# A Chaotic Image Encryption Method Based on the Artificial Fish Swarms Algorithm and the DNA Coding

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## Abstract

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## 1. Introduction

## 2. Relevant Knowledge

#### 2.1. Chen’s Hyper-Chaotic System

#### 2.2. Artificial Fish Swarm Algorithm

- Foraging behavior: It is generally believed that fish sense food or food concentration in water through vision or taste to choose the direction of action. Set the state of the current artificial fish, and randomly select another state within its visual range. If the objective function of the obtained state is larger than the current state, it will move one step closer to the latter state; otherwise, it will reselect the new state.Algorithm description: Equation (4) describes that the artificial fish ${X}_{i}$ randomly selects a state ${X}_{j}$ in its field of view.$${X}_{j}={X}_{i}+Visuanl\times Rand(),$$Calculate the objective function values ${Y}_{i}$ and ${Y}_{j}$ of ${X}_{i}$ and ${X}_{j}$ respectively. If ${Y}_{j}$ is found to be better than ${Y}_{i}$, ${X}_{i}$ moves one step in the direction of ${X}_{j}$:$${X}_{i}^{t+1}={X}_{i}^{t}+\frac{{X}_{j}-{X}_{i}^{t}}{||{X}_{j}-{X}_{i}^{t}||}\times Step\times Rand(),$$
**Algorithm 1**Foraging behavior Pseudo-code.for i = 1:N for j = 1:Try_number Xj = x(i) + Visual × rand(); If f(X(j)) < f(x(i)) X_next = x(i) + rand()× step ×(Xj − x(i))/norm(Xj − x(i)); break; else X_next = x(i) + step × rand(); end end end - Group behavior: the artificial fish explores the number of partners in the current neighborhood, calculates the central position of the partners, and then compares the newly obtained objective function of the central position with the objective function of the current position. If the objective function of the central position is better than the objective function of the current position and is not very crowded, the current position moves one step toward the central position, otherwise, the foraging behavior is executed.Three rules must be observed when fish flock together: first, try to move towards the center of the neighboring partners; second, avoid overcrowding; and third, try to be consistent with the average direction of the neighboring partners.Algorithm description: the artificial fish ${X}_{i}$ searches for the number of partners nf and the center position ${X}_{c}$ in the current field of view $\left({d}_{ij}<visual\right)$, if $\raisebox{1ex}{${Y}_{C}$}\!\left/ \!\raisebox{-1ex}{$nf$}\right.<\delta \times {Y}_{i}$ (where ${Y}_{C}$ and ${Y}_{i}$ are the fitness values of ${X}_{c}$ and ${X}_{i}$, respectively), it indicates that the partner’s central location is better and less crowded, and ${X}_{i}$ moves one step toward the partner’s central location:$${X}_{i}^{t+1}={X}_{i}^{t}+\frac{{X}_{j}-{X}_{i}^{t}}{||{X}_{j}-{X}_{i}^{t}||}\times Step\times Rand(),$$Otherwise, the foraging behavior will be executed. Algorithm 2 shows the pseudo-code snippets for the group behavior.
**Algorithm 2**Group behavior Pseudo-code.nf = 0; X_inside = 0; for i = 1:N for j = 1:N if norm(x(j) − x(i)) < Visual nf = nf + 1; X_inside = X_inside + x(j); end X_inside = X_inside − x(i); nf = nf − 1; Xc = X_inside/nf; if f(Xc)/nf < δ× f(x(i)) x_next = x(i) + rand * Step * (Xc − x(i))/norm(Xc − x(i));

