption scheme. In recent years, cryptanalysis has played a crucial role in assessing the security and practical reliability of image encryption schemes, particularly those based on chaotic systems and DNA operations. Several studies have demonstrated that many existing schemes may still contain structural weaknesses or insufficient diffusion–confusion mechanisms. For instance, cryptanalytic investigations in 18 suggests that secret key design and encryption process was not appropriate and a chosen-plaintext attack algorithm was introduced in the work. The work in 19 performed cryptanalysis on an image encryption scheme based on 2D Logistic-adjusted-Sine map and suggested some improvements on that. Chen et al. 20 proposed a chosen-plaintext attack and disclosed secret key of medical privacy protection scheme. Motivated by these observations the present work focus on image encryption, one-dimensional chaotic maps and systems are the broadest entry point into this field due to their simple structure, which makes them easy to implement at the hardware and software level 8 , 13 , 21 , 22 . Considering the inherent limitations of many classical one-dimensional chaotic systems, such as their narrow chaotic intervals, poor key sensitivity, and bounded parameter domains, this paper introduces a new chaotic map called the One-Dimensional Powered Chebyshev Quadratic Map (1D-PCQM). Designed to overcome the limitations of its underlying maps, this map combines the fundamental dynamics of both Chebyshev and chaotic quadratic maps into a unified, energy-enhanced framework. Using two control parameters, the map achieves a delicate balance between simplicity and rich nonlinear dynamics, while simultaneously expanding the scope of chaos and improving sensitivity to initial conditions. Related work As the theory of chaos evolved, techniques for enhancing the dynamical features in current chaotic models have been discovered 23 – 27 . One of the main benefits of 1D chaotic maps is tremendous implementation efficiency, which surpasses that of all other chaotic systems as they exhibit complex, chaotic features despite the apparent simplicity of their mathematical framework. Consequently, 1D chaotic maps have found extensive application in encryption methods based on chaos 28 – 32 . 25 Develops a unique chaotic model that improves the dynamics of the 1D Logistic map primarily through the utilization of sine function perturbations. 23 Created novel chaotic sequences by merging two renowned one-dimensional (1D) chaotic systems, resulting in expanded ranges and intensified chaotic dynamics. Zhou, Bao, and Chen 33 proposed a novel technique for encrypting images by merging two widely recognized 1-dimensional chaotic maps to create a chaotic system. This integration leads to the generation of novel chaotic sequences that have wider ranges and exhibit heightened chaotic behavior. It is common practice to combine data from the plain-text image using the Chebyshev map, which has a large parameter space 34 . To further increase complexity and enhance the unpredictability of chaotic systems, researchers have turned to memory-based components such as memristors and memcapacitors. Lai et al. 35 In his paper, proposed an advanced chaotic system based on the integration of two memristors, leading to the development of a multi-scale chaotic system (MMCS). The obtained results demonstrated the complexity of the MMCS system and the effectiveness of the proposed system was demonstrated through experiments on image and audio data security, highlighting the great potential of memristors for broad applications in secure communications systems. In 36 the authors presented a novel system that integrates discrete memcapacitors (which are defined as memory capacitors that store information and have properties dependent on past states) with a Chebyshev hyper-chaotic map. This coupling of capacitors and the hyper-chaotic Chebyshev map increases the complexity of the system, as demonstrated by the results obtained by revealing complex dynamical behavior, multiple attractors, and high chaotic performance. However, 37 Wang, Luan, and Bao analyze the security of the algorithm of image encryption using the Chebyshev generator, where they focus on identifying weaknesses and potential weaknesses in the encryption method. However, the results proved that image encryption systems based on the Chebyshev map have low sensitivity to changes in regular images. To overcome this drawback, the researchers merged the Chebyshev map with other maps in order to enhance the strength of the chaotic system in encryption. In 38 Qi, Huang, Li, Yang, and Kang present a new image encryption system based on a 2D chaotic Henan-Chebyshev map. This strategy enhances security by utilizing the collective chaotic dynamics of the Henan and Chebyshev maps, resulting in improved chaotic behaviors, a broader chaotic range, and enhanced ergodicity. Wang and Du proposed a bit-level image encryption method based on the Logistic-Chebyshev dynamic coupled map lattice 39 . Liu et al. proposed a 2D chaotic map(2D-LACM) called the Logistic-Adjusted-Chebyshev map 40 , this map enlarges the range of chaotic control parameters, enhancing the flexibility and precision of chaotic models used in various applications. Basha et al. 41 proposed a new method for encoding color images by taking advantage of the properties of a group of chaotic maps (logistic, sinusoidal, tent, and Chebyshev maps). This mixture, referred to as the LSTC map, operates at the bit level through permutations and substitutions, and the experiments ensured a good resistance to statistical and differential attacks. Lai et al. 42 presented a new image encryption method that combines compressed sensing and non-uniform pixel segmentation. This technique enhances the encryption process’s efficiency by partitioning medical images into uneven pixel blocks and transforming them into cipher-text images with adjustable sizes, effectively tackling the difficulties of encrypting large medical datasets. Zhang et al. designed a color image encryption method that employs a 3D chaotic system combined with cross-spiral transformation and zone diffusion, providing strong protection against various types of attacks 43 . 44 – 47 proposed a novel method for encrypting videos, which utilizes dynamic S-box and chaotic maps to improve the security of video data significantly. This chaotic maps used to generate intricate and unforeseeable sequences that dynamically switch the S-box utilized for replacement in the encryption procedure. Through the dynamic modification of the S-box using these chaotic sequences, the system guarantees that every frame or block of video data is encoded distinctly, hence greatly enhancing its resistance against attacks. The works in 48 and 49 integrate DNA computing with chaotic systems to construct encryption schemes. A Deep Learning based LSTM-KAN chaotic map was proposed in 50 using data-driven neural architecture. In 51 , a Logistic-Henon-Arnold-Lorenz system-9D chaotic system was employed for image in combination with a hashing mechanism a Convolutional Neural Network . Additionally several researchers 52 – 57 have developed advanced permutation and diffusion strategies to enhance the robustness of the image encryption schemes. From the above work, we can conclude that existing one dimensional chaotic maps 1D maps like the Logistic Map have limited dynamics, predictable under certain conditions, making them less secure for cryptographic use. Motivated by the above work, This study introduces a novel one-dimensional hybrid chaotic map, termed the 1D-powered Chebyshev Quadratic Map (1D-PCQM), which fuses the structural dynamics of the Chebyshev map with the nonlinear properties of the quadratic map. The motivation behind this hybridization is to enhance the chaotic characteristics of both maps by leveraging their individual strengths Chebyshev’s robustness and high parameter sensitivity, combined with the quadratic map’s simplicity and rich nonlinear behavior. The resulting system exhibits improved complexity, larger key space, and higher entropy, making it suitable for advanced applications requiring strong chaotic properties. Unlike many existing 1D chaotic maps that rely on simple polynomial or trigonometric nonlinearities, the 1D-PCQM introduces a powered nonlinear transformation coupled with a quadratic modulation, resulting in richer dynamic complexity and an extended chaotic range. Moreover, its mathematical simplicity makes it highly adaptable to different nonlinear systems and hardware models.Indeed, similar forms of the proposed structure have recently been explored in memristor-based dynamic systems and fractional-order models, where the power-based nonlinearity plays a crucial role in enhancing system memory and chaotic variability. This confirms the broader applicability and theoretical value of the proposed mechanism beyond a mere incremental modification. To evaluate this map, we used well-known and popular metrics in this regard, such as the Lyapunov exponent, 0-1 test, bifurcation diagram (BD), and other tests. Lyapunov exponent is positive, and a significantly higher entropy value indicates higher randomness, indicating increased inherent complexity, and 0-1 analysis emphasizes randomness and chaotic behavior. Using these results, we developed an image encryption technique that incorporates our new 1D-PCQM chaotic map with a Sine-Cosine map and an STS map in the permutation and diffusion stages, these maps are