Constructing Keyed Strong S-Box Using an Enhanced Quadratic Map

2021 ◽  
Vol 31 (10) ◽  
pp. 2150146
Author(s):  
Yuanyuan Si ◽  
Hongjun Liu ◽  
Yuehui Chen
Keyword(s):  
Phase Diagram ◽  
Fixed Point ◽  
Lyapunov Exponent ◽  
Uniform Distribution ◽  
Bifurcation Diagram ◽  
Experimental Results ◽  
Kolmogorov Entropy ◽  
Symmetric Cryptography ◽  
Quadratic Map ◽  
Nonlinear Component

As the only nonlinear component for symmetric cryptography, S-Box plays an important role. An S-Box may be vulnerable because of the existence of fixed point, reverse fixed point or short iteration cycles. To construct a keyed strong S-Box, first, a 2D enhanced quadratic map (EQM) was constructed, and its dynamic behaviors were analyzed through phase diagram, Lyapunov exponent, Kolmogorov entropy, bifurcation diagram and randomness testing. The results demonstrated that the state points of EQM have uniform distribution, ergodicity and better randomness. Then a keyed strong S-Box construction algorithm was designed based on EQM, and the fixed point, reverse fixed point, and short cycles were eliminated. Experimental results verified the algorithm’s feasibility and effectiveness.

HYPERCHAOS Qi SYSTEM

2010 ◽  
Vol 24 (24) ◽  
pp. 4771-4778 ◽  
Author(s):  
XING-YUAN WANG ◽  
YONG-FENG GAO ◽  
YAO-XIAN ZHANG
Keyword(s):  
Phase Diagram ◽  
Lyapunov Exponent ◽  
Numerical Simulations ◽  
Bifurcation Diagram ◽  
Nonlinear Term ◽  
Linear Term ◽  
Nonlinear Controller ◽  
Chaos System ◽  
Lyapunov Exponent Spectrum

This paper presents a four-dimensional hyperchaos Qi system, obtained by adding linear term and nonlinear term of nonlinear controller to Qi chaos system. The hyperchaos Qi system is studied by bifurcation diagram, Lyapunov exponent spectrum and phase diagram. Numerical simulations show that the new system's behavior can be periodic, chaotic and hyperchaotic as the parameter varies.

Realization of a Novel Logarithmic Chaotic System and Its Characteristic Analysis

2019 ◽  
Vol 29 (02) ◽  
pp. 1930004 ◽  
Author(s):  
Xiaoyuan Wang ◽  
Xiaotao Min ◽  
Jun Yu ◽  
Yiran Shen ◽  
Guangyi Wang ◽  
...  
Keyword(s):  
Phase Diagram ◽  
Lyapunov Exponent ◽  
Theoretical Analysis ◽  
Chaotic System ◽  
Nonlinear Term ◽  
Experimental Results ◽  
Characteristic Analysis ◽  
Chaotic Parameter ◽  
Hardware Circuit ◽  
Lyapunov Exponent Spectrum

To further improve the complexity of the chaotic system and broaden the chaotic parameter range, a novel logarithmic chaotic system was constructed by adding a nonlinear term of logarithm. The dynamic characteristics of the chaotic system were analyzed by chaotic phase diagram, bifurcation diagram, Lyapunov exponent spectrum, Poincaré mapping and dynamical map, etc. The system was digitized by DSP simulation, and the corresponding experimental results are completely consistent with the theoretical analysis. Furthermore, the equivalent hardware circuit was designed and theoretical analysis was verified by its experimental results.

A Switched Hyperchaotic System and Analysis of Dynamical Properties

2012 ◽  
Vol 452-453 ◽  
pp. 511-515
Author(s):  
Bian Li ◽  
Ai Xue Qi ◽  
Wei Bing Li
Keyword(s):  
Lyapunov Exponent ◽  
Numerical Simulations ◽  
Bifurcation Diagram ◽  
Experimental Results ◽  
Hyperchaotic System ◽  
Symbolic Function ◽  
Dynamical Properties ◽  
Lyapunov Exponent Spectrum

This paper introduces a new switched hyperchaotic system by utilizing symbolic function. The switched hyperchaotic system converts its states from one subsystem to another via symbolic functions. By the analysis of its dynamics, symmetry, dissipativity, Lyapunov exponent spectrum and bifurcation diagram of the system. The system is implemented by the FPGA hardware. It is shown that the experimental results are identical with numerical simulations, and the chaotic trajectories are much more complex.

