# Secret Communication Systems Using Chaotic Wave Equations with Neural Network Boundary Conditions

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Grayscale Images as Transmission Objects

#### 2.1. Wave Equation with the van der Pol Boundary Conditions

#### 2.2. Synchronization System

#### 2.3. Proposed Secret Communication System

- (1)
- F has a sensitive dependence on initial conditions within the defined region.
- (2)

**Modulation**

**Demodulation**

**(Condition 1:)**For any positive number $\epsilon $, there exists $\delta $ such that if

**Remark**

**1.**

**Remark**

**2.**

#### 2.4. Numerical Experiments

**Modulation**

**Demodulation**

## 3. Numerical Experiments with Color Images

**Modulation**

**Demodulation**

## 4. Security Evaluation of the Encrypted Images by the Proposed Method

#### 4.1. Randomness Testing of Encrypted Images

#### 4.2. Distinguishability of Encrypted Images

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Cuomo, K.M.; Oppenheim, A.V. Circuit Implementation of Synchronized Chaos with Applications to Communications. Phys. Rev. Lett.
**1993**, 1, 65–68. [Google Scholar] [CrossRef] - Kocarev, L.; Parlitz, U. General Approach for Chaotic Synchronization with Applications to Communication. Phys. Rev. Lett.
**1995**, 25, 5028–5031. [Google Scholar] [CrossRef] - Lin, C.H.; Hu, G.H.; Chan, C.Y.; Yan, J.J. Chaos-Based Synchronized Dynamic Keys and Their Application to Image Encryption with an Improved AES Algorithm. Appl. Sci.
**2021**, 11, 1329. [Google Scholar] [CrossRef] - Ushio, T. Chaotically Synchronizing Control and Its Application to Secure Communication. Trans. Inf. Process. Soc. Jpn.
**1995**, 3, 525–530. [Google Scholar] - Yoshimura, K. Multichannel Digital Communications by the Synchronization of Globally Coupled Chaotic Systems. Phys. Rev. E
**1999**, 2, 1648–1657. [Google Scholar] [CrossRef] - Li, C.; Zhao, F.; Liu, C.; Lei, L.; Zhang, J. A Hyperchaotic Color Image Encryption Algorithm and Security Analysis. Secur. Commun. Netw.
**2019**, 2019, 8132547. [Google Scholar] [CrossRef] - Sbiaa, F.; Kotel, S.; Zeghid, M.; Tourki, R.; Machhout, M.; Baganne, A. High-Level Implementation of a Chaotic and AES Based Crypto-System. J. Circuits, Syst. Comput.
**2017**, 26, 1750122. [Google Scholar] [CrossRef] - Arab, A.; Rostami, M.J.; Ghavami, B. An image encryption method based on chaos system and AES algorithm. J. Supercomput.
**2019**, 75, 6663–6682. [Google Scholar] [CrossRef] [Green Version] - Suri, S.; Vijay, R. An AES–CHAOS-Based Hybrid Approach to Encrypt Multiple Images. In Recent Developments in Intelligent Computing, Communication and Devices; Springer: Singapore, 2017; pp. 3–43. [Google Scholar]
- Liu, Y.; Tong, X.; Hu, S. A family of new complex number chaotic maps based image encryption algorithm. Signal Process. Image Commun.
**2013**, 28, 1548–1559. [Google Scholar] [CrossRef] - Huang, X.; Ye, G. An image encryption algorithm based on hyper-chaos system and DNA plane. Multimed. Tools Appl.
**2014**, 72, 57–70. [Google Scholar] [CrossRef] - Kuo, C.L. Design of a fuzzy sliding-mode synchronization controller for two different chaos systems. Comput. Math. Appl.
**2011**, 61, 2090–2095. [Google Scholar] [CrossRef] [Green Version] - Moon, S.; Baik, J.-J.; Seo, J.M. Chaos synchronization in generalized Lorenz systems and an application to image encryption. Commun. Nonlinear Sci. Numer. Simul.
