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In this paper, definition and properties of logistic map along with orbit and bifurcation diagrams, Lyapunov exponent, and its histogram are considered. In order to expand chaotic region of Logistic map and make it suitable for cryptography, two modified versions of Logistic map are proposed. In the First Modification of Logistic map (FML), vertical symmetry and transformation to the right are used. In the Second Modification of Logistic (SML) map, vertical and horizontal symmetry and transformation to the right are used. Sensitivity of FML to initial condition is less and sensitivity of SML map to initial condition is more than the others. The total chaotic range of SML is more than others. Histograms of Logistic map and SML map are identical. Chaotic range of SML map is fivefold of chaotic range of Logistic map. This property gave more key space for cryptographic purposes.

In order to explain simple chaotic dynamical systems, one-dimensional map is used. Tent, Bernoulli and Logistic maps are common examples of them. The return map of Tent and Bernoulli are linear, while Logistic map is nonlinear [

One dimensional map is simple as far as hardware implementation is concerned. In the other hand, security of nonlinear maps is usually more than linear functions. Logistic map is generally used in most of cryptosystems and pseudo random generators. It is used in chaos-based secure communication system and for generations of binary numbers. Li, Mou, and Cai proposed statistical properties of digital piecewise linear chaotic maps and their roles in cryptography and pseudo-random coding [

In this paper, with the aim of expanding chaotic range of Logistic map, two modified versions of Logistic map are proposed. Definition and properties of Logistic map are reviewed in Section 2. The first and second modified versions of Logistic map are proposed in Section 3. Return maps, orbit diagrams, bifurcation diagrams and Lyapunov exponents were also concerned. Comparison of the modified map with Logistic with respect to their sensitivity to initial conditions, their chaotic range and histograms is considered in Section 4. Finally, conclusion and references are integrated.

Logistic map is one-dimensional map which uses to model simple nonlinear discrete systems. Logistic map explain by a recursive function as follows:

where r is its parameter and

Sensitivity of Logistic map to initial condition could be observed by plotting orbit diagrams with respect to two initial conditions with small difference. The corresponding orbit diagrams with respect to two initial conditions 0.350 and 0.351 for fixed values of

In order to view chaotic properties of Logistic map, bifurcation diagram and Lyapunov exponent of it should be calculated and plotted. Bifurcation diagram of Logistic mapwith respect to “r” are calculated and plotted in

Lyapunov exponent of Logistic mapwith respect to “r” are also calculated and plotted in

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A discrete dynamic process is said to be two-segmental if there exists a partitioning point. The general equation

of the process could be defined as depicted in Equation (2), such that

where

The necessary condition for a two segmental function

As far as Logistic map is concerned, its equation could be separated as follows with respect to

Considering Equation (4), the derivatives of

We modified Logistic map, by obtaining vertical symmetry of

where

Orbit diagrams of FML map with respect to two initial conditions 0.350 and 0.351 for fixed values of

Bifurcation diagram of FML map with respect to “r” are calculated and plotted in

We make another modification to

where n is a time index,

In order to explain the performance of Equation (6), the return map of the result is drawn in

The orbit diagrams of SML map with respect to two initial conditions 0.350 and 0.351 for fixed value of

Bifurcation diagram of SML map with respect to ‘r’ are calculated and plotted in

With the purpose of expanding chaotic range, two modified version of Logistic map are proposed. In order to compare the performance of the proposed maps with respect to applications, orbit diagrams, bifurcation diagram, Lyapunov exponent and histogram of outputs are considered. Orbit diagram shows the sensitivity of the map to initial conditions. Bifurcation diagram and Lyapunov exponent is used to evaluate chaotic behavior of the maps. Histogram of outputs which could be plotted by observing outputs of large number of iterations, simulate probability density function of the maps.

In order to compare the sensitivity of the proposed maps, FML and SML, with Logistic map to initial condition, their orbit diagrams with respect to two initial conditions with small difference are considered. They are shown in

Bifurcation diagram and Lyapunov exponent of Logistic mapwith respect to “r” are plotted in plotted

ted in

Simulation of probability density function could be performed to show the statistical characteristics of the map. This simulation is run for 10,000 iterations on the map and draws its histogram.

The probability density function of the SML map is also simulated with 10,000 iterations on SML map.

It is appearing that histograms of Logistic map and SML map are identical, while FML map could not perform acceptable result.

In order to evaluate the performance of Logistic map, after considering definition and properties of it, orbit diagrams, Lyapunov exponent and histogram of Logistic map were considered. Orbit diagram showed that the sensitivity of Logistic map to initial condition was medium. Bifurcation diagram and Lyapunov exponent were used to evaluate chaotic properties of the map and recognize the range of parameters. The total chaotic range of Logistic map was small.

With the purpose of expanding chaotic range, two modified versions of Logistic map are proposed. They are one-dimensional and two-segmental nonlinear maps. We called them First and Second Modified Logistic (FML & SML). We found vertical symmetry of first segment, transformed the result to right for the second segment, and called it FML map. Definition and properties of FML map were also considered. Sensitivity of FML map to

Map | Chaotic range (out of 4) | Chaotic range ratio (%) |
---|---|---|

Logistic | 0.4 | 10% |

FML | 1.2 | 36% |

SML | 2 | 50% |

initial condition is not suitable. However, FML map is chaotic when parameter “r” lies in intervals [2.6, 2.9] or [3.2, 4] according to the graphs of bifurcation diagram and Lyapunov exponent. To define a second versionmodified Logistic map, we found vertical and horizontal symmetry of its first segment and transformed the result to right. Recursive equation of Second Modified Logistic (SML) map is forming. Sensitivity of SML map to initial condition is observed in the figure. There is superior sensitivity to initial condition. According to

It was concluded that sensitivity of FML to initial condition was less and sensitivity of SML map to initial condition was more than the others. Histograms of Logistic map and SML map are identical, while FML map cannot perform acceptable results. The total chaotic range of SML is more than the others. Chaotic range of SML map is fivefold of chaotic range of Logistic map. This property expands key space for cryptographic purposes.