# High-Security Image Encryption Based on a Novel Simple Fractional-Order Memristive Chaotic System with a Single Unstable Equilibrium Point

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

^{648}, NPCR = 0.99866, UACI = 0.49963, H(s) = 7.9993, and time efficiency = 0.3 s. The obtained numerical simulation results and the security metrics investigations demonstrate the accuracy, high-level security, and time efficiency of the used cryptosystem which exhibits high robustness against different types of pirate attacks.

## 1. Introduction

## 2. Preliminaries

^{q}) in Equation (4) [30].

## 3. Memristor Model

#### 3.1. Integer-Order Case

_{M}and f are the voltage amplitude and frequency, respectively. The ${v}_{M}-{i}_{M}$ characteristics curve of a memristor is specified in Figure 1 for different amplitudes and frequencies.

#### 3.2. Fractional-Order Case

#### 3.3. Circuit Realization of the Fractional-Order Memristor

^{0.98}can be calculated as in Equation (9) [37].

_{0}represents a unit parameter, selecting C

_{0}= 1 µF and H

_{0.98}(s). C

_{0}= 1/s

^{0.98}, that reached the resistor (R

_{f}

_{1}and R

_{f}

_{2}) and capacitor values (C

_{1}and C

_{2}) in Figure 3.

_{eq}acts as the fractional-order impedance equivalent to fractance cell in Figure 3, which is responsible for verifying the fractional-order integrator with order (q = 0.98). Therefore, the equivalent circuit corresponding to Equation (11) has been realized as shown in Figure 4. Figure 5 displays the results of ${v}_{M}-{i}_{M}$ characteristics corresponding to the fractional-order memristor which is illustrated in Figure 4. Multisim has been used to realize the electronic circuit of the fractional order. Figure 6 depicts the fractional-order memristor symbolic diagram.

## 4. Fractional-Order Memristive-Based Simple Chaotic Circuit

_{0}, y

_{0}, z

_{0}) = (0.8, 0.8, 0) and different fractional orders (q = 0.95 and q = 0.98). The chaotic behavior of the fractional-order memristive chaotic system (14) corresponding to these parameters, initial conditions, and fractional orders is displayed in Figure 8 by a form of phase portrait chaotic attractors in two-dimensional (2D) and 3D arrangements.

**Definition**

**1.**

_{i}(i = 1,2,3 … n) of the Jacobian matrix $J=\partial f\left(x\left(t\right)\right)/\partial x\left(t\right)$ evaluated at the equilibrium points satisfy $\left|arg\left({\lambda}_{i}\right)\right|>q\frac{\pi}{2}$.

_{1}= −5, λ

_{2},

_{3}= 0.25 ± 1.3919i). According to Definition 1, the stability of the equilibrium point (E(0, 0, 0)) depends on the used value of the fractional order (q). In this work, since we used the fractional order (q = 0.98), thus, the equilibrium point (E(0, 0, 0)) can be considered as an unstable equilibrium point. Furthermore, this equilibrium point can be defined as the saddle point of index 2, that is because it has one real eigenvalue in the stable region and two complex conjugates in the unstable region [40]. This single unstable equilibrium point is responsible for exciting the chaotic behavior of the proposed fractional-order memristive chaotic system described by Equation (14).

## 5. Complex Dynamics of the System

#### 5.1. Bifurcation Diagrams

_{0}, y

_{0}, z

_{0}), and fractional order (q). As can be seen in Figure 9, the proposed system displays chaotic behavior for a not small range of parameter α (0 < α < 35).

#### 5.2. Lyapunov Exponents

## 6. Microcontroller Implementation

_{0}, y

_{0}, z

_{0}) = (0.8, 0.8, 0) and fractional-order derivative value (q = 0.98).

