# A Novel Method to Identify Initial Values of Chaotic Maps in Cybersecurity

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries and Related Work

#### 2.1. Related Work

#### 2.2. Lorenz System

#### 2.3. Logistic Map

## 3. Auto-Correlation of Chaotic Maps

## 4. Identification of Initial Values

#### 4.1. Initial Values for Logistic Map

#### 4.1.1. Setting Number of Iterations n

#### 4.1.2. Variance against Parameter r

#### 4.2. Initial Values of Lorenz System

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The illustration of the application considered in this work; basic communication model of symmetric cryptography.

**Figure 2.**Numerical solution of Lorenz system with initial values of $({x}_{0},{y}_{0},{z}_{0})\to (-8,8,27)$ and constant values of $a=10,b=8/3$ and $r=28$ for: (

**a**) $x-y$ structure; (

**b**) $x-z$ structure; and (

**c**) $x-y-z$ structure.

**Figure 3.**(

**a**) The individual x-sequence against number of iterations with initial conditions of $({x}_{0},{y}_{0},{z}_{0})\to (-8,8,27)$. (

**b**) Sensitive dependence of initial conditions: the plot for x-sequence with initial conditions of $({x}_{0},{y}_{0},{z}_{0})\to (-8.00000000,8,27)$ as well as the plot for x-sequence with initial conditions of $({x}_{0},{y}_{0},{z}_{0})\to (-8.00000001,8,27)$. It can be seen that the two sequences completely differ apart after 1300 iterations, despite of the difference in one of the initial conditions by a margin of 0.00000001.

**Figure 4.**(

**a**) The individual x-sequence against number of iterations with initial conditions of $({x}_{0},r)\to (0.5,3.7)$. (

**b**) Sensitive dependence of initial conditions; shows the plot for x-sequence with initial conditions of $({x}_{0},r)\to (0.5,3.700000000)$ as well as the plot for x-sequence with initial conditions of $({x}_{0},r)\to (0.5,3.700000001)$. It can be seen that the two sequences completely differ apart after 75 iterations despite the difference in one of the initial conditions by a margin of 0.000000001.

**Figure 5.**Proposed framework for identifying the initial values of chaotic maps using three steps: accessing the long term chaotic sequence, identification of the type of chaotic map using auto-correlation graphs and identification of the initial values using different statistical analysis.

**Figure 6.**Auto-correlation graphs of different chaotic maps with their respective initial conditions and parameters taken on 1000 iterations: (

**a**) Henon chaotic map x-sequence; (

**b**) Henon chaotic map y-sequence; (

**c**) Ikeda chaotic map q-sequence; (

**d**) Ikeda chaotic map w-sequence; (

**e**) Logistic map; and (

**f**) Quadratic map.

**Figure 7.**Auto-correlation graphs of different chaotic maps with their respective initial conditions and parameters taken on 1000 iterations: (

**a**) Lorenz chaotic map x-sequence; (

**b**) Lorenz chaotic map y-sequence; (

**c**) Lorenz chaotic map z-sequence; (

**d**) Rossler chaotic map e-sequence; (

**e**) Rossler chaotic map q-sequence; and (

**f**) Rossler chaotic map w-sequence.

**Figure 9.**(

**a**)–(

**l**) Median values of logistic map for different initial values of x against r with the range of 3.6–4.0. Although the median values fluctuate in the range of 0.45–0.75 against r, there is no linear trend (increasing or decreasing), making it a difficult task to assign distinct median values to distinct values of r.

**Figure 10.**(

**a**)–(

**l**) Mode values of logistic map for different initial values of x against r with the range of 3.6–4.0. The condition is even worse in this case as compared to median analysis; not only is there no linear trend but the mode values are also not constant for different initial values of x.

**Figure 11.**(

**a**) Plot of variance values of 5000 logistic sequence iterations for 3.7 value of r against different initial values of x, where the variance remains almost the same for each and every initial value of x. (

**b**) The plot of variance values of logistic sequences of 50 iterations for a 3.7 value of r against different initial values of x. It can be seen that there is a variation between the variance values in this case.

