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According to its complex properties like ergodicity, unpredictability, and sensitivity to its initial states, chaotic systems are attracting more and more attention and are widely used for security purposes. Moreover, the chaotic signals are considered suitable for spread spectrum modulation due to their wideband properties. It is able to reduce the peak to average power ratio (PAPA). This paper presents a new Dynamic DiffeoDifference MultiDimensional (DDDMD) system. It is used as a key for a new cryptosystem designed based on the chaotic stream segmentation (CSS) method. The proposed system provides the best tradeoff between efficiency, robustness, and high data rate transmission. The behavior of the proposed chaotic system is evaluated numerically by analyzing the Lyapunov exponent spectrum, complexity, and attractor phase diagram. Besides, it is practically assessed based on the principle of system circuit design; the circuit diagram of the system is prepared and simulated by Multisim, which shows high consistency with the numerical simulation. These evaluations show that the proposed system has a rich dynamics behavior to be realized in an encryption system and provides a foundation for many engineering and physical applications.
chaotic system, modulation, secure communication, multisim, encryption process
Some emerging technologies, such as chaotic systems, are considered a suitable choice for constructing secure encryption algorithms [15]. This is due to their promising features, namely, ergodicity, unpredictability, sensitivity to the initial condition, and generating randomlike behavior sequences. These properties help enhance the security of the encryption/decryption system designed via chaotic systems. Therefore, such systems can guarantee to satisfy the security services, which are authentication, confidentiality, integrity, and nonrepudiation. Furthermore, besides the security issue, chaotic systems are considered efficient because they can be implemented with high speed and low cost.
The information technology and communication era required sharing of digital data through public and private networks. So, protecting these data from vulnerable access is a challenge and an essential task. Encryption method based on some emerging techniques such as chaotic systems is one of the essential resources to ensure the protection of such documents. Most of the previous strategies were based on optical code [69] until a chaotic laser emerged that plays an important role in enhancing the security of optical communication [1013]. In chaotic communication, the carrier is used to embed the message before sending it to the receiver side. Since then, many types of single and multicarrierbased chaotic systems have appeared in the literature to achieve highspeed and broadband capability due to their wideband characteristic [1416]. Besides, it produces a good synchronous signal, in which a big message is embedded into a chaotic carrier. Three categories for message encoding and decoding using a laser system are introduced. These are; Chaos Masking (ChM), Chaos modulation (CM), and chaos shift keying (ChSK) [1720]. Laser rate equations are used to describe and study the system's dynamic behavior. It is also used to formulate a mathematical equation in the transmitter and receiver sides. This paper proposes a new secure and highspeed optical communication scheme. It is based on designing a new 5D continuous chaotic system. The obtained chaotic signals are used as a key for the newly proposed encryption method using the principle of segmentation. The proposed cryptosystem was implemented for image encryption in this work. It shows high performance using some evaluation criteria.
The significant contributions in this work are as follows:
(1) Proposing a new 5D continuous chaotic system to be used as a key generation.
(2) Some numerical evaluation assesses the efficiency of the proposed chaotic system to investigate its dynamic properties
(3) A circuit diagram of the proposed chaotic system is sketched based on the circuit design principle. Then it was simulated using Multisim software. It is used as a practical evaluation to consist the numerical simulation results and to demonstrate the realization of the proposed system for many applications.
(4) A new cryptosystem is designed based on chaotic masking between the input and keystream.
(5) The proposed cryptosystem is used in an optical communication system to ensure data security that transfers through this channel.
(6) The images are used as a type of data encrypted before transmission. The performance evaluation for the encrypted images is assessed based on some statistical analysis to prove the quality and efficiency of the proposed cryptosystem.
The organization of the remaining work is arranged as follows: Section 2 introduced a new hyperchaotic continuous system. Its dynamical evaluations to generate keys for the proposed encryption method are discussed in Section 3. Section 4 introduced the encryption and decryption algorithms. The workflow of the optical communication system is presented in Section 5. The results and their analysis utilizing different metrics used to evaluate the proposed method's efficiency and security are discussed in Section 6. Finally, the paper is concluded in Section 7.
A new 5D dynamical system is introduced based on the NDLorenz design [21]. This system is specified as follows: For i=1, ..., N>=4:
$\frac{d x_i}{d t}=\left(x_{i+1}x_{i2}\right) x_{i1}x_i+C x_i$ (1)
where, it is expected that x_{0}=x_{N} and x_{1}=x_{N+1} at this point, x_{i} is the formal of the model, and C is an imposing constant (just like fixed force). It is a joint value known to affect chaotic conduct. Therefore, the proposed system is constructed mathematically by the following differential equations given in system (1). This system is called Dynamic DiffeoDifference of Multiple Dimension (DDDMD).
$f(1)=(yw) ux+a x$
$f(2)=(zu) xy+b y$
$f(3)=(wx) yz+c z$
$f(4)=(uy) zw+d w$
$f(5)=(xz) wu+e u$ (2)
where, x, y, z, w, and u are the state variables, and a, b, c, d, and e are positive parameters.