**2021**, 96, 105708. [Google Scholar] [CrossRef] - Njah, A.N. Tracking control and synchronization of the new hyperchaotic Liu system via backstepping techniques. Nonlinear Dyn.
**2010**, 61, 1–9. [Google Scholar] [CrossRef] - Pai, M.C. Global synchronization of uncertain chaotic systems via discrete-time sliding mode control. Appl. Math. Comput.
**2014**, 228, 663–671. [Google Scholar] [CrossRef] - Yau, H.T.; Kuo, C.L.; Yan, J.J. Fuzzy sliding mode control for a class of chaos synchronization with uncertainties. Int. J. Nonlinear Sci. Numer. Simul.
**2006**, 7, 333–338. [Google Scholar] [CrossRef] - Yu, Y.; Li, H.X. Adaptive hybrid projective synchronization of uncertain chaotic systems based on backstepping design. Nonlinear Anal. Real World Appl.
**2011**, 12, 388–393. [Google Scholar] [CrossRef] - Pecora, L.M.; Carroll, T.L. Synchronization of chaotic systems. Chaos
**2015**, 25, 097611. [Google Scholar] [CrossRef] [PubMed] - Sano, H.; Wakaiki, M.; Yaguchi, T. Secure Communication Systems Using Distributed Parameter Chaotic Synchronization. Trans. Soc. Instrum. Control. Eng.
**2021**, 2, 78–85. [Google Scholar] [CrossRef] - Chen, G.; Hsu, S.B.; Zhou, J. Chaotic Vibrations of the One-Dimensional Wave Equation due to a Self-Excitation Boundary Condition. Part I: Controlled Hysteresis, Appendix C by G.R. Chen and G. Crosta. Trans. Am. Math. Soc.
**1998**, 11, 4265–4311. [Google Scholar] [CrossRef] [Green Version] - Chen, G.; Hsu, S.B.; Zhou, J. Chaotic Vibration of the Wave Equation with Nonlinear Feedback Boundary Control: Progress and Open Questions. In Chaos Control: Theory and Applications; Chen, G.R., Yu, X., Eds.; Springer: Berlin/Heidelberg, Germany, 2003; pp. 25–50. [Google Scholar]
- Li, L.; Huang, Y.; Xiao, M. Observer Design for Wave Equations with Van Der Pol Type Boundary Conditions. SIAM J. Control. Optim.
**2012**, 3, 1200–1219. [Google Scholar] [CrossRef] - Naoe, H.K. Tanaka and Y. Takefuji, Information Security Techniques Based on Artificial Neural Network. J. Jpn. Soc. Artif. Intell.
**1995**, 5, 577–585. [Google Scholar] - Tariq, M.I.; Memon, N.A.; Ahmed, S. A Review of Deep Learning Security and Privacy Defensive Techniques. Mob. Inf. Syst.
**2020**, 2020, 6535834. [Google Scholar] [CrossRef] - Evans, L.C. Partial Differential Equations; Amer Mathematical Society: Providence, RI, USA, 2010. [Google Scholar]
- Guan, K. Important Notes on Lyapunov Exponents. arXiv
**2014**, arXiv:1401.3315. [Google Scholar] - Inoue, Y. Chaos:Definition and Characterization. Jpn. J. Multiph. Flow
**1997**, 2, 157–162. [Google Scholar] [CrossRef] [Green Version] - Kingma, D.P.; Ba, J. Adam: A Method for Stochastic Optimization. arXiv
**2014**, arXiv:1412.6980. [Google Scholar] - Preishuber, M.; Hütter, T.; Katzenbeisser, S.; Uhl, A. Depreciating Motivation and Empirical Security Analysis of Chaos-Based Image and Video Encryption. IEEE Trans. Inf. Forensics Secur.
**2018**, 13, 2137–2150. [Google Scholar] [CrossRef] - Wu, Y.; Noonan, J.P.; Agaian, S. NPCR and UACI Randomness Tests for Image Encryption. Cyber J. Multidiscip. J. Sci. Technol.
**2011**, 1, 31–38. [Google Scholar] - Ali, F.; Mohammed, A.H. Content Based Image Retrieval (CBIR) by Statistical Methods. Baghdad Sci. J.
**2020**, 17, 694–700. [Google Scholar] [CrossRef] - Seetharaman, K.; Jeyakarthic, M. Statistical distributional approach for scale and rotation invariant color image retrieval using multivariate parametric tests and orthogonality condition. J. Vis. Commun. Image Represent.
**2014**, 25, 727–739. [Google Scholar] [CrossRef]

