# Medical Images Encryption Based on Adaptive-Robust Multi-Mode Synchronization of Chen Hyper-Chaotic Systems

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## Abstract

**:**

## 1. Introduction

## 2. Formulation of the Theorem

#### 2.1. **Comparative Synchronization between More Response Systems and a Master System**

**Definition**

**1.**

**Theorem**

**1.**

**Proof.**

**Note**

**1.**

#### 2.2. **Circular Synchronization of Multiple Chaotic Systems with Unknown Parameters**

**Theorem**

**2.**

- (A)
- If there is a transmission and circular synchronization with the u(t) and m(t) controllers. Then:$$\forall i,j\text{}:\text{}\underset{t\to \infty}{lim}||{x}_{i}\left(t\right)-{x}_{j}\left(t\right)||\text{}\to 0$$
- (B)
- If transmission synchronization is established, circular synchronization is also realized and vice versa.

**Proof.**

**Proof.**

**Theorem**

**3.**

**Proof.**

#### 2.3. **Synchronization with the Presence of Disturbance and Uncertainty in the System**

_{i}are constant but indefinite and ${g}_{i}({x}_{i})$ is a definite function and in the special case$\text{}{g}_{i}({x}_{i})=\left|{x}_{i}\right|$. The dynamics of the errors are as follows:

**Theorem**

**4.**

**Proof.**

**Note**

**2.**

**Note**

**3.**

**Note**

**4.**

**Note**

**5.**

**Note**

**6.**

## 3. Application in Secure Communication Based on Chaotic Masking

## 4. Implementation of the Proposed Synchronization Method on Chen Hyper-Chaotic System

**Case A.**Multi-mode synchronization with time-varying parameters (without disturbance and uncertainty).

_{11}(0), x

_{12}(0), x

_{13}(0), x

_{14}(0)) = (10, 10, 10, 10)

(x

_{21}(0), x

_{22}(0), x

_{23}(0), x

_{24}(0)) = (2, 2, 2, 2)

(x

_{31}(0), x

_{32}(0), x

_{33}(0), x

_{34}(0)) = (3, 3, 3, 3)

_{11}= 35,

_{12}θ = 7,

_{21}θ = 35,

_{22}θ = 7,

_{31}θ = 35,

_{32}θ = 7

_{13}θ = 12,

_{14}θ = 3,

_{15}θ = 0.3,

_{23}θ = 12,

_{24}θ = 3, θ

_{25}= 0.3,

_{33}θ = 12,

_{34}θ = 3,

_{35}θ = 0.3,

_{11}θ = 33,

_{12}θ = 6,

_{21}θ = 32,

_{22}θ = 5,

_{31}θ = 34,

_{32}θ = 5,

_{13}θ = 13

_{14}θ = 1,

_{15}θ = 0.1,

_{23}θ = 12,

_{24}θ = 5, θ

_{25}= 0.2,

_{33}θ = 10,

_{34}θ = 2,

_{35}θ = 0.2,

**Case B.**Multi-mode synchronization with time-varying parameters despite disturbance and uncertainty. In this case, disturbance and uncertainties are applied to master and slave systems as follows:

**Case C**. Multi-mode synchronization, taking into account the disturbance and uncertainties bounded with the function:

## 5. Statistical Metrics

#### 5.1. **Histogram Analysis**

_{i}and I

_{j}indicate the number of pixels whose gray values are equal to i and j, respectively [24].

