# Efficient Colour Image Encryption Algorithm Using a New Fractional-Order Memcapacitive Hyperchaotic System

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Preliminaries

^{q}) in 1847, which is defined as follows [27]:

## 3. Memcapacitor Model

#### 3.1. Integer-Order Situation

_{M}and f present memcapacitor charge amplitude and its frequency, respectively. The ${q}_{M}-{v}_{M}$ hysteresis loop characteristics of the memcapacitor specified in Figure 1 are obtained with different values of amplitudes and frequencies.

#### 3.2. Fractional-Order Situation

#### 3.3. Electronic Circuit of the Fractional-Order Memcapacitor

^{0.99}can be derived using Equation (12) [33]:

_{0}signifies a unit limit, choosing Co = 1 µF and H

_{0.99}(s)·C

_{0}= 1/s

^{0.99}. These electronic component properties result from using Equations (12) and (13), where the comparison was used. Thus, the values of the resistors R

_{f}

_{1}, R

_{f}

_{2}, and R

_{f}

_{3}were obtained to be 95.082 MΩ, 6.441 kΩ, and 6.436 kΩ, respectively, while the values of the capacitors C

_{1}, C

_{2}, and C

_{3}were calculated to be 13.982, 13.05, and 1.011 μF, as illustrated in Figure 3.

_{eq}is the fractional-order impedance comparable to the fractance cell, and it corresponds to proving the fractional integrator with order value (q = 0.99). As a result, the equivalent electronic circuit consistent with Equation (14) has been realized as illustrated in Figure 4.

## 4. Fractional-Order Memcapacitive-Based Chaotic Circuit

_{1}= d, and 1/R

_{2}= g; thus, the fractional-order memcapacitive chaotic model can be defined as:

#### 4.1. Chaotic Behaviours of the Memcapacitive System

_{0}, y

_{0}, z

_{0}, u

_{0}) = (0.001, 0, 0, 0) and two different fractional-order derivative values (q = 0.97 and q = 0.99). Figure 8 depicts the phase portrait attractors of a fractional-order memcapacitive chaotic model (17) consistent with these designated setting values. A layout of phase portrait chaotic attractors is showed in two-dimensional (2D) and three-dimensional (3D) topologies.

#### 4.2. Equilibria and Stability

**Theorem**

**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}= 0, λ

_{2,3}= −0.6926 ± 0.1105i, and λ

_{4}= 1.3668). Based on Theorem 1, it’s clear that the fractional-order derivative value (q) limits the stability of the equilibrium point. Because we used fractional derivative order value (q = 0.99) in this study, the first and fourth eigenvalues have |arg(λ

_{1,4})|= 0. That indicates that the stability condition specified in Theorem 1 was not obeyed. Therefore, the equilibrium point P(0,0,0,0) is classified as an unstable equilibrium point. As a result, an excitation from this unstable equilibrium point P(0,0,0,0) might be used to generate a self-excited attractor. As a result, the suggested fractional-order memcapacitive system defined by Equation (17) is excited by this equilibrium point, which is accountable for its chaotic performance.

## 5. Dynamic Analysis

#### 5.1. Bifurcation Diagrams

_{0}, y

_{0}, z

_{0}, u

_{0}) being (0.001,0,0,0), with the system (17) parameters and fractional-order derivative value (q) as shown in Table 2.

#### 5.2. Lyapunov Exponents

_{0}, y

_{0}, z

_{0}, u

_{0}) being (0.001, 0, 0, 0), with the system (17) parameters and fractional-order derivative value (q) as shown in Table 3.

