Chaosbased encryption algorithms offer many advantages over conventional cryptographic algorithms, such as speed, high security, affordable overheads for computation, and procedure power. In this paper, we propose a novel perturbation algorithm for data encryption based on double chaotic systems. A new image encryption algorithm based on the proposed chaotic maps is introduced. The proposed chaotification method is a hybrid technique that parallels and combines the chaotic maps. It is based on combination between Discrete Wavelet Transform (DWT) to decompose the original image into subbands and both permutation and diffusion properties are attained using the chaotic states and parameters of the proposed maps, which are then concerned in shuffling of pixel and operations of substitution, respectively. Security, statistical test analyses, and comparison with other techniques indicate that the proposed algorithm has promising effect and it can resist several common attacks. Namely, the average values for UACI and NPCR metrics were 33.6248% and 99.6472%, respectively. Additionally, unscrambling quality can fulfill security and execution prerequisites as evidenced by PSNR (9.005955) and entropy (7.999275) values. In sum, the proposed method has enough ability to achieve low residual intelligibility with high quality recovered data, high sensitivity, and high security performance compared to some other recent literature approaches.
With the fast development of innovations in data communication, it can end up crucial for private information security from prohibitive actions or attackers. Data exchange is closely related to existence, such as instruction, commerce, financial matters, military, elearning, phone keeping money, and news telecasting. With the modern telecommunication and multimedia technologies progression, a huge amount of critical information voyages in a daily monotony through the shared and open networks. In order to keep security, sensitive and critical information ought to be secured before conveyance [
Chaosbased encryption is one of the foremost important security technologies within the advanced encryption zone. Chaos hypothesis is created by mathematicians and physicists. Chaos hypothesis has qualified features as nonlinearity, deterministically, abnormality, and affectability to beginning conditions. Security investigative community receives chaos hypothesis in modern cryptography. A function that has some kind of chaotic behavior is defined as a work or a chaotic map. Within the following we discuss numerous sorts of proposed chaotic maps that are utilized in this paper. To apply a chaos map, there are two ways in a cipher system: (i) produce pseudorandom stream utilizing chaotic maps, and (ii) utilize the plain or secret key(s) as control parameters and the introductory conditions [
The assessment of literature work finds that some chaosbased image encryption algorithms have security vulnerabilities, including (i) standing up to chosenplaintext attack; (ii) sensitivity to all the chaotic secret keys; (iii) decoding of primary pixel within the decryption process; and (iv) reversing rectangular transform system. To outdo the abovementioned shortcomings and security defect, we propose an improved encryption algorithm utilizing twodimensional alteration models. The main objective of our work is to propose a data encryption system with key sensitivity, low residual clarity, and keeping up great quality of information reproduced by chaotic maps. Security analysis and experimental results suggest that proposed map could encrypt digital images with powerful capability and high security to resist different attacks.
The remainder of this paper is organized as follows. Within the next section, the proposed chaotic systems details are fully explained. The proposed encryption and decryption frameworks are presented in Section
We propose a novel chaotic system for improving encryption quality and execution, which is described below. Our system is a TwoDimensional (2D), nonlinear, discretetime technique that provides dynamical chaotic behavior. Due to the nonrepeatability and ergodicity of chaos in these algorithms, they can accomplish general searches at higher speeds than stochastic searches that depend on probabilities [
//Chaotic proposed algorithm
Begin
(1) It could be a system of a discrete time that maps point
(2) Define the initial value of maximum number of iterations
(3) Randomly initialize the positions of map.
(4) Begin iteration (
(5) Select one of the four proposed finance dynamical models.
(6) End for
End
The first proposed chaotic map can be considered as 2D growth of the traditional logistic map. It has a mathematical expression similar to Hénon map. The modified map gives a thought of chaotic nature which is given by condition (
Numerical simulation of the 2D phase plot in
The second proposed chaotic map that is utilized in our technique is a new finance model. It is a discretetime dynamical system that exhibits chaotic behavior. It takes a point
The above set of equations is a dynamical nonlinear system with 2D nonlinearities. Within the finance dynamical illustration, the state factors
The third proposed chaotic map utilized in our pipeline is a 2D chaos map that includes generation of a permuted image which includes the change within the position of the pixel in unique image to some new position utilizing the taking after the following condition:
Finally, the fourth proposed chaotic map that is introduced is obtained using the iterative function introduced by
The proposed characteristic types of the modern finance models are obtained using MATLAB for the financial parameters, e.g., initial state values as
Chaotic performance can be evaluated using different techniques such as Lyapunov exponent, bifurcation, and trajectory. A quick overview of those methods are given below; then evaluation of the chaotic behavior for the proposed maps based on their bifurcation diagram, iteration function diagram, and Lyapunov exponent are detailed in the next section.
Lyapunov exponent represents the highlights of a disordered framework and can generally communicate the general execution of chaotic maps. It is utilized as a quantitative measure for the sensitive reliance on initial conditions. For a discrete system
Quadratic map is a fundamental case of a disorderly framework. It might give the wellknown and broadly utilized OneDimensional (1D) disordered logistic map which is portrayed by scientific iterative [
Logistic map: (a) Bifurcation diagram and (b) Lyapunov exponent.
The bifurcation graph of the first proposed turbulent map is introduced in Figure
The Bifurcation diagram (a) and Lyapunov exponent (b) for the first proposed map at
The iteration and trajectory examinations are introduced in Figure
Iteration and trajectory analyses of the first proposed chaotic map at
The conduct of the second proposed map is introduced through Figure
The Bifurcation diagram (a) and Lyapunov exponent (b) for the second proposed map, at
The iteration and trajectory analyses for the second proposed map are introduced in Figure
Iteration and trajectory analyses of the second proposed chaotic map at
Figures
The bifurcation diagram (a) and Lyapunov exponent (b) for the third proposed map, at
The emphasis and trajectory analyses for the third guide are introduced in Figure
Iteration and trajectory analyses of the third proposed chaotic map at
At long last, the disorderly conduct of the fourth guide is shown in Figures
The bifurcation diagram (a) and Lyapunov exponent (b) for the fourth proposed map at
Iteration and trajectory analyses of the fourth proposed chaotic map at
The iteration and trajectory examinations for the fourth proposed chaotic map are introduced in Figure
Table
Comparison between the classical and proposed quadratic maps. Note: MLE is Maximal Lyapunov Exponent.
Chaotic map  Equation  Chaotic parameter range  MLE 

