# Fractional-order 4D hyperchaotic memristive system and application in color image encryption

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## Abstract

In this paper, some properties of the fractional-order four-dimensional (4D) hyperchaotic memristive system are analyzed by the phase diagram, Lyapunov exponent spectrum and bifurcation diagram according to the Adomian decomposition method. Based on the chaotic system, a color image encryption scheme is proposed through combining the DNA sequence operation. The algorithm simulation results and security feature analysis show that the encryption scheme has good encryption effect and high safety performance, which provides an experimental basis and theoretical guidance for the safe transmission of image information.

## Keywords

Color image encryption Security analysis DNA sequence operations Fractional-order 4D hyperchaotic memristive system## Abbreviations

- 4D
4-Dimensional

- ADM
Adomian decomposition method

- AES
Advanced encryption standard

- ATCG
Adenine, thymine, cytosine, guanine

- DES
Data encryption standard

- DNA
Deoxyribonucleic acid

- NPCR
Number of pixels change rate

- UACI
Unified average changing intensity

## 1 Introduction

Nowadays, digital image is an improtant carrier of information, because of the inherent performance of digital images, including bulk data capacity, high redundancy and extremely strong correlation in adjacent pixels, which make digital image processing a research hotspot. For example, prediction error preprocessing for image compression [1], histogram equalization of images [2], image compression and reconstruction [3], and so on. To achieve the requirement of digital image safety transmission, researchers are interested in an encryption algorithm based on a chaotic system. Chaos is a random or uncertain movement in a particular system. It has inherent properties of ergodicity, sensitivity of initial value and parameters, and complex dynamic characteristic [4, 5]. Especially, chaotic attractors coexist [6, 7]. Therefore, a chaos system could be used in the image encryption fields.

Up to now, all kinds of image encryption algorithms through chaotic system are proposed [1, 8, 9, 10, 11, 12, 13]. For example, Hua et al. [8] proposed an image encryption scheme using 2D Logistic-adjusted-Sine map. Yang et al. [9] presented novel quantum image encryption through 1D quantum cellular automata. Because low-dimensional chaotic maps have fewer system parameters, the structures are simple. The system parameters and initial value may be predicted by using chaotic signal estimation technologies. On the contrary, high-dimensional chaotic maps, especially hyperchaotic maps, possess excellent chaotic performance and complex structure. Therefore, Natiq et al. [10] designed a new hyperchaotic map and its application for image encryption. Luo Y and his research team [11] proposed a parallel image encryption algorithm through two chaotic maps.

Recently, an encryption scheme using DNA addition in combination with chaotic system was proposed by Zhang et al. [14]. Soon afterwards, some cryptosystems were applied to DNA sequence operations and chaotic systems [2, 3, 4, 5, 7, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. These schemes applied DNA encoding and DNA sequence operation to encrypt images. An idea of DNA subsequence operation, rather than complex biological operation of image encryption scheme, was introduced by Zhang et al. [25]. Liu and his research team [26] employed a chaotic map and the DNA complementary rule in an image encryption algorithm. SaberiKamarposhti et al. [27] proposed hybrid image encryption algorithm through DNA sequences and a logistic map. However, compared with the general chaotic system, the fractional-order system has nonlocal character and high nonlinearity, and the encryption algorithm of fractional-order chaotic has higher security features [20, 28]. Compared with the general chaotic system, dynamic features of the memristor chaotic system depend not only on system parameters but also on the initial conditions of memristor retention internal state variables [29, 30]. However, the memristor chaotic systems are not widely used for image and data encryption algorithms. Therefore, to improve the safety performance of image encryption algorithm, in this paper, a color image encryption using a fractional-order 4D hyperchaotic memristive system and DNA sequence operations is proposed.

The following is the architecture of this paper. Preliminary materials are described in Section 2. The encryption and decryption scheme and the simulation results are presented in Section 3. In Section 4, security performance is analyzed. Finally, the conclusion is given in Section 5.

