Image compression and encryption based on integer wavelet transform and hybrid hyperchaotic system

Abstract

Compression and encryption of images are emerging as recent topics in the area of research to improve the performance of data security. A joint lossless image compression and encryption algorithm based on Integer Wavelet Transform (IWT) and the Hybrid Hyperchaotic system is proposed to enhance the security of data transmission. Initially, IWT is used to compress the digital images and then the encryption is accomplished using the Hybrid Hyperchaotic system. A Hybrid Hyperchaotic system; Fractional Order Hyperchaotic Cellular Neural Network (FOHCNN) and Fractional Order Four-Dimensional Modified Chua’s Circuit (FOFDMCC) is used to generate the pseudorandom sequences. The pixel substitution and scrambling are realized simultaneously using Global Bit Scrambling (GBS) that improves the cipher unpredictability and efficiency. In this study, Deoxyribonucleic Acid (DNA) sequence is adopted instead of a binary operation, which provides high resistance to the cipher image against crop attack and salt-and-pepper noise. It was observed from the simulation outcome that the proposed Hybrid Hyperchaotic system with IWT demonstrated more effective performance in image compression and encryption compared with the existing models in terms of parameters such as unified averaged changed intensity, a number of changing pixels rate, and correlation coefficient.

Keywords

Deoxyribonucleic acid computing global bit scrambling hyperchaotic system image compression image encryption integer wavelet transform

1. Introduction

Due to the fast-paced development of communication networks and digital technologies, several multimedia data are developed and transmitted over the networks [1]. Protecting information from unauthorized access is a primary objective that leads to the development of several data encryption algorithms. Usually, the encryption algorithms are categorized into two types that are named as: transform domain encryption algorithms and spatial domain encryption algorithms [2, 3]. Due to the intrinsic properties of digital images, such as high information redundancy, high data capacity, and strong inter-pixel correlation, conventional encryption algorithms have proven to be ineffective in image encryption [4, 5]. Further, multimedia-based applications such as commercial, military, medical, and political fields require large bandwidth for data communication [6]. Hence, the combination of compression and encryption is useful for real-time applications, in which the compression saves the bandwidth and the encryption protects the privacy of the data [7, 8]. However, the combination of compression and encryption of the image makes the operation difficult for implementation. Several types of research were undertaken on the joint operation of image compression and encryption to achieve a more secure data transmission such as sine chaotification [9], 2D logistic adjusted sine map [10], Convolutional Neural Network (CNN) [11], cross-coupled chaotic maps [12], 5D chaotic map [13], phase-truncated short-time fractional Fourier transform [14], Knuth-Durstenfeld algorithm [15], etc. In the aforementioned algorithms, complexity and time consumption of the compression and encryption of the images are higher. So, the motivation of this research is to develop a hybrid algorithm for ensuring high security, fast speed, and a promising compression ratio for protecting the content of the image and to diminish the amount of data required for transmission. the major contributions of the research work are listed below;

•
Initially, the IWT is applied to digital images such as cameraman, Lena, circuit, Barbara, peppers, aerial, etc. for data compression. The coefficients of IWT are similar to the input images, hence it is very easy to compress the digital images that significantly diminishes the complexity of the system.
•
In addition, the Deoxyribonucleic Acid (DNA) and Hyperchaotic sequences are utilized to encrypt the images and a Hybrid Hyperchaotic system (FOHCNN and FOFDMCC) is used to generate the pseudorandom sequences. The image pixel intensity values are transformed into serial binary digital streams and then the bit-streams are scramble by Hyperchaotic sequence. The complementation and DNA algebraic sequence are applied between the DNA and Hyperchaotic sequence to achieve a significant encryption performance.
•
In the experimental segment, a few performance measures such as the number of Changing Pixel Rate (NPCR), Unified Averaged Changed Intensity (UACI), and correlation coefficient are used to analyze the performance of the proposed image compression and encryption model.

Abbreviations used in the paper are as follows.

Abbreviation Description

CDCP Cipher Diffusion in Crisscross Pattern

CHC Class based Hyper Chaos

CNN Convolutional Neural Network

CT Computerized Tomography

DNA Deoxyribonucleic Acid

FOFDMCC Fractional Order Four Dimensional Modified Chua’s Circuit

FOHCNN Fractional Order Hyperchaotic Cellular Neural Network

GBS Global Bit Scrambling

HC-DNA Hyper Chaotic and Deoxyribonucleic Acid

IWT Integer Wavelet Transform

MRI Magnetic Resonance Imaging

MSE Mean Square Error

NIST National Institute of Standards and Technology

NPCR Number of Changing Pixel Rate

PSNR Peak Signal to Noise Ratio

PWLCM Piecewise Linear Chaotic Map

SSPIHT Secure Set Partitioning in Hierarchical Trees

UACI Unified Averaged Changed Intensity

The structure of this paper is given as follows; Section 2 reviews some recent research papers on “image compression and encryption”. Section 3 details the proposed model with mathematical expressions and the simulation result of the proposed model are stated in Section 4. Section 5 details the conclusion of the research work.
2. Literature review

Abbreviation	Description
CDCP	Cipher Diffusion in Crisscross Pattern
CHC	Class based Hyper Chaos
CNN	Convolutional Neural Network
CT	Computerized Tomography
DNA	Deoxyribonucleic Acid
FOFDMCC	Fractional Order Four Dimensional Modified Chua’s Circuit
FOHCNN	Fractional Order Hyperchaotic Cellular Neural Network
GBS	Global Bit Scrambling
HC-DNA	Hyper Chaotic and Deoxyribonucleic Acid
IWT	Integer Wavelet Transform
MRI	Magnetic Resonance Imaging
MSE	Mean Square Error
NIST	National Institute of Standards and Technology
NPCR	Number of Changing Pixel Rate
PSNR	Peak Signal to Noise Ratio
PWLCM	Piecewise Linear Chaotic Map
SSPIHT	Secure Set Partitioning in Hierarchical Trees
UACI	Unified Averaged Changed Intensity

Zhang and Tong [16] developed new lossless image encryption and compression method based on Secure Set Partitioning in Hierarchical Trees (SSPIHT) and IWT. The developed SSPIHT encryption algorithm does not affect the compression performance. Furthermore, a non-linear inverse operation, hyperchaotic system, plaintext-based keystream, and secure hash algorithm 256 were utilized in this research to enhance the security. Simulation outcome proves that the developed method has good lossless compression and higher security performance. If the external key size was high, then the developed lossless image encryption and compression methodology has the problem of underflow/overflow. Zhou et al. [17] presented a new image compression and encryption algorithm based on nonlinear fractional Mellin transformation. The original image was measured in two directions using measurement matrices to accomplish both compression and encryption, and the measurement matrices were controlled by Chaos map. Then, the resultant image was re-encrypted by applying nonlinear fractional Mellin transformation. Finally, the newton smoothed $l_{0}$ normalization technique was used to decrypt the image. The simulation outcome proves the reliability and validity of the developed algorithm. However, the developed algorithm resists only common attacks; chosen-plaintext attack, known-plaintext attack and chosen-ciphertext attack, where it failed in the complex brute force attacks.