Classical quadratic map [ 


0.6923 
The first proposed chaotic map 


1.225 
The second proposed chaotic map 


3.317 
The third proposed chaotic map 


4.499 
The fourth proposed chaotic map 


3.091 
An iterative handle to scramble arrangement of bytes that is 1D changed form of the 2D original image can be used in the suggested scheme. As given in equations (
Within the presented cryptosystem for encryption and decryption forms, four of the proposed maps are utilized. The initial conditions and control parameters (key states) are extracted from the secret key and used to produce chaotic sequences from the proposed maps.
The proposed image encryption plot dependent on chaos structure is delineated in Figure
Schematic illustration: (a) the encryption processes, and (b) the decryption processes. Note that DWT, LL, LH, HL, HH Inv. Diffusion, Inv. Permutation, and IDWTstand for, Discrete Wavelet Transform, lowlow, lowhigh, highlow, highhigh, inverse diffusion, inverse permutation, and inverse Discrete Wavelet Transform, respectively.
The simulation steps: (a) LL component; (b) LH component; (c) HL component; (d) HH component; (e) LL component confusion; (f) LH component confusion; (g) HL component confusion; (h) HH component confusion; (i) original image; (j) overall confusion; (k) encrypted image; and (l) decrypted image.
DWT is famous in many image/video applications because of its multigoal portrayal. The fundamental thought of the DWT for a twodimensional image is depicted as follows. With the pyramidorganized wavelet change, the original image will experience various blends of a lowpass filter and a highpass filter and afterward dependent on the convolution with these channels to produce the LL, LH, HL, and HH subgroups. To acquire the following coarser scaled wavelet coefficients, the subband LL is additionally disintegrated and fundamentally subexamined. This procedure can rehash several times, which is controlled by the application. With the pyramidorganized wavelet transform, the size of the original image is identical to adding all the decayed subimages up. Utilizing this decay structure, there will be no data lost when the disintegrated pieces are reproduced. This remaking procedure is called IDWT [
We utilize the proposed chaotic maps to produce tumultuous groupings and afterward sort that confused numbers in rising or plunging order for the age of the change key. We sort the chaotic sequences in the record network utilized in rearranging the original image to acquire the permuted image. In the wake of acquiring the rearranged image, the relationship among the neighboring pixels is totally upset and the image is totally unrecognizable. In this way, the permuted orderly conduct of the fourth g image is frail against factual assault, and realized plaincontent assault [
The dissemination step in the proposed encryption plot is performed by the key identified with the plain image calculation which utilized just one round dispersion activity and its key relies upon the initial key and the original image [
Input: plain image P
Output: cipher image C
Step 1: examine the plain image P in size
Step 2: decompose the image into four level subbands (LL, LH, HL, and HH) by the selected DWT.
Step 3: choose a twodimensional chaotic system and generalize it by introducing the initial values
Step 4: generate the chaotic sequences using the proposed chaotic maps and set the appropriate values of the secret keys. Can use the 1^{st} proposed chaotic map.
Step 5: change the chaotic sequence, with the same method, into a consistently dispersed grouping by altering the initial values and parameters.
Step 6: iterate the chaotic sequence for LL subband for scrambling LL_{P} row by row and column by column (starting from the first row and the first column)
Step 7: like step 3, compute the next quantized chaotic pair using the 2^{nd}, 3^{rd}, and 4^{th} proposed chaotic maps to scramble the next subbands of LH, HL, and HH, respectively, and reiterate this step total times. (When the last row or the last column has been scrambled, switch to the first row or the first column over again.)
Step 8: combine the chaotic vectors (LL_{P}, LH_{P}, HI_{P}, and HH_{P}) into one vector with
Step 9: make the new vector of mistook pixels for
Step 10: adjust and change the vector
Step 11: create the diffused vector with
Step 12: create the final matrix with cipher image C as follows:
The decryption procedure is the opposite activity of the encryption procedure. The schematic representation of the structure of the decoding forms is shown in Figure
Input: cipher image C
Output: plain image P
Step 1: produce the deshuffled vector as follows:
Step 2: produce the permutated each vector as follows:
Step 3: obtain the permutation subbands (LL_{P}, LH_{P}, HI_{P}, and HH_{P})
Step 4: opposite stage and reshape vector components utilizing the chaotic index sequence to get subbands (LL, LH, HI, and HH)
Step 5: use IDWT recovers to obtain the original image
The quantitative performance of proposed techniques compared with traditional techniques could be measured using different metrics. The latter include (i) statistical parameters, (ii) differential parameters, and (iii) efficiency parameters [
Good cipher must have strong resistance against any measurable examination. To confirm the security of any encryption technique, the following statistical examinations should be performed [
An image histogram depicts the conveyance of image pixels by plotting the number of pixels at each gray scale level. The redundancy of plaintext should be hidden in the distribution of cipher text and this distribution logically needs to be uniform [
The relationship between two variables is called correlation coefficient (