else

Execute foraging behavior;end end end - Rear-end behavior: It refers to the behavior of a fish moving in the optimal direction within its field of view. The artificial fish ${X}_{i}$ searches for the individual ${X}_{j}$ with the highest fitness in its field of view (${d}_{ij}<visual$), and its fitness value is ${Y}_{j}$, and explores the number of partners nf in the field of view of the artificial fish ${X}_{i}$, if $\raisebox{1ex}{${Y}_{j}$}\!\left/ \!\raisebox{-1ex}{$nf$}\right.<\delta \times {Y}_{i}$, indicating that ${X}_{j}$ is in a better state and is not too crowded, ${X}_{i}$ moves one step toward ${X}_{j}$, otherwise it executes foraging behavior. Algorithm 3 shows the pseudo-code snippets for the Rear-end behavior.
**Algorithm 3**Rear-end behavior Pseudo-code.Y_max = inf; nf = 0; for i = 1:N for j = 1:N if norm(x(j) − x(i)) < Visual && f(x(j)) < Y_max X_max = x(j); Y_max = f(x(j)); end end for j = 1:N if(norm(x(j) − X_max) < Visual) nf = nf + 1; end end nf = nf − 1; if Y_max/nf < delta × f(x(i)) x_next = x(I,:) + rand × Step. × (temp_maxX − x(i,:)) ./ norm(temp_maxX − x(i,:)); else Execute foraging behavior; end end - Random behavior: it is the default behavior of foraging behavior, which refers to the random movement of artificial fish in the field of vision. When food is found, it will move rapidly in the direction of gradually increasing food. This is to find food points or partners in a wider range.$${X}_{i}^{t+1}={X}_{i}^{t}+Visuanl\times Rand(),$$

#### 2.3. DNA Coding and Decoding Rules

#### DNA XOR Algebraic Operation

## 3. Encryption and Decryption Process

#### 3.1. The Generation of Initial Values of the Hyper-Chaotic System

#### 3.2. Artificial Fish Swarm Algorithm

#### 3.3. Substitution

#### 3.3.1. Intra Block Permutation

Algorithm 4 Pseudo code of built-in commutation in sub image block. |

Input: TB matrix A, sub block image B |

Output: displacement sub block B’ |

1. convert A and B into one-dimensional sequences |

2. for i = 1: $\left(M\times N\right)/\left(m\times n\right)$ |

3. for j = 1:m × n |

4. ${B}_{i}^{\prime}\leftarrow {B}_{i}^{\prime}\left(j\right)={B}_{i}\left(A\left(j\right)\right)$ |

5. end |

6. end |

#### 3.3.2. Inter Block Permutation

Algorithm 5 Pseudocode for permutation between sub image blocks. |

Input: permutation image block B’, chaotic sequence w |

Output: displacement image D |

1. $\mathrm{W}\leftarrow \mathrm{floor}\left(\mathrm{w}\times {10}^{13}\right)$ |

2. $\mathrm{for}\mathrm{i}$= 1:$\left(\mathrm{M}\times \mathrm{N}\right)/\left(\mathrm{m}\times \mathrm{n}\right)$ |

3. $\mathrm{for}\mathrm{j}=$1: $\mathrm{m}\times \mathrm{n}$ |

4. $\mathrm{k}\leftarrow (\mathrm{W}\mathrm{mod}\left(\frac{\mathrm{M}\times \mathrm{N}}{\mathrm{m}\times \mathrm{n}}-\mathrm{i}\right)+1$ |

5. exchange ${B}_{{i}^{\prime}}^{\prime}\left(j\right)and{B}_{i}^{\prime}\left(j\right)$ |

6. end |

7. end |

8. Reconstruct the $\left(\mathrm{M}\times \mathrm{N}\right)/\left(\mathrm{m}\times \mathrm{n}\right)$ sub block int an D |

#### 3.4. Spread

#### 3.5. Decryption Process

## 4. Simulation Results and Security Analysis

#### 4.1. Simulation Results

#### 4.2. Key Space Analysis

#### 4.3. Key Sensitivity Analysis

#### 4.4. Statistical Attack Analysis

#### 4.4.1. Histogram Analysis

#### 4.4.2. Information Entropy Analysis

#### 4.4.3. Correlation Coefficient Analysis

#### 4.5. Differential Attack Analysis

#### 4.6. Other Attack Analysis

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The phase diagram of the proposed hyper-chaotic system: (

