The Mathematics Special Issue “Chaos-Based Secure Communication and Cryptography, 1st edition” brings together eighteen outstanding contributions that showcase significant advancements in the theory and application of chaotic systems for cryptography and secure communications. Among the submissions received, the editors selected seventeen articles and one review paper that successfully passed the rigorous peer-review process, each offering original insights that push the boundaries of chaos-based security. The collection spans a diverse range of topics, including novel chaotification techniques for enhancing the complexity of dynamical systems, the development of hyperchaotic maps with wide parameter ranges, the integration of chaotic systems with traditional block ciphers, DNA computing for multi-layered encryption, optimized synchronization methods for secure data transmission, and practical hardware implementations using Internet of Things (IoT) devices. Collectively, these papers underscore a strong and consistent focus on developing efficient, robust, and theoretically grounded methods for protecting digital information through the unique properties of chaos, such as sensitivity to initial conditions, ergodicity, and pseudo-random behavior.
Contribution 1 constructs an N-Dimensional Chaotic Modular Model (ND-CMCM). In terms of dynamical analysis, simulation results—including phase diagrams, bifurcation diagrams, Lyapunov exponent plots, and sample entropy—verify that the generated chaotic system exhibits a wide and continuous chaotic parameter range, as well as excellent ergodicity and parameter sensitivity. Building on this foundation, a multi-image encryption algorithm is further designed. Within the proposed encryption framework, a novel look-up table operation based on Latin squares is defined, and an overlapping block permutation–diffusion mechanism is introduced to enhance nonlinear coupling characteristics. This algorithm is not restricted by image type, quantity, or size and allows for the design of chaotic systems with varying dimensions based on the required security level. Experimental results demonstrate that the proposed scheme can effectively encrypt multiple images with high security and computational efficiency.
Contribution 2 introduces a novel delayed coupling method that not only perturbs the control coefficients of chaotic maps but also modifies their functional structure. To validate the efficacy of this approach, the authors constructed and analyzed a time-delay coupled Logistic model. Using the classic Logistic map as a benchmark, they demonstrated a significant enhancement in dynamical complexity through bifurcation diagrams, Lyapunov exponents, and statistical distribution analyses. Numerical simulation results confirm that the delayed coupling method substantially strengthens the dynamical properties of chaotic maps. Based on this improved model, a robust image encryption algorithm was developed; security analyses indicate that the scheme exhibits high key sensitivity and excellent resilience against differential attacks.
Contribution 3 develops a color image encryption algorithm utilizing cross-spiral transformation and zone diffusion. The scheme leverages Chen’s system and piecewise linear chaotic maps (PWLCMs), treating the three-color channels as a three-dimensional matrix. During the permutation phase, the authors designed a cross-spiral transformation mechanism wherein chaotic sequences determine the spiral starting points for each plane, effectively disrupting inter-pixel correlations. In the diffusion stage, a zone-partitioning algorithm was designed to operate simultaneously across all three planes. By randomly determining dividing lines via chaotic sequences and applying bit-level diffusion, the algorithm bolsters its resistance to differential attacks. Experimental results demonstrate that the encrypted images feature uniform histograms and information entropy values nearing the theoretical limit, while the extensive key space and high processing speed ensure the algorithm’s immunity to brute-force and statistical attacks.
Contribution 4 constructs a novel two-dimensional (2D) discrete hyperchaotic map. Through dynamical analyses, including bifurcation behavior and Lyapunov exponents, the authors demonstrated that the system exhibits a broader parameter range and higher complexity than traditional chaotic maps. Building upon this, an efficient pseudo-random number generator (PRNG) was designed and validated using the NIST statistical test suite, showing strong randomness in the generated bit sequences. Furthermore, the authors proposed a Cross-Channel Parallel Permutation–Diffusion (CCPPD) image encryption scheme. This scheme achieved synchronized cross-channel pixel rearrangement and modulo 256 XOR diffusion through two rounds of parallel operations. Simulation and security analyses confirm its robustness against differential, statistical, and interference attacks, highlighting its potential for real-time secure color image communication.
