# Chaotic Discrete Fractional-Order Food Chain Model and Hybrid Image Encryption Scheme Application

^{1}

^{2}

^{3}

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^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Theorem**

**1**

**.**The delta fractional difference equation

**Remark**

**1.**

**Theorem**

**2.**

## 3. Discrete-Time Fractional-Order Food Chain Model

#### 3.1. Stability of Fixed Points

#### 3.1.1. Stability Analysis of ${E}_{0}$

#### 3.1.2. Stability Analysis of ${E}_{1}$

#### 3.1.3. Stability Analysis of ${E}_{2}$

#### 3.1.4. Stability Analysis of ${E}_{3}$

## 4. Numerical Simulations

## 5. Hybrid Image Encryption Scheme

## 6. Security Analysis of the Proposed Scheme

#### 6.1. Numerical Simulations

#### 6.2. Keyspace Analysis

#### 6.3. Analysis of Key Sensitivity

#### 6.4. Analysis of Information Entropy

#### 6.5. Differential Attacks Analysis

#### 6.6. Resistance against Other Attacks

## 7. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Region of stability of fixed point ${E}_{1}$ in (a, $\alpha $) plane for (

**a**) $\alpha =0.8$ and (

**b**) $\alpha =0.99$.

**Figure 3.**Stability region of fixed point ${E}_{2}$ in $(a,b)$ plane for (

**a**) $\alpha =0.8$, (

**b**) $\alpha =0.9$ and also in $(a,c)$ plane for (

**c**) $\alpha =0.8$, (

**d**) $\alpha =0.9$.

**Figure 4.**Stability region of fixed point ${E}_{3}$ in $(a,b)$ plane for $\alpha =0.9$ and (

**a**) $c=3.9$ and also in $(a,c)$ plane for $\alpha =0.9$ and (

**b**) $b=3$.

**Figure 5.**(

**a**–

**c**) Time series of the state variables $x,y$ and z of the stable fixed point ${E}_{0}$ at $a=0.5,$ $b=1,c=1$ and $\alpha $$=0.9$.

**Figure 6.**(

**a**–

**c**) Time series of the state variables $x,y$ and z of the stable fixed point ${E}_{1}$ at $a=2,$ $b=1,c=1$ and $\alpha $$=0.9$.

**Figure 7.**(

**a**–

**c**) Time series of the state variables $x,y$ and z of the stable fixed point ${E}_{2}$ at $a=3,$ $b=2.5,c=0.9$ and $\alpha $$=0.8$.

**Figure 8.**(

**a**–

**c**) Time series of the state variables $x,y$ and z of the stable fixed point ${E}_{3}$ at $a=4,$ $b=3,c=3.5$ and $\alpha $$=0.9$.

**Figure 9.**Bifurcation diagrams of parameters a and b vs. state variable x of (2) at (

**a**) $b=4.6,c=3$ and $\alpha =0.95,$ (

**b**) $a=2.1,c=9.14$ and $\alpha =0.95,$ respectively.

**Figure 10.**Bifurcation diagrams of (

**a**) parameter $\alpha $ when $a=2,b=3.35,c=9.15,$ (

**b**) parameter b when $a=2,c=9.15,$ $\alpha =0.95,$ (

**c**) parameter b when $a=2,c=9.15,\alpha =0.85,$ (

**d**) parameter $\alpha $ when $a=2.1,b=4.6,c=3,$ (

**e**) parameter a when $b=4.6,c=3,\alpha =0.95$ and (

**f**) parameter b when $a=2.1,c=3,\alpha =0.95$ vs. state variable x of (2).

**Figure 11.**Examples of the two and three dimensional phase portraits of the models (1) and (2) obtained at (

**a**–

**c**) $a=2;$ $b=4.6;c=3;\alpha =1$, (

**d**–

**f**) $a=2;b=3.1;c=9.15;\alpha =1$, (

**g**–

**i**) $a=2;b=3.18;$ $c=9.15;\alpha =0.99,$ (

**j**–

**l**) $a=2;b=3.25;$ $c=9.15;\alpha =0.99$ and finally (

**m**–

**o**) $a=2;b=3.35;c=9.15;\alpha =0.99$.

**Figure 12.**The plain, ciphered and deciphered King Tut images are shown in (

**a**). The corresponding histograms of each color value are illustrated in (

**b**) in the way that the first, second and third rows are representing, respectively, red, green and blue colors.

**Figure 13.**(

**a**) The original, encrypted and decrypted image of a baboon face. (

**b**) Histograms for those images given in (

**a**).

**Figure 14.**(

**a**) The original, encrypted and decrypted image of a pepper. (

**b**) Histograms for those images given in (

**a**).

**Figure 15.**(

**a**) The original, encrypted and decrypted image of the Egyptian pyramids. (b) Histograms for those images given in (

**a**).

**Figure 16.**Decrypted (

**a**) King Tut image, (

**b**) baboon image, (

**c**) pepper image and (

**d**) pyramids image for a mismatch in the value of b.

Image | ${\mathit{v}}_{\mathit{R}}$ | ${\mathit{v}}_{\mathit{G}}$ | ${\mathit{v}}_{\mathit{B}}$ |
---|---|---|---|

