# Network Synchronization of MACM Circuits and Its Application to Secure Communications

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## Abstract

**:**

## 1. Introduction

## 2. Brief Review on Synchronization of Complex Networks

#### 2.1. Synchronization of Complex Network

**synchronization**if [30]:

#### 2.2. Star Coupled Networks

#### 2.3. Synchronization Analysis Based on Master Stability Function Approach

## 3. MACM Circuit like Node

## 4. Star Network Synchronization of MACM’s Circuits

#### 4.1. Synchronization Analysis Based on Master Stability Function Approach and Its Simulation

#### 4.2. Star Network Electronic Circuit Synchronization

## 5. Application to Image Encryption

- 1.
**Binary string**. The 8-bit gray-scale digital image with $M\left(row\right)\times N\left(columns\right)$ pixels are placed row-by-row in a binary string with $M\times N\times 8$ bits.- 2.
**Synchronization of star network**. We used a coupling constant of $k=10$ between the master and slave MACM systems; different initial conditions are used for each MACM system (see Table 1); the control parameters are the same in all MACM systems, i.e., $a=2,\phantom{\rule{3.33333pt}{0ex}}b=2,\phantom{\rule{3.33333pt}{0ex}}c=0.5$, and $d=4$. After 50 time units (transient time), the star network is synchronized as shown in Figure 16.- 3.
**Extended plain binary data**. Since synchronization is achieved after a transient time and to avoid data loss in the receptors, the plain binary string is mounted over 400 time units for each bit producing an extended plain binary data of $M\times N\times 8\times 400$. As an example, Figure 16a–d show the first two bytes of the plain image transmitted, which are defined as 1010010010100011 with a length of 6400 time units (dashed line).- 4.
**Switching parameter d of master MACM**. The parameter d of the master node is switched between $d=4$ and $d=4.05$, for 0 and 1 in the extended plain binary data, respectively. During this time, the absolute synchronization error is determined in $e2$, $e3$, $e4$, and $e5$, which are shown in Figure 16a–d with a blue line. Since initial conditions are considerably different at the start communication, the error is bigger in the first time units.- 5.
**Processing the error**. The recovered binary string in the receptor is calculated with the sum of the last 100 data in each error signal considering windows of 400 data; if the sum is greater than 0.7, a bit of 1 is defined for such window or bit of 0 in other case. Figure 16e–h presents the first recovered binary string in each slave MACM system (receptor).- 6.
**Image construction**. The digital image is constructed using the recovered binary string and the inverse process of step 1; the string is separated into 8-bit segments and assigned to rows and columns to form the corresponding digital image. Figure 16i–l present the difference between the plain image and recovered image at the bit level (first 8000 bits) for slaves 2–5, respectively.

**Figure 16.**First bytes transmitted and recovered data: (

**a**) absolute of ${e}_{2}$ (blue line) and plain binary data (dashed line); (

**b**) absolute of ${e}_{3}$ (blue line) and plain binary data (dashed line); (

**c**) absolute of ${e}_{4}$ (blue line) and plain binary data (dashed line); (

**d**) absolute of ${e}_{5}$ (blue line) and plain binary data (dashed line); (

**e**) recovered data in slave 2; (

**f**) recovered data in slave 3; (

**g**) recovered data in slave 4; (

**h**) recovered data in slave 5; (

**i**) error between plain image end recovered image in slave 2; (

**j**) error between plain image end recovered image in slave 3; (

**k**) error between plain image end recovered image in slave 4; and (

**l**) error between plain image end recovered image in slave 5.

#### 5.1. Security Analysis

#### 5.1.1. Histograms

#### 5.1.2. Statistics of Histogram

#### 5.1.3. Structural Similarity Index

#### 5.1.4. Correlation Analysis

#### 5.1.5. Information Entropy

#### 5.1.6. Decryption Error Test

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

3D | Three-dimensional |

MACM | Méndez–Arellano–Cruz–Martínez |

IC | Integrated circuit |

OA | Operational amplifier |

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**Figure 1.**Electronic circuit of the MACM’s system (10).