Contribution 5 proposes a color image encryption and decryption algorithm based on a hybrid approach combining the Modular Discrete Derivative (MDD), a novel technique for hiding visual information, with Langton’s Ant and deterministic noise. The scheme exploits the high key space of Langton’s Ant and the significant visual impact of the MDD, generating encrypted images with near-ideal entropy values (up to 7.9999), high NPCR/UACI scores (exceeding 99.60% and 33.44% respectively), and a vast key space greater than 5.8459 × 10,725.8459 × 1072.
Contribution 6 proposes a provably secure and lightweight three-factor session initiation protocol tailored for the human-centered IoT (HC-IoT) environment. It leverages extended fractional Chebyshev chaotic maps (FCCMs), combined with smart cards and user biometrics, to achieve efficient remote authentication. Security was formally proven under the Random Oracle (ROR) model, with a theoretical foundation established based on the FCCM-CDH hardness assumption. Complementary informal security analyses verify the protocol’s resilience against various known attacks. Performance evaluations indicate that the scheme outperforms existing protocols in terms of computational overhead, communication costs, and smart card storage requirements, demonstrating excellent lightweight characteristics.
Contribution 7 presents an image hiding scheme based on stochastic geometric Moiré gratings. Utilizing the optical properties of Moiré gratings in time-averaged mode, the encryption logic relies on designing two distinct random gratings with nearly identical static statistical characteristics to achieve image concealment. Decryption is entirely visual, requiring the cover image to oscillate in a predefined direction and amplitude for direct decoding. Experimental results show that the scheme maintains high decryption quality while significantly enhancing robustness against statistical algorithm attacks, providing a new theoretical foundation for dynamic visual cryptography.
Contribution 8 introduces an efficient image encryption scheme utilizing a novel one-dimensional multiparametric dynamical tent map (MTM) combined with parallel computing. The MTM extends the classical tent map by incorporating six control parameters defined over an unlimited range, significantly enhancing the key space and chaotic complexity. A simple yet effective histogram equalization method is proposed to unify the distribution of chaotic sequences.
Contribution 9 proposes an image encryption scheme based on a newly designed one-dimensional (1D) fractional chaotic map and parallel DNA coding. The mathematical model integrates a Sine map with fractional operations, where the fractional structure separates adjacent orbits in the state space while the Sine function ensures system boundedness. Building on the demonstrated complex chaotic characteristics, the authors developed a parallel DNA encoding algorithm that effectively overcomes the efficiency bottlenecks of traditional DNA-based schemes. Security analyses indicate that the scheme provides excellent performance and low computational overhead, with strong robustness against noise and data-loss attacks.
Contribution 10 designs a quantum color image encryption algorithm based on the QIRHSI representation, incorporating geometric transformations and intensity channel diffusion. The process involves spatial domain permutation using generalized Logistic maps and two-point exchange, followed by intensity transformation via bit-plane cross-exchange and XOR/XNOR operations. Finally, quantum Logistic maps are utilized for intensity diffusion to generate the encrypted quantum image. Experimental results show high key sensitivity and an extensive key space, with metrics such as Shannon entropy, correlation coefficients, and NPCR nearing theoretical limits. The algorithm’s robustness against occlusion attacks and its success in NIST randomness tests further validate its superior performance.
Contribution 11 proposes a chaotification technique for one-dimensional maps based on remainder operator addition, demonstrating that through appropriate parameter choice, the modified map achieves higher Lyapunov exponents and eliminates all stable periodic orbits, leading to robust chaos across wide parameter ranges.
Contribution 12 presents a novel wide-range discrete hyperchaotic map (3D-SCC) based on the Sine map mathematical model, incorporating one Sine and two Cosine functions. The proposed image encryption algorithm integrates this map with an image-sensitive function based on convolution and a dynamic cyclic shift scrambling technique, achieving an average NPCR of 99.60%, UACI of 33.46%, and local Shannon entropy of 7.9029, demonstrating robust resistance against cutting and noise attacks.