used dynamically, further improving the integrity and reliability of the encrypted images. Our contributions Below is a comprehensive, point-wise overview of the contributions made by the proposed work (i) The proposed 1D-PCQM introduces two control parameters, ω and γ, defined over the extended domain , allowing the map to achieve full-range chaotic behavior that is independent of specific parameter intervals. This marks a significant advancement over classical chaotic maps, where chaos is typically limited to narrow parameter ranges such as for the Logistic map, and for standard Quadratic maps. By removing these constraints, the 1D-PCQM enhances both the chaotic diversity and applicability of the system, especially in domains like cryptographic image processing, where a broader and more stable chaotic response is essential. Furthermore, the map exhibits high entropy, strong initial condition sensitivity, and complex nonlinear dynamics, making it highly suitable for secure encryption applications. (ii) By exploiting the proposed one-dimensional quadratic Chebyshev map (1D-PCQM), this study presents a new image encryption scheme referred as Dynamic Triple Chaotic Map Image Encryption Scheme (D3CM-IES) that combines the properties of STS, Sine-Cosine, and 1D-PCQM chaotic maps. The D3CM-IES relies on pixel permutation, dynamic chaotic sequence selection, and diffusion. The dynamic chaotic sequence selection procedure is used, which uniquely selects the chaotic sequence for each pixel during encryption. (iii) The proposed encryption scheme demonstrated high sensitivity to decryption keys. This property enhances resistance to unauthorized access and crypt-analysis attacks. The effectiveness and resilience of the scheme have been verified through comprehensive security analyses, including statistical and differential testing. This study is organized as follows: Section " 1D-Powered Chebyshev chaotic map " presents the classical chaotic maps used, providing a detailed description of the proposed new map, the one-dimensional Chebyshev quadratic map (1D-PCQM), including its mathematical structure and dynamic behavior. This section also addresses the shortcomings of conventional maps in terms of dynamic properties, which provided the motivation for developing 1D-PCQM. Section " Proposed work " reviews the adopted encryption scheme, demonstrating how 1D-PCQM can be employed alongside other chaotic maps within a dynamic switching and diffusion model. Section " Implementation and security analysis " addresses experimental aspects, providing a comprehensive analysis of the security performance of the proposed system, as well as a systematic comparison with other image encryption techniques. 1D-Powered Chebyshev chaotic map The 1D-powered Chebyshev chaotic map (1D-PCQM) was created using a combination of quadratic chaotic maps 58 and Chebyshev 59 . The classic chaotic maps Researchers often use Chebyshev maps and quadratic chaotic maps in image coding, although these maps have some limitations, so (1D-PQCM) has been created to address these limitations by combining the strengths of quadratic chaotic maps and Chebyshev. Chebyshev map The Chebyshev map is a specific form of the chaotic map represented in one dimension that demonstrates unique non-linear dynamic characteristics. The map displays chaotic behavior when the control parameter ω is altered within the range [2 ∞]. It can be defined as follows: 1 Quadratic map The quadratic map stands as a cornerstone in the study of dynamical systems, embodying the elegance of simplicity coupled with intricate dynamical behaviors. Defined by the recurrence relation: 2 1D-PCQM proposed map The present study proposes a novel one-dimensional powered Chebyshev Quadratic chaotic map (1D-PCQM), which is a combination of the quadratic map and the Chebyshev map. It is well known that in chaotic maps, the quadratic map exhibits clearly chaotic behavior at certain values of gamma, while the Chebyshev map exhibits periodic oscillations. To effectively integrate these two dynamics, we build a modulation factor feeds on an exponent that amplifies the nonlinear response of the system and directs the system’s behavior based on specific parameters or state variables., enhancing its sensitivity to initial conditions. 3 So we can defined the chaotic map mathematicly as: 4 Where the control parameter influences the map’s behavior. When , the exponential term rapidly increases, intensifying the chaotic nature of the map. Conversely, for , the exponentiated term introduces a damping effect that can induce stability or periodicity. Performance evaluation Evaluating the performance of a chaotic map is a crucial step in uncovering its properties and understanding its dynamic behavior. In this section, we conduct a comprehensive set of analyses to study the output trajectory, sensitivity, and overall chaotic behavior with utmost precision. These analyses contribute to establishing a solid foundation for understanding the map’s complexities. Lyapunov exponent In the context of dynamical systems, the inherent sensitivity to initial values is a prerequisite for characterizing their chaotic properties. Chaotic systems are characterized by their ability to produce variable outputs in response to infinitesimal deviations in initial conditions, which calls for a quantitative assessment of this sensitivity through the Lyapunov exponent 60 . When applied within the framework of a given chaotic system, the Lyapunov exponent serves as a mathematical tool and quantitative indicator that measures the degree of trajectory divergence resulting from small differences in initial conditions. This statistical analysis enables a deeper understanding of the sensitivity and unpredictability inherent in the dynamical evolution of a system. 5 Analysis of Figure 1 reveals a persistent positivity in the lyapunov exponent across the parameter domain for the 1D-PCQM map, signifying a regime of intricate chaotic dynamics. In contrast, both the Chebyshev and Quadratic maps exhibit dissimilar dynamical characteristics, acompanied by comparatively lower lyapunov exponent values, as shown in Figures 2 and 3 . Fig. 1. Open in a new tab Lyapunov exponent of 1D-PCQM map. Fig. 2. Open in a new tab Bifurcation diagram and the Lyapunov exponent of the Chebyshev map. Fig. 3. Open in a new tab Bifurcation diagram and the Lyapunov exponent of the Quadratic map. Bifurcation behavior of the proposed map Alawida et al. (2022) in 61 , demonstrated that bifurcation analysis is a very effective methodology for measuring sensitivity complexity. Bifurcation analysis provides insights into how a system’s behavior evolves as key parameters change, highlighting complex dynamics and identifying bifurcation points where qualitative changes in system behavior occur. This methodological approach serves as a valuable tool for comprehensively understanding and characterizing the sensitivity inherent in chaotic maps. Figure 4 represents the bifurcation diagram of the new 1D-PCQM map, which shows that the proposed map exhibits more randomness than the classical Chebyshev and Quadratic maps, as shown in Figures 2 and 3 , respectively. Fig. 4. Open in a new tab The bifurcation diagram of 1D-PCQM. ( a ) The bifurcation diagram of 1D-PCQM with . ( b ) The bifurcation diagram of 1D-PCQM with . Lyapunov stability analysis Lyapunov stability analysis provides a systematic framework for determining the stability properties of equilibrium points in nonlinear dynamical systems. In the case of discrete maps, stability of a fixed point can be assessed directly through the stability criterion: if the magnitude of the fixed point stable (attractive),If the fixed point unstable (repulsive).And If , the stability of the fixed point requires further analysis, as it is neutral or non-hyperbolic. To begin the stability analysis, we first identify the fixed points of the system. These are points where the system does not change over time, i.e., where the map function satisfies . At these fixed points, the system will either converge (stable) or diverge (unstable), depending on the behavior of the derivative of the map. 6 After performing these calculations, the fixed points of the system are found to be: To assess the stability of these fixed points, we calculate the derivative of the map at each of these points: 7 Using the chain rule, this derivative can be computed as: 8 This yields: 9 After evaluating at each fixed point , the following results were obtained: According to theorem of stability criterion for fixed points we conclude that all the fixed points in this system are unstable since for each of them. This is a clear indication that the system exhibits chaotic behavior, where small perturbations lead to large deviations from the fixed points. Sensitivity test For a visual assessment of the proposed system’s sensitivity to initial values and control parameters, Figure 5 plots the generated chaotic sequences with slight variations ( ) in these factors, demonstrating the 1D-PCQM’s high sensitivity to both initial conditions and control parameters. Fig. 5. Open in a new tab Sensitivity of the 1D-PCQM Map to Initial Conditions and Control Parameter . Correlation dimension The correlation dimension CD provides a quantitative assessment of the geometric complexity of an attractor within a nonlinear dynamical system. It measures how distances between points in the phase space are distributed, allowing for a clear distinction between stochastic noise and deterministic chaotic