**Figure 1.**A diagrammatic representation of the research ideas and methods used in this study. It succinctly shows the overall structure of the confidential communication system as well as the design principles, in which the input portrait was created manually by the author and is only used here as an example to assist in illustration.

**Figure 2.**The modulated secrecy image and the restored image after the leakage of some of the parameters of the confidential communication system. The figure on the left is the confidential information after it has been encrypted, and we can see that we have no way of knowing the information about the original image, so we can say that it was a successful secret communication. The figure on the right is the restored image after the leakage of some of the parameters of the confidential communication system; we can almost see that it is a picture of a woman wearing a hat, that is to say, the hacker can restore the transmitted information to some extent after breaking some of the parameters.

**Figure 3.**Reflection of the characteristic lines. The waves are reflected at $x=0$ and $x=1$, where the two functions $v=qu$ and $u={F}_{\alpha ,\beta}\left(v\right)$ are applied.

**Figure 5.**Time evolution of $u\left(k\right)$ and $v\left(k\right)$ in the discretized systems with the boundary conditions given by the neural networks.

**Figure 11.**Examples of training error (blue line) and testing error (red line) of the neural network during the learning process. The figure on the left is the result of error training without using BN; the figure on the right is the result of error training using BN.

**Figure 12.**Examples of training error drop and test error drop.The 1st figure is training error without BN; the 2nd figure is testing error without BN; the 3rd figure is training error with BN; the 4th figure is testing error with BN.

**Figure 14.**Encrypted grayscale images and restored images in the proposed method of this study. The figure on the left is the encrypted grayscale image; the figure on the right is the restored image.

**Figure 15.**The effect of changing the m value in the modulation and demodulation portions on the image encryption effect. Six different values of m (m = 0.8, 1.25, 2.5, 3.0, 4.5, 6.0) were tried, and the corresponding encryption effects were observed for each, where the larger the value of m, the better the secrecy performance of the image, within the computable range.

**Figure 17.**Examples of training error (blue line) and testing error (red line) of the neural network during the learning process.

**Figure 19.**The effect of changing the m value in the modulation and demodulation portions on the image encryption effect. Six different values of m (m = 1.0, 2.0, 5.0, 6.0, 7.5, 8.8) were tried, and the corresponding encryption effects were observed for each, where the larger the value of m, the better the secrecy performance of the image, within the computable range.

**Figure 20.**Encrypted color image and restored image in the proposed method of this study. The figure on the left is the encrypted color image; the figure on the right is the restored color image.

**Figure 21.**The grayscale images and the histograms of the encrypted images. The first row shows three different grayscale images (

**a**–

**c**). In the second, third and fourth rows, the three figures in each row represent the histograms of the grayscale values of the three original images encrypted by the proposed approach under the settings of $T=1,2,3$ and $m=6.0$, respectively (

**d**–

**l**). The last row shows the histogram after encryption by the AES. The horizontal coordinate represents the tonal range and the vertical coordinate represents the absolute frequency (

**m**–

**o**).

**Figure 22.**The color images and the histogram of the encrypted image. The first row shows three different color images (

**a**–

**c**). In the second, third and fourth rows, the three figures in each row represent the histograms of the three original images encrypted by the proposed approach under the settings of $T=1,2,3$ and $m=6.0$, respectively. The last row shows the histogram after encryption by the AES. The red, green and blue colors correspond to the histograms of each channel of R, G and B, respectively (