Contribution 13 presents an efficient image encryption scheme based on a mix of chaos and a modified SHARK block cipher, utilizing the Duffing chaotic map for enhanced random sequence generation. The algorithm employs a substitution–permutation network structure with plaintext-dependent random keys derived from the Duffing map, incorporating multiple phases to achieve highly random sequences with minimal correlation.
Contribution 14 proposes an adaptive color image encryption scheme based on Multiple Distinct Chaotic Maps (MDCMs) and DNA computing, separately encrypting the red, green, and blue channels using Logistic, tent, and 2D Henon maps respectively. The scheme adaptively modifies map parameters utilizing statistical characteristics (mean, variance, and median) of the plain image, making secret keys plain-image-dependent and thus robust against chosen and known plaintext attacks.
Contribution 15 presents an image encryption scheme synchronizing optimized chaotic systems implemented on Raspberry Pis for IoT applications using the MQTT protocol. The Chen system is optimized using NSGA-II to maximize Lyapunov exponents and the Kaplan–Yorke dimension, with generated binary sequences passing NIST and TestU01 randomness tests after XOR post-processing. Two synchronization methods—Hamiltonian forms and OPCL—are implemented to connect publisher and subscriber devices wirelessly.
Contribution 16 develops image encryption schemes based on a class of uniformly distributed one-dimensional discrete chaotic systems, rigorously proven to be chaotic in the Li–Yorke sense using Marotto’s theorem. By defining a generalized distance function to control boundedness, the constructed chaotic systems achieve uniform distribution through parameter adjustment. The proposed encryption scheme combines these uniformly distributed chaotic sequences with DNA encoding, where DNA coding and decoding rules are dynamically determined by the plaintext image itself.
Contribution 17 proposes an image encryption algorithm that combines an enhanced 2D chaotic map, magic square scrambling, and octree diffusion technology. The improved Logistic map enhances chaotic performance through delayed coupling, parameter perturbation, and feedback control, and the selective bit-swapping operation further increases the unpredictability of the sequence. The encryption process uses a magic square for pixel shuffling, combined with octree-based bit-level diffusion and XOR operations, featuring strong resistance to differential and statistical attacks.
Finally, Contribution 18 provides a systematic review of the development and technical trajectory of chaos-based image encryption. The review categorizes existing methods from multiple perspectives, including symmetric vs. asymmetric encryption, block vs. stream ciphers, and integration with emerging technologies such as neural networks, DNA coding, blockchain, and elliptic curve cryptography. The authors highlight how the unique attributes of chaos—sensitivity to initial conditions, topological transitivity, and pseudo-randomness—facilitate cross-disciplinary evolution. They also discuss practical applications in medical, IoT, and satellite communications, while identifying critical challenges such as resistance to modern cryptanalysis and the efficient processing of encrypted data, offering comprehensive theoretical guidance for future research.
Looking ahead, future research should prioritize the following key directions: (1) developing unified frameworks for counteracting dynamical degradation in digital chaotic systems through hybrid perturbation feedback mechanisms that address finite-precision effects; (2) designing adaptive synchronization methods for heterogeneous IoT and wireless environments using lightweight observer designs and open-plus-closed-loop control; (3) advancing scalable chaotification techniques for high-dimensional systems that provide provable robust chaos across infinite parameter ranges without periodic windows; (4) creating encryption architectures that seamlessly fuse traditional cryptographic structures with chaotic dynamics for robust resistance against classical and quantum-threat scenarios; (5) extending chaos-based security frameworks to multi-modal data—including video streams, medical images, and real-time sensor data—through parallel computing, DNA integration, and hardware–software co-design on embedded platforms. Such efforts will bridge the gap between theoretical advances and practical deployment in security-critical applications.
The Guest Editor would like to express their sincere gratitude to all authors for their valuable contributions to this Special Issue. We also extend our profound thanks to the anonymous reviewers for their insightful and professional assessment reports, which greatly improved the quality of the submitted manuscripts. In addition, we recognize and appreciate the excellent collaboration with the publisher, as well as the consistent support provided by the MDPI associate editors throughout the successful completion of this project.
Conflicts of Interest
The author declares no conflicts of interest.