## 7. Image Encryption Application

**Step 1**. Read an original grayscale image to obtain its pixels as grayscale values matrix I_{M*N}(where M and N denote the row and column of the image pixels) and change this matrix to1D vector as I = {I_{1}, I_{2}, …, I_{MN}}.**Step 2.**Before using the obtained grayscale values 1D vector in the encryption process, shuffle this grayscale values 1D vector by arbitrarily moving these values. The histogram will not change as a result of this process, but it will make it more difficult for a burglar to decrypt the image without knowing the exact shuffling method.**Step 3**. Set the initial values of the fractional-order memristive chaotic system (14) (x_{0}, y_{0}, z_{0}), select its fractional order (q), and its parameters (d, g, α, β, a, b, and k).**Step 4**. Simulate the simple fractional-order memristive chaotic system (14), iterate constantly, and randomly choose MN set of solutions to generate the chaotic sequence. S = {S_{1}, S_{2},…, S_{MN}}. (These solution sets are selected randomly from the obtained values of the system (14) variables (x, y, and z)).**Step 5.**To obtain secret keys K = {K_{1}, K_{2},…, K_{MN}}, preprocess the sequence S = {S_{1}, S_{2},…, S_{MN}}_{.}These secret keys are gained according to the following mathematical operations applied to the obtained system (14) chaotic sequence in Step 4 [51].

**Step 6.**Encrypt the pixels of the original image I = {I_{1}, I_{2,}…, I_{MN}} using the obtained code in step code as:$${E}_{i}={I}_{i}\oplus {K}_{i}$$_{1}, E_{2}, … E_{MN}} is the obtained 1D vector representation of the encrypted image.

**Step 7.**Reform the 1D encrypted vector E = {E_{1}, E_{2}, … E_{MN}} to obtain 2D pixels of the encrypted image.

## 8. Numerical Simulation Results

_{0}, y

_{0}, z

_{0}) = (0.8, 0.8, 0) and fractional order (q = 0.98) for generating 262,144 samples corresponding to the total number of the original image pixels (512 × 512). These 262,144 samples are responsible for generating the secret keys K = {K

_{1}, K

_{2},…, K

_{MN}} which are mentioned in Step 5 in the above section (Section 4), where MN = 262,144. Figure 17a–c show the original image, encrypted image (cipher image), and the recovered image, respectively. The results of Figure 17 show that the applied cryptosystem approach is effective: it is clear that the encrypted image in Figure 17b is completely different from the original image in Figure 17a; also, the cipher image is successfully decrypted to give the recovered image in Figure 17c, which is exactly identical to the original image.

## 9. Cryptanalysis

#### 9.1. Histogram Analysis

#### 9.2. Keyspace Analysis

_{1}, K

_{2},…, K

_{MN}}. Thus, the system (14) fractional order (q), its initial values (x

_{0}, y

_{0}, z

_{0}), and its parameters (d, g, α, β, a, b, and k) can be considered as the secret keys. As mentioned in the introduction section, the fractional-order chaotic systems exhibit very high sensitivity to small changes in the used fractional order, initial conditions, and the system parameters. We assume that each entered key has a 10

^{−15}step-change, then the total keyspace can be calculated as (10

^{15})

^{13}= 10

^{195}≈ 2

^{648}. As a result, the used encryption method’s keyspace is large enough for resisting all types of pirate force attacks.

#### 9.3. Key Sensitivity Analysis

_{0}, y

_{0}, z

_{0}), and the parameters (d, g, α, β, a, b, and k) of the fractional-order memristive chaotic system (14) determine the key sensitivity in the applied encryption approach. In this work, the key sensitivity was measured using the net pixels change rate (NPCR) and the unified average changing intensity (UACI). They calculate the impact of small changes to a secret key in recovering the original image. Higher NPCR and UACI scores indicate that the encryption approach is more robust to different attacks [55]. The NPCR determines the difference in pixel’s absolute number change rate between two images as a percentage. On the other hand, the average intensity of discrepancies between the two images is computed by UACI. The NPCR and UACI can be calculated by the following Equations (19) and (20), respectively [56].