**a**) x–y phase diagram; (

**b**) x–z phase diagram; (

**c**) y–z phase diagram; (

**d**) y–w phase diagram; (

**e**) x–y–z phase diagram; (

**f**) x–y–w phase diagram.

**Figure 5.**Encryption and decryption results: (

**a**) plain images; (

**b**) cipher images; (

**c**) decrypted images.

**Figure 6.**Encryption secret key sensitivity test: (

**a**) ordinary image; (

**b**) original key encrypted image; (

**c**) modified key encrypted image; (

**d**) differences between the two cipher-texts; (

**e**) use the modified key to solve the cipher-text image encrypted by the original key.

**Figure 7.**(

**a**) horizontal of plain image; (

**b**) vertical of plain image; (

**c**) diagonal of plain image; (

**d**) horizontal of cipher image; (

**e**) vertical of cipher image; (

**f**) diagonal of cipher image.

**Figure 8.**Experimental results of special images: (

**a**,

**d**) special image; (

**b**,

**e**) encrypted image; (

**c**,

**f**) cipher-text histogram.

Rules | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|

A | 00 | 00 | 01 | 01 | 10 | 10 | 11 | 11 |

T | 11 | 11 | 10 | 10 | 01 | 01 | 00 | 00 |

C | 01 | 10 | 00 | 11 | 00 | 11 | 01 | 10 |

G | 10 | 01 | 11 | 00 | 11 | 00 | 10 | 01 |

XOR | A | G | C | T |
---|---|---|---|---|

A | A | G | C | T |

G | G | A | T | C |

C | C | T | A | G |

T | T | C | G | A |

Image | Ours | Ref. [55] | Ref. [58] | Ref. [60] |
---|---|---|---|---|

Lena | 7.9993 | 7.9978 | 7.9993 | -- |

Butterfly | 7.9996 | 7.994 | -- | -- |

Terrace | 7.9998 | -- | -- | -- |

Bridge (color image) | 7.9992 | -- | -- | 7.9896 |

Level | Vertical | Diagonal | ||||
---|---|---|---|---|---|---|

Image | Plain Image | CIPHER-TEXT | Plain Image | Cipher-Text | Plain Image | Cipher-Text |

lean | 0.9738 | −0.0024 | 0.9547 | −0.0054 | 0.9277 | −0.0129 |

bridge | 0.9717 | 0.0036 | 0.9650 | 0.0471 | 0.9589 | 0.0105 |

terrace | 0.9783 | −0.0248 | 0.9733 | 0.0016 | 0.9535 | 0.0057 |

butterfly | 0.9456 | −0.0135 | 0.9509 | −0.0090 | 0.9164 | −0.0089 |

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

Ours | 99.62 | 33.69 |

Ref. [55] | 99.59 | 33.41 |

Ref. [58] | 99.61 | 33.47 |

Ref. [59] | 99.46 | 33.10 |

Ref. [60] | 99.62 | 31.83 |

Information Entropy | Relevance | |||
---|---|---|---|---|

Level | Vertical | Diagonal | ||

All black cipher-text image | 7.9973 | −0.0094 | 0.0098 | −0.0016 |

All white cipher-text image | 7.9974 | −0.0169 | −0.0094 | 0.0056 |

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## Share and Cite

**MDPI and ACS Style**

Zhu, Y.; Wang, C.; Sun, J.; Yu, F.
A Chaotic Image Encryption Method Based on the Artificial Fish Swarms Algorithm and the DNA Coding. *Mathematics* **2023**, *11*, 767.
https://doi.org/10.3390/math11030767

**AMA Style**

Zhu Y, Wang C, Sun J, Yu F.
A Chaotic Image Encryption Method Based on the Artificial Fish Swarms Algorithm and the DNA Coding. *Mathematics*. 2023; 11(3):767.
https://doi.org/10.3390/math11030767

**Chicago/Turabian Style**

Zhu, Yue, Chunhua Wang, Jingru Sun, and Fei Yu.
2023. "A Chaotic Image Encryption Method Based on the Artificial Fish Swarms Algorithm and the DNA Coding" *Mathematics* 11, no. 3: 767.
https://doi.org/10.3390/math11030767