## 2 Preliminary materials

### 2.1 Adomian decomposition method

*Dq to*(

*t*) =

*f*(

*x*(

*t*)), here

*x*(

*t*) = [

*x*

_{1}(

*t*),

*x*

_{2}(

*t*),…,

*x*

_{n}(

*t*)]

^{T}, *

*Dq to*are variables and *

*Dq to*is the Caputo derivative operator of order

*q*((

*m*− 1) <

*q*≤

*m*,

*m*∈

*N*). The following initial value is obtained by making

*f*(

*x*(

*t*)) been separated into three parts [31, 32]:

*L*and

*N*are linear and nonlinear parts of system functions,

*g*(t) = [

*g*

_{1}(

*t*),

*g*

_{2}(

*t*),…,

*g*

_{n}(

*t*)]

^{T}are constants for autonomous systems, and

*b*

_{k}is a specified constant. On both sides of Eq. (3) perform

*Jq to*operators, the following equation is obtained [33]:

*Jq to*is fractional integral operator of order

*q*based on Riemann-Liouville. For

*t*∈ [

*t*

_{0},

*t*

_{1}],

*q*≥ 0,

*r*≥ 0,

*γ*> − 1 and real constant

*C*, the fundamental properties of

*Jq to*are described by [34]:

*i*= 0,1,…,∞,

*j*= 1,2,…

*n*. Then the nonlinear terms are expressed as

*x*= ∑

*∞ i = 0*is derived from

### 2.2 Fractional-order 4D hyperchaotic memristive system

where *x*, *y*, *z* and *w* are the stateful variables of chaotic system, *q*(0 < *q* ≤ 1) is the order of fractional-order differential equation, where *W*(*w*) is defined as *W*(*w*) = *a* + 3*bw*^{2}, and *a*, *b*, *α*, *β*, *γ*, and *ρ* are the system parameters.

In order to evaluate the chaotic system for image encryption, the dynamic characteristics of the fractional-order 4D hyperchaotic memristive system by the phase diagram, Lyapunov exponent spectrum, and bifurcation diagram are analyzed according to Adomian decomposition method.

*a*= 4,

*b*= 0.01,

*α*= 36,

*β*= 20,

*γ*= 3,

*q*= 0.85 and

*ρ*= 3, the initial value of the Eq. (9) is (1, 0, 1, 0). We get the phase diagram shown Fig. 1a. Then make parameters

*a*= 4,

*b*= 0.01,

*α*= 36,

*β*= 20,

*γ*= 3,

*ρ*= 3 and versus

*q*∈[0.75, 1]. The Lyapunov exponent spectrum and bifurcation diagram of the fractional-order 4D hyperchaotic memristive system are obtained as shown in Fig. 1b and c. Obviously, the phase diagram, Lyapunov exponent spectrum and bifurcation diagram of the fractional-order 4D hyperchaotic memristive system distribute in a large region. This means that the system has good randomness, large key space and pseudorandom sequence generator.

### 2.3 DNA encoding and decoding rules

DNA encoding rules

Rule | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|

00 | A | A | T | T | G | G | C | C |

01 | C | G | C | G | T | A | T | A |

10 | G | C | G | C | A | T | A | T |

11 | T | T | A | A | C | C | G | G |

### 2.4 DNA addition and subtraction rules

Addition rules and subtraction rules

+ | A | C | G | T | – | A | C | G | T |
---|---|---|---|---|---|---|---|---|---|

A | A | C | G | T | A | A | T | G | C |

C | C | G | T | A | C | C | A | T | G |

G | G | T | A | C | G | G | C | A | T |

T | T | A | C | G | T | T | G | C | A |

### 2.5 DNA complementary rule

*x*

_{i}.

where *L* (*x*_{i}) and *x*_{i} are basic pairs and they are complementary, the basic pairs are satisfied with the injective map.

- (1)
*L*_{1}(A) = T,*L*_{1}(T) = C,*L*_{1}(C) = G,*L*_{1}(G) = A; - (2)
*L*_{2}(A) = T,*L*_{2}(T) = G,*L*_{2}(G) = C,*L*_{2}(C) = A; - (3)
*L*_{3}(A) = C,*L*_{3}(C) = T,*L*_{3}(T) = G,*L*_{3}(G) = A; - (4)
*L*_{4}(A) = C,*L*_{4}(C) = G,*L*_{4}(G) = T,*L*_{4}(T) = A; - (5)
*L*_{5}(A) = G,*L*_{5}(G) = T,*L*_{5}(T) = C,*L*_{5}(C) = A; - (6)
*L*_{6}(A) = G,*L*_{6}(G) = C,*L*_{6}(C) = T,*L*_{6}(T) = A,

where *L*_{i} (*i* = 1,2,...,6) represents the *i*th complement rule.