**Figure 2.**Electronic circuit simulation of the chaotic 3D MACM system (1): (

**a**) time evolution of states $x\left(t\right)$, $y\left(t\right)$, and $z\left(t\right)$; (

**b**) phase plane $x\left(t\right)$ versus $y\left(t\right)$; (

**c**) phase plane $x\left(t\right)$ versus $z\left(t\right)$; and (

**d**) phase plane $y\left(t\right)$ versus $z\left(t\right)$.

**Figure 4.**Maximum Lyapunov exponent ${\lambda}_{max}$ applying coupling matrix $\mathbf{A}$ for $0\le k\le 35$.

**Figure 5.**Master MACM system in chaotic regime: (

**a**) x versus time; (

**b**) y versus time; (

**c**) z versus time; (

**d**) x versus y; (

**e**) x versus z; and (

**f**) y versus z.

**Figure 6.**Time series of the errors for each MACM system in the star network without coupling: (

**a**) $e2$; (

**b**) $e3$; (

**c**) $e4$; and (

**d**) $e5$.

**Figure 7.**Phase graphics between the master MACM and slaves in the star network without coupling: (

**a**) ${x}_{11}$ versus ${x}_{21}$; (

**b**) ${x}_{11}$ versus ${x}_{31}$; (

**c**) ${x}_{11}$ versus ${x}_{41}$; and (

**d**) ${x}_{11}$ versus ${x}_{51}$.

**Figure 8.**Time series of the errors for each MACM system in the star network with coupling constant $k=10$: (

**a**) $e2$; (

**b**) $e3$; (

**c**) $e4$; and (

**d**) $e5$.

**Figure 9.**Phase graphics between the master MACM and slaves in the star network with coupling constant $k=10$: (

**a**) ${x}_{11}$ versus ${x}_{21}$; (

**b**) ${x}_{11}$ versus ${x}_{31}$; (

**c**) ${x}_{11}$ versus ${x}_{41}$; and (

**d**) ${x}_{11}$ versus ${x}_{51}$.

**Figure 10.**Experimental set-up for network synchronization of five coupled MACM electronic circuits in star topology: (

**a**) master node ${N}_{1}$; (

**b**) slave node ${N}_{2}$; (

**c**) slave node ${N}_{3}$; (

**d**) slave node ${N}_{4}$; and (

**e**) slave node ${N}_{5}$.

**Figure 11.**Chaotic trajectories of the system of the Equations (14) and (23): (

**a**) state ${x}_{11}\left(t\right)$, error $e2\left(t\right)={x}_{11}\left(t\right)-{x}_{21}\left(t\right)$, and state ${x}_{21}\left(t\right)$; (

**b**) state ${x}_{11}\left(t\right)$, error $e3\left(t\right)={x}_{11}\left(t\right)-{x}_{31}\left(t\right)$, and state ${x}_{31}\left(t\right)$; (

**c**) state ${x}_{11}\left(t\right)$, error $e4\left(t\right)={x}_{11}\left(t\right)-{x}_{41}\left(t\right)$, and state ${x}_{41}\left(t\right)$; and (

**d**) state ${x}_{11}\left(t\right)$, error $e5\left(t\right)={x}_{11}\left(t\right)-{x}_{51}\left(t\right)$, and state ${x}_{51}\left(t\right)$.

**Figure 12.**Plane phase of the system of Equations (14) and (23): (

**a**) ${x}_{21}\left(t\right)$ versus ${x}_{11}\left(t\right)$; (

**b**) ${x}_{11}\left(t\right)$ versus ${x}_{31}\left(t\right)$; (

**c**) ${x}_{11}\left(t\right)$ versus ${x}_{41}\left(t\right)$; and (

**d**) ${x}_{11}\left(t\right)$ versus ${x}_{51}\left(t\right)$.

**Figure 13.**Chaotic trajectories of Equations (14) and (23): (

**a**) state ${x}_{11}\left(t\right)$, error $e2\left(t\right)={x}_{11}\left(t\right)-{x}_{21}\left(t\right)$, and state ${x}_{21}\left(t\right)$; (

**b**) state ${x}_{11}\left(t\right)$, error $e3\left(t\right)={x}_{11}\left(t\right)-{x}_{31}\left(t\right)$, and state ${x}_{31}\left(t\right)$; (

**c**) state ${x}_{11}\left(t\right)$, error $e4\left(t\right)={x}_{11}\left(t\right)-{x}_{41}\left(t\right)$, and state ${x}_{41}\left(t\right)$; and (

**d**) state ${x}_{11}\left(t\right)$, error $e5\left(t\right)={x}_{11}\left(t\right)-{x}_{51}\left(t\right)$, and state ${x}_{51}\left(t\right)$.

