# A Novel Plaintext-Related Color Image Encryption Scheme Based on Cellular Neural Network and Chen’s Chaotic System

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Cellular Neural Network

#### 2.2. Chen’s Chaotic System

## 3. Encryption and Decryption on Color Images

#### 3.1. Key Generation

#### 3.1.1. Initial Key

#### 3.1.2. Switching Key

#### 3.2. Encryption Scheme Design

#### 3.2.1. Scrambling

#### 3.2.2. Diffusion

#### 3.3. Decryption Scheme Design

#### 3.3.1. First Decryption

#### 3.3.2. Second Decryption (Reverse Scrambling)

#### 3.4. Step to Encrypt and Decrypt Color Images

#### 3.4.1. Steps to Encrypt

- Step 1: Initial keys ${k}_{1}$, ${k}_{2}$, …, ${k}_{6}$, switching key d, and control parameters are inputted into the CNN hyperchaotic system, and the position matrix $\mathbf{ICS}$ will be obtained. Then, the matrix $\mathbf{ICS}$ is utilized to scramble the plaintext image $\mathbf{PI}$ and complete the first encryption.
- Step 2: Input the initial values ${x}_{1}$, ${x}_{2}$, ${x}_{3}$ into Chen’s chaotic system. Then, the diffusion matrix $\mathbf{DM}$ can be obtained, which is XORed respectively with the three components matrices of the plaintext image obtained after the first encryption to complete the second encryption.
- Step 3: For higher security, the above steps can be repeated n times.

#### 3.4.2. Steps to Decrypt

- Step 1: Initial keys ${k}_{1}$, ${k}_{2}$, …, ${k}_{6}$; switching key d; and control parameters are inputted into the CNN hyperchaotic system, the position matrix $\mathbf{ICS}$ and the initial values ${x}_{1}$, ${x}_{2}$, ${x}_{3}$ of Chen’s system will be obtained.
- Step 2: Input the initial values ${x}_{1}$, ${x}_{2}$, ${x}_{3}$ into the Chen’s system, and the diffusion matrix $\mathbf{DM}$ will be obtained. The matrix $\mathbf{DM}$ is XORed respectively with the three components matrices of the ciphertext image $\mathbf{CI}$. After the integration, the first decryption is completed and the scrambled image $\mathbf{SI}$ is obtained.
- Step 3: The position matrix $\mathbf{ICS}$ in step1 is carried out the inverse scrambling on the RGB component of the scrambled image $\mathbf{SI}$. Integrate it into the RGB form, the second stage of decryption process is completed.

## 4. Experimental Results and Security Analysis

#### 4.1. Histogram Analysis

#### 4.2. Correlation of Adjacent Pixels

#### 4.3. Information Entropy

#### 4.4. Sensitivity Analysis

#### 4.4.1. Key Sensitivity

#### 4.4.2. Plaintext Sensitivity

#### 4.5. Key Space

#### 4.6. Robustness Analysis

#### 4.7. Known and Chosen-Plaintext Attacks

#### 4.8. Randomness Tests Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

CNN | Cellular Neural Network |

UAV | Unmanned Aerial Vehicle |

VLSI | Very Large Scale Integration |

NPCR | Number of Pixel of Change Rate |

PRNG | Pseudorandom Number Generator |

UACI | Unified Average Changing Intensity |

NIST | National Institute of Standards and Technology |

KPA | Known-Plaintext Attack |

CPA | Chosen-Plaintext Attack |

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**Figure 3.**Partial 6-order CNN hyperchaotic attractor distribution: (