In the diffusion of pixels, used DNA complementary rule bases complementary replacement, and we can randomly select one of the six kinds of complementary combination rules are complementary to replace, which achieve the goal of pixel diffusion.

## 3 Method - image encryption and decryption algorithm

### 3.1 The key design

*x*

_{0},

*y*

_{0},

*z*

_{0},

*w*

_{0}), parameters

*a*,

*b*,

*α*,

*β*,

*γ*,

*q*,

*ρ*, cycle numbers

*m*,

*n*, starting acid base

*c*

_{0}(

*c*

_{0}∈A, T, C, G) and DNA encoding rules in Table 1

*α*1,

*β*1(

*α*1,

*β*1∈ [1, 8]).

### 3.2 Image encryption algorithm

#### 3.2.1 Pixel position scrambling

The pixel location is scrambled in order to destroy correlation of the original image, and an image is rearranged, which makes the image become disturbed. The random sequences through the fractional-order 4D hyperchaotic memristive system are generated, and the image is permutated. The detailed confusion process can be presented as the following steps.

**Step****1.**The input is color original image

*I*with the size of

*M*×

*N*× 3. Setting secret key values

*a*,

*b*,

*α*,

*β*,

*γ*,

*q*,

*ρ*,

*x*

_{0},

*y*

_{0},

*z*

_{0},

*w*

_{0}. New initial conditions of the fractional-order 4D hyperchaotic memristive system are generated by

**Step****2.**Setting the

*L*= max(

*M*,

*N*). Let the chaotic system (9) iterate for (

*m*+

*L*) times based on new initial conditions, and then throw out the former

*m*values to improve initial value sensitivity. The four chaotic sequences {

*x*

_{i}}

*L i = 1*, {

*y*

_{i}}

*L i = 1*, {

*z*

_{i}}

*L i = 1*and {

*w*

_{i}}

*L i = 1*are obtained by Eq. (9). The following shift step numbers are used for scrambling:

*Bri*means that the cyclic step size of row

*i*, and

*Bcj*is the cyclic step number of column

*j*. Here,

*i*= 1,2,...,

*M*,

*j*= 1,2,...,

*N*.

**Step****3*** .* Color image

*I*is decomposed into

*R*,

*G*,

*B*parts, and then

*R*,

*G*,

*B*parts are converted into three matrices and the rows are shifted. The shift results

*TR*1,

*TG*1 and

*TB*1 are obtained by the following rules. Assumption

*x*

_{i}> 0, let the row

*i*of

*R*would be moved to left and step number is

*Bri*; otherwise, the row

*i*of

*R*would be moved to the right with step number

*Bri*, where

*i*= 1,2,...,

*M*. The same rules are used as in the

*G*and

*B*channels.

**Step****4.** The columns shift results *TR*, *TG* and *TB* are obtained as follows. When *y*_{j} > 0, the column *j* of *TR*1 would be moved up with the size of step is *Bcj*, or else the column *j* of *TR*1 would be moved down with the size of step is *Bcj*, where *j* = 1,2,...,*N*. The same rules are used as in the *TG*1 and *TB*1.

#### 3.2.2 DNA sequence operation

The pixel values are diffused according to DNA operations and include addition and complementary operations. Specific steps are as follows.

**Step****1.** The *M* × 8 *N* binary matrices *R*, *G* and *B* are obtained by *TR*, *TG* and *TB*. Then the matrices *R*, *G* and *B* are encoded through the DNA encoding Rule α, and then the *M* × 4 *N* DNA matrix *S*1, *S*2 and *S*3 are obtained.

**Step****2.**Setting the chaotic system initial values of

*x*

_{0},

*y*

_{0},

*z*

_{0},

*w*

_{0}and getting chaotic sequences {

*x*

_{i}}

*MN i = 1*, {

*y*

_{i}}

*MN i = 1*, {

*z*

_{i}}

*MN i = 1*, {

*w*

_{i}}

*L i = 1*by iterating system (1) (

*n*+

*M*×

*N*) times and discarding the former

*n*values. Three sequences

*k*1,

*k*2 and

*k*3 are obtained by

where *i* = 1,2,….,*MH*.