## 6. Image Encryption Algorithm

**Step****1.**- Read a coloured plain image to obtain its pixel values as a matrix I
_{M*N}, where M and N represent the rows and columns of the image pixels, respectively. **Step****2.**- Decompose this image into its three basic bands, which are R (red), G (green), and B (blue).
**Step****3.**- Read these three bands, R, G, and B, to obtain their pixel values as matrices IR
_{M*N}, IG_{M*N}, and IB_{M*N}, respectively. Then shuffle these three matrixes, where the histogram will remain unchanged, whereas it will be further difficult for an intruder to decode the image unless he knows the specific shuffling procedure. **Step****4.**- Each shuffled pixel matrix of the bands R, G, and B is split to four nonoverlapped submatrices (KP
_{1}, KP_{2}, KP_{3}, KP_{4}; P = R,G,B), as shown in Figure 13. In other words, the original band matrix is divided into four blocks, taking into account the total number of elements in the obtained four submatrices equivalent to the pixel number of the basic band matrix. The size of these submatrices is determined as follows:Size(KP_{1}) = Round(M/2) × round(N/2)Size(KP_{2}) = Round(M/2) × floor(N-N/2)Size(KP_{3}) = Floor (M-M/2) × round(N/2)Size(KP_{4}) = Floor (M-M/2) × floor(N-N/2) **Step****5.**- For the fractional-order memcapacitive hyperchaotic system defined by Equation (17), set the following values: initial conditions (x
_{(0)}, y_{(0)}, z_{(0)}, u_{(0)}), fractional-order derivative value (q), and parameters, which are a, b, d, g, α, and β. **Step****6.**- Use these determined values in step 5 for simulating the fractional-order memcapacitive hyperchaotic system (17). Consequently, iterate the solving process with fixed steps to ensure the iteration solution set coverage of the submatrix size of the generated chaotic sequence for each state variable (x, y, z, and u). Then randomly select elements from the solution set for each state variable of the system (17) with a number equivalent to the decomposed four blocks in step 4, where x, y, z, and u state variables are responsible for generating matrices with element numbers equivalent to these four blocks, KP
_{1}, KP_{2}, KP_{3}, and KP_{4}, respectively. **Step****7.**- To determine the secret keys S
_{x,y,z,u}, preprocess the chaotic sequences of the state_{.}variables obtained in step 6. The following mathematical operations are used to obtain these secret keys:$${S}_{xi}=\left|round\left(mod\left(\left|({x}_{i}-floor\left(\left|{x}_{i}\right|\right)\right|\right)\ast 5\ast {10}^{5}\right),256))\right|;i\phantom{\rule{0ex}{0ex}}=1,2,\dots ,Round\left(M/2\right)round\left(N/2\right).$$$${S}_{yi}=\left|round\left(mod\left(\left|({y}_{i}-floor\left(\left|{y}_{i}\right|\right)\right|\right)\ast 5\ast {10}^{5}\right),256))\right|;i\phantom{\rule{0ex}{0ex}}=1,2,\dots ,Round\left(M/2\right)floor\left(N-N/2\right).\phantom{\rule{0ex}{0ex}}{S}_{zi}=\left|round\left(mod\left(\left|({z}_{i}-floor\left(\left|{z}_{i}\right|\right)\right|\right)\ast 5\ast {10}^{5}\right),256))\right|;i\phantom{\rule{0ex}{0ex}}=1,2,\dots ,Floor\left(M-M/2\right)round\left(N/2\right).\phantom{\rule{0ex}{0ex}}{S}_{ui}=\left|round\left(mod\left(\left|({u}_{i}-floor\left(\left|{u}_{i}\right|\right)\right|\right)\ast 5\ast {10}^{5}\right),256))\right|;i\phantom{\rule{0ex}{0ex}}=1,2,\dots ,Floor\left(M-M/2\right)floor\left(N-N/2\right).$$ **Step****8.**- Reshape these obtained secret keys in step 7 to form the matrices S
_{x}, S_{y}, S_{z}, and S_{u}, where their sizes as round(M/2) × round(N/2), round(M/2) × floor(N-N/2), floor(M-M/2) × round(N/2), and floor(M-M/2) × floor(N-N/2), respectively. **Step****9.**- Encrypt the pixels in the four blocks of each band (R, G, and B) of the plain image using the obtained secret key in step 8 by the flowing operations:$${E}_{R1}=K{R}_{1}\oplus {S}_{x};{E}_{R2}=K{R}_{2}\oplus {S}_{y};{E}_{R3}=K{R}_{3}\oplus {S}_{z};{E}_{R4}=K{R}_{4}\oplus {S}_{u}\phantom{\rule{0ex}{0ex}}{E}_{G1}=K{G}_{1}\oplus {S}_{x};{E}_{G2}=K{G}_{2}\oplus {S}_{y};{E}_{G3}=K{G}_{3}\oplus {S}_{z};{E}_{G4}=K{G}_{4}\oplus {S}_{u}\phantom{\rule{0ex}{0ex}}{E}_{B1}=K{B}_{1}\oplus {S}_{x};{E}_{B2}=K{B}_{2}\oplus {S}_{y};{E}_{B3}=K{B}_{3}\oplus {S}_{z};{E}_{B4}=K{G}_{4}\oplus {S}_{u}$$
_{Ri}_{(i = 1,2,3,4)}, E_{Gi}_{(i = 1,2,3,4)}, and E_{Bi}_{(i = 1,2,3,4)}are the encrypted blocks of the R, G, and B bands, respectively. **Step****10.**- Rearrange (reshape) these encrypted blocks obtained in step 9 to form the encrypted matrix bands (encrypted R, encrypted G, and encrypted B) of the original image.
**Step****11.**- Recompose the encrypted bands obtained in step 10 to give the encrypted image corresponding to the coloured plain image.

