# A Novel Hyperchaotic 2D-SFCF with Simple Structure and Its Application in Image Encryption

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## Abstract

**:**

## 1. Introduction

## 2. 2D-SFCF

#### 2.1. Attractor

#### 2.2. NIST Statistical Test Suite

#### 2.3. Lyapunov Exponents

## 3. SFCF-IE Algorithm

#### 3.1. Function Declaration

- (1)
- Set a key processing function $[A,B,C,D]=F(a,b,c,d)$, the function F can be converted $a,b,c,d$ into a new value $A,B,C,D$ by Equation (4),$$\left\{\begin{array}{c}A=\left(a\phantom{\rule{0.277778em}{0ex}}\mathrm{mod}\phantom{\rule{0.277778em}{0ex}}1\right)\times 100+1,\hfill \\ B=floor(b\times {10}^{10})\phantom{\rule{0.277778em}{0ex}}\mathrm{mod}\phantom{\rule{0.277778em}{0ex}}35+2,\hfill \\ C=c\phantom{\rule{0.277778em}{0ex}}\mathrm{mod}\phantom{\rule{0.277778em}{0ex}}1,\hfill \\ D=d\phantom{\rule{0.277778em}{0ex}}\mathrm{mod}\phantom{\rule{0.277778em}{0ex}}1.\hfill \end{array}\right.$$
- (2)
- Set to a keystream generation function $[SX,SY]=C(a,b,c,d,N)$, where a is the parameter $\alpha $ of the 2D-SFCF, b is the parameter $\beta $ of the 2D-SFCF, c is the initial value ${x}_{0}$ of the 2D-SFCF, d is the initial value ${x}_{0}$ of the 2D-SFCF, and N is the number of iterations. Output sequences are $SX\in {\mathcal{M}}_{1\times N}$ and $SY\in {\mathcal{M}}_{1\times N}$. Note that when generating a chaotic sequence, some initial values of iterations need to be discarded, so that the generated sequence is sufficiently chaotic. Here, the first 100 iteration values are set to be discarded.
- (3)
- Set a sorting function $B=Sort\left(a\right)$, which can sort the one-dimensional vector a and find the position of the sorted vector in the vector a, and the return value is B. An example of a sorting function. If $A=\left[\begin{array}{cccccc}0.2785& 0.5469& 0.9575& 0.9649& 0.1576& 0.9706\end{array}\right]$, and then $B=\left[\begin{array}{cccccc}4& 5& 2& 3& 6& 1\end{array}\right]$.