A Novel Four-Order Chaotic System with N-Attractor Multi-Direction Multi-Scroll Attractors

2014 ◽  
Vol 678 ◽  
pp. 81-88
Author(s):  
Wen Shuang Yin ◽  
Dai Jun Wei ◽  
Shi Qiang Chen
Keyword(s):  
Lyapunov Exponent ◽  
Bifurcation Diagram ◽  
Chaotic System ◽  
Operational Amplifier ◽  
Experimental Results ◽  
Order System ◽  
Chaotic Attractors ◽  
Maximum Lyapunov Exponent ◽  
Nonlinear Functions ◽  
New System

In this paper, a novel four-order system is proposed. It can generate N-attractor multi-direction multi-scroll attractor by adding simple nonlinear functions. We analyze the new system by using means of maximum Lyapunov exponent, bifurcation diagram and Poincaré maps of the system. Moreover, an minimum operational amplifier circuit is designed for realizing 2×(3×3 ×3) scroll chaotic attractors, and experimental results are also obtained, which verify chaos characteristics of the system.

CHAOTIC FEATURE OF MARTIN PROCESS IMPOSED ON THE COSINE FUNCTION

Fractals ◽  
2009 ◽  
Vol 17 (02) ◽  
pp. 191-195
Author(s):  
CHUANHOU GAO ◽  
ZHIMIN ZHOU ◽  
JIUSUN ZENG ◽  
JIMING CHEN
Keyword(s):  
Phase Diagram ◽  
Lyapunov Exponent ◽  
Bifurcation Diagram ◽  
Chaotic Attractor ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
Cosine Function ◽  
Maximum Lyapunov Exponent ◽  
System Parameters ◽  
Stable Focus

By analyzing the phase diagram of Martin process on the cosine function, it is shown that with the change of system parameters the system will eventually converge to a chaotic attractor. The process is repeated and stable focus, period doubling bifurcation occurs during this process. Further computation gives the maximum Lyapunov exponent of the system and meanwhile, the bifurcation diagram is drawn. Thus it is proved from theory that the system exhibits strong chaotic properties.

Hamiltonian Chaos in a Model Alveolus

2008 ◽  
Vol 131 (1) ◽  
Author(s):  
F. S. Henry ◽  
F. E. Laine-Pearson ◽  
A. Tsuda
Keyword(s):  
Reynolds Number ◽  
Fixed Point ◽  
Lyapunov Exponent ◽  
Fine Particles ◽  
Hamiltonian Dynamics ◽  
Maximal Lyapunov Exponent ◽  
Poincaré Sections ◽  
Pulmonary Acinus ◽  
Hamiltonian Chaos ◽  
Poincare Sections

In the pulmonary acinus, the airflow Reynolds number is usually much less than unity and hence the flow might be expected to be reversible. However, this does not appear to be the case as a significant portion of the fine particles that reach the acinus remains there after exhalation. We believe that this irreversibility is at large a result of chaotic mixing in the alveoli of the acinar airways. To test this hypothesis, we solved numerically the equations for incompressible, pulsatile, flow in a rigid alveolated duct and tracked numerous fluid particles over many breathing cycles. The resulting Poincaré sections exhibit chains of islands on which particles travel. In the region between these chains of islands, some particles move chaotically. The presence of chaos is supported by the results of an estimate of the maximal Lyapunov exponent. It is shown that the streamfunction equation for this flow may be written in the form of a Hamiltonian system and that an expansion of this equation captures all the essential features of the Poincaré sections. Elements of Kolmogorov–Arnol’d–Moser theory, the Poincaré–Birkhoff fixed-point theorem, and associated Hamiltonian dynamics theory are then employed to confirm the existence of chaos in the flow in a rigid alveolated duct.