_{1}, K

_{2},…, K

_{MN}} are generated from the system (14) with elected parameters as d = 4, g = 0.5, α = 1, β = 1, a = 0.25, b = 5, and k = 4, initial conditions (x

_{0}, y

_{0}, z

_{0}) = (0.8, 0.8, 0), and fractional order (q = 0.98). By these encryption keys, the 512 × 512 grayscale “Lena.png” original image is encrypted. Consequently, in the NPCR and UACI tests for checking the key sensitivity, only the fractional-order has been very slightly changed to be q = 0.98 + 10

^{−15}for the decryption process, and the corresponding NPCR and UACI are found to be 0.99866 and 0.49963, respectively. The obtained simulation results of the recovered image with the abovementioned very slight change in the decryption keys are illustrated in Figure 19.

#### 9.4. Correlation Analysis

#### 9.5. Entropy Analysis

_{i}) is the probability of symbol (s

_{i}). The calculated information entropy values of the original grayscale Lena image and its corresponding encrypted images are 7.5946 and 7.9993, respectively. As is clear, the obtained results show that all of the encrypted image’s entropy values are very near to the ideal theoretical value of 8. Thus, our cryptosystem shows good performance in resisting the entropy attack.

#### 9.6. Time Efficiency Analysis

#### 9.7. Comparison Summary

## 10. Conclusions

^{648}, NPCR = 0.99866, UACI = 0.49963, H(s) = 7.9993, and time efficiency = 0.3 s. The obtained numerical simulation results and the comprehensive security analyses confirm the effectiveness, high-level security, and excellent time efficiency of the used cryptosystem, as well as its great resistance against various sorts of pirate attacks.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**${v}_{M}-{i}_{M}$ characteristics of memristor (6): (

**a**) with f = 1 Hz and different amplitudes; (

**b**) with A

_{m}= 10 V different frequencies.

**Figure 2.**${v}_{M}-{i}_{M}$ characteristics of fractional-order memristor described by Equation (8).

**Figure 4.**The realized circuit of the fractional-order memristor that is described by Equation (11).

**Figure 5.**${v}_{M}-{i}_{M}$ characteristics of the realized circuit of the fractional-order memristor.

**Figure 8.**The chaotic attractors phase portraits of the fractional-order memristive chaotic system (14): (

**a**) x-y; (

**b**) y-z; (

**c**) x-z; (

**d**) 3D arrangement (x-y-z).

**Figure 10.**The system (14) bifurcation diagram with respect to changing system fractional order (q).

**Figure 13.**The hardware setup for implementing the new fractional-order memristive chaotic system via a microcontroller (Arduino Due).

**Figure 14.**The obtained experimental results of the ${v}_{M}-{i}_{M}$ characteristics of the fractional-order memristor.

**Figure 15.**The phase portraits of chaotic attractors of the proposed fractional-order memristive chaotic system obtained from the microcontroller implementation: (

**a**) x-y; (

**b**) y-z; (

**c**) x-z.

**Figure 16.**Image encryption and decryption block diagram based on the proposed fractional-order memristive chaotic system.

**Figure 17.**Simulation results of encryption and decryption process of “Lena.png” 512 × 512 grayscale image: (

**a**) the original image; (

**b**) encrypted image; (

**c**) the decrypted image.

**Figure 18.**The histogram: (

**a**) the original image histogram; (

**b**) encrypted image histogram; (

**c**) the decrypted image histogram.

**Figure 19.**Key sensitivity analysis: (

**a**) the original image; (

**b**) the encrypted image; (

**c**) the recovered image with very slight change (10

^{−16}in the fractional order only) in decryption keys; (

**d**) the difference between (

**a**,

**c**).

**Figure 20.**Correlation distribution between two adjacent pixels of original Lena image and its corresponding encrypted image: (