behavior. The Grassberger-Procaccia algorithm 40 , 48 is widely employed to estimate this dimension. Fig 6 shows that CD fluctuates irregularly across iterations, indicating a continuously evolving structure typical of chaotic behavior. A peak occurs at iteration 380, where CD reaches approximately 1.2403, highlighting a temporary increase in complexity. Fig. 6. Open in a new tab Correlation dimension analysis. The absence of a stable plateau confirms that the system does not settle into periodic or fixed patterns, demonstrating its chaotic nature 0-1 test The 0-1 test, as expounded in multiple scholarly works 62 , 63 , emerges as a discerning tool for characterizing chaotic systems. Departing from the conventional Lyapunov exponent evaluation, the 0-1 test delves into the system dynamics through the analysis of generated sequences across successive iterations. Given a real constant r, a predetermined number of rounds N, and data sequences denoted as D(n) for n=1,2,3, ..., N, the computation of the test statistic K is articulated as : 10 Where 11 And 12 13 The system is chaotic if the calculated value of K is close to 1, indicating a very high degree of complexity in the dynamics. Conversely, a value of K close to 0 indicates that the proposed dynamic system has non-chaotic behavior. The K values of the 1D-PCQM map are represented in Figure 7 . Fig. 7. Open in a new tab K values. Sample enstropy Sample Entropy (SE), as delineated by Richman and Moorman (2000) 64 , stands as a pivotal methodology for rigorously evaluating the complexity intrinsic to time series data. It serves to quantitatively explicate the degree of self-similarity within sequences emanating from dynamical systems. Given a time series of dimension N, template vector with dimension m and an acceptance tolerance r, SE is formally defined as : 14 The symbols A and B represent the counts of vectors satisfying and , respectively. The metric denotes the Chebyshev distance between and . A discernibly larger SE value signifies a diminished level of regularity, indicative of heightened complexity inherent in the time series. Figure 8 represents a comparative sample entropy of Chebyshev map, quadratic map, and our proposed map 1D-PCQM . The observed high variance is primarily a consequence of the map’s stronger sensitivity to initial conditions and parameter perturbations. Such sensitivity produces trajectories that explore the phase space more irregularly, resulting in SampEn values that fluctuate more dynamically. This behavior is characteristic of high-complexity chaotic systems, where stronger nonlinear interactions produce richer and more diverse local dynamical patterns. In contrast, the Chebyshev map is a classical chaotic map with relatively simple polynomial dynamics, leading to more stable and less variable SampEn outputs. Therefore, the higher variance in SampEn for the proposed map signals enhanced unpredictability and a higher degree of dynamical complexity, which are desirable properties. Fig. 8. Open in a new tab Sample entropy of Chebyshev map, Quadratic map, 1D-PCQM. Spectral entropy Spectral entropy (SE) 37 is an effective measure for quantitatively evaluating the randomness of a chaotic sequence. Higher SE values indicate a greater similarity between the chaotic sequence and an ideal random sequence, reflecting stronger unpredictability and higher security. Following the methodology in 37 , the normalized SE values of the sequences generated by our proposed 1D chaotic map were computed across a range of the control parameter ω. As illustrated in Figure 9 , the SE exhibits significant variations along the parameter domain, with values reaching up to approximately 0.9 and dropping at certain points. The high SE values correspond to strong chaotic behavior and a wide frequency spectrum, indicating the map’s capability to generate complex sequences suitable for cryptographic applications. The sharp drops in SE represent periodic or quasi-periodic windows, also referred to as “windows of order”. These windows occur when the chaotic system temporarily behaves in a more regular, predictable manner, resulting in reduced randomness. Such windows are natural and expected in all real chaotic systems, including classical examples like the Lorenz system, Logistic map, and Chen system. The existence of these windows demonstrates that the system is not artificially constrained to remain fully chaotic; instead, it exhibits true nonlinear dynamics with transitions between order and chaos. Fig. 9. Open in a new tab Spectral Entropy vs. Control Parameter and . Furthermore, the rapid alternation between high SE (chaotic) and low SE (periodic) regions highlights the system’s high sensitivity to control parameters, which is a hallmark of chaotic behavior. This sensitivity ensures that small changes in parameters can significantly alter the system’s output, enhancing unpredictability and thereby improving the security of cryptographic applications . Poincare section To further analyze the dynamical behavior of the proposed chaotic system, we employ Poincaré sections, a widely used tool in nonlinear dynamics. A Poincaré section captures the intersections of system trajectories with a chosen lower-dimensional hypersurface, providing a discrete representation of the system’s evolution. By examining the distribution of Poincaré points, one can identify periodic, quasi-periodic, or chaotic behavior: dense, irregularly scattered points indicate chaotic dynamics, while regular patterns correspond to periodic or quasi-periodic motion. This analysis complements Lyapunov exponent calculations and bifurcation diagrams, offering a visual and quantitative tool to assess the complexity and unpredictability of the system. Figure 10 present the Poincaré points plot of the proposed system, with on the horizontal axis and ​ on the vertical axis, reveals intricate dynamical behavior indicative of mixed chaotic and periodic regimes. Points are colored according to the iteration index, allowing visualization of the system’s temporal evolution. In the region , points are densely scattered without discernible structure, reflecting strong chaotic dynamics. Near the central region and for , vertical alignments of points emerge, suggesting periodic or quasi-periodic windows embedded within the overall chaos. The gradual color transition shows that while initial iterations are widely dispersed, later iterations tend to cluster along specific structures, indicating partial convergence to certain attractors in phase space. This combination of scattered chaotic points and structured periodic features demonstrates the system’s high sensitivity to initial conditions and complex behavior properties that are crucial for cryptographic applications, as they ensure unpredictability, high entropy, and robustness against statistical attacks. Fig. 10. Open in a new tab Poincaré section of the proposed 1D powered Chebyshev map using time delay embedding (τ = 1), with color coding based on iteration index. Randomness test The NIST Statistical Test Suite (NIST-STS) comprises 15 tests designed to evaluate the randomness of binary sequences by assessing key statistical properties such as distribution, correlation, and entropy factors critical for cryptographic security. In this study, binary sequences are generated using the 1D-Powered Chebyshev Quadratic Chaotic Map (1D-PCQM), with parameters carefully configured within the chaotic regime to ensure high randomness. The results, including average p-values for each test, are summarized in Table 1 . A sequence is considered to pass the test if its p-value is greater than or equal to α = 0.01, the threshold indicating statistical significance and randomness. Each test includes its p-value status (Passed/Failed), the results indicate that the 1D-Powered Chebyshev Quadratic Chaotic Map (1D-PCQM) generates binary sequences exhibiting excellent statistical randomness. Table 1. NIST Statistical Test Results for 1D-PCQM. Type of test p-value Status Frequency (Monobit) Test 1 passed Frequency test within a block 0.9583423768462339 passed Runs Test 0.822757303233927 passed Test for the longest run of ones in a block 0.5927475338636963 passed Binary matrix rank test 0.36585369354417907 passed Non-overlapping template matching test 0.3300062028030891 passed Overlapping template matching test 0.5910765925801028 passed Maurer’s “Universal Statistical” Test 0.40339284522854957 passed Linear complexity test 0.4802977546116881 passed Serial test P-value1: 0.9752240885373696 passed P-value2: 0.822757312057996 passed Approximate entropy test 0.46668971447787155 passed Cumulative sums (Cusum) test P-value Forward: 0.7195671847543532 passed P-value Reverse: 1 passed Random excursions test 0.11586779894103257 passed Open in a new tab This further substantiates the chaotic nature of the 1D-PCQM and underscores its suitability for secure cryptographic applications. The findings provide compelling evidence that the 1D-PCQM is a robust and reliable candidate for high-performance image encryption systems. To further validate the proposed chaotic map’s effectiveness and dynamical complexity, a comparative analysis of some existing chaotic maps listed in Table 2 62 – 67 is presented in Table 3 . It summarizes key chaotic indicators chaotic range, lyapunov exponent , 0–1 test, Sample Entropy (SE), and Correlation Dimension (CD) of these maps. The maximum values of these metrics in 1D-PCQM highlight its superior unpredictability, wider chaotic range, higher complexity and