**d**–

**l**). The last row shows the histograms after encryption by the AES. The horizontal coordinates represent the tonal range and the vertical coordinates represent the absolute frequency (

**m**–

**o**).

Epoch | Data Set | Mean ± Standard |
---|---|---|

1000 times | train set | 0.000005 ± 0.000001 |

test set | 0.000005 ± 0.000001 | |

Lyapunov exponent | −0.067282 ± 0.102745 |

Epoch | Data Set | Mean ± Standard |
---|---|---|

1000 times | train set | 0.000002 ± 0.000001 |

test set | 0.000275 ± 0.000394 | |

Lyapunov exponent | 0.024377 ± 0.148675 |

Epoch | Data Set | Mean ± Standard |
---|---|---|

1000 times | train set | 0.000694 ± 0.000035 |

test set | 0.000919 ± 0.000033 | |

Lyapunov exponent | 0.1144 ± 0.1469 | |

0.0038 ± 0.0917 | ||

−0.1717 ± 0.2333 |

Object | Correlation | UACI | ||||
---|---|---|---|---|---|---|

Horizontal | Vertical | Diagonal | ||||

lena | m = 6 | T = 1 | 0.972642 | 0.972993 | 0.922492 | 36.141486 |

T = 2 | 0.962627 | 0.962568 | 0.890740 | 39.410799 | ||

T = 3 | 0.968880 | 0.969089 | 0.915038 | 35.114740 | ||

m = 7.5 | T = 1 | 0.968823 | 0.969244 | 0.908995 | 39.877836 | |

T = 2 | 0.954396 | 0.954471 | 0.874803 | 39.730604 | ||

T = 3 | 0.976311 | 0.976517 | 0.940879 | 35.622222 | ||

AES | 0.002630 | 0.008785 | 0.000658 | 49.996347 | ||

boat | m = 6 | T = 1 | 0.980538 | 0.980823 | 0.943327 | 36.728787 |

T = 2 | 0.961264 | 0.961193 | 0.885866 | 41.783973 | ||

T = 3 | 0.973791 | 0.973994 | 0.930897 | 39.928194 | ||

m = 7.5 | T = 1 | 0.972897 | 0.973300 | 0.923706 | 36.761638 | |

T = 2 | 0.957479 | 0.957473 | 0.874213 | 41.551513 | ||

T = 3 | 0.973904 | 0.974118 | 0.936444 | 40.359627 | ||

AES | 0.000163 | 0.000445 | 0.000502 | 50.000055 | ||

clock | m = 6 | T = 1 | 0.921742 | 0.922687 | 0.806450 | 48.089881 |

T = 2 | 0.858309 | 0.858372 | 0.664064 | 58.067992 | ||

T = 3 | 0.888560 | 0.889011 | 0.722784 | 58.823428 | ||

m = 7.5 | T = 1 | 0.899274 | 0.900966 | 0.770316 | 47.179087 | |

T = 2 | 0.853312 | 0.957473 | 0.874213 | 56.637741 | ||

T = 3 | 0.880133 | 0.881703 | 0.754349 | 57.388910 | ||

AES | 0.005367 | 0.004363 | 0.003360 | 49.929277 |

Object | Correlation | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

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

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

lena | m = 7.5 | T = 1 | 0.97 | 0.99 | 0.98 | 0.97 | 0.99 | 0.98 | 0.95 | 0.99 | 0.96 |

T = 2 | 0.99 | 0.99 | 0.98 | 0.99 | 0.99 | 0.98 | 0.98 | 0.98 | 0.97 | ||

T = 3 | 0.93 | 0.94 | 0.81 | 0.93 | 0.94 | 0.81 | 0.82 | 0.89 | 0.59 | ||

m = 8.8 | T = 1 | 0.96 | 0.99 | 0.97 | 0.96 | 0.99 | 0.97 | 0.88 | 0.88 | 0.88 | |