Chen et al. [18] used a diffusion mask and a permutation vector for image encryption and a 3D cat map was introduced to generate the measurement matrix of compression sensing. Furthermore, security, histogram, image reconstruction, information entropy, and keyspace performances were comprehensively analyzed in this literature study. The developed model achieved satisfactory performance in the encryption and compression of public networks. Hence, the input of diffusion was uncontrollable or unknown, due to the encryption effect in the permutation and the unpredictability of compression sensing outputs. Zhu et al. [19] has developed a hybrid algorithm for image compression and encryption. Initially, a random scrambling matrix and gauss random matrix were generated utilizing logistic mapping and Chebyshev mapping. A compression approach was developed for cipher-text images based on random scrambling matrix and gauss random matrix. The developed compression approach consists of three phases; the First phase: a random scrambling matrix that was used for permutation-based encryption, the second phase: a gauss random matrix that was used for encoding, and the third phase: joint decoding and decryption. The simulation outcome proves that the developed hybrid algorithm attained better image encryption performance in terms of peak signal-to-noise ratio value under conditions such as plaintext attack and noise. The developed hybrid algorithm includes two major problems which were: high computational complexity, and blocking artifacts.

Brindha and Gounden, [20] developed a new image encryption algorithm based on compression and Chaos based on the Chinese remainder theorem and hash table structure to compress and encrypt the images. At first, the Henon map and Arnold cat map were used for generating the scramble blocks in the input image. Furthermore, the scrambled image was permuted using a hash table structure to decrease the complexity of Chaos systems. Hence, the permuted image was divided into blocks and then diffusion and compression were performed using the Lorenz equation and the Chinese remainder theorem, respectively. In this literature, keyspace of the developed algorithm was small, which was considered a major concern. Mou et al. [21] developed a modular operation algorithm, Arnold matrix transformation algorithm, and 3D hyperchaotic map for compressing and encrypting the images to improve the security of data transmission and diminish the amount of data required for storage and transmission. It was observed in the Safety performance analysis that the developed algorithm attained better performance against statistical, brute force, and other such attacks. The Arnold matrix transformation algorithm was applied only to the square area of the image, where it may degrade the performance of encryption.

Liu et al. [22] developed a new Chaos-based image compression and encryption approach for multi-modal images. At first, the sparse images were identified using a key-controlled pseudorandom measurement matrix, which was constructed by a logistic map that decreases the amount of data to be processed. Then, an adaptive weighted fusion rule was utilized for fusing the obtained measurements that were further encrypted into cipher text using fractional Fourier transform and the improved random pixel changing approach. By using a recovery algorithm, the fused images were decrypted, in which the reconstructed image reveals major information about the source images. It was observed from the experimental outcome that the reliability and validity of the developed approach was degraded. Zhan et al. [23] implemented a new simple encryption algorithm based on the Hyperchaotic and Deoxyribonucleic Acid (HC-DNA) sequence. In this literature, the 4D hyperchaotic sequence was used to generate the pseudorandom sequence. The DNA addition operation was used instead of the binary operation to increase the effectiveness of the 4D hyperchaotic sequence and cipher unpredictability. It was observed in the experimental evaluation that the developed algorithm (HC-DNA) was robust against differential attacks and resisted the linear attacks, cropping attacks, and noise effectively. Some of the issues faced by the researchers in these studies were; small keyspace and security vulnerability.

Wang and Liu [24] developed a novel image encryption approach based on DNA encoding rules and Chaos encryption. The Logistic map and Piecewise Linear Chaotic Map (PWLCM) were utilized to generate the parameters for the DNA encoding rule. The developed algorithm consists of three phases; (i) PWLCM was used to develop a key image, whose image pixels was generated by Chaos encryption, (ii) To encode the key and plain image using DNA rules, where the rules were decided by logistic map, and (iii) To finally decode the intermediate image as the plain image. Hence, it was observed from the experimental results that the developed encryption approach withstands typical attacks, but showed limited performance towards security. Zhu et al. [25] presented a parallel encryption algorithm based on Crisscross Pattern and improved hyperchaotic sequence. At first, the hyperchaotic sequences were modified for generating the chaotic key streams. Then, the plaintext and chaotic key streams were correlated with the results in plaintext and key sensitivity. The plain image was categorized into two sub-images, which were encrypted in a parallel manner. It was observed in the security analysis and performance test that the developed encryption algorithm has a better ability to resist key sensitivity tests and differential attacks. Furthermore, Zhu and Sun [26] presented an effective encryption algorithm based on hyper-chaos, which includes two rounds of the encryption operation. It was observed from the simulation outcome that the developed algorithm obtained better cryptographic performance against differential attacks, but the encryption speed need to be improved. The developed hyper-chaos algorithm does not make use of the perceptual relevance and obtained poor performance while compressing and encrypting the digital images with homogeneous background. To highlight the above-stated issues, a new compression and encryption algorithm is proposed in this research paper.

3. Hybrid image compression and encryption model

The network has become a common technology that is used to acquire and transmit information resources, due to the increased development of multimedia data. Recently, information security has become an important factor since it plays a vital role in the applications such as commercial, military, medical, and political fields as these domains require high confidentiality for their data [27, 28]. The image encryption has a unique practical application value compared with conventional information carries, in which the digital image has the following properties; (1) redundancy of big data, (2) capacity of big data, and (3) correlation between neighborhood pixels [29, 30]. The conventional encryption algorithms do not possess these inherent properties, therefore, an effective hyperchaotic system is proposed in this research along with IWT to achieve better performance in image compression and encryption. The workflow of the proposed model majorly includes three phases such as data collection: multimedia images and medical images from MedPix ${}^{\@setsize{\scriptsize}{9.5pt}{\viiipt}{\@viiipt}\textregistered}$ dataset, image compression: IWT and image encryption: Hybrid hyper-chaotic system, which is graphically depicted in Fig. 1.

Figure 1.

Whorkflow of the proposed model.