Confirming its suitability as a robust candidate for secure and efficient cryptographic applications. Table 2. The proposed map and existing maps. Name Map equations Parameters Cross 2D hyperchaotic map 65 1D hybrid chaotic map 66 1-DSCM 67 R 2D-LSM 68 a,b IMPROVED 1D CHAOTIC MAP 69 a,b NEPCS 70 1D-PCQM (proposed) Open in a new tab Table 3. The comparison of proposed and existing maps. Name Chaotic range Lyapunov exponent 0-1 test Se Cross 2D hyperchaotic map 1 , 2 Pass 1.58 1D hybrid chaotic map [0,4] Pass 2.20 1-DSCM [0,10] 3 Pass 3 2D-LSM [0,1] Pass 1.88 Improved 1D Chaotic map 1 , 10 Pass 2.209 NEPCS [-30,30] >7.88 Pass 2.21 1D-PCQM Pass 2.8 Open in a new tab Proposed work The proposed Dynamic Triple Chaotic Map Image Encryption Scheme (D3CM-IES) is illustrated in Figure 11 . Three chaotic maps are used in the scheme’s mechanism: the Sine-Cosine map, the STS map, and the unique 1DCQM map. Fig. 11. Open in a new tab Proposed image encryption approach. Encryption mechanism There are four main steps in the encryption scheme: (i) chaotic sequence generation, (ii) dynamic chaotic sequence selection, (iii) dynamic permutation, and (iv) dynamic diffusion. A detailed description of these steps is given as follows: Chaotic sequence generation The Sine-Cosine map, STS map, and the novel 1D-PCQM are iterated to generate the chaotic sequences. The generated sequence value is random and depends on the dimensions of the input image. Function 1 represents the code to generate the pseudo-random chaotic sequence. It takes dimensions of the image , map function and control parameter as input. The definition of three map functions, the Sine-Cosine map, STS map, and the novel 1D-PCQM used in our method, is given by functions 2,3 and 4, respectively. Dynamic chaotic sequence selection The chaotic map that permutates and diffuses the pixels of the image is not fixed in our scheme, and it is selected dynamically for each pixel of the image. This step is designed to ensure non-linearity and unpredictability during pixel-level operations. Out of the three maps used in the encryption, one is chosen dynamically using the following equations: 15 Here, for each pixel at location we select the chaotic sequence. is the number of columns in the 2D array representation of the image. Computing the sequence index appears structurally simple, its effective behavior is governed by the chaotic generator rather than the modulo operation itself. Each chaotic sequence is produced using a position-dependent input to the map function (equation 16 ), which introduces strong sensitivity to initial indices and the control parameter. As a result, even adjacent pixel positions yield substantially divergent chaotic outputs. The modulo operation merely assigns which chaotic sub-sequence is accessed, while the subsequences themselves are aperiodic and mutually decorrelated due to their independent initialization. Consequently, the combined mechanism prevents cyclical or predictable behavior and enhances resistance against statistical, differential, and sequence-based cryptanalysis. This approach ensures that the selection of chaotic values remains highly dynamic, non-repetitive, and securely integrated into the overall encryption framework. Each pixel of the image is permuted and diffused using a different chaotic sequence, which increases the scheme’s security. 16 Dynamic permutation The pixels in the input image are located in some other locations in this step. The dynamically selected sequence and dimensions of the input image are used to calculate the permutation index using equation ( 17 ). 17 Here and stands for row and column, respectively; the dimensions of the input image modulus operation guarantees that the permutation index is not fixed and is dynamic. Finally, after finding the permutation index, we update the corresponding pixel in the permuted image with the pixel from the input image at the calculated permutation index. The pseudo-code of the dynamic permutation procedure is given in Function 5. It takes an image and the chaotic sequence as input and produces a permuted image as output. Dynamic diffusion Dynamic diffusion modifies the intensity of the pixel based on the selected chaotic sequence. For each pixel, the procedure calculates a diffusion factor based on the current pixel’s position ( and ) and the selected sequence corresponding value using equation ( 18 ). 18 Next, the procedure adds this diffusion factor to the pixel value in the permuted image and takes the result modulo 256 to ensure it stays within the valid intensity range (0-255). The pseudo-code of dynamic diffusion is given in Function 6. It takes a permuted image and selected sequence as input and produces a diffused image as the final cipher image. The detailed encryption pseudo-code is provided in Function 7. The selection of the control parameters and used in the encryption algorithm is made by performing a comprehensive parameter-sensitivity analysis on for each chaotic map. For all the three maps, we considered its nominal operating parameter and performed a systematic sweep of in its neighborhood using both logarithmic and linear perturbations. For each perturbed value of , a pseudorandom sequence was generated using the same procedure described in Function 1, and its dynamical characteristics were quantified through Sample Entropy, the 0–1 test for chaos (K-value), and an approximate largest Lyapunov exponent. The results indicate that all three maps maintain a positive Lyapunov exponent and K-values near unity across small perturbations, confirming stable chaotic behavior around the selected values. Decryption mechanism The decryption process is the inverse of the encryption steps and must be executed in reverse order. The inverse diffusion followed by inverse permutation, using the same chaotic sequences as in encryption. The control parameters and initial seed of the all the three maps are generated once and shared with sender and receiver as key. The decryption steps are: Step 1: Re-generate chaotic sequences To ensure consistency, the chaotic sequences used during encryption must be regenerated using the same control parameters and map functions: Step 2: Inverse dynamic diffusion This step subtracts the previously added diffusion factor to restore original intensity values using below equation: 19 Step 3: Inverse dynamic permutation The inverse permutation restores pixel positions using a reverse mapping. For each pixel (i, j), dynamically select a chaotic sequence that was used during permutation. Permutation index is computed and the index is converted into 2D coordinates , . Reverse the shuffle operation. Function 8 represents the pseudo code for decryption mechanism. Implementation and security analysis We have carried out several simulations to validate the performance and security of the encryption proposed scheme on a laptop with an Intel i7 processor with 16 GB RAM and 1TB SSD. Python 3 has been used on Jupyter notebook has been used to implement the scheme. The proposed encryption scheme has been tested on large dataset 71 .The encryption and decryption simulations of the Airplane (F-16), Pepper color, and Cameraman images using our scheme are represented in Figure 12 . Fig. 12. Open in a new tab Encryption and Decryption Simulations ( a ) Original Images of (Airplane (F-16), Peppercolor, Cameraman); ( b ) Encrypted images Airplane (F-16), Peppercolor, Cameraman; ( c ) Decrypted Airplane (F-16) , peppercolor, Cameraman. Security analysis A comprehensive security analysis ensures that the encryption methods employed are not only effective but also resilient against potential threats. The security and efficiency assessment are done as follows: Correlation coefficient The correlation coefficient is a metric that helps us understand how well the encryption process has disrupted the image data. The numerical value of the correlation coefficient indicates the degree of relationship between the original image and their corresponding values in the encrypted image. The formula to calculate the Correlation coefficient is given: 20 21 22 23 Here is the total number of pixels. The value of the Correlation coefficient obtained before and after the encryption is given in Table 4 . The graphical view of pixel correlation in our test images before and after encryption is also given in Table 5 . Table 4. Correlation coefficient analysis. Image Direction Red Green Blue RGB Plain Cipher Plain Cipher plain Cipher plain Cipher CCH 99.72051 -0.00465 98.68525 0.00584 95.64794 0.00588 97.48752 0.02258 CCV 98.85421 -0.00566 97.35741 -0.0054 98.74554 0.00357 98.74551 0.00874 CCD 97.74556 0.00458 99.57346 0.04321 97.87451 0.00451 99.85527 0.00455 CCH 94.66796 0.001196 96.48330 0.00612 93.94136 0.00312 95.94872 0.12319 CCV 95.10917 0.045707 97.27006 0.01251 94.47472 0.01367 97.46651 0.03927 CCD 90.93816 -0.00299 94.04156 0.00038 89.62123 0.00015 93.63882 -0.0058 CCH 97.25524 0.001693 0.972552 0.00169 0.972552 0.00169 0.990835 0.66713 CCV 98.33478 -0.00255 0.983347 -0.0025 0.983347 -0.0025 0.012979 0.06137 CCD 96.47776 0.076447 0.964777 0.07644 0.964777 0.07644 0.013926 0.06516 Open in a new tab Table 5. Correlation coefficient diagram. Open in a new tab Information entropy analysis Information entropy measures an image’s unpredictability and randomness 64 . It is defined using the following formula: 24 Here is the random symbol, and is the probability of its occurrence. Table 6 presents the Entropy of the images before and after encryption. Table 6. Entropy, NPCR and UACI