scholarly journals Approximation of planar Sobolev $W^{2,1}$ homeomorphisms by Piecewise Quadratic Homeomorphisms and Diffeomorphisms

2021 ◽  
Author(s):  
Daniel Campbell ◽  
Stanislav Hencl
Keyword(s):  
Quadratic Map

Given a Sobolev homeomorphism $f\in W^{2,1}$ in the plane we find a piecewise quadratic homeomorphism that approximates it up to a set of $\epsilon$ measure. We show that this piecewise quadratic map can be approximated by diffeomorphisms in the $W^{2,1}$ norm on this set.

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.

scholarly journals Some criteria for chaos and no chaos in the quadratic map of the plane

2009 ◽  
Vol 22 (1) ◽  
pp. 105-117 ◽  
Author(s):  
Zeraoulia Elhadj ◽  
Clinton Sprott
Keyword(s):  
Chaotic Attractors ◽  
Quadratic Map

This paper gives some criteria for the existence and the non-existence of chaotic attractors in the general 2-D quadratic map. .

scholarly journals Dynamics of a quasi-quadratic map

2013 ◽  
Vol 20 (1) ◽  
pp. 36-48
Author(s):  
Assis Azevedo ◽  
Maria Carvalho ◽  
António Machiavelo
Keyword(s):  
Quadratic Map

Automatic generation of general quadratic map basins

1998 ◽  
pp. 341-345
Author(s):  
Julien C. Sprott ◽  
Clifford A. Pickover
Keyword(s):  
Automatic Generation ◽  
Quadratic Map

SEARCH-BASED CHAOTIC PSEUDORANDOM BIT GENERATOR

2009 ◽  
Vol 19 (12) ◽  
pp. 4227-4235 ◽  
Author(s):  
ALI KANSO
Keyword(s):  
Stream Cipher ◽  
Chaotic Map ◽  
Statistical Test ◽  
Search Technique ◽  
Real Numbers ◽  
Quadratic Map ◽  
Tent Map ◽  
Numerical Orbit ◽  
High Level ◽  
Theoretical Results

This paper proposes the construction of a new chaotic pseudorandom bit generator, which forms the main building block of a chaotic stream cipher. The design of the algorithm is based on a single chaotic map whose numerical orbit indirectly contributes towards the generation of the keystream. The latter is produced from the numerical orbit by applying a technique that searches for iterates in specific intervals [a,b], for some real numbers a and b, and outputs 0 or 1 based on the iterate preceding the targeted iterate. The generator suggested here is built up from a quadratic map. We analyze the cycle length of the keystreams and investigate the resistance of the generator to well-known cryptanalytic attacks. Furthermore, the statistic characteristics of the keystreams are examined numerically using the NIST statistical test suite. The numerical and theoretical results demonstrate that the proposed technique results in generating keystreams possessing very good cryptographic properties and high level of security against existing cryptanalytic attacks. Empirical results show that the search technique leads to the generation of keystreams possessing good randomness properties when applied to any chaotic map whose orbits have good randomness properties such as the quadratic map, tent map and sawtooth map.

The Quadratic Map

2010 ◽  
pp. 179-208
Author(s):  
Stan Wagon
Keyword(s):  
Quadratic Map

CHAOS AND FRACTAL OF THE GENERAL TWO-DIMENSIONAL QUADRATIC MAP

2008 ◽  
Vol 22 (20) ◽  
pp. 3461-3471
Author(s):  
XINGYUAN WANG
Keyword(s):  
Fractal Dimension ◽  
General Feature ◽  
Strange Attractors ◽  
Two Dimensional ◽  
Control Parameters ◽  
Self Similarity ◽  
Boundary Equation ◽  
Quadratic Map ◽  
Fractal Images ◽  
Stable Points

The nature of the stable points of the general two-dimensional quadratic map is considered analytically, and the boundary equation of the first bifurcation of the map in the parameter space is given out. The general feature of the nonlinear dynamic activities of the map is analyzed by the method of numerical computation. By utilizing the Lyapunov exponent as a criterion, this paper constructs the strange attractors of the general two-dimensional quadratic map, and calculates the fractal dimension of the strange attractors according to the Lyapunov exponents. At the same time, the researches on the fractal images of the general two-dimensional quadratic map make it clear that when the control parameters are different, the fractal images are different from each other, and these fractal images exhibit the fractal property of self-similarity.

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