3.1 Integer wavelet transform

In this research, digital images such as cameraman, Lena, circuit, Barbara, peppers, aerial, etc. are used for experimental analysis. The spatial frequency characteristics and multi-resolution analysis are the advantages of discrete wavelet transform (DWT) that decomposes the input image into higher and lower frequency components. The higher frequency components provide detailed information about the image and the lower frequency components comprises the signal energy of the image. The digital image is decomposed into 4 frequency sub-bands are Low Low (LL), Low High (LH), High Low (HL), and High High (HH) by applying DWT. During image reconstruction, the wavelet transform known as DWT returns floating-point variables, during which the truncation of the floating-point variables leads to errors while extracting the secret information. The IWT approach maps integer to integer for perfect re-construction and decomposition of wavelet transform without getting affected by the truncation errors. In this research study, the input image is initially decomposed by IWT to ensure the secret bits precisely. Among the available 4 frequency sub-bands, LL is considered in IWT which looks similar to the input image. Mathematically, the IWT coefficients are denoted in Eqs (1)–(4).

$\displaystyle LL_{i,j}=\left|{\frac{({P_{2i,2j}+P_{2i+1,2j}})}{2}}\right|$ (1) $\displaystyle HL_{i,j}=P_{2i+1,2j}-P_{2i,2j}$ (2) $\displaystyle LH_{i,j}=P_{2i,2j+1}-P_{2i,2j}$ (3) $\displaystyle HH_{i,j}=P_{2i+1,2j+1}-P_{2i,2j}$ (4)

Mathematically, the inverse IWT coefficients are denoted in Eqs (5)–(8).

$\displaystyle P_{2i,2j}=LL_{i,j}-\left|{\frac{HL_{i,j}}{2}}\right|$ (5) $\displaystyle P_{2i,2j+1}=LL_{i,j}+\left|{{\left({HL_{i,j+1}}\right)}\mathord{% \left/{\vphantom{{\left({HL_{i,j+1}}\right)}2}}\right.\kern-1.2pt}2}\right|$ (6) $\displaystyle P_{2i+1,2j}=P_{2i,2j+1}+LH_{i,j}-L_{i,j}$ (7) $\displaystyle P_{2i+1,2j+1}=P_{2i+1,2j}+HH_{i,j}-LH_{i,j}$ (8)

where an input image is denoted as, $P$ , $1\leqslant i\leqslant\frac{X}{2}$ , and $1\leqslant j\leqslant Y\mathord{\left/{\vphantom{Y2}}\right.\kern-1.2pt}2$ are stated as floor values, and the level of each pixel is indicated as $({i,j})$ .

3.2 Hybrid hyperchaotic system

Hyper-Chaos is an extension of the Chaos algorithm, which includes more positive Lyapunov exponents. The hyperchaotic system demonstrates dynamic behaviors that are less complex compared with a chaotic system. Furthermore, the uncertainty and randomness are significantly improved in the hyperchaotic system compared with a chaotic system. Due to less state variables, the hyperchaotic system is linear and has a smaller key space, therefore it is easily predictable with limited system complex. The combined FOHCNN and FOFDMCC methods are used as a hyperchaotic system, where the FOHCNN method is mathematically represented in Eqs (3.2) and (10).

$\displaystyle D^{a_{01}}x_{1}(t)=-x_{3}(t)-x_{4}(t),$ $\displaystyle D^{a_{02}}x_{2}(t)=h_{1}x_{2}(t)+x_{3}(t),$

(9) $\displaystyle D^{a_{03}}x_{3}(t)=h_{2}({x_{1}(t)-x_{2}(t)}),$ $\displaystyle D^{a_{04}}x_{4}(t)=h_{3}({x_{1}(t)-G({x_{4}(t)})})$

where

$\displaystyle G({x_{4}(t)})=x_{4}(t)-({|{x_{4}(t)-0.4}|-|{x_{4}(t)-0.8}|-|{x_{% 4}(t)+0.4}|+|{x_{4}(t)+0.8}|})$ (10)

where $a_{01}$ , $a_{02}$ , $a_{03}$ , and $a_{04}$ are denoted as fractional orders, $h_{1}$ , $h_{2}$ , and $h_{3}$ are stated as system parameters and $x_{1}(t)$ , $x_{2}(t)$ , $x_{3}(t)$ , and $x_{4}(t)$ are represented as system state variables. The fractal orders are selected as, $a_{01}=a_{02}=a_{03}=a_{04}=0.97$ , initial conditions are selected as $x_{1}(0)=0.2$ , $x_{2}(0)=0.5$ , $x_{3}(0)=0.3$ , and $x_{4}(0)=0.3$ and system parameters are chosen as $h_{1}=2$ , $h_{2}=14$ , and $h_{3}=100$ . Then, the FOFDMCC approach is applied in Eq. (3.2) based on integral order 4D modified Chua’s model, as indicated in Eq. (3.2).

$\displaystyle D^{a_{01}}x_{1}(t)=-x_{3}(t)-x_{4}(t)+h_{1}x(t),$ $\displaystyle D^{a_{02}}x_{2}(t)=h_{1}x_{2}(t)+x_{3}(t),$ (11) $\displaystyle D^{a_{03}}x_{3}(t)=h_{2}({x_{1}(t)-x_{2}(t)}),$ $\displaystyle D^{a_{04}}x_{4}(t)=h_{3}({x_{1}(t)-G({x_{4}(t)})})$

where $h_{1}({x(t)})$ is represented as a piecewise linear function, which is denoted in Eq. (12).

$\displaystyle h_{1}({x(t)})=m_{1}x_{1}(t)+1\mathord{/{\vphantom{12}}.\kern-1.2% pt}2({m_{0}-m_{1}})({|{x_{1}(t)+1}|-|{x_{1}(t)-1}|})$ (12)

where $m_{0}<-1<m_{1}<0$ , $m_{1}=-0.6724$ , and $m_{0}=-1.246$ . By utilizing the master system Eq. (3), the slave system is mathematically defined in Eq. (3.2).

$\displaystyle D^{a_{01}}y_{1}(t)=-y_{3}(t)-y_{4}(t)+h_{1}y(t)+\textit{sat}({u_% {1}(t)}),$ $\displaystyle D^{a_{02}}y_{2}(t)=h_{1}y_{2}(t)+y_{3}(t)+\textit{sat}({u_{2}(t)% }),$ (13) $\displaystyle D^{a_{03}}y_{3}(t)=h_{2}({y_{1}(t)-y_{2}(t)})+\textit{sat}({u_{3% }(t)}),$ $\displaystyle D^{a_{04}}y_{4}(t)=h_{3}({y_{1}(t)-G({y_{4}(t)})})+\textit{sat}(% {u_{4}(t)})$

where $u_{1}(t)$ , $u_{2}(t)$ , $u_{3}(t)$ and $u_{4}(t)$ are represented as synchronization control inputs and $y_{1}(t)$ , $y_{2}(t)$ , $y_{3}(t)$ , and $y_{4}(t)$ are indicated as system state variables.