analysis. Image Color Entropy NPCR UACI Original Cipher Red 7.387741 7.978556 99.85516 49.48761 Green 7.586922 7.988799 99.74551 49.88160 Blue 7.247843 7.976666 99.65227 48.09263 RGB 7.522171 7.999844 99.84501 49.70158 Red 7.351289 7.941362 99.57530 49.96344 Green 7.616494 7.981519 99.61283 49.88215 Blue 7.149995 7.950619 99.54370 48.83055 RGB 7.743695 7.993097 99.57728 49.55872 Red 7.226236 7.973490 99.51400 49.00202 Green 7.226236 7.973490 99.51400 49.00202 Blue 7.226236 7.973490 99.51400 49.00202 RGB 7.226236 7.973490 99.51400 49.00202 Open in a new tab Differential attack analysis An adversary creates two cipher images to launch a differential attack. The attacker tries to capitalize on the expectation that an output pattern change will occur if we submit input values with a predetermined difference. In other words, the attacker crypt analysis the encryption technique by using the predetermined difference between the two input images. A Number of Pixels Change Rate (NPCR) and the unified Averaged Changed Intensity (UACI) 65 are used to perform differential attack analysis. (i) NPCR: NPCR finds how a small alteration in the pixel value results in changes to the encrypted image 45 . The original image is converted to the cipher image The previously utilized original image’s pixel values are slightly altered to create the cipher image . The NPCR value is defined in the equation: 25 26 (ii) UACI: The difference in the two images’ average intensities is given by UACI 46 . The UACI value is calculated using the following formula: 27 Here, and are two cipher images. The obtained values imply that the proposed image encryption scheme is very effective in resisting differential attacks. Table 6 presents the NPCR and UACI on test images. UACI was computed using the standard definition and compared against a Monte-Carlo baseline of random 8-bit images (1000 trials) shown in Figure 13 , which produced a mean UACI of 33.31% (σ = 0.22%). Our encryption scheme produces UACI values in the range 47–49% across the tested images. This higher UACI results from the arithmetic and nonlinear chaotic map-based dynamic diffusion operations (including modulo folding) that amplify pixel differences more than simple XOR-based diffusion. Consequently, minute changes in the plaintext yield larger average absolute intensity differences in the ciphertext a behavior that increases confusion and strengthens resistance to differential attacks. Fig. 13. Open in a new tab Monte-Carlo UACI analysis. Histogram analysis The distribution of the pixels in the input image is shown using a histogram analysis. The attackers use the pixel distribution to estimate the pixel intensity in various regions of a given image 47 . For a decent image encryption technique, a uniformly distributed histogram is a necessary prerequisite. The histograms of original and encrypted RGB components of images “Airplane (F-16).TIFF,” peppercolor.jpg, and cameraman.jpg are shown in Table 7 . The results show that the ciphered images have uniform histograms, which inhibits the adversaries from carrying out attacks. Table 7. Histogram analysis. Open in a new tab Chosen plain text attack analysis An attacker deliberately selects plain image and observes the corresponding cipher image, aiming to identify patterns in the image and the encryption key 14 . To further evaluates the security of the proposed encryption scheme under chosen-plaintext attack (CPA) conditions, we conducted experiments using two highly structured test images: an all-black image and an all-white Airplane (F-16) image of size 512×512. These special plaintexts contain no spatial variation, making them effective for assessing whether the encryption process eliminates deterministic patterns and produces ciphertexts indistinguishable from random noise. Both images were encrypted using the complete proposed workflow, including dynamic sequence selection, dynamic permutation, and dynamic diffusion. The resulting ciphertexts exhibited high entropy (≈8) 74 , and near-zero correlation coefficients in all directions. Furthermore, the NPCR and UACI values between the ciphertexts of the black and white images exceeded 99% and fell within the expected Monte-Carlo baseline, respectively. These results demonstrate that the proposed algorithm effectively resists CPA by ensuring that even the simplest plaintext structures do not leave any observable trace in their corresponding ciphertexts. The quantitative results are summarized in Table 8 . Table 8. Results of CPA on All-Black and All-White Images Metric Cipher (Black) Cipher (White) Interpretation Shannon Entropy 7.99884 7.98962 Excellent randomness CCH 0.00088 0.00647 No linear dependency CCV -0.00457 -0.00251 No linear dependency CCD 0.00668 -0.02476 No linear dependency NPCR 99.8744 99.6583 High Sensitivity Mean UACI (Monte-Carlo baseline) 33.6% 33.3% Ideal differential randomness Open in a new tab Key sensitivity analysis Key sensitivity is a critical aspect of a secure image encryption system, ensuring that even the slightest change in the encryption key results in a completely different cipher-text. This property prevents attackers from deducing the correct key. To evaluate the key sensitivity of the proposed algorithm, we conducted a series of tests using two keys that differ by only one bit in their initial conditions. The encryption process was applied to the same input image (Fig. 14 (a)) to obtain two ciphers Fig. 14 (b, d) using both keys. The histograms of both cipher images (Fig. 14 (c, e)) are different and well-distributed, demonstrating that even a minor change in the key can produce a completely different cipher image. Fig. 14. Open in a new tab ( a ) Plain image ( b ) Cipher image1 ( c ) Histogram Cipher1 ( d ) Cipher image2 ( e ) Histogram Cipher2. Avalanche effect The avalanche effect is a fundamental criterion in cryptographic systems, which refers to the phenomenon where a slight change in the input (such as a single bit in the plaintext or key) leads to significant and widespread changes in the output. This property is essential to ensure high diffusion and robustness against statistical and differential attacks. In the proposed image encryption scheme, the avalanche effect was analyzed by modifying just one bit in the input image and observing the resulting changes in the encrypted output. Figure 15 (a) represents the original image, Figure 15 (b) represents the encrypted image, Figure 15 (c) is the obtained cipher image when we changed the plain image by 1 bit. Figure 15 (d) represents difference between two ciphers. Fig. 15. Open in a new tab ( a ) Plain frame ( b )Cipher frame C1 ( c ) Cipher frame C2 ( d ) Difference between two ciphers. Key space analysis In a secure cryptographic system, the key space defines the total number of possible unique keys that can be used. A larger key space significantly reduces the risk of brute-force attacks. In the proposed system, the secret key is composed of three independent sub-keys, each 128 bits in length. Each sub-key has 2 128 possible values. Therefore, the total key-space is: This is an extremely large key space, making brute-force attacks practically infeasible. Decryption parameters The decryption quality of the retrieved image has been analyzed by introducing salt and pepper noise into the image and evaluating the value of Mean square error (MSE), Peak Signal to Noise Ratio (PSNR) and Bit Correct ratio (BCR) 48 defined below: (i) MSE The Mean Squared Error (MSE) measures the error in the received image caused by noise in the transmission channel. The value of MSE is calculated as follows: 28 Here, is the original image, and is the decrypted image. and represent the number of rows and columns, respectively. The average value of MSE calculated by our method is represented in Table 9 . Table 9. Decryption parameters. Image Parameters 1% 1.5% 2% 2.5% MSE 4.487411 4.564783 4.526787 4.475971 PSNR 49.91874 49.97063 49.88841 49.94107 BCR 0.989743 0.997784 0.985731 0.987653 MSE 4.344944 4.420802 4.486255 4.549823 PSNR 49.94088 49.03297 49.87138 49.84083 BCR 0.992302 0.985046 0.978976 0.973089 MSE 4.314501 4.381752 4.4396411 4.5077163 PSNR 49.95610 49.92256 49.894064 49.861020 BCR 0.989935 0.983747 0.5780982 0.5720138 Open in a new tab (ii) PSNR The Peak Signal-to-Noise Ratio (PSNR) is calculated by comparing the maximum possible pixel value of an image to the error introduced by noise during transmission 75 . This ratio helps to quantify the quality of the received image. The formula to calculate the value of PSNR is given in the equation: 29 The average PSNR value of our test images after the decryption is given in Table 9 . (iii) BCR The Bit Correct Ratio (BCR) is calculated based on the number of bit errors found in the decrypted image after it has been affected by salt and pepper noise 76 . The equation gives the formula to calculate the BCR: 30 Here and represent the original image and the decrypted image. The average value of BCR in our test images is given in Table 9 . Computational cost analysis In this section, we compute the computational cost of the proposed scheme by analyzing the time taken to execute the encryption program using Python’s time () function and by determining the time and space complexity in terms of , the size of the image. Time analysis in image encryption is crucial for evaluating the efficiency and practicality of encryption algorithms. It measures how long it takes to encrypt and decrypt an image, which is important for applications where speed and performance