**a**) Distribution in the direction of ${x}_{3}$-${x}_{6}$-${x}_{4}$. (

**b**) Distribution in the direction of ${x}_{1}$-${x}_{3}$-${x}_{4}$. (

**c**) Distribution in the direction of ${x}_{5}$-${x}_{4}$-${x}_{3}$. (

**d**) Distribution in the direction of ${x}_{2}$-${x}_{4}$-${x}_{6}$. (

**e**) Distribution in the direction of ${x}_{2}$-${x}_{3}$-${x}_{4}$. (

**f**) Distribution in the direction of ${x}_{4}$-${x}_{5}$- ${x}_{6}$. (

**g**) Distribution in the direction of ${x}_{1}$-${x}_{3}$-${x}_{5}$. (

**h**) Distribution in the direction of ${x}_{1}$-${x}_{2}$-${x}_{3}$.

**Figure 4.**Chaotic attractors of Chen’s system: (

**a**) Distribution in the direction of x-y-z. (

**b**) Distribution in the direction of x-y. (

**c**) Distribution in the direction of y-z. (

**d**) Distribution in the direction of x-z.

**Figure 7.**An example of a scrambling algorithm: (

**a**) The 2D chaotic matrix. (

**b**) The sorted chaotic matrix. (

**c**) The position matrix. (

**d**) The plaintext matrix. (

**e**) The scrambled plaintext matrix.

**Figure 8.**The experimental results of encryption and decryption algorithm: (

**a**) The original plaintext image. (

**b**) The scrambled image; (

**c**) The encrypted ciphertext image. (

**d**) The decrypted image.

**Figure 9.**Histogram tests: (

**a**) The plaintext image. (

**b**) Histogram for RGB components of the plaintext image. (

**c**) The ciphertext image. (

**d**) Histogram for RGB components of ciphertext image.

**Figure 10.**Distribution of pixels: (

**a**) Original image distribution of horizontal. (

**b**) Cipher image distribution of horizontal. (

**c**) Original image distribution of vertical. (

**d**) Cipher image distribution of vertical. (

**e**) Original image distribution of diagonal. (

**f**) Cipher image distribution of diagonal.

**Figure 11.**The test results of the key sensitivity: (

**a**) The plaintext image. (

**b**) The correctly encrypted image. (

**c**) The incorrectly encrypted image. (

**d**) The difference between the two encrypted images. (

**e**) The incorrectly decrypted image. (

**f**) The correctly decrypted image.

**Figure 12.**Experimental results of adding noise: (

**a**) Cipher House image with Pepper & Salt noise of 0.1. (

**b**) Cipher House image with Pepper & Salt noise of 0.15. (

**c**) Cipher House image with Pepper & Salt noise of 0.5. (

**d**) Decrypted image from cipher with Pepper & Salt noise of 0.1. (

**e**) Decrypted image from cipher with Pepper & Salt noise of 0.15. (

**f**) Decrypted image from cipher with Pepper & Salt noise of 0.5. (

**g**) Cipher House image with Gaussian white noise with variance value 0.01. (

**h**) Cipher House image with Gaussian white noise with variance value 0.1. (

**i**) Cipher House image with Gaussian white noise with variance value 0.15. (

**j**) Decrypted image from cipher with Gaussian white noise with variance value 0.01. (

**k**) Decrypted image from cipher with Gaussian white noise with variance value 0.1. (

**l**) Decrypted image from cipher with Gaussian white noise with variance value 0.15.

**Figure 13.**Experimental results of decryption from cropped cipher images: (

**a**) Cipher image with 6.25% cropped. (

**b**) Cipher image with 25% cropped. (

**c**) Cipher image with 50% cropped. (

**d**) Decrypted image from 6.25% cropped cipher. (

**e**) Decrypted image from 25% cropped cipher. (

**f**) Decrypted image from 50% cropped cipher.

**Table 1.**Correlation of adjoining pixels of the original plaintext image and the encrypted ciphertext image.

Image | Horizontal | Vertical | Diagonal | ||||||
---|---|---|---|---|---|---|---|---|---|

R | G | B | R | G | B | R | G | B | |

Lena($512\times 512$) | 0.9737 | 0.9740 | 0.8418 | 0.9963 | 0.9902 | 0.9555 | 0.9743 | 0.9615 | 0.9219 |

Encrypted Lena | 0.0073 | 0.0084 | 0.0072 | –0.0134 | –0.0133 | –0.0107 | 0.0034 | 0.0055 | 0.0035 |

Baboon($512\times 512$) | 0.4301 | 0.4717 | 0.5558 | 0.8568 | 0.7619 | 0.8969 | 0.8588 | 0.7307 | 0.8461 |

Encrypted Baboon | 0.0019 | –0.0030 | –0.0117 | 0.0530 | –0.0115 | –0.0116 | 0.0051 | 0.0056 | 0.0043 |

Peppers($512\times 512$) | 0.9727 | 0.9929 | 0.9847 | 0.9710 | 0.9927 | 0.9812 | 0.9587 | 0.9706 | 0.9467 |

Encrypted Peppers | 0.0069 | 0.0130 | 0.0041 | –0.0014 | 0.0076 | 0.0124 | 0.0069 | 0.0075 | 0.0067 |

Airplane($512\times 512$) | 0.9851 | 0.9575 | 0.9575 | 0.9605 | 0.9941 | 0.9941 | 0.9355 | 0.9327 | 0.9327 |

Encrypted Airplane | 0.0062 | –0.0041 | –0.0016 | 0.0399 | 0.0446 | 0.0333 | 0.0013 | 0.0018 | 0.0003 |