## 7. Experimental Results

_{0}, y

_{0}, z

_{0}, u

_{0}) = (0.001, 0, 0, 0) and a fractional-order derivative value (q = 0.99). The system (17) was numerically solved with iterations, which ensures that the number of solution sets of the total state variables (x, y, z, and u) can cover 262,144 samples. These samples match the whole number of pixels in the original image (MN = 512 × 512).

_{x}, S

_{y}, S

_{z}, and S

_{u}) highlighted in step 7 (Section 6). Figure 15 displays the visual investigation of the employed hyperchaotic-based cryptosystem on a “Lena.png” colour plain image of size 512 × 512. Figure 15a displays the plain image. R, G, and B bands of the original image are illustrated in Figure 15b–d, respectively. On the other hand, the encrypted (ciphered) images of the respective original R, G, and B are exposed in Figure 15e,f. The recovered (decrypted) images corresponding to the encrypted images are displayed in Figure 15i–l in their respective arrangements. As can be seen in Figure 15, it demonstrates the exact identicalness between the plain and recovered image, which shows the high accuracy of the proposed cryptosystem in recovering the original images. Furthermore, the encrypted images are wholly different from their respective plain images, and these images do not display any pattern similar to the plain images. As a result, the attacker will be unable to extract any information or patterns from the encrypted images. That shows the robust resistance of the cryptosystem against attacks.

## 8. Cryptanalysis

#### 8.1. Histogram Check

^{8}) different possible brightnesses of each of these three bands, which are from 0 to 255. As a result, the histogram will display 256 numbers representing the distribution of the image pixels signifying their intensity levels [41]. The histogram of an encrypted image must be statistically and visually dissimilar from the histogram of the plain image. In order to resist statistical pirate attacks, the histogram of the encrypted image and its R, G, and B bands must have a reasonably consistent (flat) shape. The flatness of the histogram specifies the randomness of the encrypted image pixel values. Figure 16a displays the histograms of the plain colour Lena image. Figure 16b–d displays the histogram of the R, G, and B bands of the plain Lena image. On the other hand, the histograms of the encrypted images of the respective original Lena, R, G, and B are shown in Figure 16e,f. The recovered images consistent with the encrypted images are shown in Figure 16i–l in their respective arrangements.

#### 8.2. Keyspace Analysis

_{(}

_{0)}, y

_{(0)}, z

_{(0)}, u

_{(0)}), parameters (a, b, d, g, α, and β), and fractional-order derivative value (q).

^{−15}step alteration, then the whole keyspace is computed to be (10

^{16})

^{14}= 10

^{224}≈ 2

^{744}. These findings indicate that the keyspace of the utilized encryption approach is large enough to resist all forms of brute force attacks.

#### 8.3. Key Sensitivity Analysis

_{0}, y

_{0}, z

_{0}, u

_{0}), its parameters (a, b, d, g, α, and β), and the fractional-order derivative value (q) control the sensitivity of the secret keys in the employed cryptosystem approach.

_{x,y,z,u}are created based on the solution set of the system (17) with the chosen parameters a = 2.2222, b = 0.1667, d = 0.45, g = 2, α = −0.75, and β = 1.724 with initial conditions (x

_{0}, y

_{0}, z

_{0,}u

_{0}) = (0.001, 0, 0, 0) and fractional-order derivative value (q = 0.99), where a 512 × 512 color “Lena.png” plain image is encrypted by these keys. Consequently, in the decryption procedure, just the fractional-order derivative value (q) was very slightly varied as q = 0.99 + 10

^{−15}in NPCR and UACI tests for determining key sensitivity. Table 4 illustrates the results of key sensitivity comparative assessments of NPCR and UACI. Furthermore, Figure 17 depicts the experimental results of a recovering image with the aforementioned very little variation at the decryption key.

