Cryptanalyzing image encryption using chaotic logistic map

Chaotic behavior arises from very simple non-linear dynamical equation of logistic map which makes it was used often in designing chaotic image encryption schemes. However, some properties of chaotic maps can also facilitate cryptanalysis especially when they are implemented in digital domain. Utilizing stable distribution of the chaotic states generated by iterating the logistic map, this paper presents a typical example to show insecurity of an image encryption scheme using chaotic logistic map. This work will push encryption and chaos be combined in a more effective way.     

Analysis of a new simple one dimensional chaotic map

In this paper, a new one-dimensional map is introduced, which exhibits chaotic behavior in small interval of real numbers. It is discovered that a very simple fraction in a square root with one variable and two parameters can lead to a period-doubling bifurcations. Given the nonlinear dynamics of one-dimensional chaotic maps, it is usually seen that chaos arises when the parameter raises up to a value, however in our map, which seems reverse, it arises when the related parameter decreases and approaches to a constant value. Since proposing a new map entails solid foundations, the analysis is originated with linear stability analysis of the new map, finding fixed points. Additionally, the nonlinear dynamics analysis of the new map also includes cobweb plot, bifurcation diagram, and Lyapunov analysis to realize further dynamics. This research is mainly consisting of real numbers, therefore imaginary parts of the simulations are omitted. For the numerical analysis, parameters are assigned to given values, yet a generalized version of the map is also introduced.     

Designing a multi-scroll chaotic system by operating Logistic map with fractal process

Recently, chaotic systems have been widely investigated in several engineering applications. This paper presents a new chaotic system based on Julia’s fractal process, chaotic attractors and Logistic map in a complex set. Complex dynamic characteristics were analyzed, such as equilibrium points, bifurcation, Lyapunov exponents and chaotic behavior of the proposed chaotic system. As we know, one positive Lyapunov exponent proved the chaotic state. Numerical simulation shows a plethora of complex dynamic behaviors, which coexist with an antagonist form mixed of bifurcation and attractor. Then, we introduce an algorithm for image encryption based on chaotic system. The algorithm consists of two main stages: confusion and diffusion. Experimental results have proved that the proposed maps used are more complicated and they have a key space sufficiently large. The proposed image encryption algorithm is compared to other recent image encryption schemes by using different security analysis factors including differential attacks analysis, statistical tests, key space analysis, information entropy test and running time. The results demonstrated that the proposed image encryption scheme has better results in the level of security and speed.     

A Simple Chaotic Circuit with Impulsive Switch Depending on Time and State

This paper presents a simple chaotic circuit consisting of two capacitors, one linear two-port VCCS and one time-state-controlled impulsive switch. The impulsive switch causes rich chaotic and periodic behavior. The circuit dynamics can be simplified into a one-dimensional return map that is piecewise linear and piecewise monotone. Using the return map, we clarify parameter conditions for existence of chaotic and periodic attractors and coexistence state of attractors.     

Stability of synchronized and clustered states in a system of coupled piecewise-linear maps

Parameter regions for different types of stability of synchronized and clustered states are obtained for two interacting ensembles of globally coupled one-dimensional piecewise-linear maps. We analyze the strong (asymptotic) and weak (Milnor) stability of the synchronized state, as well as its instability. We establish that the stability and instability regions in the phase space depend only on parameters of the individual skew tent map and do not depend on the ensemble size. In the simplest nontrivial case of four coupled chaotic maps, we obtain stability regions for coherent and two-cluster states. The regions appear to be large enough to provide an efficient control of coherent and clustered chaotic regimes. The transition from desynchronization to synchronization is identified to be qualitatively different in smooth and piecewise-linear models.Published in Neliniini Kolyvannya, Vol. 7, No. 2, pp. 217–228, April–June, 2004.     