Scheme | Horizontal | Vertical | Diagonal |
---|---|---|---|

Ours | 0.0076 | –0.0125 | 0.0101 |

Reference [35] | 0.0076 | 0.0130 | 0.0138 |

Reference [26] | 0.0445 | –0.0168 | –0.0022 |

Reference [27] | 0.0195 | 0.0086 | –0.0260 |

Reference [10] | 0.0113 | 0.0173 | 0.0099 |

Reference [36] | 0.1410 | 0.1967 | 0.1116 |

Reference [12] | 0.0172 | 0.0039 | 0.0277 |

Reference [37] | –0.0909 | 0.2389 | 0.0126 |

Plaintext Image | Ciphertext Image | |||||
---|---|---|---|---|---|---|

Images | R | G | B | R | G | B |

Lena ($512\times 512$) | 7.2710 | 7.5310 | 6.9603 | 7.9997 | 7.9937 | 7.9976 |

Peppers ($512\times 512$) | 7.3255 | 7.3912 | 6.9469 | 7.9932 | 7.9824 | 7.9969 |

Baboon ($512\times 512$) | 7.6833 | 7.4466 | 7.6874 | 7.9941 | 7.9701 | 7.9838 |

Airplane ($512\times 512$) | 6.9113 | 6.9853 | 6.5128 | 7.9933 | 7.9742 | 7.9994 |

Barbara ($787\times 576$) | 7.7581 | 7.6078 | 7.5774 | 7.9915 | 7.9730 | 7.9806 |

**Table 4.**The number of pixel of change rate (NPCR) and unified average changing intensity (UACI) of RGB component.

NPCR(%) | UACI(%) | |||||
---|---|---|---|---|---|---|

Images | R | G | B | R | G | B |

Peppers ($512\times 512$) | 99.6059 | 99.6124 | 99.6197 | 33.4151 | 33.4492 | 33.3812 |

Baboon ($512\times 512$) | 99.6128 | 99.5903 | 99.5781 | 33.3810 | 33.4133 | 33.3467 |

Airplane ($512\times 512$) | 99.6166 | 99.5796 | 99.6128 | 33.4021 | 33.4243 | 33.3585 |

Barbara ($787\times 576$) | 99.6098 | 99.6042 | 99.5877 | 33.4288 | 33.4573 | 33.3858 |

Reference [13] | 99.6338 | 99.6094 | 99.5795 | 33.2918 | 33.6839 | 33.3533 |

Reference [14] | 99.6000 | 99.6100 | 99.6100 | 33.5600 | 33.4500 | 33.4900 |

Reference [11] | 99.6300 | 99.5900 | 99.6700 | 33.4300 | 33.3900 | 33.5100 |

Reference [37] | 99.335 | - | - | 33.600 | - | - |

Scheme | Key Space |
---|---|

Ours | $6\times {2}^{192}$ |

Reference [35] | ${2}^{159}$ |

Reference [26] | ${10}^{58}$ |

Reference [27] | ${10}^{56}$ |

Reference [39] | ${10}^{45}$ |

Reference [12] | ${10}^{56}$ |

Reference [40] | ${2}^{128}$ |

Reference [24] | ${2}^{128}$ |

Reference [37] | ${10}^{37}$ |

Tests | P-Value | Results |
---|---|---|

Frequency test | 0.441300 | Pass |

Block Frequency test | 0.810465 | Pass |

Cusum-forward test | 0.387693 | Pass |

Cusum-reverse test | 0.832407 | Pass |

Runs test | 0.290305 | Pass |

Longest run test | 0.644892 | Pass |

Rank test | 0.908125 | Pass |

FFT test | 0.720427 | Pass |

Non-Overlapping template test | 0.546890 | Pass |

Overlapping template test | 0.577412 | Pass |

Universal test | 0.911848 | Pass |

Approximate entropy test | 0.276530 | Pass |

Random-excursions test (x = –1) | 0.672039 | Pass |

Random-excursions variant test (x = 1) | 0.643912 | Pass |

Serial1 test | 0.380598 | Pass |

Serial2 test | 0.117241 | Pass |

Linear complexity test | 0.633098 | Pass |

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## Share and Cite

**MDPI and ACS Style**

Zhang, R.; Yu, L.; Jiang, D.; Ding, W.; Song, J.; He, K.; Ding, Q.
A Novel Plaintext-Related Color Image Encryption Scheme Based on Cellular Neural Network and Chen’s Chaotic System. *Symmetry* **2021**, *13*, 393.
https://doi.org/10.3390/sym13030393

**AMA Style**

Zhang R, Yu L, Jiang D, Ding W, Song J, He K, Ding Q.
A Novel Plaintext-Related Color Image Encryption Scheme Based on Cellular Neural Network and Chen’s Chaotic System. *Symmetry*. 2021; 13(3):393.
https://doi.org/10.3390/sym13030393

**Chicago/Turabian Style**

Zhang, Renxiu, Longfei Yu, Donghua Jiang, Wei Ding, Jian Song, Kuncheng He, and Qun Ding.
2021. "A Novel Plaintext-Related Color Image Encryption Scheme Based on Cellular Neural Network and Chen’s Chaotic System" *Symmetry* 13, no. 3: 393.
https://doi.org/10.3390/sym13030393