COMPLEX BIFURCATION STRUCTURES IN THE HINDMARSH–ROSE NEURON MODEL

2007 ◽  
Vol 17 (09) ◽  
pp. 3071-3083 ◽  
Author(s):  
J. M. GONZÀLEZ-MIRANDA
Keyword(s):  
Phase Diagram ◽  
Bifurcation Diagram ◽  
Parameter Space ◽  
Neuron Model ◽  
Two Dimensional ◽  
Dimensional Parameter ◽  
Rapid Responses ◽  
Bifurcation Structures

The results of a study of the bifurcation diagram of the Hindmarsh–Rose neuron model in a two-dimensional parameter space are reported. This diagram shows the existence and extent of complex bifurcation structures that might be useful to understand the mechanisms used by the neurons to encode information and give rapid responses to stimulus. Moreover, the information contained in this phase diagram provides a background to develop our understanding of the dynamics of interacting neurons.

Research on Distribution Regularity of Nano-Sized WC in Cast Steel

2014 ◽  
Vol 668-669 ◽  
pp. 52-55
Author(s):  
Yang Han ◽  
Ai Ling Zhang ◽  
Lin Yang ◽  
Ya Ling Han
Keyword(s):  
Wear Resistance ◽  
Uniform Distribution ◽  
Cast Steel ◽  
Experimental Results ◽  
Steel Matrix ◽  
Gravity Segregation ◽  
Resistance Alloy ◽  
Segregation Phenomenon

The microstructure of 40CrNi2Mo steel matrix strengthened with wear resistance alloy is observed by the optical scope, SEM with EDS and FESEM. Analysis emphasis is lied on the distribution regulation of nanosized WC particulates in the microstructure of the steel matrix. Experimental results show that the method of adding wear resistance alloy in steel matrix can avoid gravity segregation phenomenon effectively and guarantee a uniform distribution of WC in steel matrix. nanosized WC particulates distributing evenly in steel matrix improve the wear resistance, and make microstructure of the steel matrix more uniform, finer and denser proved by its high-expansion micrograph.

scholarly journals A Hidden Chaotic System with Multiple Attractors

Entropy ◽  
2021 ◽  
Vol 23 (10) ◽  
pp. 1341
Author(s):  
Xiefu Zhang ◽  
Zean Tian ◽  
Jian Li ◽  
Xianming Wu ◽  
Zhongwei Cui
Keyword(s):  
Phase Diagram ◽  
Lyapunov Exponent ◽  
Numerical Methods ◽  
Chaotic System ◽  
Time Domain ◽  
Chaotic Attractor ◽  
Largest Lyapunov Exponent ◽  
Spectral Entropy ◽  
Multiple Attractors ◽  
System Parameters

This paper reports a hidden chaotic system without equilibrium point. The proposed system is studied by the software of MATLAB R2018 through several numerical methods, including Largest Lyapunov exponent, bifurcation diagram, phase diagram, Poincaré map, time-domain waveform, attractive basin and Spectral Entropy. Seven types of attractors are found through altering the system parameters and some interesting characteristics such as coexistence attractors, controllability of chaotic attractor, hyperchaotic behavior and transition behavior are observed. Particularly, the Spectral Entropy algorithm is used to analyze the system and based on the normalized values of Spectral Entropy, the state of the studied system can be identified. Furthermore, the system has been implemented physically to verify the realizability.

A Chaotification model based on sine and cosecant functions for enhancing chaos

2021 ◽  
Author(s):  
Yuqing Li ◽  
Xing He ◽  
Dawen Xia
Keyword(s):  
Fixed Point ◽  
Lyapunov Exponent ◽  
Chaotic Maps ◽  
Dynamic Properties ◽  
Chaotic Map ◽  
Sample Entropy ◽  
One Dimensional ◽  
Model Based ◽  
Chaotic Region ◽  
Definition Of

Chaotic maps with higher chaotic complexity are urgently needed in many application scenarios. This paper proposes a chaotification model based on sine and cosecant functions (CMSC) to improve the dynamic properties of existing chaotic maps. CMSC can generate a new map with higher chaotic complexity by using the existing one-dimensional (1D) chaotic map as a seed map. To discuss the performance of CMSC, the chaos properties of CMSC are analyzed based on the mathematical definition of the Lyapunov exponent (LE). Then, three new maps are generated by applying three classical 1D chaotic maps to CMSC respectively, and the dynamic behaviors of the new maps are analyzed in terms of fixed point, bifurcation diagram, sample entropy (SE), etc. The results of the analysis demonstrate that the new maps have a larger chaotic region and excellent chaotic characteristics.

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