有界噪声激励下Josephson系统的混沌运动

利用随机Melnikov方法分析了有界噪声激励下Josephson系统的运动,并运用均方准则得到了系统产生混沌的临界值。结果表明:有界噪声对系统混沌行为的产生起到了加速的作用;且有界噪声的强度越大,混沌吸引子的发散程度就越大。最后利用数值模拟得到系统的庞加莱映射,分析了在不同参数组合下系统庞加莱映射的特征。结果显示:当有界噪声中的一个参数发生改变,系统的庞加莱映射也会发生相应的改变;特别是有界噪声的激励强度增大时,系统庞加莱映射的发散程度也会随之增大。这从侧面验证了理论结果的正确性。     

Difference map and its electronic circuit realization

In this paper we study the dynamical behavior of the one-dimensional discrete-time system, the so-called iterated map. Namely, a bimodal quadratic map is introduced which is obtained as an amplification of the difference between well-known logistic and tent maps. Thus, it is denoted as the so-called difference map. The difference map exhibits a variety of behaviors according to the selection of the bifurcation parameter. The corresponding bifurcations are studied by numerical simulations and experimentally. The stability of the difference map is studied by means of Lyapunov exponent and is proved to be chaotic according to Devaney’s definition of chaos. Later on, a design of the electronic implementation of the difference map is presented. The difference map electronic circuit is built using operational amplifiers, resistors and an analog multiplier. It turns out that this electronic circuit presents fixed points, periodicity, chaos and intermittency that match with high accuracy to the corresponding values predicted theoretically.     

Construction of a novel nth-order polynomial chaotic map and its application in the pseudorandom number generator

Chaotic maps with good chaotic performance have been extensively designed in cryptography recently. This paper gives an nth-order polynomial chaotic map by using topological conjugation with piecewise linear chaos map. The range of chaotic parameters of this nth-order polynomial chaotic map is large and continuous. And the larger n is, the greater the Lyapunov exponent is and the more complex the dynamic characteristic of the nth-order polynomial chaotic map. The above characteristics of the nth-order polynomial chaotic map avoid the disadvantages of one-dimensional chaotic systems in secure application to some extent. Furthermore, the nth-order polynomial chaotic map is proved to be an extension of the Chebyshev polynomial map, which enriches chaotic map. The numerical simulation of dynamic behaviors for an 8th-order polynomial map satisfying the chaotic condition is carried out, and the numerical simulation results show the correctness of the related conclusion. This paper proposed the pseudorandom number generator according to the 8th-order polynomial chaotic map constructed in this paper. Using the performance analysis of the proposed pseudorandom number generator, the analysis result shows that the pseudorandom number generator according to the 8th-order polynomial chaotic map can efficiently generate pseudorandom sequences with higher performance through the randomness analysis with NIST SP800-22 and TestU01, security analysis and efficiency analysis. Compared with the other pseudorandom number generators based on chaotic systems in recent references, this paper performs a comprehensive performance analysis of the pseudorandom number generator according to the 8th-order polynomial chaotic map, which indicates the potential of its application in cryptography.

     

A new pseudo-random number generator based on CML and chaotic iteration

In this paper, we propose a new algorithm of generating pseudorandom number generator (PRNG), which we call (couple map lattice based on discrete chaotic iteration (CMLDCI)) that combine the couple map lattice (CML) and chaotic iteration. And we can prove that this method can be written in a form of chaos map, which is under the sense of Devaney chaos. In addition, we test the new algorithm in NIST 800-22 statistical test suits and we use it in image encryption.     

