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Some weaknesses of 1D chaotic maps, such as lacking of ergodicity, multiple bifurcations, dense periodic windows, and short iteration period, limit their practical applications in cryptography. A higher-dimensional chaotic map with ergodicity can solve these problems. Based on 1D quadratic map, a 3D exponential hyperchaotic map (3D-EHCM) is constructed, and its dynamic behaviors, such as phase diagram, Lyapunov exponent spectrum, Kolmogorov entropy (KE), correlation dimension, approximate entropy and randomness, are analyzed and tested. The results demonstrate that the 3D-EHCM has ergodicity in a larger range of control parameter, and its state points have a longer period. To counteract dynamical degradation and make it suitable for a PRNG, the periodic point detection and random impulsive perturbation are applied to lengthen the aperiodic time sequence, and statistical results demonstrate that a full-period sequence can be obtained.

The chaotic map has complex dynamics under ideal conditions however it suffers from the problem of performance degradation in the case of finite computing precision. In order to prevent the dynamics degradation, in this paper the continuous Chen chaotic system is used to perturb both the inputs and parameters of Chebyshev map to minimize the chaotic degradation phenomenon under finite precision. Experimental evaluations and corresponding performance analysis demonstrate that the Chebyshev chaotic map has a good randomness and complex dynamic performance by using the proposed perturbation method, and some attributes of the proposed system are stronger than the original system (e.g. chaos attractor and approximate entropy). Finally, the corresponding pseudorandom number generator (PRNG) is constructed by this method and then its randomness is evaluated via NIST SP800-22 and TestU01 test suites, respectively. Statistical test results show that the proposed PRNG has high reliability of randomness, thus it can be used for cryptography and other potential applications.

This paper introduces an FPGA implementation of a pseudo-random number generator (PRNG) using Chen’s chaotic system. This paper mainly focuses on the development of an efficient VLSI architecture of PRNG in terms of bit rate, area resources, latency, maximum length sequence, and randomness. First, we analyze the dynamic behavior of the chaotic trajectories of Chen’s system and set the parameter’s value to maintain low hardware design complexity. A circuit realization of the proposed PRNG is presented using hardwired shifting, additions, subtractions, and multiplexing schemes. The benefit of this architecture, all the binary multiplications (except $X_i\cdot Y_i$ and $X_i\cdot Z_i)$ operations are performed using hardwired shifting. Moreover, the generated sequences pass all the 15 statistical tests of NIST, while it generates pseudo-random numbers at a uniform clock rate with minimum hardware complexity. The proposed architecture of PRNG is realized using Verilog HDL, prototyped on the Virtex-5 FPGA (XC5VLX50T) device, and its analysis has been done using the Matlab tool. Performance analysis confirms that the proposed Chen chaotic attractor-based PRNG scheme is simple, secure, and hardware efficient, with high potential to be adopted in cryptography applications.

Based on the mathematical model of the elliptical cylinder, we design a new hyperchaotic map with an elliptical cylinder or a cylinder attractor. The dynamical analysis results indicate the proposed system is globally hyperchaotic, and has large Lyapunov Exponents (LEs), and high Permutation Entropy (PE) complexity. Interestingly, the hyperchaotic system exhibits the offset boosting coexistence attractors with respect to the system parameters. In addition, three Multicavity Hyperchaotic Maps (MHCM) are constructed by introducing a symmetric staircase function, which expands greatly the phase space of the system. The MHCM have more complex topological structures and maintain the chaotic performance of the original map. To illustrate the feasibility of the hyperchaotic systems further, we apply them to design a Pseudo-Random Number Generator (PRNG), and implement them on the DSP platform.

To address the problem of dynamic degradation over a finite-precision platform of chaotic maps and the reversibility of linear chaotic maps, we propose an improved model over GF($2^n$) that is called the nondegenerate m-Dimensional $(m\ge 2)$ Integer-Domain Chaotic Maps (mD-IDCMs). This model incorporates modular exponentiation operation, and is capable of constructing nondegenerate IDCMs of any dimension. Moreover, we prove the irreversibility of mD-IDCM and analyze its chaotic behaviors in terms of positive Lyapunov Exponents (LEs). The results of theoretical analysis show that the proposed mD-IDCM model can obtain the desired positive LEs by appropriately configuring its coefficient matrix. Then, we present two instances, and analyze their LEs, Kolmogorov entropy, Sample entropy, Correlation dimension, and the dynamic analysis indicates that the chaotic map constructed by mD-IDCM has ergodicity within a sufficiently large chaotic range. Finally, we design a Pseudo-Random Number Generator (PRNG) with a key to verify the practicability of the mD-IDCM.

This paper explores the deep-zoom properties of the chaotic $k$-logistic map, in order to propose an improved chaos-based cryptosystem. This map was shown to enhance the random features of the Logistic map, while at the same time reducing the predictability about its orbits. We incorporate its strengths to security into a previously published cryptosystem to provide an optimal pseudo-random number generator (PRNG) as its core operation. The result is a reliable method that does not have the weaknesses previously reported about the original cryptosystem.

A novel 3D chaotic oscillator that incorporates quadratic and absolute-function nonlinearities is introduced in this paper. The system dynamics are explored using the Lyapunov direct method, phase plane trajectories, time response, Lyapunov exponents, bifurcation diagrams, and basins of attraction. The uniqueness and existence of the system solution have been proven. The analytical investigations show that the system has two stable equilibrium points along with one unstable equilibrium point and a line of equilibria. The positive half of this line represents unstable equilibria, while the negative half is associated with stable equilibria. Additionally, it is found that the oscillator exhibits a chaotic basin of attraction centered along the line of equilibria and surrounded by a fixed-point attractor. An electronic circuit using Multisim software is designed to demonstrate the possibility of physical implementation for the considered mathematical model. Chaos control is addressed using adaptive and sliding mode control strategies. The performance of both control methods is compared, with the sliding mode control demonstrating superior results in both fast response and small transient overshoot. Furthermore, a novel sliding mode controller for master–slave synchronization is introduced, showing high performance compared to other sliding-mode and adaptive control methods applied in the literature. Finally, the proposed oscillator is employed as a Pseudo Random Number Generator (PRNG) for image encryption applications using a new encryption model. The experimental results confirm the high security and robustness of the proposed encryption algorithm against various attack methods.