Discrete chaotic maps in the real number field have been widely investigated and applied to various applications. However, there has been limited focus on constructing discrete chaotic maps with complicated dynamics in the complex field. In light of this, this article proposes a 1-D complex-variable chaotic model (1-D-CCM), which can produce a multitude of 1-D complex-variable chaotic maps by combining unbounded analytic functions and locally bounded analytic functions. To illustrate the effectiveness of 1-D-CCM, we construct two 1-D complex-variable chaotic maps by combining inverse trigonometric functions and hyperbolic trigonometric functions. We provide theoretical proof of a new 1-D complex-variable chaotic map as one example to demonstrate that the generated chaotic maps satisfy the chaos definition in terms of Lyapunov exponent. Property analysis reveals distinct strange attractors and hyperchaotic behaviors for the two 1-D complex-variable chaotic maps. Performance evaluations show that the example maps of 1-D-CCM model can achieve a 0–1 test value of 1.0017, a$C_{0}$complexity of 0.7253, a correlation dimension of 2.0242, and a sample entropy of 0.8288. Experimental results demonstrate superior performance indicators compared to other representative chaotic maps. We construct a hardware platform using a microcontroller to implement the attractors of the two new complex-variable chaotic maps. Finally, we design pseudorandom number generators to demonstrate the potential applications of the two 1-D complex-variable chaotic maps.

With the proliferation of cloud computing, outsourcing databases has become a common strategy for reducing on premise storage and computation costs. However, this approach raises serious privacy concerns, as sensitive data and query information may be exposed to the cloud. While existing top-$k$query methods have made progress in performance and privacy protection, they offer limited support for the more advanced requirement of diversity constrained queries. In light of this, we present SecDiv, the first privacy-preserving query system that supports diversity constraints over ciphertext in the cloud. SecDiv is built on a two-server distributed trust model and lightweight additive secret sharing, and hides data contents under an honest-but-curious, non-colluding adversary model to ensure that cloud servers learn no sensitive information. SecDiv comprises three customized secure components: SecDMap maps the structured database of the data owner into two secret-shared tables; SecQMap translates each SQL statement and its diversity constraints into vectors whose lengths match the database attributes, thereby hiding targeted attributes and literal values; and SecCQ performs secure filtering, ordering, and top-$k$selection in the cloud, centered on a secure most significant bit comparison implemented by a parallel-prefix adder. A formal security analysis is conducted to provide theoretical guarantees for the security of SecDiv. SecDiv is evaluated on three real datasets, with diversity constraints configured using top-$k$and category count conditions to emulate practical ranking scenarios. Compared with a plaintext baseline, SecDiv achieves identical results with 100% accuracy. Query latency remains practical, with second-level response times in typical settings, and communication overhead increases as expected. Overall, experimental results demonstrate that SecDiv attains a balanced trade-off among privacy, accuracy, and efficiency in real-world cloud service environments.

Secure communication techniques can protect data confidentiality during transmission through public channels. Chaotic systems are commonly used in secure communication due to their random-like behavior, unpredictability, and ergodicity. However, existing chaos-based secure communication schemes have some drawbacks concerning the chaotic systems used and the communication structures, so they cannot achieve satisfactory performance to resist transmission channel noise. In light of this, in this paper, we propose a two-dimensional (2D) cyclic chaotic system (2D-CCS) and design a novel chaos-based secure communication scheme called noise-reduced orthogonal frequency division multiplexing based differential chaos shift keying (NR-OFDM-DCSK). The 2D-CCS is a general framework that can generate a large number of new 2D chaotic maps using existing one-dimensional (1D) chaotic maps as seed maps. Theoretical analysis and experiment results demonstrate its robust chaotic behaviors. The NR-OFDM-DCSK employs a new chaotic map generated by 2D-CCS as the chaos generator, and its structure exhibits a strong ability to resist channel noise, as demonstrated by formulaic analysis. Our extensive experiments show that our developed 2D chaotic maps are more suitable for secure communication applications than existing 2D chaotic maps, and our NR-OFDM-DCSK can achieve a lower bit-error-rate (BER) than state-of-the-art secure communication schemes.

Chaotic systems have attracted extensive research due to their pseudorandomness, ergodicity, and unique properties. Most studies focus on chaotic systems in the real number domain, but recent research has explored the design of complex chaotic systems. However, the chaotic behaviors of previous complex chaotic systems can only be observed through experiments and lack theoretical proof. In this article, we construct a 2-D coupled complex chaotic (2D-CCC) map using two nonlinear functions in the complex number domain. We theoretically prove the robust and complex chaotic behavior of the 2D-CCC map using the Lyapunov exponent. In addition, we conduct extensive experiments to demonstrate the map's intricate dynamics and high performance indicators. Comparison results highlight its superiority over previous chaotic systems. We also implement our 2D-CCC map on a hardware platform to validate its implementation feasibility on hardware devices. Finally, we investigate the 2D-CCC map's application in pseudorandom number generation and the testing results validate the high degree of randomness in the generated pseudorandom numbers.