$\displaystyle D^{a_{01}}w_{1}(t)=-w_{3}(t)-w_{4}(t)+({h_{1}y(t)-h_{1}x(t)})+% \textit{sat}({u_{1}(t)}),$ $\displaystyle D^{a_{02}}w_{2}(t)=h_{1}w_{2}(t)+w_{3}(t)+\textit{sat}({u_{2}(t)% }),$ (14) $\displaystyle D^{a_{03}}w_{3}(t)=h_{2}({w_{1}(t)-w_{2}(t)})+\textit{sat}({u_{3% }(t)}),$ $\displaystyle D^{a_{04}}w_{4}(t)=h_{3}({w_{1}(t)-G({w_{4}(t)})})+\textit{sat}(% {u_{4}(t)})$

Based on master and slave systems, the synchronization error system ${w}$ is mathematically stated in Eq. (3.2). Hence, the hyperchaotic behavior of the hybrid hyperchaotic system is graphically indicated in Fig. 2.
3.2.1 DNA encoding

The DNA sequence comprises 4 nucleic acid bases as “T” (thymine), “A” (adenine), “G” (guanine), and “C” (cytosine). The “G” and “C” are complementary to one another same as “T” and “A” because the binary digits “1” and “0” are complementary to each other. Here, two binary digits represents a DNA base, where twenty-four rules are generated for data representation, in that only 8 rules satisfy the Watson crick complement rule. These 8 coding rules are stated in Table 1.

Table 1
Rules of DNA coding

Rule	1	2	3	4	5	6	7	8
00	A	A	C	C	G	G	T	T
01	C	G	A	T	A	T	C	G
10	G	C	T	A	T	A	G	C
11	T	T	G	G	C	C	A	A

Figure 2.

Hybrid hyperchaotic system behavior, a) $x_{1}$ , $x_{3}$ the plane, b) $x_{1}$ , $x_{2}$ the plane, c) $x_{1}$ , $x_{4}$ plane, and d) $x_{1}$ , $x_{2}$ , $x_{3}$ space.

3.2.2 Algebraic operation of DNA sequence

In DNA computation, the DNA subtraction and addition are carried out based on traditional binary subtraction and addition. The DNA subtraction and addition rules are described in Tables 2 and 3.

Table 2
DNA subtraction sequence rules

$--$	A	G	C	T
A	A	T	C	G
G	G	A	T	C
C	C	G	A	T
T	T	C	G	A

Table 3

DNA addition sequence rules

$++$	A	G	C	T
A	A	G	C	T
G	G	C	T	A
C	C	T	A	G
T	T	A	G	C

3.3 Image encryption technique

3.3.1 Hyperchaotic sequence generation

The hyperchaotic sequence has better statistical characteristics to generate the pseudorandom sequences. The process involved in the generation of a pseudorandom sequence is given as follows:

Step 1:
The hyperchaotic system is iterated $N_{0}$ times to improve the security and to eliminate the adverse effect.
Step 2:
The system is iterated another $m\times n$ times after the iteration of $N_{0}$ times. In each iteration, four state values $\{{x_{1}^{j},x_{2}^{j},x_{3}^{j},x_{4}^{j}}\}$ are stored, where $j$ is denoted as iteration index.
Step 3:
During iteration, every state value is utilized for generating two dissimilar key values such as $({s_{i}^{b}})^{j}\in[{0,255}]$ and $({s_{i}^{a}})^{j}\in[{0,255}],({i=1,2,3,4})$ , which are calculated by Eqs (15) and (16).

$\displaystyle({s_{i}^{b}})^{j}=\textit{mod}({\lfloor{\textit{mod}\{{[{({|{x_{i% }^{j}}|-\lfloor{|{x_{i}^{j}}|}\rfloor})\times 10^{15}}],10^{8}}\}}\rfloor,256}% )i=1,2,3,4$ (15) $\displaystyle({s_{i}^{a}})^{j}=\textit{mod}\left\{{\left\lfloor{\frac{[{({|{x_% {i}^{j}}|-\lfloor{|{x_{i}^{j}}|}\rfloor})\times 10^{15}}]}{10^{8}}}\right% \rfloor,256}\right\},i=1,2,3,4$ (16)

where $\lfloor.\rfloor$ is indicated as flooring operation, and it rounds the nearest integer element towards $-\infty$ and mod (.) is stated as modulo operation. The key values are concatenated with Eq. (17) to obtain a vector $s^{j}$ .

$\displaystyle s^{j}=[{({s_{1}^{a}})^{j},({s_{2}^{a}})^{j},({s_{3}^{a}})^{j},({% s_{4}^{a}})^{j},({s_{1}^{b}})^{j},({s_{2}^{b}})^{j},({s_{3}^{b}})^{j},({s_{4}^% {b}})^{j}}]$ (17)
Step 4:
After the entire iteration, the sequences are concatenated with Eq. (18) to obtain $k$ .

$\displaystyle k=[{s^{1},s^{2},\ldots,s^{m\times n}}],\textit{ where }k_{i},i% \in[{1,8mn}]$ (18)

3.3.2 Global-bit scrambling

The original image $P$ has eight bits with an intensity value in the range of [0, 255]. The intensity value of the image is scrambled globally for reducing the correlation among adjacent pixel values. In GBS, pixel substitution is accomplished by changing the intensity value of every pixel. In this scenario, GBS is implemented in two phases,

•
The pixel value is represented as binary bits for obtaining a 1D binary sequence $b^{0}$ . Hence, the hyperchaotic sequence $k$ is in ascending order to achieve index sequences $k^{x}$ .
•
The $b^{0}$ is scrambled globally based on index sequence $k^{x}$ to achieve a 1D binary sequence, as mathematically represented in Eq. (19).

$\displaystyle b_{i}^{1}=b_{k_{i}^{x}}^{0},i\in[{1,8mn}]$ (19)

The outcome of GBS is a non-linear relation between the cipher and the original image that significantly improves security. The image encryption technique consists of seven steps, which are given below.