T = 2 | 0.98 | 0.99 | 0.98 | 0.98 | 0.99 | 0.97 | 0.96 | 0.98 | 0.96 | ||

T = 3 | 0.89 | 0.95 | 0.82 | 0.90 | 0.95 | 0.83 | 0.75 | 0.88 | 0.57 | ||

AES | 5 ×${10}^{-3}$ | 1 ×${10}^{-3}$ | −1 ×${10}^{-3}$ | 1 ×${10}^{-2}$ | 6 ×${10}^{-3}$ | 8 ×${10}^{-3}$ | 1 ×${10}^{-3}$ | −8 ×${10}^{-5}$ | 3 ×${10}^{-3}$ | ||

boat | m = 7.5 | T = 1 | 0.96 | 1.00 | 0.98 | 0.97 | 1.00 | 0.98 | 0.94 | 0.99 | 0.96 |

T = 2 | 0.99 | 0.99 | 0.98 | 0.99 | 0.99 | 0.98 | 0.98 | 0.98 | 0.98 | ||

T = 3 | 0.92 | 0.97 | 0.92 | 0.93 | 0.97 | 0.92 | 0.82 | 0.94 | 0.86 | ||

m = 8.8 | T = 1 | 0.96 | 0.99 | 0.97 | 0.96 | 0.99 | 0.97 | 0.93 | 0.99 | 0.95 | |

T = 2 | 0.98 | 0.99 | 0.97 | 0.98 | 0.99 | 0.97 | 0.96 | 0.98 | 0.96 | ||

T = 3 | 0.92 | 0.96 | 0.92 | 0.93 | 0.97 | 0.93 | 0.82 | 0.93 | 0.87 | ||

AES | 6 ×${10}^{-4}$ | −1 ×${10}^{-3}$ | −1 ×${10}^{-4}$ | −3 ×${10}^{-3}$ | −1 ×${10}^{-3}$ | −3 ×${10}^{-3}$ | 1 ×${10}^{-3}$ | −2 ×${10}^{-3}$ | −3 ×${10}^{-3}$ | ||

veg | m = 7.5 | T = 1 | 0.97 | 0.99 | 0.98 | 0.97 | 0.99 | 0.98 | 0.94 | 0.99 | 0.96 |

T = 2 | 0.99 | 0.99 | 0.98 | 0.99 | 0.99 | 0.88 | 0.98 | 0.98 | 0.97 | ||

T = 3 | 0.92 | 0.97 | 0.92 | 0.93 | 0.97 | 0.92 | 0.82 | 0.94 | 0.86 | ||

m = 8.8 | T = 1 | 0.96 | 0.99 | 0.97 | 0.97 | 0.99 | 0.98 | 0.94 | 0.99 | 0.96 | |

T = 2 | 0.98 | 0.99 | 0.97 | 0.98 | 0.99 | 0.97 | 0.96 | 0.98 | 0.96 | ||

T = 3 | 0.92 | 0.96 | 0.92 | 0.93 | 0.96 | 0.92 | 0.82 | 0.96 | 0.86 | ||

AES | 2 ×${10}^{-4}$ | 2 ×${10}^{-3}$ | 5 ×${10}^{-3}$ | −8 ×${10}^{-4}$ | 2 ×${10}^{-3}$ | 6 ×${10}^{-4}$ | 4 ×${10}^{-4}$ | 4 ×${10}^{-3}$ | 7 ×${10}^{-4}$ |