The entropy is a perfect feature to evaluate the degree of randomness. The entropy of a message source could be computed as [
Encrypted image needs to be sensitive to tiny changes in plain image. Attacker can change some features in the plain image to get changes within the encrypted one. If a small unsettling influence within the original image comes about in a significant change in the encrypted one, then differential attacks lose their efficiency and become useless [
The Mean Square Error (MSE) is used in this paper to measure difference between the plain and encrypted images. The high value of MSE corresponds to a high difference between plain and encrypted images. It can present as in equation (
Another popular performance measurement related to MSE is Normalized Mean Square Error (NMSE) which equals MSE divided by the maximum MSE as in equation (
The peak signaltonoise ratio (PSNR) measures the conformity between the original and decrypted images [
The Number of Pixels Change Rate (NPCR) is utilized to measure the percentage of different pixel numbers between the original and decrypted images and is assessed as within the following condition [
NPCR evaluates the rate of pixels change in the coded image after modification in one pixel of an original one; as with higher value for NPCR, more effective performance is got [
The Unified Average Changing Intensity (UCI) measures the average intensity of difference between plain and decrypted images. It could be computed through the following equation [
Efficiency and high speed are additionally imperative issues for a successful cryptosystem, particularly for realtime Internet application. Generally, encryption speed is highly dependent on the CPU/MPU structure, size of RAM, operation system, the programming language, and compiler option. So, there is no need to compare the encryption speeds of two ciphers image using two different devices [
Most encryption algorithms are tested by utilizing measurable examination. Those analyses are utilized to find a relation between the encrypted and the original image. All of our experiments have been conducted utilizing a core i52400 Windows 7 machine with a 4 GB RAM, 160 GB HDD, and the same version of MATLAB programming environment. Our device was connected to the web most of time. All tests have been connected more than one time and thus the elapsed time represents the average simulation time for all trials for each test. The execution of proposed algorithm is tested using MATLAB R2017a where it is inspected through an arrangement of tests.
The proposed approach is implemented using the proposed maps for encryption and decryption of an image. We used the benchmark images Lena, Cameraman, Baboon, etc. (each of which is 512 × 512 pixels) as plain (original) images. With multimap orbit key, the proposed maps are performed. The foremost direct technique to choose the disorderly degree of the encrypted image is by the sense of sight. On the other hand, the stochasticity of encrypted images can be quantitatively calculated by the connection coefficient. Appling the proposed maps, the parameters
Four pictures are utilized to test the encryption algorithm, “Lena,” “Cameraman,” “Baboon,” and “Peppers.” From the simulation results shown in Figure
Encryption and decoding results: (a) Lena plain image; (b) Lena scrambled picture; (c) Lena decrypted image with right keys; (d) Cameraman plain image; (e) Cameraman encrypted image; (f) Cameraman decrypted image with right keys; (g) Baboon plain image; (h) Baboon encrypted image; (i) Baboon decrypted image with right keys; (k) Peppers plain image; (l) Peppers scrambled image; and (m) Peppers decrypted image with right keys.
Conveyances of information values in a system comprised the histogram. Histogram investigation can be made by looking at information distributions in numerous diverse fields. In encryption practices, in case the conveyances of numbers that represent encrypted data are near, this implies encryption is performing well. The closer the encrypted data distributions, the higher their encryption level. The histogram investigation for the chosen sample images is shown in Figure
Simulation results of sample test images: (a) Lena, (b) Cameraman, (c) Baboon, and (d) Peppers, respectively; (e)(h) histogram of original images; and (i)(l) histogram of cipher images.
The key space is the all out number of various keys that can be utilized in the encryption procedure. The proposed calculation comprises two procedures: permutation and diffusion. In permutation process, we utilize the four proposed maps with autonomous factors
In addition to histogram analysis, we employed another critical feature of chaos encryption, which is key sensitivity. During the decryption, any little alteration within the key leads to diverse results. Even if only one parameter has been changed, encrypted data cannot be unscrambled. Additionally, the information cannot be decrypted with knowing all the keys since the decryption does not occur within the correct order. Figure
Key sensitivity of encrypted process: (a) Cameraman plain image; (b) Cameraman encrypted image with first key; and (c) Cameraman encrypted image with another key.
The decrypted image is shown in Figure
Decrypted process with key sensitivity (a) using same keys of encrypted and (b,c) using the error keys.
Moreover, to assess the robustness of the proposed system the statistical analysis is conducted. Table
Parameters of the encryption quality for different test image. Please note that MSE, PSNR (dB), and ET (sec) stand for minimum mean square error, peak signaltonoise ratio, and elapsed time, respectively.
Image name  MSE  PSNR  ET  Entropy 