**Step****3.** The sequence *k*1, *k*2 and *k*3 are transformed into binary matrix, and the matrixes are encoded according to the same DNA rule α to get three *M* × 4 *N* matrixes *K*1, *K*2 and *K*3.

**Step****4.**According to the DNA complementary rule, the middle encryption result of DNA formulation matrix

*C*= {

*c*

_{i}}

*4MN i = 1*is obtained as follows:

**Step****5.**The encrypted image of DNA sequence

*D*= {

*d*

_{i}}

*4MN i =*1 is calculated by

Here, *i* = 1,2,….,*MH*. “+” means that the DNA addition operation, and *d*_{0} = *c*_{4MN}. Three DNA matrices *D*1, *D*2 and *D*3 are obtained by Eq. (18).

**Step****6.** The matrices *D*1, *D*2, and *D*3 are decoded by DNA Rule *β* and then recovering three binary formulations *C*1, *C*2 and *C*3. Finally, the encrypted image *C* by combination *C*1, *C*2 and *C*3 is obtained.

### 3.3 Decryption algorithm

*C*is decomposed

*C*1,

*C*2 and

*C*3, and then

*C*1,

*C*2 and

*C*3 are encoded as matrices

*D*1,

*D*2 and

*D*3 through DNA rule

*β*, and then the middle encryption result of DNA formulation matrix

*C*= {

*c*

_{i}}

*4MH i = 1*is recovered as

where *i* = 1,2,….,*MH*. “–” is the DNA subtraction, and *d*_{0} = *c*_{4MH}. The matrices *K*1, *K*2 and *K*3 are generated by doing Step 3 of the DNA sequence operation. Second, the image of DNA sequence matrices *S*1, *S*2 and *S*3 is recovered. The same iteration as Step 3 and Step 4 of pixel position scrambling is performed. Finally, the encrypted image is recovered.

### 3.4 Simulation result

*a*= 4,

*b*= 0.01,

*α*= 36,

*β*= 20,

*γ*= 3,

*q*= 0.855,

*ρ*= 2.67,

*x*

_{0}= 1,

*y*

_{0}= 0,

*z*

_{0}= 1,

*w*

_{0}= 0,

*m*= 1000,

*n*= 5000,

*c*

_{0}=

*A*,

*α*= 1,

*β*= 3. The encrypted Lena image can be obtained as in Fig. 3b and the corresponding decryption image as shown in Fig. 3c.

## 4 Results and discussion

### 4.1 Key space

As a good image encryption algorithm, it should have large enough key space to resist the brute-force attack. In our encryption scheme, the keys are *x*_{0}, *y*_{0}, *z*_{0}, *w*_{0}, *a*, *b*, *α*, *β*, *γ*, *q*, *ρ*; if the calculation precision is 10^{− 15}, the key space will be 2^{548}. For the other part of key *c*_{0}, *α*1, *β*1, *b*_{1}, *b*_{2},...*b*_{8}, because DNA has four acid base, eight kinds of encoding and decoding rules and six DNA complementary rules, and get the key space 2^{2} × 2^{6} × 2^{20} = 2^{28}. So, all the key space would be 2^{576}, which shows that the algorithm key space is large enough and can resist the brute-force attack.

### 4.2 Key sensitivity analysis

*x*

_{0}+ 10

^{− 16}), (

*y*

_{0}+ 10

^{− 16}), (

*z*

_{0}+ 10

^{− 16}), (

*a*+ 10

^{− 15}), (

*b*+ 10

^{− 15}) and (

*c*+ 10

^{− 15}), to decrypt the encrypted Lena image shown in Fig. 3b and the sensitivity test shown in Fig. 4. Obviously, these restored images are completely different from the correct decrypted image in Fig. 3c. Therefore, the proposed algorithm is very sensitive to its key.