The Fast Norm Vector Indicator (FNVI) method: a new dynamical parameter for detecting order and chaos in Hamiltonian systems

In the present article, we introduce and also deploy a new, simple, very fast, and efficient method, the Fast Norm Vector Indicator (FNVI) in order to distinguish rapidly and with certainty between ordered and chaotic motion in Hamiltonian systems. This distinction is based on the different behavior of the FNVI for the two cases: the indicator after a very short transient period of fluctuation displays a nearly constant value for regular orbits, while it continues to fluctuate significantly for chaotic orbits. In order to quantify the results obtained by the FNVI method, we establish the dFNVI, which is the quantified numerical version of the FNVI. A thorough study of the method??s ability to achieve an early and clear detection of an orbit??s behavior is presented both in two and three degrees of freedom (2D and 3D) Hamiltonians. Exploiting the advantages of the dFNVI method, we demonstrate how one can rapidly identify even tiny regions of order or chaos in the phase space of Hamiltonian systems. The new method can also be applied in order to follow the time evolution of sticky orbits. A detailed comparison between the new FNVI method and some other well-known dynamical methods of chaos detection reveals the great efficiency and the leading role of this new dynamical indicator.     

An efficient approach for the construction of LFT S-boxes using chaotic logistic map

In this paper, we synthesize substitution boxes by the use of chaotic logistic maps in linear fractional transformation. In order to introduce randomness in the construction of S-boxes, the data from the chaotic system is used in linear fractional transformation to add additional unpredictable behavior. The proposed S-box is tested for its strength in encryption applications. The nonlinearity characteristic of the proposed S-box is studied, and the strength of the cipher is quantized in terms of this property. In addition, the behavior of bit changes at the output of the cipher in comparison with the input is also studied. Similarly, the input/output differential is also evaluated for different bit patterns. The results of statistical analyses show superior performance of the proposed S-boxes.     

A pseudo-random numbers generator based on a novel 3D chaotic map with an application to color image encryption

Nonlinear Dynamics - In this work, we propose a novel 3D chaotic map obtained by coupling the piecewise and logistic maps. Showing excellent properties, like a high randomness, a high complexity...     

Bifurcation and chaos of a simple walking model driven by a rhythmic signal

The walk of animals is achieved by the interaction between the dynamics of their mechanical system and the central pattern generator (CPG). In this paper, we analyze dynamic properties of a simple walking model of a biped robot driven by a rhythmic signal from an oscillator. In particular, we examine the long-term global behavior and the bifurcation of the motion that leads to chaotic motion, depending on the model parameter values. The simple model consists of a hip and two legs connected at the hip through a rotational joint. The joint is driven by a rhythmic signal from an oscillator, which is an open loop. In order to analyze the bifurcation, we first obtained approximate solutions of the walking motion and then constructed discrete dynamics using the Poincaré map. As a result, we found that consecutive period-doubling bifurcations occur as the model parameter values change, and that the walking motion leads to chaotic motion over the critical value of the model parameters. Moreover, we approximately obtained the period-doubling solutions and the critical value by employing a Newton-Raphson method. Our analytical results were verified by the numerical simulations.     

Bifurcations and chaos in parametrically excited single-degree-of-freedom systems

The behavior of single-degree-of-freedom systems possessing quadratic and cubic nonlinearities subject to parametric excitation is investigated. Both fundamental and principal parametric resonances are considered. A global bifurcation diagram in the excitation amplitude and excitation frequency domain is presented showing different possible stable steady-state solutions (attractors). Fractal basin maps for fundamental and principal parametric resonances when three attractors coexist are presented in color. An enlargement of one region of the map for principal parametric resonance reveals a Cantor-like set of fractal boundaries. For some cases, both periodic and chaotic attractors coexist.     

On fractional difference logistic maps: Dynamic analysis and synchronous control

This paper investigates a logistic map derived from a difference equation in the framework of discrete fractional calculus. Through the Poincaré plots and Julia sets, the map’s chaotic and fractal characteristics are studied comparing with those of a quadratic map to be proposed. The memory effect of fractional difference maps is reflected in these dynamics, and some reasonable explanations are given by combining with quantitative analysis. A coupled controller is designed to realize synchronization between fractional difference logistic map and fractional difference quadratic map.