Step 1:
As mentioned previously, $m\times n$ represents the input image size. At first, GBS is applied to the image $P$ for obtaining the binary sequence $b^{1}$ .
Step 2:
Based on the DNA coding rule, $b^{1}$ is encoded into a DNA sequence $d^{1}$ . The DNA addition operation is performed on every element of $d^{1}$ for obtaining $d^{2}$ , which is mathematically stated in Eq. (20).

$\displaystyle\left\{{{\begin{array}[]{l}{d_{1}^{2}=d_{0}++d_{1}^{1},}\\ {d_{i}^{2}=d_{i-1}^{2}++d_{i}^{1},i\in[{2,4mn}],}\\ \end{array}}}\right.$ (20)

where $d_{0}$ is denoted as an initial value and $++$ is indicated as DNA addition operation.
Step 3:
A sequence $k^{s}=[{k_{1},\ldots,k_{mn}}]$ is extracted from $k$ , and then the sequence $k^{s}$ is transformed into binary digits $b^{k}$ , where $b^{k}$ is encoded into $d^{k}$ using the 3rd DNA encoding rule. The DNA addition operation is performed between $d^{k}$ and $d^{2}$ for obtaining a sequence $d^{3}$ .
Step 4:
A threshold function $f(z)$ is defined in Eq. (21).

$\displaystyle f(z)=\left\{{{\begin{array}[]{*{20}c}{0,}\hfill&{0\leqslant\frac% {z}{255}\leqslant 0.5,}\\ {1,}\hfill&{0.5<\frac{z}{255}\leqslant 1}\\ \end{array}}}\right.$ (21)

The cut sequence of $k,[{k_{1},k_{2},\ldots,k_{4mn}}]$ is converted into a mask sequence $w$ using Eq. (21). The $d^{3}$ and a mask sequence $w$ are used to construct $d^{4}$ . If $w_{i}=1$ , $d_{i}^{3}$ is complemented to achieve $d_{i}^{4}$ , or else there will not be any change.
Step 5:
The 1 ${}^{\text{st}}$ DNA coding rule is utilized for decoding $d^{4}$ to attain a binary sequence $b^{2}$ .
Step 6:
To obtain the cipher binary sequence $b^{3}$ , bitwise XOR is carried out between $b^{2}$ and $b^{k}$ .
Step 7:
The binary sequence $b^{3}$ is transformed into a cipher image $Q$ . The decryption procedure is the same as an encryption procedure that is carried out in reverse order. The graphical depiction of the workflow of the proposed model is denoted in Fig. 3, in which each image represents each stage of the workflow.

Figure 3.
a) original image, b) compressed image using IWT, c) encrypted image using hyperchaotic system, d) decrypted image, e) decompressed image using inverse IWT.

4. Experimental study

In this research, the environment of the MATLAB 2019 tool was applied for experimental simulation with a windows 10 operating system, 16 GB RAM, 3 TB hard disk, and Intel core i9 processor. The performance of the proposed model was compared with a few existing benchmark models such as HC-DNA [23], Chaotic sequence-based DNA (C-DNA) [24], Cipher Diffusion in Crisscross Pattern (CDCP) [25], and Class-based Hyper-Chaos (CHC) [26] to justify the effectiveness of the proposed model. Furthermore, the performance of the proposed Hybrid hyperchaotic system with IWT was evaluated in terms of NPCR, UACI, and correlation coefficient, under the conditions that involved salt and pepper noise and crop attack. Further, the mathematical equations of correlation coefficient $\gamma$ , NPCR and UACI are represented in Eqs (22)–(24), respectively.

In image compression and encryption, the correlation coefficient is a crucial performance measure, which is utilized to measure the correlation among adjacent image pixels. The better correlation coefficient is closer to zero in a cipher image, where the neighboring pixel values are selected for calculating the correlation coefficient [31, 32].

Mathematically, the correlation coefficient is stated in Eq. (22).

$\displaystyle\gamma=\frac{\sum_{i=1}^{N}{({P_{i}^{a}-\overline{P}^{a}})({P_{i}% ^{b}-\overline{P}^{b}})}}{\sqrt{\left[{\sum_{i=1}^{N}{({P_{i}^{a}-\overline{P}% ^{a}})^{2}}}\right]\left[{\sum_{i=1}^{N}{({P_{i}^{b}-\overline{P}^{b}})^{2}}}% \right]}}$ (22)

where

$\displaystyle\overline{P}^{a}=1\mathord{/{\vphantom{1{N\sum_{i=1}^{N}{P_{i}^{a% }}}}}.\kern-1.2pt}{N\sum_{i=1}^{N}{P_{i}^{a}}}$ $\displaystyle\overline{P}^{b}=1\mathord{/{\vphantom{1{N\sum_{i=1}^{N}{P_{i}^{b% }}}}}.\kern-1.2pt}{N\sum_{i=1}^{N}{P_{i}^{b}}}$

where the intensity value of $i^{\text{th}}$ adjacent pixel pairs $a$ and $b$ is represented as $P_{i}^{a}$ and $P_{i}^{b}$ , the average intensity value of the adjacent pixel pair is denoted as $\overline{P}^{a}$ , and the average intensity value of the latter adjacent pixel pair is indicated as $\overline{P}^{b}$ . The correlation coefficient is evaluated in three dissimilar directions; diagonal $\gamma_{d}$ , horizontal $\gamma_{h}$ , and vertical $\gamma_{v}$ . Furthermore, the adjacent pixel pairs of the input image $P$ are evaluated with $m\times n$ size.

In Table 4, the performance of the proposed and existing models is calculated in terms of the correlation coefficient. In Table 4 it was observed that the correlation coefficient of the cipher images is near to zero and the correlation coefficient of the input images is near to one. It states that the adjacent image pixel values of the cipher images are uncorrelated. By investigating Table 4, the proposed hybrid hyperchaotic system with IWT demonstrated significant performance compared with the existing models such as CNN, HC-DNA [23], C-DNA [24], CDCP [25], and CHC [26] in terms of the correlation coefficient in all three directions; diagonal $\gamma_{d}$ , horizontal $\gamma_{h}$ , and vertical $\gamma_{v}$ .