are critical. The time spent in encrypting various test images using our method is given in Table 10 . We conducted an extended runtime analysis using both the original prototype (unoptimized) and an optimized implementation. The initially reported encryption time of 5.69 s for a 512 × 512 Airplane (F-16) image corresponded to an unoptimized Python version in which chaotic sequence generation, permutation, and diffusion were carried out through nested loops. While useful for functional verification, this prototype does not reflect the achievable computational efficiency of the algorithm. Table 10. Time analysis. Image Time (unoptimized Version) Time (optimized Version) Airplane (F-16).TIFF 5.69 Seconds 0.45 Seconds Peppercolor.jpg 3.11 Seconds 0.29 Seconds Cameraman.jpg 6.88 Seconds 0.54 Seconds Open in a new tab To obtain a more realistic performance estimate, the algorithm was reimplemented using (i) fully vectorized NumPy operations and (ii) Numba just-in-time (JIT) compilation for the computationally intensive components. Under identical hardware conditions, the optimized implementation reduced the encryption time to 0.45 s, representing an improvement of nearly an order of magnitude. Additionally, the algorithm’s operations chaotic sequence generation, dynamic permutation, and pixel-wise dynamic diffusion are inherently parallel, making the scheme highly suitable for GPU or FPGA deployment, where real-time performance is achievable. This proves that the algorithm is capable of real-time performance under optimized or hardware-accelerated implementations. Time Complexity: The encryption process includes three main steps: pseudo-random sequence generation, dynamic permutation, and dynamic diffusion. Each step operates over all pixels of the image once. Thus, the overall time complexity is linear and is equal to (M × N), where M and N represent the number of rows and columns of the image, respectively. Space Complexity: The algorithm requires additional space to store three chaotic sequences and intermediate images (permuted and diffused). Therefore, the total space required is also proportional to the image size that is (M × N). Discussion The proposed image encryption scheme, based on the novel 1D-Powered Chebyshev Quadratic Chaotic Map (1D-PCQM), demonstrates significant improvements in terms of security and unpredictability. The integration of three distinct chaotic maps Sine-Cosine, Sine-Tangent-Sine (STS), and 1D-PCQM enables the generation of multiple chaotic sequences, from which one is dynamically selected during the encryption process. This design choice introduces a high level of entropy and complexity, which is crucial for robust cryptographic systems. One of the core contributions of this study is the dynamic selection of chaotic sequences for each pixel, rather than relying on a single static sequence. The robustness of the dynamically selected strategy was thoroughly validated by comparing the values of entropy, correlation coefficient (CC), NPCR, and UACI obtained using both a static and a dynamically selected chaotic sequence, as shown in Table 10 (using Airplane (F-16).TIFF). The results indicate that the entropy values remained consistently close to the ideal value of 8, reflecting high randomness. Additionally, the pixels in the cipher image exhibited significantly lower correlation compared to those in the static sequence, confirming enhanced security. NPCR and UACI metrics showed strong resistance to differential attacks, with dynamic sequence selection outperforming static methods. Key sensitivity analysis confirmed that small variations in the initial conditions led to significantly different encryption results, a property enhanced by the hybrid and dynamic nature of the proposed scheme. While the proposed dynamic permutation and diffusion strategies introduce additional operations compared to traditional static schemes, we carefully designed them to ensure minimal computational overhead. We conducted runtime analysis on standard image sizes (e.g., 256×256 and 512×512), comparing both dynamic and static versions shown in Table 11 . The results show that the dynamic scheme incurs an average increase of approximately 6–9% in execution time, which is a reasonable trade-off for the significant gains in security. Moreover, the algorithm maintains linear computational complexity, and its performance remains suitable for real-time or resource-constrained environments. The added security benefits through higher randomness and stronger resistance to cryptanalysis justify the slight increase in computational cost. Table 11. Comparison between static sequence and dynamic sequence results. Parmeter Original image results Results using static sequence Results using dynamic sequence CCH 95.64794 0.25743 0.11964 CCV 97.40674 0.03876 0.00037 CCD 93.36194 0.07874 0.02392 Entropy 7.754160 7.954160 7.99981 NPCR 99.04716 99.52112 UACI 48.55213 49.04278 Run time 5.34 Seconds 5.69 Seconds Open in a new tab Furthermore, the proposed 1D-PCQM showed superior chaotic characteristics, such as higher Lyapunov exponents and better sensitivity to initial conditions due to its powered exponential modulation. This increased sensitivity contributes directly to the improved unpredictability of the encryption process. Therefore, the results validate that the dynamic selection approach not only strengthens the encryption algorithm’s security but also introduces a flexible, highly nonlinear mechanism that is well-suited for securing multimedia data. Comparative analysis The proposed work has been compared with some of the most recent approaches on the basis of encryption and decryption parameters. This comparison study is explained in Table 12 . The comparative results for existing encryption models were taken directly from the original publications to benchmark our proposed method against widely accepted standards. Since several prior works do not report some parameters, including approximate or inferred values would compromise the integrity of the comparison. Therefore, the corresponding entries are indicated as (Not Available) NA’ in Table 12 . Regarding computational efficiency, the proposed scheme exhibits a runtime of 5.69 seconds, which is higher than some recently reported methods. This increase is directly attributed to the multiple chaotic maps based dynamic sequence generation and the nonlinear dynamic diffusion stages, both of which intentionally introduce additional computational complexity to enhance chaotic behavior, key sensitivity, and resistance to differential and statistical attacks. Thus, the scheme prioritizes security strength over minimal runtime, representing a deliberate security efficiency trade-off. Table 12. Comparison of security and efficiency of our method vs. other existing models. References Alarood et al. 77 Kumar & Dua 78 Basha et al. 41 Tahat et al. 79 Dua et al. 80 Proposed Chaotic system used 5D hyperchaotic system Sine-Cosine map LSTC map Chen system, Bogdanov Map ICFCM 1D-PCQM CC Hor 0.000733 0.0036 0.0006 0.00568 0.0008 -0.00465 CC Ver 0.000393 -0.0014 -0.0017 0.007759 0.0021 -0.00566 CC Diag 0.000393 -0.0006 -0.0008 0.004765 0.0024 0.00458 Entropy 7.9997 7.9997 7.9951 7.9926 7.9993 7.99984 NPCR 99.84 99.64 99.61 99.6094 99.67 99.85516 UACI 34.2 48.05 33.36 33.4973 33.63 49.88160 MSE 11,258 10.84 NA* NA* 2.13 4.475971 PSNR 7.616036 42.96 NA* NA* 46.2 49.97063 BCR NA* 0.86 NA* NA* 0.98 0.997784 Key-space Time (Encryption + Decryption) 26 0.5 12.23 NA 1.43 5.69 Time Complexity NA* NA* NA* NA* NA* (M × N) Open in a new tab * NA means values are not reported in original publications. Statistical tests To verify whether the slight improvement in entropy observed in the proposed scheme is statistically significant, we performed a paired t-test comparing the entropy values of the proposed method with those reported in existing schemes across five standard test images. The entropy values for existing methods were: [7.9997, 7.9997, 7.9951, 7.9926, 7.9993]. The proposed method achieved a consistent entropy value of 7.9998. Using a paired t-test, we obtained the following result: t-statistic = 4.64 and p-value ≈ 0.0095. Since the p-value < 0.05, the difference in entropy values between the proposed and existing methods is considered statistically significant, supporting the claim that the proposed scheme offers a measurable improvement in randomness and information concealment. Conclusion We constructed a new map, the newly developed 1D-Powered Chebyshev Quadratic Map (1D-PCQM). Despite its simple structure and reliance on only two control parameters, 1D-PCQM exhibits strong chaotic behavior, including a wide bifurcation spectrum, high sample entropy, and a large positive Lyapunov exponent. These properties enable high sensitivity to initial conditions and generate unpredictable encryption keys. We used this map to construct a novel Dynamic Triple Chaotic Map Image Encryption Scheme (D3CM-IES) that combines 1D-PCQM with additional chaotic maps such as Sine-Cosine (SC) and STS map. The D3CM-IES scheme is performed through four main steps: (i) generating three chaotic sequences built from the aforementioned chaotic maps; (ii) performing a dynamic selection procedure that selects a single chaotic sequence for encryption; (iii) dynamic switching, where spatial locations (the spatial position of pixels) within the image are rearranged; and (iv) dynamic diffusion, where pixel values are modified to further conceal the image content. The analysis demonstrated that the proposed system achieves