     

Hardware implementation of pseudo-random number generators based on chaotic maps

We show the usefulness of bifurcation diagrams to implement a pseudo-random number generator (PRNG) based on chaotic maps. We provide details on the selection of the best parameter values to obtain high entropy and positive Lyapunov exponent from the bifurcation diagram of four chaotic maps, namely: Bernoulli shift map, tent, zigzag, and Borujeni maps. The binary sequences obtained from these maps are analyzed to implement a PRNG both in software and in hardware. The software implementation is realized using 32 and 64 bits microprocessor architectures, and with floating point and fixed point computer arithmetic. The hardware implementation is done by using a field-programmable gate array (FPGA) architecture. We developed a serial communication interface between the PRNG on the FPGA and a personal computer to obtain the generated sequences. We validate the randomness of the generated binary sequences with the NIST test suite 800-22-a both in floating point and fixed point arithmetic. At the end, we show that those chaotic maps are suitable to implement a PRNG but according to the hardware resources, the one based on the Bernoulli shift map is better. In addition, another advantage is that the required initial value for the sequences can be within the whole interval \([-1,1]\), including its bounds.     

Efficient method for designing chaotic S-boxes based on generalized Baker’s map and TDERC chaotic sequence

The theory of chaos is applied to the construction of substitution boxes used in encryption applications. The synthesis process of the proposed substitution boxes is presented, which is based on chaotic Baker’s map and TDERC chaotic sequences. The objectives of the new substitution box are to provide enhanced resistance against differential and linear cryptanalysis. The constructed substitution boxes uses Galois field elements and relies on discrete chaotic maps while keeping differential and linear approximation probabilities to desired levels.     

裂纹转子在支承松动时的振动特性研究

以具有支承松动的Jeffcott裂纹转子为研究对象，分析了支承松动和轴上横向裂纹对转子系统刚度的影响，建立了转子系统振动的微分方程，并用数值方法分析了其振动特性。分析表明，转子在裂纹和支承松动这两种非线性因素的作用下，表现出复杂的非线性行为。     

The simple chaotic model of passive dynamic walking

Recent findings on the dynamical analysis of human locomotion characteristics such as stride length signal have shown that this process is intrinsically a chaotic behavior. The passive walking has been defined as walking down a shallow slope without using any muscular contraction as an active controller. Based on this definition, some knee-less models have been proposed to present the simplest possible models of human gait. To maintain stability, these simple passive models are compelled to show a wide range of different dynamics from order to chaos. Unfortunately, based on simplifications, for many years the cyclic period-one behavior of these models has been considered as the only stable response. This assumption is not in line with the findings about the nature of walking. Thus, this paper proposes a novel model to demonstrate that the knee-less passive dynamic models also have the ability to model the chaotic behavior of human locomotion with some modifications. The presented novel model can show chaotic behavior as a stable and acceptable answer using a chaotic function in heel-strike condition. The represented chaotic model is also able to simulate different types of motor deficits such as Parkinson’s disease only by manipulating the value of chaotic parameter. Our model has extensively examined in complexity and chaotic behavior using different analytical methods such as fractal dimension, bifurcation and largest Lyapunov exponent, and it was compared with conventional passive models and the stride signal of healthy subjects and Parkinson patients.     

On parameter characteristics of chaotic synchronization in two nonlinear gyroscope systems

In this paper, the partial and full chaotic synchronizations of two nonlinear gyroscope systems with/without noise are investigated. From analytical conditions for synchronization and non-synchronization of two gyroscope systems, the parameter characteristic study is completed for a better understanding of the synchronization dynamics of two gyroscope dynamical systems. The boundaries of the parameter map for synchronization are determined by the onset and vanishing conditions of synchronization. The simple feedback control can make the noised gyroscope system synchronizing with chaotic behaviors of the expected gyroscope system. The methodology presented in this paper is different from other techniques for synchronization. The partial synchronization is an important phenomenon to be observed in engineering applications.