#### 8.4. Correlation Coefficients Analysis

#### 8.5. Entropy Evolution

#### 8.6. Time Efficiency

#### 8.7. Comparison with Related Works

## 9. Conclusions

^{744}, NPCR = 0.99814, UACI = 0.336251, H(s) = 7.9996, and time efficiency = 0.45 s. The acquired experimental findings and detailed security assessments support the utilized cryptosystem’s effectiveness, high-level security, and good time efficiency, and show high robust resistance to various types of attacks.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**${q}_{M}-{v}_{M}$ characteristic curve of memcapacitor (9): (

**a**) f = 1 Hz and various amplitude values; (

**b**) A

_{m}= 10 C and various frequency values.

**Figure 2.**The ${q}_{M}-{v}_{M}$ hysteresis loop characteristics of the fractional-order memcapacitor (11).

**Figure 5.**The ${q}_{M}-{v}_{M}$ characteristic curve of fractional-order memcapacitor realized circuit.

**Figure 8.**Chaotic attractors of the memcapacitive chaotic system: (

**a**) x-y; (

**b**) y-z; (

**c**) x-z; (

**d**) x-u; (

**e**) y-u; (

**f**) 3-D layout (x-y-z).

**Figure 10.**Bifurcation diagram, influence of the fractional-order derivative value (q) on the system state variable x(t).

**Figure 12.**The system (17) Lyapunov exponents in contradiction to varying the system fractional-order derivative value (q).

**Figure 13.**Block illustration of the image encryption algorithm using the fractional-order memcapacitive hyper chaotic model (17).

**Figure 15.**Experimental results of a plain “Lena.png” 512 × 512 color image: (

**a**) the original image, (

**b**) R band, (

**c**) G band, (

**d**) B band, (

**e**) encrypted Lena image, (

**f**) encrypted R band, (

**g**) encrypted G band, (

**h**) encrypted B band, (

**i**) recovered (decrypted) Lena image, (

**j**) recovered R band, (

**k**) recovered G band, (

**l**) recovered B band.

**Figure 16.**The histogram of a Lena image: (

**a**) plain image, (

**b**) R band, (

**c**) G band, (

**d**) B band, (

**e**) encrypted image, (

**f**) encrypted R band, (

**g**) encrypted G band, (

**h**) encrypted B band, (

**i**) recovered image, (

**j**) recovered R band, (

**k**) recovered G band, (

**l**) recovered B band.

**Figure 17.**Sensitivity of key test: (

**a**) original image, (

**b**) encrypted image, (

**c**) recovered image with variation (10

^{−15}is added to q) of the decryption keys, (

**d**) difference between (

**a**,

**c**).

**Figure 18.**Correlation analysis of a plain Lena image and its consistent encrypted image: (

**a**,

**d**) horizontal correlation, (

**b**,

**e**) vertical correlation, (

**c**,

**f**) diagonal correlation.

**Figure 19.**Correlation analysis of the R band of a plain Lena image and its consistent encrypted image: (

**a**,

**d**) horizontal correlation, (

**b**,

**e**) vertical correlation, (

**c**,

**f**) diagonal correlation.

**Figure 20.**Correlation analysis of the G band of a plain Lena image and its consistent encrypted image: (

**a**,

**d**) horizontal correlation, (

**b**,

**e**) vertical correlation, (

**c**,

**f**) diagonal correlation.

**Figure 21.**Correlation analysis of the B band of a plain Lena image and its consistent encrypted image: (

**a**,

**d**) horizontal c arrangement, (

**b**,

**e**) vertical arrangement, (

**c**,

**f**) diagonal arrangement.

**Figure 22.**Test of “macaws.jpg”, 300 × 309: (

**a**) original image, (

**b**) encrypted image, (

**c**) recovered image, (

**d**–

**f**) histograms consistent with (

**a**–

**c**), respectively.

**Figure 23.**Test of “fruits.jpg”, 236 × 235: (

**a**) plain image, (

**b**) encrypted image, (

**c**) recovered image, (

**d**–

**f**) histograms consistent with (

**a**–

**c**), respectively.

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

Ref. [14] | 2^{256} | 0.99602 | 0.3348 | 0.0019 | 0.0069 | 0.0087 | 7.9976 | N/A |

