# A New Algorithm for Digital Image Encryption Based on Chaos Theory

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Chaos and Transformation Theories

#### 2.2. Chaotic Sequence Based on Logistic Map

_{1}, r

_{2}…r

_{m×n}}, and put P

_{1}= {p

_{1}, p

_{2}…p

_{m×n}} as the encrypted 1D matrix. The procedure of the encrypted algorithm will be as follows:

- Step NO.1:
- In the first step, two chaotic sequences, x = {x
_{1}, x_{2}… x_{m×n}} are produced by two one-dimensional logistic maps. Place the two logistic maps system parameter as a primary value as x_{1}(0) and x_{2}(0), respectively. - Step NO.2:
- In the second step, for every iteration, compare x
_{1}(i), and x_{2}(i), i = 1, 2, m × n and choose one that is numerically larger. - Step NO.3:
- In the next step, perform the Exclusive NOR (XNOR) operation for sequences produced by Step NO.2 with the original image’s pixels.
- Step NO.4:
- In the last step, change the encrypted one-dimensional matrix, namely P, into a two-dimensional matrix. Set the size of this modified matrix to m × n. In this process, a two-dimensional data matrix R2 is generated. Thus, a diffused image is obtained.

#### 2.3. Kinetics of Coupled Map Lattice

- Step NO.1:
- In the first step, the chaotic sequences x
_{1}, x_{2}= {x_{1}, x_{2}… x_{m}} are produced with the length of m, and y_{1}, y_{2}= {y_{1}, y_{2}… y_{n}} with the length of n similar to CML chaos mapping. - Step NO.2:
- In the second step, x, y chaotic sequences are arranged in rising sequences, producing position sequences w
_{2}, w_{3}. - Step NO.3:
- In the last step, the pixel confusion is performed, using w
_{2}, w_{3}as the row, and column sequences of the data matrix R.$$R\left(i,1\right)=R\left({w}_{2}\left(i,{w}_{3}\left(j\right)\right)\right).$$

#### 2.4. Wavelet Transform

## 3. Proposed Algorithm

- Step NO.1:
- In the first step, a grayscale image G is arranged. The image’s size is set to m × n. Moreover, data matrix R is placed. By evaluating two logistic maps, a chaotic sequence is generated. Making XNOR with the primary image, the diffusion is terminated.
- Step NO.2:
- In this step, for the diffused image in step NO.1, the wavelet decomposition is performed and then the wavelet coefficient is extracted, registered as ca1.
- Step NO.3:
- Utilizing a two-dimensional hyper-chaotic map CML, the chaotic sequence is produced, and with ca1 established in step NO.2, the position confusion is performed.
- Step NO.4:
- In the last step, the confused image can be rebuilt by wavelet. After all, the encrypted image is obtained. The inverse operations of the encryption are known as the decryption algorithm. System parameters and the primary value of the chaotic sequences in the image encryption and image decryption are consistent.

#### 3.1. Encryption Assessments Metrics

#### 3.2. Peak Signal to Noise Ratio (PSNR)

#### 3.3. Number of Pixels Change Rate (NPCR)

_{1}and C

_{2}as the two images with N × M size, we defined an array, D, with the sizes similar to images C

_{1}and C

_{2}as:

#### 3.4. Unified Average Changing Intensity (UACI)

_{1}and C

_{2}), using the below expression [60]. It is applied to evaluate the encryption method’s strength. Its value is based on the image’s format and size [61,62]. Through UACI, the average variation in intensity between the ciphered and original images is assessed. The greatest UACI indicates that the suggested technique has resistance against various attacks. UACI is determined for the grayscale image of size M × N as follows:

#### 3.5. Correlation Coefficient

## 4. Experimental and Numerical Results

#### 4.1. Histogram Analysis

#### 4.2. Complexity

#### 4.3. Robustness

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The proposed algorithm for image encryption: CWT: Continuous Wavelet Transform; DWT: Discrete Wavelet Transforms.

**Figure 3.**The visual results for applying the proposed algorithm to some images. (

**a**) Original images. (

**b**) Encrypted images. (

**c**) Reconstructed images.

**Figure 5.**Plots of pixel horizontal, vertical and diagonal, correlation for input (

**Left**) and encrypted images (

**Right**) in Lena image.

Image | Type of Image | PSNR | NPCR | UACI | NC |
---|---|---|---|---|---|

Lena Image | Jpg | 42.612 | 99.757 | 33.120 | 0.9548 |

Peppers Image | Jpg | 39.220 | 99.787 | 33.621 | 0.9934 |

Barbara Image | Jpg | 36.841 | 99.626 | 33.126 | 0.9809 |

Baboon Image | Jpg | 39.134 | 99.881 | 33.415 | 0.9137 |

Boat Image | Jpg | 38.223 | 99.625 | 33.671 | 0.9001 |

x1(1) | x2(1) | µ 1 | µ 2 |

0.5 | 0.5 | 4 | 3.9 |

x3(1) | y3(1) | µ 1 | µ 2 |

0.3 | 0.3 | 4 | 3.9 |

Image | Median Filter | Histogram Equalization | Rotation | Gaussian Noise |
---|---|---|---|---|

Lena Image | 0.984 | 0.987 | 0.999 | 0.999 |

Peppers Image | 0.704 | 0.280 | 0.923 | 0.964 |

Barbara Image | 0.914 | 0.497 | 0.980 | 0.991 |

Baboon Image | 0.960 | 0.629 | 0.991 | 0.996 |

Boat Image | 0.976 | 0.746 | 0.995 | 0.998 |

**Table 4.**The numerical results of the proposed algorithm in comparison with state-of-the-art methods.

Reference | Image | NPCR | UACI |
---|---|---|---|

Presented model | Lena Image | 99.757 | 33.120 |

Presented model | Peppers Image | 99.787 | 33.621 |

Presented model | Barbara Image | 99.626 | 33.126 |

Presented model | Baboon Image | 99.881 | 33.415 |

Presented model | Boat Image | 99.625 | 33.671 |

Amina et al. [69] | Lena Image | 99.646 | 33.625 |

Amina et al. [69] | Peppers Image | 99.632 | 33.507 |

Amina et al. [69] | Baboon Image | 99.602 | 33.629 |

Yavuz et al. [70] | Lena Image | 99.620 | 33.410 |

Zhang and Zhao [71] | Lena Image | 99.605 | 33.411 |

Assad and Farajallah [72] | Lena Image | 99.607 | 33.463 |

Assad and Farajallah [72] | Boat Image | 99.615 | 33.465 |

Kari et al. [38] | Lena Image | 99.646 | 33.625 |

Kari et al. [38] | Peppers Image | 99.713 | 33.541 |

Kari et al. [38] | Baboon Image | 99.623 | 33.416 |

Kari et al. [38] | Boat Image | 99.619 | 33.556 |

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**MDPI and ACS Style**

Pourasad, Y.; Ranjbarzadeh, R.; Mardani, A.
A New Algorithm for Digital Image Encryption Based on Chaos Theory. *Entropy* **2021**, *23*, 341.
https://doi.org/10.3390/e23030341

**AMA Style**

Pourasad Y, Ranjbarzadeh R, Mardani A.
A New Algorithm for Digital Image Encryption Based on Chaos Theory. *Entropy*. 2021; 23(3):341.
https://doi.org/10.3390/e23030341

**Chicago/Turabian Style**

Pourasad, Yaghoub, Ramin Ranjbarzadeh, and Abbas Mardani.
2021. "A New Algorithm for Digital Image Encryption Based on Chaos Theory" *Entropy* 23, no. 3: 341.
https://doi.org/10.3390/e23030341