**Figure 14.**Plane phase of Equations (14) and (23): (

**a**) ${x}_{21}\left(t\right)$ versus ${x}_{11}\left(t\right)$; (

**b**) ${x}_{11}\left(t\right)$ versus ${x}_{31}\left(t\right)$; (

**c**) ${x}_{11}\left(t\right)$ versus ${x}_{41}\left(t\right)$; and (

**d**) ${x}_{11}\left(t\right)$ versus ${x}_{51}\left(t\right)$.

**Figure 17.**Experimental results of image encryption with the coupling constant $k=10$: (

**a**) plain Lena image; (

**b**) cryptogram; (

**c**) decrypted image in slave 2 MACM; (

**d**) decrypted image in slave 3 MACM; (

**e**) decrypted image in slave 4 MACM; and (

**f**) decrypted image in slave 5 MACM.

**Figure 18.**Experimental results of image encryption with coupling constant $k=5$: (

**a**) plain Lena image; (

**b**) cryptogram; (

**c**) decrypted image in slave 2 MACM; (

**d**) decrypted image in slave 3 MACM; (

**e**) decrypted image in slave 4 MACM; and (

**f**) decrypted image in slave 5 MACM.

**Figure 19.**Histograms of images with the coupling constant $k=10$: (

**a**) plain Lena histogram; (

**b**) histogram of cryptogram; (

**c**) histogram in slave 2 MACM; (

**d**) histogram in slave 3 MACM; (

**e**) histogram in slave 4 MACM; and (

**f**) histogram in slave 5 MACM.

**Figure 20.**Histograms of images with coupling constant $k=5$: (

**a**) plain Lena histogram; (

**b**) histogram of cryptogram; (

**c**) histogram in slave 2 MACM; (

**d**) histogram in slave 3 MACM; (

**e**) histogram in slave 4 MACM; and (

**f**) histogram in slave 5 MACM.

**Figure 21.**Graphic correlation of images with the coupling constant $k=10$: (

**a**) plain Lena histogram; (

**b**) cryptogram; (

**c**) correlation in slave 2 MACM; (

**d**) correlation in slave 3 MACM; (

**e**) correlation in slave 4 MACM; (

**f**) correlation in slave 5 MACM.

**Figure 22.**Graphic correlation of images with coupling constant $k=5$: (

**a**) plain Lena histogram; (

**b**) correlation of cryptogram; (

**c**) correlation in slave 2 MACM; (

**d**) correlation in slave 3 MACM; (

**e**) correlation in slave 4 MACM; (

**f**) correlation in slave 5 MACM.

Initial | Master 1 | Slave 2 | Slave 3 | Slave 4 | Slave 5 |
---|---|---|---|---|---|

Condition | MACM | MACM | MACM | MACM | MACM |

${x}_{i1}\left(0\right)$ | −4.0 | 2.0 | 2.5 | −4.5 | 2.2 |

${x}_{i2}\left(0\right)$ | −4.0 | 2.0 | 2.5 | −4.5 | 2.2 |

${x}_{i3}\left(0\right)$ | −3.0 | 4.0 | 4.5 | −3.5 | 4.2 |

**Table 2.**Hardware description of the coupled-star-network to achieve network synchronization, as depicted in Figure 10.

Component or IC | Value or Description |
---|---|

C1, C2, C3, C4, C5, C6, C7, C8, C9, C10, C11, C12, C13, C14, C15 | 10 nF |

R1, R25, R38, R51, R64 | 500 k$\mathrm{\Omega}$ |

R2, R37, R63 | 47 k$\mathrm{\Omega}$ |

R3, R4, R7, R8, R12, R13, R14, R15, R19, R20, R23, R26, R27, R28, R32, R33, R36, R39, R40, R41, R45, R46, R49, R52, R53, R54, R58, R59, R62, R66, R66, R67, R68, R69, R70, R71, R72, R73, R74, R75, R76, R77, R78, R79, R80, R81, R82, R83, R84, R85, R86, R87 R88, R89, R90, R91, R92, R93 | 10 k$\mathrm{\Omega}$ |

R5, R9, R11, R17, R18, R22, R31, R30, R35, R43, R44, R48, R56, R57, R61, R94, R95, R96, R97 | 1 M$\mathrm{\Omega}$ |