#### 5.2. **Correlation Analysis**

#### 5.3. **Differential Attack Analysis**

#### 5.4. **PSNR Analysis**

#### 5.5. **Information Entropy Analysis**

## 6. Experiment Results

#### 6.1. **Image Benchmarks**

#### 6.2. **Simulation**

## 7. Advantages and Disadvantages

## 8. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 7.**Estimation error curves of bounds related to uncertainties (

**right**) and disturbances (

**left**)—(case B).

**Figure 8.**Curves of synchronization errors in case of disturbance and structural uncertainty of case (C).

**Figure 10.**Phase curves for master chaotic system (

**green**), Slave 1 (

**blue**) and slave 2 (

**red**) chaotic systems.

Images | Histogram | Correlation | Differential Attack | PSNR | Information Entropy | ||
---|---|---|---|---|---|---|---|

Standard | Encrypted | NPCR (%) | UACI (%) | ||||

Image 1 | 1,555,164 | 3,375,508 | 0.9970 | 99.611 | 33.461 | 34.623 | 5.5081 |

Image 2 | 1,942,777 | 5,880,577 | 0.9964 | 99.610 | 33.460 | 33.186 | 5.2665 |

Image 3 | 2,926,366 | 6,511,850 | 0.9976 | 99.609 | 33.463 | 34.982 | 4.8647 |

Image 4 | 1,969,829 | 5,299,018 | 0.9954 | 99.611 | 33.459 | 33.182 | 5.077 |

Image 5 | 2,776,087 | 5,805,249 | 0.9967 | 99.610 | 33.460 | 34.945 | 4.6828 |

Image 6 | 1,983,221 | 5,394,168 | 0.9956 | 99.610 | 33.462 | 33.198 | 5.0558 |

Image 7 | 3,108,704 | 6,811,427 | 0.9974 | 99.611 | 33.460 | 35.053 | 4.5683 |

Image 8 | 2,783,671 | 8,248,927 | 0.9958 | 99.610 | 33.460 | 33.483 | 4.5248 |

Image 9 | 2,887,118 | 6,433,437 | 0.9976 | 99.612 | 33.461 | 34.979 | 4.7115 |

Image 10 | 1,168,385 | 3,379,135 | 0.9956 | 99.610 | 33.463 | 32.850 | 5.6934 |

Images | Histogram | Correlation | Differential Attack | PSNR | Information Entropy | ||
---|---|---|---|---|---|---|---|

Standard | Encrypted | NPCR (%) | UACI (%) | ||||

Image 1 | 371,790.95 | 414,854.68 | 0.9965 | 99.611 | 33.459 | 34.146 | 6.4966 |

Image 2 | 637,672.20 | 775,806.22 | 0.9958 | 99.609 | 33.461 | 32.599 | 5.9531 |

Image 3 | 704,174.55 | 770,799.32 | 0.9969 | 99.610 | 33.462 | 34.284 | 5.8406 |

Image 4 | 655,672.19 | 823,316.52 | 0.9941 | 99.612 | 33.459 | 32.583 | 5.7021 |

Image 5 | 711,500.39 | 800,070.88 | 0.9959 | 99.608 | 33.460 | 34.286 | 5.5608 |

Image 6 | 658,916.62 | 837,271.82 | 0.9944 | 99.611 | 33.461 | 32.603 | 5.6814 |

Image 7 | 764,189.21 | 849,713.85 | 0.9966 | 99.612 | 33.463 | 34.304 | 5.5387 |

Image 8 | 910,672.69 | 1,120,143.86 | 0.9942 | 99.611 | 33.462 | 32.728 | 5.3159 |

Image 9 | 712,326.25 | 774,208.57 | 0.9969 | 99.613 | 33.458 | 34.276 | 5.6761 |

Image 10 | 377,977.28 | 471,233.22 | 0.9948 | 99.611 | 33.461 | 32.426 | 6.4675 |

Images | Histogram | Correlation | Differential Attack | PSNR | Information Entropy | ||
---|---|---|---|---|---|---|---|