**Figure 12.**(

**a**)–(

**l**) Plot of variance values of 1200 logistic sequence iterations for 0.5 value of x against different values of parameter r. The variance for different values of r are different and partially unique.

**Figure 14.**Illustration of periodicity of logistic map for (

**a**) $r=3.83$; (

**b**) $r=3.84$ and (

**c**) $r=3.85$.

**Figure 15.**Standard deviation of Lorenz system for x sequence, with initial values of $({y}_{0},{z}_{0})\to (8,27)$ and constant values of $a=10,b=8/3$ and $r=28$, showing different values of standard deviation for different values of r, for $\left({x}_{0}\right)\to (-3)\phantom{\rule{4pt}{0ex}}to\phantom{\rule{4pt}{0ex}}(-8)$ in (

**a**–

**f**), respectively.

**Figure 16.**Standard deviation of Lorenz system for y sequence, with initial values of $({x}_{0},{z}_{0})\to (-8,27)$ and constant values of $a=10,b=8/3$ and $r=28$, showing different values of standard deviation for different values of r, for $\left({y}_{0}\right)\to \left(3\right)\phantom{\rule{4pt}{0ex}}to\phantom{\rule{4pt}{0ex}}\left(8\right)$ in (

**a**–

**f**), respectively.

**Figure 17.**Standard deviation of Lorenz system for y sequence, with initial values of $({x}_{0},{y}_{0})\to (-8,8)$ and constant values of $a=10,b=8/3$ and $r=28$, showing different values of standard deviation for different values of r, for $\left({z}_{0}\right)\to \left(24\right)\phantom{\rule{4pt}{0ex}}to\phantom{\rule{4pt}{0ex}}\left(28\right)$ in (

**a**–

**f**), respectively.

Region | Variance Range | Parameter, r Range |
---|---|---|

1 | 0.039–0.050 | 3.60–3.73 |

2 | 0.050–0.085 | 3.73–3.82 |

3 | 0.085–0.110 | 3.82–3.86 |

4 | 0.085–0.118 | 3.86–3.99 |

**Table 2.**Standard deviation values of Lorenz system for x, y and z sequences against parameter r range.

r | x | y | z | r | x | y | z |
---|---|---|---|---|---|---|---|

24 | 5.9721 | 6.8049 | 7.5644 | 44 | 10.4401 | 13.7855 | 14.8617 |

26 | 6.5186 | 7.2033 | 6.1305 | 46 | 10.6639 | 13.7069 | 13.3215 |

28 | 7.9851 | 9.1540 | 8.2105 | 48 | 10.9514 | 14.2571 | 14.6004 |

30 | 7.7617 | 9.0118 | 7.8162 | 50 | 11.5522 | 14.7872 | 13.4296 |

32 | 8.8601 | 10.3146 | 9.1604 | 52 | 11.6983 | 15.5916 | 14.7479 |

34 | 8.9962 | 10.5397 | 9.4619 | 54 | 12.0774 | 15.6581 | 13.7141 |

36 | 9.4806 | 11.0827 | 9.3204 | 56 | 12.1455 | 15.9730 | 13.9568 |

38 | 9.7408 | 11.4435 | 9.7575 | 58 | 12.3523 | 16.5226 | 14.4226 |

40 | 10.0324 | 12.6311 | 12.7988 | 60 | 12.8113 | 17.6593 | 16.2303 |

42 | 10.2070 | 13.1060 | 13.7087 | 62 | 13.0964 | 18.1885 | 16.5554 |

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**MDPI and ACS Style**

Anees, A.; Hussain, I.
A Novel Method to Identify Initial Values of Chaotic Maps in Cybersecurity. *Symmetry* **2019**, *11*, 140.
https://doi.org/10.3390/sym11020140

**AMA Style**

Anees A, Hussain I.
A Novel Method to Identify Initial Values of Chaotic Maps in Cybersecurity. *Symmetry*. 2019; 11(2):140.
https://doi.org/10.3390/sym11020140

**Chicago/Turabian Style**

Anees, Amir, and Iqtadar Hussain.
2019. "A Novel Method to Identify Initial Values of Chaotic Maps in Cybersecurity" *Symmetry* 11, no. 2: 140.
https://doi.org/10.3390/sym11020140