Hierarchy of random chaotic maps with an invariant measure

Jafarizadeh, M. A.; Behnia, S.

doi:10.1063/1.1610240

Cited by 8 publications

(28 citation statements)

References 26 publications

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AbstractWe present hierarchy of one and many-parameter families of elliptic chaotic maps of cn and sn types at the interval [0,1]. It is proved that for small values of k the parameter of the elliptic function, these maps are topologically conjugate to the maps of references [1,2], where using this we have been able to obtain the invariant measure of these maps for small k and thereof it is shown that these maps have the same Kolmogorov-Sinai entropy or equivalently Lyapunov characteristic exponent of the maps [1,2].

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confidence: 99%

Hierarchy of one- and many-parameter families of elliptic chaotic maps of cn and sn types

Jafarizadeh¹,

Behnia²

2003

Physics Letters A

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We present hierarchy of one and many-parameter families of elliptic chaotic maps of cn and sn types at the interval [0,1]. It is proved that for small values of k the parameter of the elliptic function, these maps are topologically conjugate to the maps of references [1,2], where using this we have been able to obtain the invariant measure of these maps for small k and thereof it is shown that these maps have the same Kolmogorov-Sinai entropy or equivalently Lyapunov characteristic exponent of the maps [1,2]. As this parameter vanishes, the maps are reduced to the maps presented in above-mentioned reference. Also in contrary to the usual family of one-parameter maps, such as the logistic and tent maps, these maps do not display period doubling or period-n-tupling cascade transition to chaos, but they have single fixed point attractor at certain parameter values where they bifurcate directly to chaos without having period-n-tupling scenario exactly at these values of parameters whose Lyapunov characteristic exponent begin to be positive.

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confidence: 99%

Hierarchy of one- and many-parameter families of elliptic chaotic maps of cn and sn types

Jafarizadeh¹,

Behnia²

2003

Physics Letters A

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“…A recent attempts in introducing the hierarchy of chaotic maps with their invariant measure [16,17,22,23,24] allows us to advance in answering to a question how to define non-ergodic maps and what are the condition for non-ergodicity in these types of system.…”

Section: Resultsmentioning

confidence: 99%

“…Similar to the calculation of the invariant measure in our pervious papers [16,17,22,23,24], we present here it for the piecewise chaotic map. In order to prove that measure satisfied equation (3-1), we consider the conjugate map;…”

Section: Appendix Amentioning

confidence: 95%

Hierarchy of piecewise non-linear maps with non-ergodic behaviour

Jafarizadeh

Behnia

Foroutan

2004

J. Phys. A: Math. Gen.

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We study the dynamics of hierarchy of piecewise maps generated by one-parameter families of trigonometric chaotic maps and one-parameter families of elliptic chaotic maps of cn and sn types, in detail. We calculate the Lyapunov exponent and KolmogorovSinai entropy of the these maps with respect to control parameter. Non-ergodicity of these piecewise maps is proven analytically and investigated numerically . The invariant measure of these maps which are not equal to one or zero, appears to be characteristic of non-ergodicity behavior. A quantity of interest is the Kolmogorov-Sinai entropy, where for these maps are smaller than the sum of positive Lyapunov exponents and it confirms the non-ergodicity of the maps.

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“…One-parameter families of chaotic maps of the interval [0, 1] with an invariant measure can be defined as the ratio of polynomials of degree N [Jafarizadeh et al, 2001]:…”

Section: Composition Of Trigonometric Chaotic Mapsmentioning

confidence: 99%

“…This paper aims to introduce a new chaotic algorithm which has the advantages of high-level security, large key space and the acceptable encryption speed. Since digital images are usually represented as twodimensional arrays, we present algorithm based on Trigonometric Chaotic Maps and their Composition [Jafarizadeh et al, 2001]. A diffusion process is performed to confuse the relationship between cipher-image and plain-image.…”

Section: Introductionmentioning

confidence: 99%

Chaotic Cryptographic Scheme Based on Composition Maps

Behnia

Akhshani

Mahmodi

et al. 2008

Int. J. Bifurcation Chaos

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In recent years, a growing number of cryptosystems based on chaos have been proposed. But most of them encountered many problems such as small key space and weak security. In the present paper, a new kind of chaotic cryptosystem based on Composition of Trigonometric Chaotic Maps is proposed.These maps which are defined as ratios of polynomials of degree N, have interesting properties such as invariant measure, ergodicity, variable chaotic region with respect to the control parameters and ability to construct composition form of maps. We have used a composition of chaotic map to shuffle the position of image pixels. Another composition of chaotic map is used * E-mail: S.behnia@iaurmia.ac.ir 1 Cryptography based on Composed Maps 2 in diffusion process. According to the performed analysis, the introduced algorithm can satisfy the required performances such as high level security, large key space and the acceptable encryption speed.

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Hierarchy of random chaotic maps with an invariant measure

Cited by 8 publications

References 26 publications

Hierarchy of one- and many-parameter families of elliptic chaotic maps of cn and sn types

Hierarchy of one- and many-parameter families of elliptic chaotic maps of cn and sn types

Hierarchy of piecewise non-linear maps with non-ergodic behaviour

Chaotic Cryptographic Scheme Based on Composition Maps

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