Standard | Encrypted | NPCR (%) | UACI (%) | ||||

Image 1 | 239,642.29 | 273,559.38 | 0.9963 | 99.611 | 33.461 | 34.052 | 6.6729 |

Image 2 | 399,917.39 | 493,824.29 | 0.9954 | 99.609 | 33.462 | 32.376 | 6.2428 |

Image 3 | 439,060.82 | 492,967.11 | 0.9967 | 99.613 | 33.458 | 34.141 | 6.1468 |

Image 4 | 453,864.31 | 566,619.14 | 0.9937 | 99.610 | 33.460 | 32.416 | 5.9343 |

Image 5 | 490,271.52 | 542,562.67 | 0.9956 | 99.611 | 33.461 | 34.115 | 5.8208 |

Image 6 | 450,636.56 | 557,774.79 | 0.9939 | 99.612 | 33.459 | 32.399 | 5.9312 |

Image 7 | 505,486.28 | 558,333.49 | 0.9963 | 99.613 | 33.463 | 34.156 | 5.8280 |

Image 8 | 585,431.25 | 723,173.69 | 0.9936 | 99.609 | 33.462 | 32.481 | 5.6384 |

Image 9 | 456,812.35 | 501,600.98 | 0.9966 | 99.611 | 33.462 | 34.120 | 5.9823 |

Image 10 | 244,061.98 | 307,606.74 | 0.9944 | 99.612 | 33.460 | 32.263 | 6.6424 |

Images | Histogram | Correlation | Differential Attack | PSNR | Information Entropy | ||
---|---|---|---|---|---|---|---|

Standard | Encrypted | NPCR (%) | UACI (%) | ||||

Image 1 | 183,606.33 | 207,963.36 | 0.9961 | 99.611 | 33.461 | 33.975 | 6.7881 |

Image 2 | 287,175.67 | 347,957.53 | 0.9951 | 99.610 | 33.459 | 32.271 | 6.4632 |

Image 3 | 318,969.51 | 351,484.97 | 0.9965 | 99.608 | 33.458 | 34.027 | 6.3680 |

Image 4 | 353,759.37 | 420,480.95 | 0.9934 | 99.611 | 33.462 | 32.302 | 6.1234 |

Image 5 | 379,975.17 | 413,399.53 | 0.9954 | 99.609 | 33.457 | 34.054 | 6.0266 |

Image 6 | 347,921.57 | 414,116.53 | 0.9936 | 99.611 | 33.456 | 32.304 | 6.1326 |

Image 7 | 381,476.11 | 422,535.34 | 0.9962 | 99.612 | 33.461 | 34.097 | 6.0471 |

Image 8 | 435,904.80 | 517,108.82 | 0.9932 | 99.610 | 33.460 | 32.354 | 5.8833 |

Image 9 | 333,557.81 | 368,586.44 | 0.9965 | 99.611 | 33.459 | 34.073 | 6.2279 |

Image 10 | 185,554.28 | 227,767.28 | 0.9942 | 99.607 | 33.454 | 32.185 | 6.7663 |

Images | Histogram | Correlation | Differential Attack | PSNR | Information Entropy | ||
---|---|---|---|---|---|---|---|

Standard | Encrypted | NPCR (%) | UACI (%) | ||||

Image 1 | 160,406.07 | 181,599.02 | 0.9961 | 99.609 | 33.459 | 33.965 | 6.8530 |

Image 2 | 234,952.50 | 276,693.14 | 0.9950 | 99.611 | 33.461 | 32.237 | 6.6058 |

Image 3 | 262,347.29 | 294,388.46 | 0.9964 | 99.608 | 33.458 | 34.007 | 6.5026 |

Image 4 | 300,806.96 | 345,570.37 | 0.9931 | 99.610 | 33.457 | 32.204 | 6.2599 |

Image 5 | 319,370.10 | 347,968.94 | 0.9953 | 99.612 | 33.462 | 34.046 | 6.1673 |

Image 6 | 295,654.72 | 341,419.34 | 0.9934 | 99.607 | 33.457 | 32.213 | 6.2688 |

Image 7 | 320,298.48 | 349,954.05 | 0.9960 | 99.611 | 33.459 | 33.995 | 6.1951 |

Image 8 | 364,283.38 | 423,633.80 | 0.9930 | 99.609 | 33.462 | 32.290 | 6.0396 |

Image 9 | 276400.42 | 307161.05 | 0.9964 | 99.610 | 33.460 | 34.000 | 6.3806 |

Image 10 | 162565.94 | 191476.22 | 0.9940 | 99.612 | 33.461 | 32.139 | 6.8390 |

Properties Encryption Method | Encryption Method | Data Types | Works | ||
---|---|---|---|---|---|