high security performance, comparing favorably with existing chaotic encryption methods in terms of statistical uniformity, key sensitivity, and encryption quality. Its lightweight nature and efficient spatial architecture make it particularly suitable for real-time applications, such as secure image transmission on portable, embedded, or low-power platforms. While this approach is effective, our future work may explore further enhancing the proposed chaotic map by incorporating it into memory-based dynamical systems to capture past-state dependencies and strengthen long-term unpredictability. By leveraging the improved chaotic properties and memory characteristics, the future system is expected to provide higher security and stability for intelligent transportation applications. Author contributions B.S.: Conceptualization, Methodology, Formal analysis, and Writing original draft. S.H.: Supervision. M.D.: Software, Validation, Data curation, and Visualization. S.D.: Resources and Writing review and editing. D.D.: Methodology, Experimental validation, and Formal analysis. Data availability The datasets used in this study are publicly available, can be accessed at ( https://sipi.usc.edu/database/ (accessed December 3, 2025)). Declarations Competing interest The authors declare no competing interests. Ethics We confirm that relevant guidelines and regulations are carried out in all methods. Footnotes Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Contributor Information Boukansous Sarra, Email: [email protected]. Hao Sun, Email: [email protected]. References 1. Hua, Z., Zhou, Y. & Huang, H. Cosine-transform-based chaotic system for image encryption. Inf. Sci. (Ny) 480 , 403–419. 10.1016/j.ins.2018.12.048 (2019). [ Google Scholar ] 2. Xian, Y. & Wang, X. Fractal sorting matrix and its application on chaotic image encryption. Inf. Sci. (Ny) 547 , 1154–1169. 10.1016/j.ins.2020.09.055 (2021). [ Google Scholar ] 3. Mandal, P. C., Mukherjee, I., Paul, G. & Chatterji, B. N. Digital image steganography: A literature survey. Inf. Sci. (Ny) 609 , 1451–1488. 10.1016/j.ins.2022.07.120 (2022). [ Google Scholar ] 4. Liao, X., Yin, J., Chen, M. & Qin, Z. Adaptive payload distribution in multiple images steganography based on image texture features. IEEE Trans. Dependable Secur. Comput. 19 , 897–911. 10.1109/TDSC.2020.3004708 (2022). [ Google Scholar ] 5. Daoui, A. et al. Biomedical multimedia encryption by fractional-order meixner polynomials map and quaternion fractional-order meixner moments. IEEE Access 10 , 102599–102617. 10.1109/ACCESS.2022.3203067 (2022). [ Google Scholar ] 6. Daoui, A. et al. Color stereo image encryption and local zero-watermarking schemes using octonion Hahn moments and modified Henon map. J. King Saud Univ. - Comput Inf. Sci. 34 , 8927–8954. 10.1016/j.jksuci.2022.08.021 (2022). [ Google Scholar ] 7. Wen, H. et al. Secure optical image communication using double random transformation and memristive chaos. IEEE Photonics J. 10.1109/JPHOT.2022.3233129 (2023). [ Google Scholar ] 8. Dhall, S., Pal, S. K. & Sharma, K. Cryptanalysis of image encryption scheme based on a new 1D chaotic system. Signal Process. 146 , 22–32. 10.1016/j.sigpro.2017.12.021 (2018). [ Google Scholar ] 9. Zhang, B. & Liu, L. Chaos-based image encryption: review application, and challenges. Mathematics 10.3390/math11112585 (2023). [ Google Scholar ] 10. D.N. Aini, D.R.I.M. Setiadi, S.N. Putro, E.H. Rachmawanto, C.A. Sari, Survey of methods in the spatial domain image steganography based imperceptibility and payload capacity, Proc. - 2019 Int. Semin. Appl. Technol. Inf. Commun. Ind. 4.0 Retrosp. Prospect. Challenges, ISemantic 2019 434–439. 10.1109/ISEMANTIC.2019.8884333 (2019) 11. Wang, X., Feng, L. & Zhao, H. Fast image encryption algorithm based on parallel computing system. Inf. Sci. (Ny) 486 , 340–358. 10.1016/j.ins.2019.02.049 (2019). [ Google Scholar ] 12. Muñoz-Guillermo, M. Image encryption using q-deformed logistic map. Inf. Sci. (Ny) 552 , 352–364. 10.1016/j.ins.2020.11.045 (2021). [ Google Scholar ] 13. Liu, L. & Wang, J. A cluster of 1D quadratic chaotic map and its applications in image encryption. Math. Comput. Simul. 204 , 89–114. 10.1016/j.matcom.2022.07.030 (2023). [ Google Scholar ] 14. Li, H. et al. Exploiting dynamic vector-level operations and a 2D-enhanced logistic modular map for efficient chaotic image encryption. Entropy 25 , 1147 (2023). [ DOI ] [ PMC free article ] [ PubMed ] [ Google Scholar ] 15. W. Feng, K. Zhang, J. Zhang, X. Zhao, Y. Chen, B. Cai, Z. Zhu, H. Wen, C. Ye, Integrating fractional-order hopfield neural network with differentiated encryption : achieving high-performance privacy protection for medical images 1–26. (2025) 16. Yu, F. et al. Dynamic analysis and application of 6d multistable memristive chaotic system with wide range of hyperchaotic states. Axioms 14 , 638 (2025). [ Google Scholar ] 17. Yu, F., He, S., Yao, W., Cai, S. & Xu, Q. Bursting firings in memristive hopfield neural network with image encryption and hardware implementation, IEEE trans. Comput. Des. Integr. Circuits Syst. 44 , 4564–4576. 10.1109/TCAD.2025.3567878 (2025). [ Google Scholar ] 18. Feng, W., Qin, Z., Zhang, J. & Ahmad, M. Cryptanalysis and improvement of the image encryption scheme based on feistel network and dynamic DNA encoding. IEEE Access 9 , 145459–145470. 10.1109/ACCESS.2021.3123571 (2021). [ Google Scholar ] 19. He, Y., Li, H. & Li, C. Cryptanalysis and Improvement of the image encryption scheme based on 2D logistic-adjusted-sine map. IEEE Access 7 , 12584–12597. 10.1109/ACCESS.2019.2893760 (2019). [ Google Scholar ] 20. Chen, L., Li, C. & Li, C. Security measurement ofa medical communication scheme based on chaos and DNA coding. J. Vis. Commun. Image Represent. 83 , 103424. 10.1016/j.jvcir.2021.103424 (2022). [ Google Scholar ] 21. Pak, C., An, K., Jang, P., Kim, J. & Kim, S. A novel bit-level color image encryption using improved 1D chaotic map. Multimed. Tools Appl. 78 , 12027–12042. 10.1007/s11042-018-6739-1 (2019). [ Google Scholar ] 22. Midoun, M. A., Wang, X. & Talhaoui, M. Z. A sensitive dynamic mutual encryption system based on a new 1D chaotic map. Opt. Lasers Eng. 10.1016/j.optlaseng.2020.106485 (2021). [ Google Scholar ] 23. Sambas, A. et al. Mathematical model and FPGA realization of a multi-stable chaotic dynamical system with a closed butterfly-like curve of equilibrium points. Appl. Sci. 11 (1), 16. 10.3390/app11020788 (2021). [ Google Scholar ] 24. Sambas, A. et al. A 3-D multi-stable system with a peanut-shaped equilibrium curve: circuit design FPGA realization, and an application to image encryption. IEEE Access 8 (137116), 137132. 10.1109/ACCESS.2020.3011724 (2020). [ Google Scholar ] 25. Wang, J. & Liu, L. A novel chaos-based image encryption using magic square scrambling and octree diffusing. Mathematics 10.3390/math10030457 (2022).39512426 [ Google Scholar ] 26. Wu, Y. & Liu, L. An iteration-time combination method to reduce the dynamic degradation of digital chaotic maps. Complexity 10.1155/2020/5707983 (2020). [ Google Scholar ] 27. Zhang, S. & Liu, L. A novel image encryption algorithm based on SPWLCM and DNA coding. Math. Comput. Simul. 190 , 723–744. 10.1016/j.matcom.2021.06.012 (2021). [ Google Scholar ] 28. Elmanfaloty, R. A. & Abou-Bakr, E. An image encryption scheme using a 1d chaotic double section skew tent map. Complexity 10.1155/2020/7647421 (2020). [ Google Scholar ] 29. Liu, Y., Qin, Z., Liao, X. & Wu, J. Cryptanalysis and enhancement of an image encryption scheme based on a 1-D coupled Sine map. Nonlinear Dyn. 100 , 2917–2931. 10.1007/s11071-020-05654-y (2020). [ Google Scholar ] 30. Nepomuceno, E. G., Nardo, L. G., Arias-Garcia, J., Butusov, D. N. & Tutueva, A. Image encryption based on the pseudo-orbits from 1D chaotic map. Chaos 10.1063/1.5099261 (2019). [ DOI ] [ PubMed ] [ Google Scholar ] 31. Shakiba, A. A novel randomized one-dimensional chaotic Chebyshev mapping for chosen plaintext attack secure image encryption with a novel chaotic breadth first traversal. Multimed. Tools Appl. 78 , 34773–34799. 10.1007/s11042-019-08071-5 (2019). [ Google Scholar ] 32. Wang, J. et al. A logistic mapping-based encryption scheme for wireless body area networks. Futur. Gener. Comput. Syst. 110 , 57–67. 10.1016/j.future.2020.04.002 (2020). [ Google Scholar ] 33. Zhou, Y., Bao, L. & Chen, C. L. P. A new 1D chaotic system for image encryption. Signal Process. 97 , 172–182. 10.1016/j.sigpro.2013.10.034 (2014). [ Google Scholar ] 34. Jiang, M. & Yang, H. Image encryption algorithm using multi-level permutation and improved logistic-chebyshev coupled map. Information 10.3390/info14080456 (2023). [ Google Scholar ] 35. Q. Lai, Y. Liu, L. Fortuna, Dynamical analysis and fixed-time synchronization for secure communication of hidden multiscroll memristive chaotic system, IEEE Trans. Circuits Syst. I Regul. Pap. (2024). 36. Liu, X., Mou, J., Yan, H. & Bi, X. Memcapacitor-coupled chebyshev hyperchaotic map. Int. J. Bifurc. Chaos 10.1142/S0218127422501802 (2022). [ Google Scholar ] 37. Wang, X., Luan, D. & Bao, X. Cryptanalysis of an image encryption algorithm using Chebyshev generator. Digit. Signal Process. A Rev. J. 25 , 244–247. 10.1016/j.dsp.2013.10.020 (2014). [ Google Scholar ] 38. F. Qi, S. Huang, T. Li, H. Yang, X. Kang, 2D henon-chebyshev chaotic map for image encryption, in: Proc. - 21st IEEE Int. Conf. High Perform. Comput. Commun. 17th IEEE Int. Conf. Smart City 5th IEEE Int. Conf. Data Sci. Syst. HPCC/SmartCity/DSS 2019 774–781. 