### 4.3 Statistical analysis

#### 4.3.1 Histogram analysis

#### 4.3.2 Correlation coefficient analysis

*r*

_{xy}of pixels

*x*and

*y*is calculated as:

*x*and

*y*are the pixel values of different image pixels, cov (

*x*,

*y*) represents the covariance, and

*D*(

*x*) means that the variance of

*x*. Similarly,

*E*(

*x*) is the average, and

*N*represents the number of all pixels. Correlation coefficients of the original Lena image and encrypted image in

*R*,

*G*, and

*B*channels are listed in Table 3. The correlation coefficients of the encrypted image with identical positions in

*R*,

*G*,

*B*components are listed in Table 4. Table 5 lists the correlation coefficients of the encrypted image with adjacent positions in

*R*,

*G*,

*B*components. The tabular data indicate that the original images have significant correlation, whereas encrypted images are very small, which shows the encryption algorithm effect is up to the mustard.

Correlation coefficients in *R*, *G*, *B* channels

Channels | Direction | Original Lena image | Our Algorithm | Ref [12] | Ref [20] |
---|---|---|---|---|---|

| Horizontal | 0.9556 | − 0.0025 | 0.0032 | 0.0031 |

Vertical | 0.9780 | 0.0913 | 0.0058 | − 0.0009 | |

Diagonal | 0.9434 | 0.0011 | 0.0133 | 0.0027 | |

| Horizontal | 0.9443 | 0.0058 | 0.0068 | − 0.0018 |

Vertical | 0.9711 | − 0.0372 | 0.0042 | 0.0079 | |

Diagonal | 0.9301 | − 0.0014 | 0.0130 | − 0.0002 | |

| Horizontal | 0.9280 | − 0.0058 | 0.0014 | 0.0033 |

Vertical | 0.9575 | 0.0036 | 0.0035 | − 0.0049 | |

Diagonal | 0.9030 | 2.1180e-04 | 0.0091 | 0.0015 |

Identical position with *R*, *G*, *B*

### 4.4 Information entropy

*p*(

*m*

_{i}) means that the probability of occurrence for symbol

*m*

_{i}, and

*L*represents all the number of symbols

*m*

_{i}. Because there are 2

^{8}states of the 256 gray-level images, so the theoretical value of information entropy is 8. The information entropy values of encryption image in

*R*,

*G*,

*B*channels, and the combination of

*R*,

*G*,

*B*components

*S*, are listed Table 6. It can be seen clearly from Table 6 that calculation of the values of the new algorithm is close to 8. Therefore, randomness of the encrypted images is good.

Information entropy of encryption image

Images | | | | |
---|---|---|---|---|

Lena | 7.9991 | 7.9973 | 7.9967 | 7.9974 |

Ref [20] | 7.9967 | 7.9974 | 7.9973 | 7.9970 |

### 4.5 Differential attack

where *L* is the number of all image pixels. *C* and *C*_{1} are pixel values before and after the same position change, respectively, and *D*(*i*, *j*) is obtained through the following rules. If *C*(*i*, *j*) ≠ *C*_{1}(*i*, *j*), then *D*(*i*, *j*) = 1, if *C*(*i*, *j*) = *C*_{1}(*i*, *j*), then *D*(*i*, *j*) = 0.

Mean values number of pixels change rates and unified average changing intensities of encryption image

### 4.6 Robustness analysis

#### 4.6.1 Noise attack

#### 4.6.2 Cropping attack

## 5 Conclusion

In this paper, we focus on studying a color image encryption algorithm through a fractional-order 4D hyperchaotic memristive system and DNA sequence operations. The dynamic analysis results show that the fractional-order 4D hyperchaotic memristive system has more complexity in dynamic characteristics and randomness; moreover, it is more suited for image encryption. Algorithm simulation test and security performance analysis indicate that our algorithm not only can effectively to encrypt image but also has excellent safety features. Therefore, image encryption algorithm based on the fractional-order 4D hyperchaotic memristive system can effectively encrypt images and has more efficiency, which provides the related theoretical basis and practical application foundation applied to cryptography, secure communication and information security and other fields.

## Notes

### Acknowledgements

The authors thank the editor and anonymous reviewers for their helpful comments and valuable suggestions.

### Funding

The research presented in this paper was supported by Provincial Natural Science Foundation of Liaoning (Grant No. 20170540060), Basic Scientific Research Projects of Colleges and Universities of Liaoning Province (Grant Nos. 2017 J045 and 2017 J046).

### Availability of data and materials

Please contact author for data requests.

### Authors’ contributions

PL made a theoretical guidance for this paper. JX designed and performed experiments. JM wrote this manuscript. FY analyzed data. All authors carefully read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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