Disturbance | Unknown Parameter | Uncertainty | |||

✕ | ✕ | ✕ | Chaos Logic Map | EEG Signals | [17] |

✕ | ✕ | ✕ | Double Chaotic Layer Encryption (DCLE) | EEG, ECG Signals | [18] |

✕ | ✕ | ✕ | Optical Chaos (Additive Chaos Masking) | EEG Signals | [19] |

✕ | ✕ | ✕ | Chaotic Modulation on the Intrinsic Mode Functions | EEG, ECG Signals | [20] |

✕ | ✕ | ✕ | Dynamic S-Boxes and Chaotic Maps | Medical Images | [21] |

✕ | ✕ | ✕ | Improvement Chaotic System | Medical Images | [22] |

✕ | ✕ | ✕ | chaotic Map + Fractional Discrete Cosine Transform (FrDCT) Coefficients | Medical Images | [23] |

✕ | ✕ | ✕ | Fourth Order Chaotic System | Medical Images | [24] |

✕ | ✕ | ✕ | Non Linear 4D Logistic Map and DNA Sequences (NL4DLM_DNA) | Medical Images | [25] |

✕ | ✕ | ✕ | Chaotic Method Based on Arnold’s Cat Map | MRI Images | [26] |

✕ | ✕ | ✕ | Latin Square + Memristive Chaotic System | Medical Images | [27] |

✕ | ✕ | ✕ | 3D Chaotic Cat Map + NCA | Medical Images | [28] |

✕ | ✕ | ✕ | Multiple Chaotic Systems + MD5 | Medical Images | [29] |

✕ | ✕ | ✕ | Double-Humped Logistic Map | MRI, X-ray Images | [30] |

✕ | ✕ | ✕ | Chaotic Map-based Remote Authentication Scheme | Medical Informatics | [31] |

✕ | ✕ | ✕ | Fused Coupled Chaotic Map (FCCM) | ECG Signals | [32] |

✕ | ✕ | ✕ | Rossler Dynamical Chaotic system + Sine Map | Medical Images | [33] |

✕ | ✕ | ✕ | DWT, DCT, SVD + Chaotic System | MRI Images | [34] |

✕ | ✕ | ✕ | Data Encryption Standard (DES), and Elliptic Curves Cryptography (ECC) | EEG Signals | [35] |

Chen Hyper-Chaotic System + Adaptive-Robust Multi-Mode Synchronization | Medical Images (CT, X-ray), Standard Benchmarks | Proposed Method |

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## Share and Cite

**MDPI and ACS Style**

Javan, A.A.K.; Jafari, M.; Shoeibi, A.; Zare, A.; Khodatars, M.; Ghassemi, N.; Alizadehsani, R.; Gorriz, J.M.
Medical Images Encryption Based on Adaptive-Robust Multi-Mode Synchronization of Chen Hyper-Chaotic Systems. *Sensors* **2021**, *21*, 3925.
https://doi.org/10.3390/s21113925

**AMA Style**

Javan AAK, Jafari M, Shoeibi A, Zare A, Khodatars M, Ghassemi N, Alizadehsani R, Gorriz JM.
Medical Images Encryption Based on Adaptive-Robust Multi-Mode Synchronization of Chen Hyper-Chaotic Systems. *Sensors*. 2021; 21(11):3925.
https://doi.org/10.3390/s21113925

**Chicago/Turabian Style**

Javan, Ali Akbar Kekha, Mahboobeh Jafari, Afshin Shoeibi, Assef Zare, Marjane Khodatars, Navid Ghassemi, Roohallah Alizadehsani, and Juan Manuel Gorriz.
2021. "Medical Images Encryption Based on Adaptive-Robust Multi-Mode Synchronization of Chen Hyper-Chaotic Systems" *Sensors* 21, no. 11: 3925.
https://doi.org/10.3390/s21113925