The work in this paper extends a memristive chaotic system with transcendental nonlinearities to the fractional-order domain. The extended system’s chaotic properties were validated through bifurcation analysis and spectral entropy. The presented system was employed in the substitution stage of an image encryption algorithm, including a generalized Arnold map for the permutation. The encryption scheme demonstrated its efficiency through statistical tests, key sensitivity analysis and resistance to brute force and differential attacks. The fractional-order memristive system includes a reconfigurable coordinate rotation digital computer (CORDIC) and Grünwald–Letnikov (GL) architectures, which are essential for trigonometric and hyperbolic functions and fractional-order operator implementations, respectively. The proposed system was implemented on the Artix-7 FPGA board, achieving a throughput of 0.396 Gbit/s. © 2023 by the authors.
FPGA realization of fractals based on a new generalized complex logistic map
This paper introduces a new generalized complex logistic map and the FPGA realization of a corresponding fractal generation application. The chaotic properties of the proposed map are studied through the stability conditions, bifurcation behavior and maximum Lyapunov exponent (MLE). A relation between the mathematical analysis and fractal behavior is demonstrated, which enables formulating the fractal limits. A compact fractal generation process is presented, which results in designing and implementing an optimized hardware architecture. An efficient FPGA implementation of the fractal behavior is validated experimentally on Artix-7 FPGA board. Two examples of fractal implementation are verified, yielding frequencies of 34.593 MHz and 31.979 MHz and throughputs of 0.415 Gbit/s, 0.384 Gbit/s. Compared to recent related works, the proposed implementation demonstrates its efficient hardware utilization and suitability for potential applications. © 2021 Elsevier Ltd
Numerical Sensitivity Analysis and Hardware Verification of a Transiently-Chaotic Attractor
We introduce a new chaotic system with nonhyperbolic equilibrium and study its sensitivity to different numerical integration techniques prior to implementing it on an FPGA. We show that the discretization method used in numerically integrating the set of differential equations in MATLAB and Mathematica does not yield chaotic behavior except when a low accuracy Euler method is used. More accurate higher-order numerical algorithms (such as midpoint and fourth-order Runge-Kutta) result in divergence in both MATLAB and Mathematica (but not Python), which agrees with the divergence observed in an analog circuit implementation of the system. However, a fixed-point digital FPGA implementation confirms the chaotic behavior of the system using Euler and fourth-order Runge-Kutta realizations. Therefore, the increased sensitivity of chaotic systems with nonhyperbolic equilibrium should be carefully considered for reproducibility. © 2022 World Scientific Publishing Company.
Software and hardware realizations for different designs of chaos-based secret image sharing systems
Secret image sharing (SIS) conveys a secret image to mutually suspicious receivers by sending meaningless shares to the participants, and all shares must be present to recover the secret. This paper proposes and compares three systems for secret sharing, where a visual cryptography system is designed with a fast recovery scheme as the backbone for all systems. Then, an SIS system is introduced for sharing any type of image, where it improves security using the Lorenz chaotic system as the source of randomness and the generalized Arnold transform as a permutation module. The second SIS system further enhances security and robustness by utilizing SHA-256 and RSA cryptosystem. The presented architectures are implemented on a field programmable gate array (FPGA) to enhance computational efficiency and facilitate real-time processing. Detailed experimental results and comparisons between the software and hardware realizations are presented. Security analysis and comparisons with related literature are also introduced with good results, including statistical tests, differential attack measures, robustness tests against noise and crop attacks, key sensitivity tests, and performance analysis. © The Author(s) 2024.
Chaotic Dynamics and FPGA Implementation of a Fractional-Order Chaotic System with Time Delay
This article proposes a numerical solution approach and Field Programmable Gate Array implementation of a delayed fractional-order system. The proposed method is amenable to a sufficiently efficient hardware realization. The system’s numerical solution and hardware realization have two requirements. First, the delay terms are implemented by employing LookUp Tables to keep the already required delayed samples in the dynamical equations. Second, the fractional derivative is numerically approximated using Grünwald-Letnikov approximation with a memory window size, L, according to the short memory principle such that it balances between accuracy and efficiency. Bifurcation diagrams and spectral entropy validate the chaotic behaviour of the system for commensurate and incommensurate orders. Additionally, the dynamic behaviour of the system is studied versus the delay parameter, ?, and the window size, L. The system is realized on Nexys 4 Artix-7 FPGA XC7A100T with throughput 1.2 Gbit/s and hardware resources utilization 15% from the total LookUp Tables and 4% from the slice registers. Oscilloscope experimental results verify the numerical solution of the delayed fractional-order system. The amenability to digital hardware realization, which is experimentally validated in this article, is added to the system’s advantages and encourages its utilization in future digital applications such as chaos control and synchronization and chaos-based communication applications. © 2020 IEEE.
Self-Reproducing Hidden Attractors in Fractional-Order Chaotic Systems Using Affine Transformations
This article proposes a unified approach for hidden attractors control in fractional-order chaotic systems. Hidden attractors have small basins of attractions and are very sensitive to initial conditions and parameters. That is, they can be easily drifted from chaotic behavior into another type of dynamics, which is not suitable for encryption applications that require quite wide initial conditions and parameters ranges for encryption key design. Hence, a systematic coordinate affine transformation framework is utilized to construct transformed systems with self-reproducing attractors. Simulation results of two three-dimensional fractional-order chaotic systems with hidden attractors validate that the proposed framework supports attractors geometric structure design and multi-wing generation. Hidden attractor size, polarity, phase, shape and position control while preserving the chaotic dynamics is indicated by strange attractors, spectral entropy, maximum Lyapunov exponent and bifurcation diagrams. Simulations demonstrate the capability of multi-wing generation from fractional-order hidden attractors with no equilibria using non-autonomous parameters as opposed to the classical equilibria extension techniques suitable only for self-excited attractors. The self-reproduced multiple wings can share the same center point or be distributed along an arbitrary line, curve or surface thanks to the non-autonomous translation parameters. Multi-wing attractors widen the basin of attraction and enlarge the state space volume. For practical applications, the proposed technique makes fractional-order systems with hidden attractors suitable for circuit implementations that require specific signal level and polarity conditions. In addition, for digital encryption applications, the relatively wide range of the extra parameters enhances the key space and hence the robustness against brute force attacks. © 2020 IEEE.
Finite precision logistic map between computational efficiency and accuracy with encryption applications
Chaotic systems appear in many applications such as pseudo-random number generation, text encryption, and secure image transfer
Design of Positive, Negative, and Alternating Sign Generalized Logistic Maps
The discrete logistic map is one of the most famous discrete chaotic maps which has widely spread applications
Design and FPGA Verification of Custom-Shaped Chaotic Attractors Using Rotation, Offset Boosting and Amplitude Control
This paper proposes a method of generating custom-shaped attractors, which depends on a planarly rotating V-shape multi-scroll chaotic system with offset boosting and amplitude control, and its FPGA verification
Trajectory control and image encryption using affine transformation of lorenz system
This paper presents a generalization of chaotic systems using two-dimensional affine transformations with six introduced parameters to achieve scaling, reflection, rotation, translation and/or shearing