Object | UACI | ||||
---|---|---|---|---|---|

R | G | B | |||

lena | m = 7.5 | T = 1 | 46.293217 | 40.515568 | 24.708355 |

T = 2 | 43.678409 | 43.262431 | 37.007906 | ||

T = 3 | 41.407206 | 42.583560 | 29.628579 | ||

m = 8.8 | T = 1 | 46.454520 | 41.186064 | 24.949227 | |

T = 2 | 44.341227 | 43.377668 | 33.920451 | ||

T = 3 | 43.072341 | 40.273322 | 40.672362 | ||

AES | 50.133255 | 50.047464 | 49.936812 | ||

boat | m = 7.5 | T = 1 | 35.143726 | 48.660561 | 52.421603 |

T = 2 | 39.571601 | 55.066602 | 56.038358 | ||

T = 3 | 47.716444 | 53.135069 | 53.060568 | ||

m = 8.8 | T = 1 | 35.245613 | 50.609906 | 55.037498 | |

T = 2 | 38.337034 | 56.456049 | 58.441035 | ||

T = 3 | 42.331758 | 55.305545 | 56.647502 | ||

AES | 49.953939 | 50.052023 | 50.175874 | ||

veg | m = 7.5 | T = 1 | 37.751291 | 54.587069 | 54.226224 |

T = 2 | 45.768913 | 47.626056 | 62.656476 | ||

T = 3 | 37.213291 | 48.628576 | 69.999379 | ||

m = 8.8 | T = 1 | 37.915899 | 54.778044 | 55.353727 | |

T = 2 | 62.202183 | 48.556424 | 40.446628 | ||

T = 3 | 33.298991 | 51.570749 | 69.901463 | ||

AES | 49.956022 | 50.026667 | 49.943955 |

**Table 7.**Similarity comparison between the grayscale encrypted images $I1$ and $I2$ with the proposed approach. Six different encrypted images are taken, the similarity is calculated between each two, and no comparison is made between encrypted images of the same original image. The result column has three values, from top to bottom, corresponding to (47)–(49). In particular, “lena” is Figure 21a, “boat” is Figure 21b, and “clock” is Figure 21c.

I2 | Lena (m = 6, T = 1) | Boat (m = 7.5, T = 2) | Clock (m = 6, T = 2) | |
---|---|---|---|---|

I1 | ||||

lena (m = 7.5, T = 1) | 0.068396 | 0.0898650 | ||

0.112057 | 0.119775 | |||

0.999999 | 0.999999 | |||

boat (m = 7.5, T = 1) | 0.042764 | 0.099446 | ||

0.077088 | 0.143051 | |||

0.999999 * | 0.999999 | |||

clock (m = 6, T = 3) | 0.106253 | 0.071040 | ||

0.107990 | 0.081516 | |||

0.999999 | 0.999999 |

**Table 8.**Similarity comparison between the color encrypted images $I1$ and $I2$ with the proposed approach. Six different encrypted images are taken, the similarity is calculated between each two, and no comparison is made between encrypted images of the same original image. The result column has three values, from top to bottom, corresponding to (47)–(49). In particular, “lena” is Figure 21a, “boat” is Figure 21b, and “veg” is Figure 21c.

I2 | Lena (m = 8.8, T = 2) | Boat (m = 7.5, T = 2) | Veg (m = 8.8, T = 1) | |
---|---|---|---|---|

I1 | ||||

lena (m = 8.8, T = 3) | 0.047194 | 0.070985 | ||

0.103503 | 0.184824 | |||

0.999999 | 0.999999 | |||

boat (m = 7.5, T = 1) | 0.084080 | 0.008566 | ||

0.185896 | 0.037862 | |||

0.999999 * | 0.999999 | |||

veg (m = 7.5, T = 1) | 0.092913 | 0.106315 | ||

0.191360 | 0.210313 | |||

0.999999 | 0.999999 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chen, Y.; Sano, H.; Wakaiki, M.; Yaguchi, T.
Secret Communication Systems Using Chaotic Wave Equations with Neural Network Boundary Conditions. *Entropy* **2021**, *23*, 904.
https://doi.org/10.3390/e23070904

**AMA Style**

Chen Y, Sano H, Wakaiki M, Yaguchi T.
Secret Communication Systems Using Chaotic Wave Equations with Neural Network Boundary Conditions. *Entropy*. 2021; 23(7):904.
https://doi.org/10.3390/e23070904

**Chicago/Turabian Style**

Chen, Yuhan, Hideki Sano, Masashi Wakaiki, and Takaharu Yaguchi.
2021. "Secret Communication Systems Using Chaotic Wave Equations with Neural Network Boundary Conditions" *Entropy* 23, no. 7: 904.
https://doi.org/10.3390/e23070904