Table 4

Correlation coefficient $\gamma$ of digital images

Image	$\gamma$	Input images	Cipher image
			Proposed	CNN	HC-DNA [23]	C-DNA [24]	CDCP [25]	CHC [26]
Lena	$\gamma_{h}$	0.9494	0.0033	$-$ 0.0055	0.0019	$-$ 0.0047	$-$ 0.0021	$-$ 0.0043
	$\gamma_{v}$	0.9667	$-$ 0.0065	$-$ 0.0131	$-$ 0.0030	0.0040	$-$ 0.0042	$-$ 0.0034
	$\gamma_{d}$	0.9366	$-$ 0.0010	0.0032	0.0018	$-$ 0.0034	$-$ 0.0022	0.0041
Cameraman	$\gamma_{h}$	0.9329	0.0039	0.0014	0.0076	$-$ 0.0022	$-$ 0.0022	0.00031
	$\gamma_{v}$	0.9566	0.0001	$-$ 0.0037	$-$ 0.0091	$-$ 0.0012	$-$ 0.0054	0.0024
	$\gamma_{d}$	0.9117	0.0038	$-$ 0.0024	$-$ 0.0012	$-$ 0.0016	0.0048	0.0042
Circuit	$\gamma_{h}$	0.9766	$-$ 0.0053	0.0015	0.0019	$-$ 0.0046	0.0026	0.0039
	$\gamma_{v}$	0.9775	0.0051	0.0002	0.0041	$-$ 0.0031	0.003	0.0022
	$\gamma_{d}$	0.9678	$-$ 0.0064	0.0005	$-$ 0.0012	$-$ 0.0020	0.0021	$-$ 0.0011
Peppers	$\gamma_{h}$	0.9733	0.0044	$-$ 0.0028	0.0009	$-$ 0.0025	$-$ 0.0015	$-$ 0.0019
	$\gamma_{v}$	0.9763	0.0019	0.0063	0.0041	$-$ 0.0025	$-$ 0.0012	0.0018
	$\gamma_{d}$	0.9650	$-$ 0.0008	0.0002	0.00079	0.0013	0.0017	0.00014
Barbara	$\gamma_{h}$	0.8271	0.0015	0.0014	0.0011	0.0016	$-$ 0.0038	$-$ 0.0023
	$\gamma_{v}$	0.9501	0.0025	$-$ 0.0022	0.00063	$-$ 0.00018	$-$ 0.0028	$-$ 0.00079
	$\gamma_{d}$	0.8310	$-$ 0.0014	$-$ 0.0003	$-$ 0.0038	$-$ 0.0011	$-$ 0.0004	$-$ 0.0014
Aerial	$\gamma_{h}$	0.9083	$-$ 0.0112	$-$ 0.0006	$-$ 0.00099	$-$ 0.00071	$-$ 0.0006	$-$ 0.0020
	$\gamma_{v}$	0.8891	$-$ 0.0058	0.0050	0.0034	0.0037	$-$ 0.0007	$-$ 0.0041
	$\gamma_{d}$	0.8502	$-$ 0.0025	$-$ 0.0012	0.0022	0.0024	0.0026	0.00052

Usually, an effective encryption algorithm distributes the input image information to the whole cipher image. The cipher image is changed completely even if a minor change occurs, therefore it can resist the attacks significantly. The NPCR is used to determine the variation between two cipher images of similar input image before and after a minor change. Similarly, UACI is used to determine the intensity variation between the two cipher images [33, 34]. Mathematically, UACI and NPCR are represented in Eqs (23) and (24).

$\displaystyle\textit{NPCR}=1\mathord{/{\vphantom{1{mn\sum_{i=1}^{m}{\sum_{j=1}% ^{n}{\delta_{ij}\times 100}}}}}.\kern-1.2pt}{mn\sum_{i=1}^{m}{\sum_{j=1}^{n}{% \delta_{ij}\times 100}}}$ (23)

where $\delta_{ij}=\left\{{{\begin{array}[]{*{20}c}{0,}&{q_{ij}^{1}=q_{ij}^{2},}\\ {1,}&{q_{ij}^{1}\neq q_{ij}^{2},}\\ \end{array}}}\right.$

$\displaystyle\textit{UACI}=1\mathord{/{\vphantom{1{mn\sum_{i=1}^{m}{\sum_{j=1}% ^{n}{\frac{|{q_{ij}^{1}=q_{ij}^{2}}|}{255}\times 100}}}}}.\kern-1.2pt}{mn\sum_% {i=1}^{m}{\sum_{j=1}^{n}{\frac{|{q_{ij}^{1}=q_{ij}^{2}}|}{255}\times 100}}}$ (24)

where $m\times n$ is denoted as the size of the input image $P$ , $Q^{1}$ and $Q^{2}$ are denoted as cipher image, $q_{ij}^{1}$ and $q_{ij}^{2}$ are represented as the intensity values located in the pixel $({i,j})$ in $Q^{1}$ and $Q^{2}$ . In this research study, 1 bit of the original input image $P$ is randomly altered ten times and the average NPCR and UACI values are evaluated, for peppers, Lena, Barbara, circuit, cameraman and aerial, respectively. The results are summarized in Tables 5 and 6. The proposed hybrid hyperchaotic system with IWT demonstrated significant performance compared with the existing models such as CNN, HC-DNA [23], C-DNA [24], CDCP [25], and CHC [26] in terms of NPCR and UACI.

Table 5

Performance investigation of proposed and existing models in terms of NPCR

NPCR (%)
Image	Proposed	CNN	HC-DNA [23]	C-DNA [24]	CDCP [25]	CHC [26]
Lena	79.65	58.92	59.7406	15.2588e-4	100	99.6605
Cameraman	97.24	37.30	49.8087	15.2588e-4	100	99.4688
Circuit	75.37	6.57	58.5667	13.1303e-4	99.6874	99.5304
Peppers	75.41	6.64	53.8002	3.8147e-4	99.662	99.5671
Barbara	75.41	6.65	51.0966	4.3950e-4	100	99.6338
Aerial	75.39	6.60	43.9338	7.5062e-4	100	99.6889

Table 6

Performance investigation of proposed and existing models in terms of UACI

UACI (%)
Image	Proposed	CNN	HC-DNA [23]	C-DNA [24]	CDCP [25]	CHC [26]
Lena	39.17	26.50	25.0487	8.9758e-6	33.5752	33.4263
Cameraman	29.89	9.35	18.5261	9.5741e-6	33.4827	33.5204
Circuit	37.73	3.24	22.4548	14.4175e-6	33.3634	33.3858
Peppers	37.82	3.31	20.6976	7.6294e-6	33.5030	33.4918
Barbara	37.84	3.36	20.5315	8.6176e-6	33.4742	33.4521
Aerial	37.69	1.64	16.7418	7.0646e-6	33.4621	33.4982

Table 7

Performance investigation of proposed and existing models under salt and pepper noise condition