Ref. [15] | N/A | 0.99602 | 0.3348 | 0.0019 | 0.0069 | 0.0087 | 7.9976 | N/A |

Ref. [16] | 2^{256} | 0.99661 | 0.33617 | 0.0046 | 0.0024 | 0.0051 | 7.9973 | 28.49 s |

Ref. [17] | 2^{600} | 0.99690 | 0.33437 | 0.0004 | 0.0019 | 0.0012 | N/A | N/A |

Ref. [18] | 2^{262} | 0.99620 | 0.33560 | 0.0023 | 0.0012 | 0.0001 | 7.9994 | N/A |

Ref. [19] | N/A | 0.99643 | 0.33502 | 0.000617 | 0.000535 | 0.000411 | 7.9914 | 0.8379 s |

Ref. [20] | 2^{279} | 0.99613 | 0.334706 | 0.000312 | 0.002088 | 0.001444 | 7.9976 | 1.708 s |

Ours | 2^{744} | 0.99814 | 0.336251 | 0.000262 | 0.000472 | 0.00013 | 7.9996 | 0.45 s |

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

Parameter | Value | Parameter | Value |

a | 2.2222 | a | 2.2222 |

b | 0.1667 | b | 0.1667 |

d | 0.45 | d | 0.45 |

g | 2 | g | 2 |

α | Variable | α | 0.75 |

β | 1.72 | β | 1.72 |

Fractional-order (q) | 0.99 | Fractional-order (q) | Variable |

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

Parameter | Value | Parameter | Value |

a | 2.2222 | a | 2.2222 |

b | 0.1667 | b | 0.1667 |

d | 0.45 | d | 0.45 |

g | 2 | g | 2 |

α | 0.75 | α | 0.75 |

β | 1.72 | β | 1.72 |

Fractional-order (q) | 0.99 | Fractional-order (q) | Variable |

Direction | Original Images | |||
---|---|---|---|---|

Lena | R Band | G Band | B Band | |

NPCR | 0.99814 | 0.99783 | 0.9982 | 0.99813 |

UACI | 0.33625 | 0.336192 | 0.33626 | 0.33620 |

Direction | Plain Images | Encrypted Images | ||||||
---|---|---|---|---|---|---|---|---|

Lena | R Band | G Band | B Band | Lena | R Band | G Band | B Band | |

Vertical | 0.9821 | 0.9712 | 0.9677 | 0.9675 | 0.000472 | 0.000466 | 0.000413 | 0.000398 |

Horizontal | 0.9743 | 0.9638 | 0.9847 | 0.9789 | 0.000262 | 0.000269 | 0.000245 | 0.000221 |

Diagonal | 0.9672 | 0.9855 | 0.9789 | 0.9813 | 0.00013 | 0.000157 | 0.000173 | 0.000141 |

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

Lena | 7.2351 | 7.9996 |

R Band | 7.1334 | 7.9994 |

G Band | 6.9541 | 7.9995 |

B Band | 7.1263 | 7.9993 |

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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.
Efficient Colour Image Encryption Algorithm Using a New Fractional-Order Memcapacitive Hyperchaotic System. *Electronics* **2022**, *11*, 1505.
https://doi.org/10.3390/electronics11091505

**AMA Style**

Rahman Z-ASA, Jasim BH, Al-Yasir YIA, Abd-Alhameed RA.
Efficient Colour Image Encryption Algorithm Using a New Fractional-Order Memcapacitive Hyperchaotic System. *Electronics*. 2022; 11(9):1505.
https://doi.org/10.3390/electronics11091505

**Chicago/Turabian Style**

Rahman, Zain-Aldeen S. A., Basil H. Jasim, Yasir I. A. Al-Yasir, and Raed A. Abd-Alhameed.
2022. "Efficient Colour Image Encryption Algorithm Using a New Fractional-Order Memcapacitive Hyperchaotic System" *Electronics* 11, no. 9: 1505.
https://doi.org/10.3390/electronics11091505