Designing chaotic maps and fractal maps with rich dynamics in the complex field presents an interesting and challenging research topic. In this paper, we propose a novel approach called the one-dimensional multi-valued model (1D-MVM) for generating 1D complex chaotic and fractal maps by levering both single-valued and multi-valued functions. To demonstrate the effectiveness of the 1D-MVM, we present two 1D complex chaotic maps and one 1D fractal map as specific examples. Theoretical analysis confirms that the chaotic maps generated by the 1D-MVM exhibit chaotic behavior, while property analysis reveals that these chaotic maps possess hyperchaotic strange attractors, with their associated Lyapunov exponents being determined by specific system parameters. We also conduct extensive experiments to demonstrate the intricate dynamics and high performance indicators of the newly generated complex chaotic maps. A hardware platform is developed and the principal value attractors of these chaotic maps are experimentally captured. In addition, we explore the application of the hyperchaotic sequences generated by these complex chaotic maps to the pseudo-random number generators. Rigorous testing results validate the high degree of randomness exhibited by the generated pseudo-random numbers. Finally, we leverage the second branch of the 1D fractal map to generate a diverse range of fractal structures, further demonstrating the versatility and potential applications of the proposed approach.

High-dimensional (HD) chaotic map has wide applications in various research fields such as neural networks and secure communication. Designing HD chaotic maps with expected dynamics and robust hyperchaotic behaviors is an interesting but challenging topic. In this article, we propose an$n$-dimensional hyperchaotic map$(n\text{D}$-HCM) generation method on the basis of the Gershgorin-type theorem. First, the general form of the proposed$n\text{D}$-HCM is built using$n$parametric polynomials. Then, the entity and coefficient parameter matrices are configured according to the Gershorin-type theorem. Theoretical analysis shows that the generated$n\text{D}$-HCM has$n$positive Lyapunov exponents and thus can show robust hyperchaotic behaviors. Two examples of hyperchaotic map with specified equations are provided and their properties are analyzed to show the availability of the proposed method. Performance evaluations display that our$n\text{D}$-HCM possesses abundant properties and complex behaviors, and it can outperform some representative HD chaotic maps. Moreover, to show the application of our$n\text{D}$-HCM, we apply it to a secure communication scheme and the experimental results exhibit that it shows much better performance than these representative HD chaotic maps in resisting transmission noise.

Designing high-dimensional chaotic maps with expected dynamic properties is an attractive but challenging task. The dynamic properties of a chaotic system can be reflected by the Lyapunov exponents (LEs). Using the inherent relationship between the parameters of a chaotic map and its LEs, this paper proposes an$n$-dimensional polynomial chaotic system ($n\text{D}$-PCS) that can generate$n\text{D}$chaotic maps with any desired LEs. The$n\text{D}$-PCS is constructed from$n$parametric polynomials with arbitrary orders, and its parameter matrix is configured using the preliminaries in linear algebra. Theoretical analysis proves that the$n\text{D}$-PCS can produce high-dimensional chaotic maps with any desired LEs. To show the effects of the$n\text{D}$-PCS, two high-dimensional chaotic maps with hyperchaotic behaviors were generated. A microcontroller-based hardware platform was developed to implement the two chaotic maps, and the test results demonstrated the randomness properties of their chaotic signals. Performance evaluations indicate that the high-dimensional chaotic maps generated from$n\text{D}$-PCS have the desired LEs and more complicated dynamic behaviors compared with other high-dimensional chaotic maps. In addition, to demonstrate the applications of$n\text{D}$-PCS, we developed a chaos-based secure communication scheme. Simulation results show that$n\text{D}$-PCS has a stronger ability to resist channel noise than other high-dimensional chaotic maps.

When high-dimensional chaotic systems are applied to many practical applications, they are required to have robust and complex hyperchaotic behaviors. In this article, we propose a novel$n$D chaotic system construction method using the Pascal-matrix theory. First, a parametric Pascal matrix is constructed. Then, an$n$D chaotic system can be generated by using the parametric Pascal matrix as the parameter matrix of the system. Theoretical analysis shows that the generated$n$D chaotic systems have robust and complex chaotic behaviors, and they become$n$D Arnold Cat maps by fixing the parameters as some special values. Performance evaluations demonstrate that the$n$D chaotic systems have more complex chaotic behaviors and better distribution of outputs compared with existing HD chaotic systems. A 4-D Arnold Cat map and a 4-D chaotic map with hyperchaotic behaviors are generated as two examples. The two chaotic maps are then simulated on a microcontroller-based hardware platform and the chaotic sequences are tested to show good randomness.