Salt and pepper noise
Performance measure	Density	Proposed	CNN	HC-DNA [23]	C-DNA [24]	CDCP [25]	CHC [26]
$\gamma_{h}$	0.05	$-$ 0.0003	$-$ 0.0063	0.6726	0.7584	0.5571	0.192
	0.10	0.0076	$-$ 0.0143	0.5135	0.6173	0.3323	0.1156
	0.15	$-$ 0.0043	$-$ 0.0049	0.4032	0.4933	0.226	0.07044
	0.20	0.0011	0.0045	0.3021	0.4164	0.1468	0.03239
	0.25	$-$ 0.0003	$-$ 0.0063	0.2335	0.3333	0.0921	0.02991
NPCR (%)	0.05	0.8065	0.8065	0.2286	0.04987	0.1356	0.458
	0.10	0.8144	0.8144	0.3953	0.09857	0.2565	0.5355
	0.15	0.8274	0.8274	0.5263	0.1511	0.365	0.6037
	0.20	0.8376	0.8376	0.6431	0.198	0.4602	0.6707
	0.25	0.8483	0.8483	0.7247	0.2494	0.5483	0.7249
UACI (%)	0.05	0.3976	0.2780	0.03073	0.01503	0.03864	0.1133
	0.10	0.4010	0.2895	0.05589	0.02942	0.07463	0.1386
	0.15	0.4084	0.3013	0.0795	0.04545	0.1052	0.1612
	0.20	0.4127	0.3094	0.1041	0.05887	0.1337	0.1839
	0.25	0.4213	0.3222	0.1237	0.07468	0.1588	0.2007

During data transmission, the cipher images are degraded by noise, hence a correct and accurate key is required to decrypt the cipher images to the original input images $P$ . The image encryption algorithm needs to resist the noise and crop attack to a greater extent because the noisy image pixels have a negative impact on the decrypted image quality. In this study, salt and pepper noise is added to the cipher images and the decryption results are indicated in Table 7 and Fig. 4.

Figure 4.

Decrypted image under salt and pepper noise condition, a) 0.05 of noise density, b) 0.10 of noise density, c) 0.15 of noise density, d) 0.20 of noise density, and e) 0.25 of noise density.

In Table 7, the NPCR, UACI and correlation coefficient of a horizontal direction $\gamma_{h}$ are calculated for both input and decrypted Lena image with different noise densities (0.05, 0.10, 0.15, 0.20, and 0.25). Compared with the existing models, the global features are effectively distinguished by the hybrid hyperchaotic system with IWT, but still, some of the decrypted images is partially blurred. The first column in Fig. 4 denotes encrypted image, second and third columns states noisy encrypted images and decrypted images, respectively.

Table 8

Performance investigation of proposed and existing models under crop attack condition

Crop attack
Performance measure	Location	Proposed	CNN	HC-DNA [23]	C-DNA [24]	CDCP [25]	CHC [26]
$\gamma_{h}$	Left	0.3142	0.3059	0.6726	0.7584	0.5571	0.192
	Middle	0.0596	0.0525	0.5135	0.6173	0.3323	0.1156
	Right	0.3321	0.3272	0.4032	0.4933	0.226	0.07044
NPCR (%)	Left	0.8311	0.7324	0.5686	0.249	0.4508	0.5284
	Middle	0.7964	0.5892	0.5512	0.249	0.2529	0.5301
	Right	0.8287	0.6563	0.6539	0.2488	0.449	0.5235
UACI (%)	Left	0.4168	0.3420	0.1008	0.07396	0.1292	0.1381
	Middle	0.3925	0.2648	0.09768	0.07302	0.07378	0.1379
	Right	0.4093	0.3029	0.1045	0.07304	0.1249	0.1347

Figure 5.

Decrypted image under cropping attack condition, a) position left, b) position right, c) position middle.

The cipher or encrypted image is cut off in a random position (left, right, and middle) and then the decryption is performed using the proposed model, where the decryption result is denoted in Table 8 and Fig. 5. In Table 8, the NPCR, UACI, and correlation coefficient of a horizontal direction $\gamma_{h}$ are estimated for both inputs and decrypted images with random positions left, right, and middle. Even when a part of the encrypted image is cropped, the decrypted image significantly reflects the input image information. Compared with the existing models, the proposed model restored the global features effectively. The first column in the Fig. 5 states encrypted images, second and third columns represents the cropped encrypted image and decrypted image, respectively.

4.1 Discussion

As seen in the Tables 4–8, the proposed hybrid algorithm obtained better performance in image compression and encryption compared to the existing algorithms by means of NPCR, UACI, and correlation coefficient under the conditions of salt and pepper noise, and crop attack. The proposed hybrid algorithm exhibits rich dynamical behaviors that includes hyperchaotic, chaotic, and periodic motions that over-comes the concerns mentioned in the literatures like underflow/overflow [16], blocking artifacts [19], small keyspace [20], and security vulnerability [23]. In addition to this, the hyper-chaotic phenomena exist within an extensive range of parametric value that significantly reduces the computational complexity of the system, which is the main issue mentioned in the literatures [19, 26].

5. Conclusion

In this study, new hybrid image compression and encryption algorithm are proposed to enhance the security of data transmission. At first, IWT is applied to the digital images for image compression and the image encryption is accomplished using the Hybrid hyperchaotic system. In this research article, a hybrid hyperchaotic system with IWT significantly improves the sensitivity of the input image and it is capable of decrypting the cipher image under the conditions that involve cropping attack and salt and pepper noise. From the experimental outcomes, the hybrid hyperchaotic system with IWT demonstrated better imperceptibility compared with the existing models such as CNN, HC-DNA, C-DNA, CDCP, and CHC. Furthermore, the proposed Hybrid hyperchaotic system with IWT achieved better performance in image compression and encryption compared with the existing models in terms of UACI, NPCR, and correlation coefficient. In future work, a new optimization algorithm is included in the proposed model to further enhance the image compression and encryption performance.

Footnotes

Author’s Bios

	R. Srinivas has completed his professional career of education in B. Tech (ECE) from JNTU, Hyderabad. He obtained M. Tech degree from Acharya Nagarjuna University. He is pursing Ph. D from University Of Technology, Jaipur as a part-time scholar and also working as an Associate Professor and Head of the ECE Department Rajamahendri Institute of Engineering and Technology, Rajamahendravaram, (AP).
	N. Mayur has completed his Doctorate Degree in area of Image processing from Jawaharlal Nehru Technological University, Anantapur. He is now working as Associate Professor in the Department of Electronics and Communication Engineering at University Of Technology, Jaipur.

References

Hanis

and Amutha

, Double image compression and encryption scheme using logistic mapped convolution and cellular automata, Multimedia Tools and Applications 77(6) (2018), 6897–6912.