R6, R21, R34, R47, R60 | 2 M$\mathrm{\Omega}$ |

R10, R16, R42 | 94 k$\mathrm{\Omega}$ |

R24 | 47.5 k$\mathrm{\Omega}$ |

R50 | 48 k$\mathrm{\Omega}$ |

R29 | 94.5 k$\mathrm{\Omega}$ |

R55 | 95 k$\mathrm{\Omega}$ |

U1, U4, U5, U6, U8, U9, U12, U13, U15, U16 | Analog-multiplier AD633 |

U2, U3, U7, U10, U11, U14, U17, U18, U19 | OA TL084 |

Plain | Encrypted | Image in | Image in | Image in | Image in | |
---|---|---|---|---|---|---|

Image | Image | Slave 2 | Slave 3 | Slave 4 | Slave 5 | |

$\alpha $ with $k=10$ | 3713.12 | 101.04 | 3711.78 | 3711.78 | 3711.78 | 3.71178 |

$\beta $ with $k=10$ | 60.93 | 10.05 | 60.92 | 60.92 | 60.92 | 60.92 |

Plain | Encrypted | Image in | Image in | Image in | Image in | |

Image | Image | Slave 2 | Slave 3 | Slave 4 | Slave 5 | |

$\alpha $ with $k=5$ | 3713.12 | 101.04 | 96.91 | 96.87 | 96.91 | 96.93 |

$\beta $ with $k=5$ | 60.93 | 10.05 | 9.84 | 9.84 | 9.84 | 9.84 |

P | E | $\mathbf{SSIM}$ with $\mathit{k}=10$ | $\mathbf{SSIM}$ with $\mathit{k}=5$ |
---|---|---|---|

Plain image | Plain image | 1 | 1 |

Plain image | Encrypted image | 0.0030 | 0.0030 |

Plain image | Image in slave 2 | 0.9998 | 0.0135 |

Plain image | Image in slave 3 | 0.9998 | 0.0134 |

Plain image | Image in slave 4 | 0.9998 | 0.0135 |

Plain image | Image in slave 5 | 0.9998 | 0.0134 |

Coupling | Plain | Encrypted | Image in | Image in | Image in | Image in |
---|---|---|---|---|---|---|

Constant | Image | Image | Slave 2 | Slave 3 | Slave 4 | Slave 5 |

$k=10$ | 0.8757 | 0.1255 | 0.8520 | 0.8520 | 0.8520 | 0.8520 |

$k=5$ | 0.8758 | 0.1256 | 0.1060 | 0.0957 | 0.1060 | 0.0957 |

Coupling | Plain | Encrypted | Image in | Image in | Image in | Image in |
---|---|---|---|---|---|---|

Constant | Image | Image | Slave 2 | Slave 3 | Slave 4 | Slave 5 |

$k=10$ | 7.5250 | 7.9903 | 7.5253 | 7.5253 | 7.5253 | 7.5253 |

$k=5$ | 7.5250 | 7.9903 | 7.9910 | 7.9910 | 7.9910 | 7.9910 |

P | D | $\mathcal{E}$ (%) with $\mathit{k}=10$ | $\mathcal{E}$ (%) with $\mathit{k}=5$ |
---|---|---|---|

Plain image | Encrypted image | 99.5688 | 99.5688 |

Plain image | Image in slave 2 | 0.0044 | 99.6488 |

Plain image | Image in slave 3 | 0.0044 | 99.6488 |

Plain image | Image in slave 4 | 0.0044 | 99.6488 |

Plain image | Image in slave 5 | 0.0044 | 99.6488 |

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**MDPI and ACS Style**

Méndez-Ramírez, R.; Arellano-Delgado, A.; Murillo-Escobar, M.Á.
Network Synchronization of MACM Circuits and Its Application to Secure Communications. *Entropy* **2023**, *25*, 688.
https://doi.org/10.3390/e25040688

**AMA Style**

Méndez-Ramírez R, Arellano-Delgado A, Murillo-Escobar MÁ.
Network Synchronization of MACM Circuits and Its Application to Secure Communications. *Entropy*. 2023; 25(4):688.
https://doi.org/10.3390/e25040688

**Chicago/Turabian Style**

Méndez-Ramírez, Rodrigo, Adrian Arellano-Delgado, and Miguel Ángel Murillo-Escobar.
2023. "Network Synchronization of MACM Circuits and Its Application to Secure Communications" *Entropy* 25, no. 4: 688.
https://doi.org/10.3390/e25040688