**a**,

**b**) vertical correlation; (

**c**,

**d**) horizontal correlation; (

**e**,

**f**) diagonal correlation.

**Figure 21.**Encryption and decryption of “Peppers.pmb” 256 × 256 grayscale image: (

**a**) the original image; (

**b**) encrypted image; (

**c**) the decrypted image; (

**d**) histogram corresponding to (

**a**); (

**e**) histogram corresponding to (

**b**); (

**f**) histogram corresponding to (

**c**).

**Figure 22.**Encryption and decryption of “penguins.jpg” 546 × 636 grayscale image: (

**a**) the original image; (

**b**) encrypted image; (

**c**) the decrypted image; (

**d**) histogram corresponding to (

**a**); (

**e**) histogram corresponding to (

**b**); (

**f**) histogram corresponding to (

**c**).

Figure 9 | Figure 10 | ||
---|---|---|---|

Parameter | Value | Parameter | Value |

d | 4 | d | 4 |

g | 0.5 | g | 0.5 |

α | Variable | α | 1 |

β | 1 | β | 1 |

a | 0.25 | a | 0.25 |

b | 5 | b | 5 |

k | 4 | k | 4 |

fractional order (q) | 0.98 | fractional order (q) | Variable |

x_{0} | 0.8 | x_{0} | 0.8 |

y_{0} | 0.8 | y_{0} | 0.8 |

z_{0} | 0 | z_{0} | 0 |

Figure 11 | Figure 12 | ||
---|---|---|---|

Parameter | Value | Parameter | Value |

d | 4 | d | 4 |

g | 0.5 | g | 0.5 |

α | 1 | α | 1 |

β | 1 | β | 1 |

a | 0.25 | a | 0.25 |

b | 5 | b | 5 |

k | 4 | k | 4 |

fractional order (q) | 0.98 | fractional order (q) | Variable |

x_{0} | 0.8 | x_{0} | 0.8 |

y_{0} | 0.8 | y_{0} | 0.8 |

z_{0} | 0 | z_{0} | 0 |

Direction | Original Image | Encrypted Image |
---|---|---|

Vertical | 0.97241 | 0.00032 |

Horizontal | 0.98671 | 0.00054 |

Diagonal | 0.96181 | 0.00011 |

Algorithm | Key Space | NPCR | UACI | Vertical r _{xy} | Horizontal r _{xy} | Diagonal r _{xy} | H(s) | Time Efficiency |
---|---|---|---|---|---|---|---|---|

Ref. [19] | 2^{449} | 0.99606 | 0.33489 | 0.0002 | 0.0046 | 0.0005 | 7.9951 | 0.9 s |

Ref. [20] | 2^{530} | 0.99640 | 0.33537 | - | - | - | 7.9978 | - |

Ref. [21] | 2^{154} | 99.6096 | 0.33459 | 0.000333 | 0.000524 | 0.000872 | 7.9993 | 0.3261 s |

Ref. [22] | 2^{598} | 0.9955 | 0.3325 | 0.0059 | 0.0082 | 0.0007 | 7.9866 | 1.02 s |

Ref. [23] | 2^{285} | 0.9964 | 0.3355 | 0.000312 | 0.002088 | 0.001444 | 7.9976 | 1.708 s |

Ours | 2^{648} | 0.99866 | 0.49963 | 0.00032 | 0.00054 | 0.00011 | 7.9993 | 0.3 s |

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## Share and Cite

**MDPI and ACS Style**

Rahman, Z.-A.S.A.; Jasim, B.H.; Al-Yasir, Y.I.A.; Abd-Alhameed, R.A.
High-Security Image Encryption Based on a Novel Simple Fractional-Order Memristive Chaotic System with a Single Unstable Equilibrium Point. *Electronics* **2021**, *10*, 3130.
https://doi.org/10.3390/electronics10243130

**AMA Style**

Rahman Z-ASA, Jasim BH, Al-Yasir YIA, Abd-Alhameed RA.
High-Security Image Encryption Based on a Novel Simple Fractional-Order Memristive Chaotic System with a Single Unstable Equilibrium Point. *Electronics*. 2021; 10(24):3130.
https://doi.org/10.3390/electronics10243130

**Chicago/Turabian Style**

Rahman, Zain-Aldeen S. A., Basil H. Jasim, Yasir I. A. Al-Yasir, and Raed A. Abd-Alhameed.
2021. "High-Security Image Encryption Based on a Novel Simple Fractional-Order Memristive Chaotic System with a Single Unstable Equilibrium Point" *Electronics* 10, no. 24: 3130.
https://doi.org/10.3390/electronics10243130