Hua

and Zhou

, Image encryption using 2D Logistic-adjusted-Sine map, Information Sciences 339 (2016), 237–253.

Duseja

and Deshmukh

, Image compression and encryption using Chinese remainder theorem, Multimedia Tools and Applications 78(12) (2019), 16727–16753.

Tong

X.J.

Zhang

Wang

and Ma

, A joint color image encryption and compression scheme based on hyper-chaotic system, Nonlinear Dynamics 84(4) (2016), 2333–2356.

Liao

and Yang

, A novel scheme for simultaneous image compression and encryption based on wavelet packet transform and multi-chaotic systems, Multimedia Tools and Applications 77(21) (2018), 28633–28663.

Wang

Hou

Liu

Zhang

and Duan

, Galois field-based image encryption for remote transmission of tumor ultrasound images, IEEE Access 7 (2019), 49945–49950.

and Lo

K.T.

, A content-adaptive joint image compression and encryption scheme, IEEE Transactions on Multimedia 20(8) (2017), 1960–1972.

Yang

Chen

Coatrieux

and Luo

, A novel chaos-based symmetric image encryption using bit-pair level process, IEEE Access 7 (2019), 99470–99480.

Liu

Xiao

Huang

and Liu

, Quantum block image encryption based on arnold transform and sine chaotification model, IEEE Access 7 (2019), 57188–57199.

10.

Feng

and Li

, Cryptanalysis and improvement of the image encryption scheme based on 2D logistic-adjusted-sine map, IEEE Access 7 (2019), 12584–12597.

11.

Chen

X.W.

and Wang

Q.H.

, Deep learning for improving the robustness of image encryption, IEEE Access 7 (2019), 181083–181091.

12.

Patro

K.A.K.

Soni

Netam

P.K.

and Acharya

, Multiple grayscale image encryption using cross-coupled chaotic maps, Journal of Information Security and Applications 52 (2020), 102470.

13.

Kaur

Singh

Sun

and Rawat

, Color image encryption using non-dominated sorting genetic algorithm with local chaotic search-based 5D chaotic map, Future Generation Computer Systems 107 (2020), 333–350.

14.

S.S.

Zhou

N.R.

Gong

L.H.

and Nie

, Optical image encryption algorithm based on phase-truncated short-time fractional Fourier transform and hyper-chaotic system, Optics and Lasers in Engineering 124 (2020), 105816.

15.

Wang

and Xu

, An image encryption algorithm based on a hidden attractor chaos system and the knuth-durstenfeld algorithm, Optics and Lasers in Engineering 128 (2020), 105995.

16.

Chen

Zhang

and Xu

, Exploiting chaos-based compressed sensing and cryptographic algorithm for image encryption and compression, Optics & Laser Technology 99 (2018), 238–248.

17.

Zhu

and Wang

, A novel image compression-encryption scheme based on chaos and compression sensing, IEEE Access 6 (2018), 67095–67107.

18.

Zhang

and Tong

, Joint image encryption and compression scheme based on IWT and SPIHT, Optics and Lasers in Engineering 90 (2017), 254–274.

19.

Brindha

and Gounden

N.A.

, A chaos-based image encryption and lossless compression algorithm using hash table and Chinese remainder theorem, Applied Soft Computing 40 (2016), 379–390.

20.

Mou

Yang

Chu

and Cao

, Image compression and encryption algorithm based on hyper-chaotic map, Mobile Networks and Applications (2019), 1–13.

21.

Zhou

Wang

Pan

and Zhou

, Image compression and encryption scheme based on 2D compressive sensing and fractional mellin transform, Optics Communications 343 (2015), 10–21.

22.

Liu

Mei

and Du

, Simultaneous image compression, fusion and encryption algorithm based on compressive sensing and chaos, Optics Communications 366 (2016), 22–32.

23.

Zhan

Wei

Shi

and Yu

, Cross-utilizing hyper-chaotic and DNA sequences for image encryption, Journal of Electronic Imaging 26(1) (2017), 013021.

24.

Wang

and Liu

, A novel and effective image encryption algorithm based on chaos and DNA encoding, Multimedia Tools and Applications 76(5) (2017), 6229–6245.

25.

Zhu

C.X.

Y.P.

and Sun

K.H.

, New image encryption algorithm based on hyper-chaotic system and ciphertext diffusion in crisscross pattern, Journal of Electronics and Information Technology 34(7) (2012), 1735–1743.

26.

Zhu

C.X.

and Sun

K.H.

, Cryptanalysis and improvement of a class of hyper-chaos-based image encryption algorithms, Acta Physica Sinica 61(12) (2012), 120503.

27.

Duan

Liu

and Zhang

, Efficient image encryption and compression based on a VAE generative model, Journal of Real-Time Image Processing 16(3) (2019), 765–773.

28.

Sun

Chen

and Hu

, Image compression and encryption scheme using fractal dictionary and Julia set, IET Image Processing 9(3) (2015), 173–183.

29.

Gao

Tang

and Li

, An image compression and encryption scheme based on deep learning, ArXiv abs/1608.05001, 2016.

30.

Chen

Zhang

Yuen

and Tong

, Image encryption and compression based on kronecker compressed sensing and elementary cellular automata scrambling, Optics & Laser Technology 84 (2016), 118–133.

31.

X.Z.

Chen

W.W.

and Wang

Y.Q.

, Quantum image compression-encryption scheme based on quantum discrete cosine transform, International Journal of Theoretical Physics 57(9) (2018), 2904–2919.

32.

Tong

X.J.

Chen

and Zhang

, A joint image lossless compression and encryption method based on chaotic map, Multimedia Tools and Applications 76(12) (2017), 13995–14020.

33.

Zhu

and Zhu

, A new image compression-encryption scheme based on compressive sensing and cyclic shift, Multimedia Tools and Applications 78(15) (2019), 20855–20875.

34.

Tong

X.J.

Wang

Zhang

and Liu

, A new algorithm of the combination of image compression and encryption technology based on cross chaotic map, Nonlinear Dynamics 72(1-2) (2013), 229–241.

Image compression and encryption based on integer wavelet transform and hybrid hyperchaotic system

Abstract

Keywords

1. Introduction

3. Hybrid image compression and encryption model

Table 1 Rules of DNA coding

Table 2 DNA subtraction sequence rules

3.3.1 Hyperchaotic sequence generation

5. Conclusion

Footnotes

Author’s Bios

References

Table 1
Rules of DNA coding

Table 2
DNA subtraction sequence rules