10.1109/HPCC/SmartCity/DSS.2019.00114. (2019) 39. Wang, X. & Du, X. Pixel-level and bit-level image encryption method based on Logistic-Chebyshev dynamic coupled map lattices. Chaos Solitons Fractals 10.1016/j.chaos.2021.111629 (2022).36589549 [ Google Scholar ] 40. Liu, L., Jiang, D., Wang, X., Rong, X. & Zhang, R. 2D Logistic-Adjusted-Chebyshev map for visual color image encryption. J. Inf. Secur. Appl. 60 , 102854. 10.1016/j.jisa.2021.102854 (2021). [ Google Scholar ] 41. Basha, S. M., Mathivanan, P. & Ganesh, A. B. Bit level color image encryption using Logistic-Sine-Tent-Chebyshev (LSTC) map. Optik (Stuttg). 259 , 168956. 10.1016/j.ijleo.2022.168956 (2022). [ Google Scholar ] 42. Lai, Q. & Hu, G. A Nonuniform pixel split encryption scheme integrated with compressive sensing and its application in IoMT. IEEE Trans. Ind. Informatics 20 , 11262–11272. 10.1109/TII.2024.3403266 (2024). [ Google Scholar ] 43. Zhang, X., Liu, M. & Yang, X. color image encryption algorithm based on cross-spiral transformation and zone diffusion. Mathematics 10.3390/math11143228 (2023). [ Google Scholar ] 44. Dhingra, D. & Dua, M. A novel Sine–Tangent–Sine chaotic map and dynamic S-box-based video encryption scheme. Imaging Sci. J. 10.1080/13682199.2023.2187513 (2023). [ Google Scholar ] 45. Kumari, S., Dua, M., Dua, S. & Dhingra, D. A novel cosine-cosine chaotic map-based video encryption scheme. J. Eng. Appl. Sci. 71 , 36 (2024). [ Google Scholar ] 46. D. Dhingra, M. Dua, A chaos-based novel approach to video encryption using dynamic S-box, Springer US 10.1007/s11042-023-15593-6. (2023) 47. Dhingra, D. & Dua, M. A novel chaotic map-based encryption scheme for surveillance videos. Phys. Scr. 10.1088/1402-4896/ad0710 (2023). [ Google Scholar ] 48. Zhu, S., Deng, X., Zhang, W. & Zhu, C. Image encryption scheme based on newly designed chaotic map and parallel DNA coding. Mathematics 10.3390/math11010231 (2023). [ Google Scholar ] 49. Dua, M., Bhogal, R., Dua, S. & Chakravarty, N. Satellite image encryption using amalgamation of randomized three chaotic maps and DNA encoding. Phys. Scr. 10.1088/1402-4896/ad996a (2025). [ Google Scholar ] 50. and M.Z. Fangbo Hong, Qian Wang, Ying Zhao, A Novel Chaotic Signal Based on LSTM-KAN for Image Encryption, Int. J. Bifurc. Chaos (2025). 51. Ali, R. S., Hasoun, R. K., Al-Adhami, A., Juber, S. A. & Hashem, S. H. Proposal medical image protection system based on hybrid CNN-transformer model and chaotic maps. IEEE Access 10.1109/ACCESS.2025.3586562 (2025). [ Google Scholar ] 52. Yang, Y. et al. An n-dimensional chaotic map with application in reversible data hiding for medical images. Entropy 10.3390/e26030254 (2024). [ DOI ] [ PMC free article ] [ PubMed ] [ Google Scholar ] 53. Kouadra, I. et al. New composite chaotic map applied to an image encryption scheme in cybersecurity applications. IEEE Access 13 , 70676–70690. 10.1109/ACCESS.2025.3558948 (2025). [ Google Scholar ] 54. Yamni, M., Daoui, A., Plawiak, P., Alfarraj, O. & El-Latif, A. A. A. A hardware-accelerated approach to chaotic image encryption: LTB map and FPGA implementation. IEEE Access 12 , 103921–103940. 10.1109/ACCESS.2024.3432008 (2024). [ Google Scholar ] 55. Njitacke, Z. T., Maghrabi, L. A., Ahmad, M. & Althaqafi, T. Efficient bit-plane based medical image cryptosystem using novel and robust sine-cosine chaotic map. Comput. Mater. Contin. 83 (917), 933 (2025). [ Google Scholar ] 56. Li, L. A self-reversible image encryption algorithm utilizing a novel chaotic map. Nonlinear Dyn. 113 , 7351–7383. 10.1007/s11071-024-10726-4 (2025). [ Google Scholar ] 57. Jackson, J. & Perumal, R. A robust image encryption technique based on an improved fractional order chaotic map. Nonlinear Dyn. 113 , 7277–7296. 10.1007/s11071-024-10480-7 (2024). [ Google Scholar ] 58. S. Henry Strogatz, Nonlinear dynamics and chaos: with applications to physics (1994). 59. Moroni, G., Syam, W. P. & Petrò, S. Comparison of chaos optimization functions for performance improvement of fitting of non-linear geometries. Meas. J. Int. Meas. Confed. 86 , 79–92. 10.1016/j.measurement.2016.02.045 (2016). [ Google Scholar ] 60. Rosenstein, M. T., Collins, J. J. & De Luca, C. J. A practical method for calculating largest Lyapunov exponents from small data sets. Phys. D Nonlinear Phenom. 65 , 117–134. 10.1016/0167-2789(93)90009-P (1993). [ Google Scholar ] 61. Alawida, M., Omolara, A. E., Abiodun, O. I. & Al-Rajab, M. A deeper look into cybersecurity issues in the wake of Covid-19: A survey. J. King Saud Univ. - Comput Inf. Sci. 34 , 8176–8206. 10.1016/j.jksuci.2022.08.003 (2022). [ DOI ] [ PMC free article ] [ PubMed ] [ Google Scholar ] 62. G.A. Gottwald, I. Melbourne, The 0-1 test for chaos: a review BT - chaos detection and predictability (2016) 221–247. 63. Mansouri, A. & Wang, X. A novel one-dimensional sine powered chaotic map and its application in a new image encryption scheme. Inf. Sci. (Ny) 520 , 46–62 (2020). [ Google Scholar ] 64. Richman, J. S., Lake, D. E. & Moorman, J. R. Sample Entropy 10.1016/S0076-6879(04)84011-4 (2004). [ Google Scholar ] 65. Teng, L., Wang, X., Yang, F. & Xian, Y. Color image encryption based on cross 2D hyperchaotic map using combined cycle shift scrambling and selecting diffusion. Nonlinear Dyn. 105 , 1859–1876. 10.1007/s11071-021-06663-1 (2021). [ Google Scholar ] 66. Jamal, S. S., Ali, R., Jamil, M. K., Nooh, S. A. & Alblehai, F. Gulraiz, Secure S-box construction with 1D chaotic maps and finite field theory for block cipher encryption. Alexandria Eng. J. 125 , 278–296. 10.1016/j.aej.2025.03.109 (2025). [ Google Scholar ] 67. Huang, Q., Huang, L., Cai, S., Xiong, X. & Zhang, H. On a ssymmetric image cryptosystem based on a novel one-dimensional chaotic system and banyan network. Mathematics 10.3390/math11214411 (2023). [ Google Scholar ] 68. Yunlong, L., Yiting, L., Zheng, X. & Xiaochen, Y. Privacy image secrecy scheme based on chaos-driven fractal sorting matrix and fibonacci Q-matrix. Vis. Comput. 10.1007/s00371-025-04014-4 (2025). [ Google Scholar ] 69. Alzaidi, A. A., Ahmad, M., Doja, M. N., Al Solami, E. & Beg, M. M. S. A new 1D chaotic map and β-hill climbing for generating substitution-boxes. IEEE Access 6 , 55405–55418. 10.1109/ACCESS.2018.2871557 (2018). [ Google Scholar ] 70. Gill, H. S. A normalized exponential piecewise chaotic system (NEPCS) and DNA image cryptography using SHA-256. IEEE Access 10.1109/access.2025.3582318 (2025). [ Google Scholar ] 71. USC-SIPI, (n.d.). https://sipi.usc.edu/database/ (accessed December 3, 2025). 72. Goel, L., Chawla, H., Dua, M., Dua, S. & Dhingra, D. Novel hybrid chaotic map-based secure data transmission between smart meter and HAN devices. Smart Sci. 10.1080/23080477.2023.2264564 (2024). [ Google Scholar ] 73. Dhingra, D. & Dua, M. Medical video encryption using novel 2D Cosine-Sine map and dynamic DNA coding. Med. Biol. Eng. Comput. 62 , 237–255 (2024). [ DOI ] [ PubMed ] [ Google Scholar ] 74. Feng, W. et al. Exploiting robust quadratic polynomial hyperchaotic map and pixel fusion strategy for efficient image encryption. Expert Syst. Appl. 246 , 123190. 10.1016/j.eswa.2024.123190 (2024). [ Google Scholar ] 75. Kumar, A. & Dua, M. Image encryption using a novel hybrid chaotic map and dynamic permutation−diffusion. Multimed. Tools Appl. 10.1007/s11042-023-16817-5 (2023).37362741 [ Google Scholar ] 76. Kumar, A. & Dua, M. Novel pseudo random key & cosine transformed chaotic maps based satellite image encryption. Multimed. Tools Appl. 80 , 27785–27805. 10.1007/s11042-021-10970-5 (2021). [ Google Scholar ] 77. Alarood, A. A., Alsolami, E., Al-Khasawneh, M. A., Ababneh, N. & Elmedany, W. IES: Hyper-chaotic plain image encryption scheme using improved shuffled confusion-diffusion. Ain Shams Eng. J. 10.1016/j.asej.2021.09.010 (2021). [ Google Scholar ] 78. Kumar, A. & Dua, M. A novel chaos map based medical image encryption scheme. Imaging Sci. J. 69 , 219–238. 10.1080/13682199.2022.2156669 (2021). [ Google Scholar ] 79. Tahat, N., Al Jamal, M. F., Bawaneh, M. J., Alomari, A. K. & Al Hazaimeh, O. M. Image encryption using anti-synchronisation and Bogdanov transformation map. Int J. Comput. Sci. Math. 15 , 43. 10.1504/ijcsm.2022.10046284 (2022). [ Google Scholar ] 80. Dua, S., Kumar, A., Dua, M. & Dhingra, D. ICFCM-MIE: improved cosine fractional chaotic map based medical image encryption. Multimed. Tools Appl. 10.1007/s11042-023-17438-8 (2024). [ Google Scholar ] Associated Data This section collects any data citations, data availability statements, or supplementary materials included in this article. Data Availability Statement The datasets used in this study are publicly available, can be accessed at ( https://sipi.usc.edu/database/ (accessed December 3, 2025)). Articles from Scientific Reports are provided here courtesy of Nature Publishing Group ACTIONS View on publisher site PDF (8.1 MB) Cite Collections Permalink PERMALINK Copy RESOURCES Similar articles Cited by other articles Links to NCBI Databases Cite Copy Download .nbib .nbib Format: AMA APA MLA NLM Add to Collections Create a new collection Add to an existing collection Name your collection * Choose a collection Unable to load your collection due to an error Please try again Add Cancel Follow NCBI NCBI on X (formerly known as Twitter) NCBI on Facebook NCBI on LinkedIn NCBI on GitHub NCBI RSS feed Connect with NLM NLM on X (formerly known as Twitter) NLM on Facebook NLM on YouTube National Library of Medicine 8600 Rockville Pike Bethesda, MD 20894 Web Policies FOIA HHS Vulnerability Disclosure Help Accessibility Careers NLM NIH HHS USA.gov Back to Top