List of Contributions
- Zhou, Z.; Xu, X.; Jiang, Z.; Sun, K. Multiple-Image Encryption Scheme Based on an N-Dimensional Chaotic Modular Model and Overlapping Block Permutation–Diffusion Using Newly Defined Operation. Mathematics 2023, 11, 3373.
- Huang, S.; Deng, G.; Liu, L.; Li, X. Technique for Enhancing the Chaotic Characteristics of Chaotic Maps Using Delayed Coupling and Its Application in Image Encryption. Mathematics 2023, 11, 3295.
- Zhang, X.; Liu, M.; Yang, X. Color Image Encryption Algorithm Based on Cross-Spiral Transformation and Zone Diffusion. Mathematics 2023, 11, 3228.
- Zhu, S.; Deng, X.; Zhang, W.; Zhu, C. Construction of a New 2D Hyperchaotic Map with Application in Efficient Pseudo-Random Number Generator Design and Color Image Encryption. Mathematics 2023, 11, 3171.
- Moya-Albor, E.; Romero-Arellano, A.; Brieva, J.; Gomez-Coronel, S.L. Color Image Encryption Algorithm Based on a Chaotic Model Using the Modular Discrete Derivative and Langton’s Ant. Mathematics 2023, 11, 2396.
- Meshram, C.; Lee, C.-C.; Bahkali, I.; Imoize, A.L. An Efficient Fractional Chebyshev Chaotic Map-Based Three-Factor Session Initiation Protocol for the Human-Centered IoT Architecture. Mathematics 2023, 11, 2085.
- Saunoriene, L.; Saunoris, M.; Ragulskis, M. Image Hiding in Stochastic Geometric Moiré Gratings. Mathematics 2023, 11, 1763.
- Daoui, A.; Yamni, M.; Chelloug, S.A.; Wani, M.A.; El-Latif, A.A.A. Efficient Image Encryption Scheme Using Novel 1D Multiparametric Dynamical Tent Map and Parallel Computing. Mathematics 2023, 11, 1589.
- Zhu, S.; Deng, X.; Zhang, W.; Zhu, C. Image Encryption Scheme Based on Newly Designed Chaotic Map and Parallel DNA Coding. Mathematics 2023, 11, 231.
- Song, X.; Chen, G.; Abd El-Latif, A.A. Quantum Color Image Encryption Scheme Based on Geometric Transformation and Intensity Channel Diffusion. Mathematics 2022, 10, 3038.
- Moysis, L.; Kafetzis, I.; Baptista, M.S.; Volos, C. Chaotification of One-Dimensional Maps Based on Remainder Operator Addition. Mathematics 2022, 10, 2801.
- Zhong, H.; Li, G.; Xu, X.; Song, X. Image Encryption Algorithm Based on a Novel Wide-Range Discrete Hyperchaotic Map. Mathematics 2022, 10, 2583.
- Rabie, O.; Ahmad, J.; Alghazzawi, D. Modified SHARK Cipher and Duffing Map-Based Cryptosystem. Mathematics 2022, 10, 2034.
- Mansoor, S.; Sarosh, P.; Parah, S.A.; Ullah, H.; Hijji, M.; Muhammad, K. Adaptive Color Image Encryption Scheme Based on Multiple Distinct Chaotic Maps and DNA Computing. Mathematics 2022, 10, 2004.
- Guillén-Fernández, O.; Tlelo-Cuautle, E.; de la Fraga, L.G.; Sandoval-Ibarra, Y.; Nuñez-Perez, J.-C. An Image Encryption Scheme Synchronizing Optimized Chaotic Systems Implemented on Raspberry Pis. Mathematics 2022, 10, 1907.
- Zang, H.; Tai, M.; Wei, X. Image Encryption Schemes Based on a Class of Uniformly Distributed Chaotic Systems. Mathematics 2022, 10, 1027.
- Wang, J.; Liu, L. A Novel Chaos-Based Image Encryption Using Magic Square Scrambling and Octree Diffusing. Mathematics 2022, 10, 457.
- Zhang, B.; Liu, L. Chaos-Based Image Encryption: Review, Application, and Challenges. Mathematics 2023, 11, 2585.
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2026 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license.