Lena  7747.309  9.23929  0.28913  7.9993 
Cameraman  9445.441  8.3785  0.17302  7.9991 
Baboon  7254.201  9.52486  0.30186  7.9993 
Peppers  8413.235  8.88117  0.20668  7.9994 
Average  8215.0465  9.005955  0.2426725  7.999275 
The proposed encryption employs distinctive midpoints when scrambling distinctive input images. This progressively can impressively increment the resistance of the cryptography system against unknown/chosen attacks and differential assaults. Security performance of the proposed algorithm is better than those results mentioned in [
The effect of speckle noise attack: (a) encrypted picture with 5% salt & pepper noise; (b) encrypted picture with 4% speckle noise; (c) cipher picture with 2% Gaussian commotion and turn of 30°; (d) decrypted image with 5% salt & pepper noise; (e) decrypted image with 4% speckle noise; and (f) decrypted image with 2% Gaussian noise and turn of 30°.
It may be a common form of cryptanalysis and a secure encryption scheme ought to have strong capacity of standing up to these attacks. For an image encryption scheme, by the number of pixels changing rate and bound together normal changed intensity can measure its capacity of standing up to differential attack. The results can be observed in Tables
The NPCR (%) of encrypted images for our approach compared with other literature algorithms. Please note that NA stands for “not applicable.”
Image name  Proposed method  Wu et al. [ 
Ben Slimane et al. [ 
Wang et al. [ 
Luo and Ge [ 
Amina and Mohamed [ 
Alawida et al. [ 

Lena  99.6641  99.6002  99.6271  99.59  99.6137  99.6452  99.620 
Cameraman  99.6523  99.6082  NA  99.59  99.6131  NA  NA 
Baboon  99.6438  99.5903  99.6145  99.56  99.6111  99.6154  99.601 
Peppers  99.6287  99.6112  NA  99.61  99.6137  99.6315  99.617 
The UACI (%) of encrypted images for our approach compared with other literature algorithms. Note that NA stands for “not applicable.”
Image name  Proposed method  Wu et al. [ 
Ben Slimane et al. [ 
Wang et al. [ 
Luo and Ge [ 
Amina and Mohamed [ 
Alawida et al. [ 