Original King Tut | $6.45\times {10}^{4}$ | $1.11\times {10}^{5}$ | $1.63\times {10}^{4}$ |

Encrypted King Tut | 289 | 293 | 278 |

Original baboon | $2.93\times {10}^{4}$ | ${10}^{5}$ | $2.68\times {10}^{4}$ |

Encrypted baboon | 252 | 248 | 228 |

Original pepper | $4.77\times {10}^{4}$ | $4\times {10}^{4}$ | $2.33\times {10}^{4}$ |

Encrypted pepper | 267 | 309 | 215 |

Original pyramids | $3.06\times {10}^{4}$ | $1.57\times {10}^{5}$ | $2.61\times {10}^{5}$ |

Encrypted pyramids | 298 | 209 | 269 |

Image | Key Value | Mismatch (%) | Average Difference (%) |
---|---|---|---|

King Tut | $a=2$ | ${10}^{-10}$ | $99.56$ |

King Tut | $b=3.35$ | ${10}^{-10}$ | $99.61$ |

King Tut | $c=9.15$ | ${10}^{-10}$ | $99.54$ |

Baboon | $a=2$ | ${10}^{-10}$ | $99.58$ |

Baboon | $b=3.35$ | ${10}^{-10}$ | $99.55$ |

Baboon | $c=9.15$ | ${10}^{-10}$ | $99.61$ |

Pepper | $a=2$ | ${10}^{-10}$ | $99.63$ |

Pepper | $b=3.35$ | ${10}^{-10}$ | $99.53$ |

Pepper | $c=9.15$ | ${10}^{-10}$ | $99.59$ |

Pyramids | $a=2$ | ${10}^{-10}$ | $99.62$ |

Pyramids | $b=3.35$ | ${10}^{-10}$ | $99.64$ |

Pyramids | $c=9.15$ | ${10}^{-10}$ | $99.66$ |

Image | Information Entropy (R) | (G) | (B) |
---|---|---|---|

King Tut | 7.9975 | 7.9973 | 7.9977 |

Baboon | 7.9968 | 7.9966 | 7.9978 |

Pepper | 7.9967 | 7.9976 | 7.9967 |

Egyptian pyramids | 7.9972 | 7.9974 | 7.9975 |

**Table 4.**Number of Pixels Change Rate (NPCR) and Unified Average Changing Intensity (UACI) for the four encrypted images.

Image | NPCR (R, G, B) % | UACI (R, G, B) % |
---|---|---|

King Tut | 99.6268, 99.6402, 99.6361 | 33.531, 33.531, 33.533 |

Baboon | 99.6332, 99.6329, 99.6241 | 33.537, 33.536, 33.536 |

Pepper | 99.6112, 99.6351, 99.6293 | 33.528, 33.506, 33.511 |

Egyptian pyramids | 99.6339, 99.6383, 99.6154 | 33.471, 33.483, 33.485 |

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**MDPI and ACS Style**

Askar, S.; Al-khedhairi, A.; Elsonbaty, A.; Elsadany, A.
Chaotic Discrete Fractional-Order Food Chain Model and Hybrid Image Encryption Scheme Application. *Symmetry* **2021**, *13*, 161.
https://doi.org/10.3390/sym13020161

**AMA Style**

Askar S, Al-khedhairi A, Elsonbaty A, Elsadany A.
Chaotic Discrete Fractional-Order Food Chain Model and Hybrid Image Encryption Scheme Application. *Symmetry*. 2021; 13(2):161.
https://doi.org/10.3390/sym13020161

**Chicago/Turabian Style**

Askar, Sameh, Abdulrahman Al-khedhairi, Amr Elsonbaty, and Abdelalim Elsadany.
2021. "Chaotic Discrete Fractional-Order Food Chain Model and Hybrid Image Encryption Scheme Application" *Symmetry* 13, no. 2: 161.
https://doi.org/10.3390/sym13020161