#### 3.2. Key Generation of SFCF-IE

#### 3.3. Scrambling of SFCF-IE

#### 3.4. Diffusion of SFCF-IE

## 4. Performance Analysis

#### 4.1. Visualization

#### 4.2. Key Analysis

#### 4.3. Histogram Analysis

#### 4.4. Information Entropy Analysis

#### 4.5. Correlation Analysis

#### 4.6. NIST for Ciphertexts

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Attractor of 2D-SFCF. (

**a**) $\alpha =1.8,\beta =2$. (

**b**) $\alpha =3,\beta =4.2$. (

**c**) $\alpha =5,\beta =9.7$. (

**d**) $\alpha =8,\beta =9.7$.

**Figure 2.**Lyapunov exponents. (

**a**) LEs of 2D-SFCF with $\beta =1$. (

**b**) LEs of 2D-SFCF with $\beta =2$. (

**c**) LEs of 2D-SFCF with $\beta =4$. (

**d**) LEs of 2D-SFCF with $\beta =6$. (

**e**) LEs of 2D-SFCF with $\beta =10$. (

**f**) LEs of 2D-SFCF with $\beta =20$. (

**g**) LEs of 1-DCF with $\beta =1$. (

**h**) LEs of 1-DCF with $\beta =10$. (

**i**) LEs of 2D-SCMCI.

**Figure 3.**Bifurcation diagram (BD). (

**a**) BD of 2D-SFCF with $\beta =4$ (X). (

**b**) BD of 2D-SFCF with $\beta =4$ (Y). (

**c**) BD of 2D-SFCF with $\beta =10$ (X). (

**d**) BD of 2D-SFCF with $\beta =10$ (Y). (

**e**) BD of 1-DCF with $\beta =4$. (

**f**) BD of 1-DCF with $\beta =10$. (

**g**) BD of 2D-SCMCI (X). (

**h**) BD of 2D-SCMCI (Y).

**Figure 5.**Visualization of SFCF-IE for image ‘1.5.01’ (512 × 512). (

**a**) plaintext of ‘1.5.01’. (

**b**) ciphertext of ‘1.5.01’. (

**c**) decrypted ‘1.5.01’.

**Figure 6.**Visualization of SFCF-IE for image ‘1.5.03’ (512 × 512). (

**a**) plaintext of ‘1.5.03’. (

**b**) ciphertext of ‘1.5.03’. (

**c**) decrypted ‘1.5.03’.

**Figure 7.**Visualization of SFCF-IE for image Black (512 × 512). (

**a**) plaintext of Black. (

**b**) ciphertext of Black. (

**c**) decrypted Black.

**Figure 8.**Key sensitivity analysis of SFCF-IE. (

**a**) decrypted by K. (

**b**) decrypted by ${K}_{1}$. (

**c**) decrypted by ${K}_{2}$. (

**d**) decrypted by ${K}_{3}$. (

**e**) decrypted by ${K}_{4}$. (

**f**) decrypted by ${K}_{5}$. (

**g**) decrypted by ${K}_{6}$. (

**h**) decrypted by ${K}_{7}$.

**Figure 9.**Spatiotemporal histogram of ‘1.5.01’. (

**a**) Plaintext spatiotemporal histogram of ‘1.5.01’. (

**b**) Ciphertext spatiotemporal histogram of ‘1.5.01’.

**Figure 10.**Spatiotemporal histogram of ‘1.5.02’. (

**a**) Plaintext spatiotemporal histogram of ‘1.5.02’. (

**b**) Ciphertext spatiotemporal histogram of ‘1.5.02’.

**Figure 11.**Correlation analysis of ‘1.5.06’. (

**a**) Horizontal of plaintext. (

**b**) Vertical of plaintext. (

**c**) Diagonal of plaintext. (

**d**) Horizontal of ciphertext. (

**e**) Vertical of ciphertext. (

**f**) Diagonal of ciphertext.

Statistical Test | a = 9.7, b = 8 | a = 2.8, b = 10 | ||||||
---|---|---|---|---|---|---|---|---|

X | Y | X | Y | |||||

p-Value | Result | p-Value | Result | p-Value | Result | p-Value | Result | |

Longest run of ones | 0.419021 | Success | 0.096578 | Success | 0.236810 | Success | 0.534146 | Success |

Overlapping template matching | 0.616305 | Success | 0.383827 | Success | 0.534146 | Success | 0.574903 | Success |

Random excursions variant | 0.671779 | Success | 0.350485 | Success | 0.976060 | Success | 0.888137 | Success |

Rank | 0.657933 | Success | 0.350485 | Success | 0.816537 | Success | 0.911413 | Success |

Frequency | 0.066882 | Success | 0.350485 | Success | 0.494392 | Success | 0.108791 | Success |

Universal | 0.657933 | Success | 0.494392 | Success | 0.350485 | Success | 0.883171 | Success |

Random excursions | 0.602458 | Success | 0.275709 | Success | 0.862344 | Success | 0.995711 | Success |

Block frequency | 0.289667 | Success | 0.911413 | Success | 0.006661 | Success | 0.779188 | Success |

Cumulative sums | 0.191687 | Success | 0.289667 | Success | 0.574903 | Success | 0.236810 | Success |

Runs | 0.213309 | Success | 0.816537 | Success | 0.816537 | Success | 0.085587 | Success |

Serial | 0.779188 | Success | 0.616305 | Success | 0.883171 | Success | 0.289667 | Success |

Spectral | 0.045675 | Success | 0.816537 | Success | 0.883171 | Success | 0.851383 | Success |

Approximate entropy | 0.955835 | Success | 0.455937 | Success | 0.816537 | Success | 0.383827 | Success |

Nonoverlapping template matching | 0.971699 | Success | 0.383827 | Success | 0.534146 | Success | 0.739918 | Success |

Linear complexity | 0.574903 | Success | 0.534146 | Success | 0.455937 | Success | 0.455937 | Success |

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

1.4.01 | 6.3291 | 7.9998 |

1.4.02 | 7.1882 | 7.9998 |

1.4.03 | 6.6188 | 7.9998 |

1.4.04 | 6.1911 | 7.9998 |

1.4.05 | 7.1177 | 7.9998 |

1.5.01 | 5.6826 | 7.9993 |

1.5.02 | 5.8145 | 7.9993 |

1.5.03 | 6.9857 | 7.9993 |

1.5.04 | 6.4154 | 7.9993 |

1.5.05 | 6.8087 | 7.9993 |

1.5.06 | 5.1332 | 7.9993 |

1.5.07 | 6.7359 | 7.9993 |

Black | 0 | 7.9993 |

White | 0 | 7.9992 |

Average | 5.5015 | 7.9995 |

Algorithms | SFACF-IE | Ref. [34] | Ref. [35] | Ref. [36] | Ref. [37] | Ref. [38] |
---|---|---|---|---|---|---|