Lena  33.6124  33.5079  33.5589  33.48  33.4594  33.6152  33.505 
Cameraman  33.6425  33.5574  NA  33.53  33.4615  NA  NA 
Baboon  33.6430  33.5281  33.4277  33.58  33.4629  33.4354  33.424 
Peppers  33.6012  33.5265  NA  33.41  33.3948  33.5073  33.391 
Using the sample images above, we compute the correlation coefficients of adjacent pixels for the original and the encrypted image, and this is done through estimating the correlation among two vertically adjacent pixels, two horizontally adjacent pixels, and two diagonally adjacent pixels in the original and the corresponding encrypted images [
As can be seen from Table
Correlation coefficient of the original images and their encrypted images using the proposed chaotic maps.
Image  Direction  Horizontal  Diagonal  Vertical 

Lena  Plan image  0.98453  0.97553  0.95271 
Encrypted image  0.00047  0.00305  −0.03911  


Cameraman  Plan image  0.92021  0.91321  0.90124 
Encrypted image  0.00212  −0.00205  0.00190  


Baboon  Plan image  0.91251  0.9029  0.89215 
Encrypted image  0.00318  −0.00294  0.00285  


Peppers  Plan image  0.97543  0.97697  0.95871 
Encrypted image  0.00198  0.02547  0.04321 
Adjacent pixels correlation test for Lena: (a) plain image by horizontal, (b) plain image by vertical, (c) plain image by diagonal, (d) cipher image by horizontal, (e) cipher image by vertical, and (f) cipher image by diagonal.
To guarantee the security of the cryptosystem, the figured picture must have properties to segregate designs for additional measurable investigation, for example, great dispersion (i.e., arrangement’s connection gets feeble), extensive stretch (i.e., long key period), and high multifaceted nature and productivity (i.e., disarray and dissemination) [
Result of the NIST (SP800) test suite.
Test name 

Result  

Frequency  0.338753490  Success  
Block frequency  0.375654387  Success  
Runs ( 
0.374565348  Success  
Long runs of ones  0.334567898  Success  
Rank  0.345345266  Success  
Spectral DFT  0.464527  Success  
No overlapping templates  0.527653  Success  
Universal ( 
0.264534567  Success  
Lempel–Ziv complexity  0.565435  Success  
Linear complexity  0.384534167  Success  
Serial 

0.492345123  Success 
Serial 

0.424355767  Success 
Approximate entropy  0.543556665  Success  
Cumulative sums forward  0.345456565  Success  
Cumulative sums reverse  0.287662009  Success  
Random excursions 

0.535435  Success 

0.675656  Success  

0.434521  Success  

0.429843  Success  

0.512344  Success  

0.576545  Success  

0.496565  Success  

0.486632  Success 
Result of DIEHARD tests suite.
Test name  Average value  Result 

Birthday spacing  0.524546  Success 
Overlapping permutation  0.486766  Success 
Binary rank 31 × 31  0.823667  Success 
Binary rank 32 × 32  0.456273  Success 
Binary rank 6 × 8  0.686388  Success 
Bitstream  0.423876  Success 
OPSO  0.4601  Success 
OQSO  0.5243  Success 
DNA  0.5561  Success 
Count the ones 01  0.480243  Success 
Count the ones 02  0.256778  Success 
Parking lot  0.638823  Success 
Minimum distance  0.467348  Success 
3DS spheres  0.327673  Success 
Squeeze  0.536561  Success 
Overlapping sum  0.476538  Success 
Runs  0.426565  Success 
Craps  0.387243  Success 
A set of novel chaotic maps based on DWT and double chaotic function have been proposed in an effort to improve encryption quality and execution. In such a way, the proposed pipeline was able to avoid many existing cryptanalysis methodologies and cryptography attacks. This has been documented using the NPCR and UACI metrics with values of 99.6472% and 33.6248%, individually. The dynamical analysis and sample entropy algorithms showed that the proposed map is overall hyperchaotic with the high sensitivity and high complexity. Thus, the proposed chaosbased image cipher can be seen as reasonable tool for applications like wireless communications. There are a few research focuses that can follow after this investigation. The key choice handle can be randomized. The number of offers superimposed can be expanded to increase the layers of security. Different sorts of chaotic maps can be connected to the same image to improve the encryption handle. The proposed chaotic maps for multimedia security algorithms can be applied based on chaotic system for fog computing.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.