Information entropy | 7.9995 | 7.993 | 7.9995 | 7.9993 | 7.9992 | 7.9972 |

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

Horizontal | Vertical | Diagonal | Horizontal | Vertical | Diagonal | ||

1.4.01 | 1024 × 1024 | 0.9468 | 0.9172 | 0.9175 | −0.0002 | −0.0005 | 0.0011 |

1.4.02 | $1024\times 1024$ | 0.9749 | 0.9347 | 0.9209 | −0.0002 | −0.00002 | 0.0005 |

1.4.03 | $1024\times 1024$ | 0.9700 | 0.9557 | 0.9503 | −0.0010 | −0.0009 | 0.0005 |

1.4.04 | $1024\times 1024$ | 0.9739 | 0.9691 | 0.9633 | 0.0006 | 0.0007 | 0.0006 |

1.4.05 | $1024\times 1024$ | 0.9894 | 0.9734 | 0.9670 | 0.0012 | 0.0010 | −0.0011 |

1.5.01 | $512\times 512$ | 0.8383 | 0.8956 | 0.7945 | −0.0003 | −0.0053 | −0.0009 |

1.5.02 | $512\times 512$ | 0.8893 | 0.8940 | 0.8043 | 0.0009 | 0.0030 | −0.0003 |

1.5.03 | $512\times 512$ | 0.8821 | 0.9214 | 0.8119 | 0.0006 | 0.0019 | −0.0004 |

1.5.04 | $512\times 512$ | 0.7553 | 0.7146 | 0.5593 | −0.0006 | −0.0014 | 0.0001 |

1.5.05 | $512\times 512$ | 0.9510 | 0.9582 | 0.9206 | −0.0015 | 0.0039 | −0.0006 |

1.5.06 | $512\times 512$ | 0.6165 | 0.5006 | 0.4633 | 0.0020 | 0.0017 | −0.0017 |

1.5.07 | $512\times 512$ | 0.7163 | 0.8144 | 0.5950 | 0.0007 | 0.0003 | −0.0024 |

Black | $512\times 512$ | 1 | 1 | 1 | 0.00003 | −0.0022 | 0.0031 |

White | $512\times 512$ | 1 | 1 | 1 | 0.0012 | −0.0019 | −0.0016 |

Average | 0.8931 | 0.8892 | 0.8334 | 0.0002 | 0.00005 | −0.0002 |

Algorithms | SFCF-IE | Ref. [34] | Ref. [35] | Ref. [36] | Ref. [37] | Ref. [38] |
---|---|---|---|---|---|---|

Horizontal | 0.0002 | 0.0048 | −0.0042 | 0.0022 | −0.0519 | −0.0016 |

Vertical | 0.00005 | −0.0025 | −0.0049 | 0.0017 | −0.0385 | −0.0026 |

Diagonal | −0.0002 | −0.0072 | −0.0045 | 0.0019 | 0.0046 | 0.0116 |

Number | Statistical Test | Plaintext | Ciphertext | ||
---|---|---|---|---|---|

p-Value | Result | p-Value | Result | ||

1 | Longest run of ones | 0 | Fail | 0.151616 | Success |

2 | Overlapping template matching | 0 | Fail | 0.611108 | Success |

3 | Random excursions variant | 0 | Fail | 0.949602 | Success |

4 | Rank | 0 | Fail | 0.016431 | Success |

5 | Frequency | 0 | Fail | 0.258961 | Success |

6 | Universal | 0 | Fail | 0.559523 | Success |

7 | Random excursions | 0 | Fail | 0.602458 | Success |

8 | Block frequency | 0 | Fail | 0.199580 | Success |

9 | Cumulative sums | 0 | Fail | 0.855534 | Success |

10 | Runs | 0 | Fail | 0.113706 | Success |

11 | Serial | 0 | Fail | 0.714660 | Success |

12 | Spectral | 0 | Fail | 0.509162 | Success |

13 | Approximate entropy | 0 | Fail | 0.258961 | Success |

14 | Non-overlapping template matching | 0 | Fail | 0.953553 | Success |

15 | Linear complexity | 0 | Success | Fail | Success |

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**MDPI and ACS Style**

Hu, Y.; Wu, H.; Zhou, L.
A Novel Hyperchaotic 2D-SFCF with Simple Structure and Its Application in Image Encryption. *Entropy* **2022**, *24*, 1266.
https://doi.org/10.3390/e24091266

**AMA Style**

Hu Y, Wu H, Zhou L.
A Novel Hyperchaotic 2D-SFCF with Simple Structure and Its Application in Image Encryption. *Entropy*. 2022; 24(9):1266.
https://doi.org/10.3390/e24091266

**Chicago/Turabian Style**

Hu, Yongsheng, Han Wu, and Luoyu Zhou.
2022. "A Novel Hyperchaotic 2D-SFCF with Simple Structure and Its Application in Image Encryption" *Entropy* 24, no. 9: 1266.
https://doi.org/10.3390/e24091266