The dynamic properties of (1) are investigated through some multifarious dynamics such as; phase space (Figure 1), Lyapunov exponent (Figure 2), and sample entropy (Figure 3). The constant parameters (a, b, c, d, e) are chosen after intensive numerical analysis to ensure a high chaosity of the proposed system.
3.1 System attractor
As an indication of the system that has rich dynamics when its attractor in the phase space is complex. In our proposed 5Dhyperchaotic system, we found that it has a rich dynamic, which is oblivious from the projections of the attractors into different plans as shown in Figure 1 for the values of the parameters are a=1, b=3, c=0.09, d=0.1, e=0.01. This figure shows that the phase space is very complex and has disordered dynamic behavior.
3.2 Lyapunov exponents of DDDMD
As an indication of the system that exhibits new behavior, we use the Lyapunov Exponents (LE), which is a quality used to measure the exponential rate of convergence or divergence of two adjacent trajectories in the dynamical system. Its positive values refer to the chaosity of the given system. When more than one positive value considers that the system is hyperchaotic, we found that our proposed chaotic system is hyperchaotic by applying this evaluator. When the parameters values are: a=1, b=3, c=0.09, d=0.1, and e=0.01.
3.3 Sample entropy
Based on the Shannon entropy, determining the amount of information needed to predict the new output in the system trajectory based on previous time output is known as sample Entropy (SamEn). Its higher value indicates the high complexity or irregularity of the given system. Figure 3 shows the reasonable complexity and irregularity of the system (1).
Figure 1. (a), (b), (c), (d), and (e) Phase space of system (1) for the parameters values a=1; b=3; c=0.09; d=0.1; e=0.01; (f) represent the plot of one axis in the face plane
Figure 2. Lyapunov exponent of system (1) for the parameters a=1; b=3; c=0.09; d=0.1; e=0.01
Figure 3. Sample Entropy of system (1) based on the parameter a, where the other parameters are: b=3; c=0.09; d=0.1; e=0.01
The circuit design of the system (1) is developed to realize the proposed chaotic system. It is designed by the modular circuits using different resistance values and capacitors amplifier LM741 and the multipliers AD633 and 1 /s^{0.96}. This can be seen obviously in Figure 4. The limited voltage of LM741 and AD633 reduces the linearity of (1) to 0.1 times the original circuit [22, 23].
The simulation of the designed analog circuit by adjusting the horizontal axis is performed using Multisim12 for the circuit simulator. Finally, it is implemented on the oscilloscope to show the realtime results, as presented in Figure 4.
Figure 4. Circuit design of the system (1)
The state equation for the design in Figure 3 is given by:
$c_1 x_1^{\prime}=\left(\frac{y u}{10 R_1}\left(\frac{w u}{10 R_2}\right)\left(\frac{R_{24}}{R_{23}}\right)\right)\left(\frac{R_6}{R_5}\right)\left(\frac{x}{R_3}\right)+\frac{a x}{R_4}$
$c_2 y_1^{\prime}=\left(\frac{z x}{10 R_7}\left(\frac{u x}{10 R_8}\right)\left(\frac{R_{30}}{R_{29}}\right)\right)\left(\frac{R_{12}}{R_{11}}\right)\left(\frac{y}{R_9}\right)+\frac{b y}{R_{10}}$
$c_3 z_1^{\prime}=\left(\frac{w y}{10 R_{13}}\left(\frac{x y}{10 R_{14}}\right)\left(\frac{R_{18}}{R_{17}}\right)\right)\left(\frac{R_{18}}{R_{17}}\right)\left(\frac{z}{R_{15}}\right)+\frac{c z}{R_{16}}$
$c_4 w_1^{\prime}=\left(\frac{u z}{10 R_{19}}\left(\frac{y z}{10 R_{20}}\right)\left(\frac{R_{12}}{R_{11}}\right)\right)\left(\frac{R_{24}}{R_{23}}\right)\left(\frac{w}{R_{21}}\right)+\frac{d w}{R_{22}}$
$c_5 u_1^{\prime}=\left(\frac{x w}{10 R_{25}}\left(\frac{z w}{10 R_{26}}\right)\left(\frac{R_{18}}{R_{17}}\right)\right)\left(\frac{R_{30}}{R_{29}}\right)\left(\frac{u}{R_{27}}\right)+\frac{e u}{R_{28}}$
$R_1=R_2=R_7=R_8=R_{13}=R_{14}=R_{19}=R_{20}=$
$R_{25}=R_{26}=1 \mathrm{k} \Omega$
$R_3=R_4=R_5=R_6=R_9=R_{11}=R_{12}=R_{15}=$
$R_{21}=R_{22}=R_{23}=R_{24}=R_{27}=R_{28}=R_{29}=$
$R_{30}=100 \mathrm{k} \Omega$
$R_{10}=33.333 \mathrm{k} \Omega$
$R_{16}=1111.1 \mathrm{k} \Omega$ (3)
Figure 5. The attractors of the chaotic system (1), such that (a) xy, (b) xz, and (c) yz, (d) xu, and (e) xw, (f) represent one axis phase plane
After setting the right hand side of the system (1) equal to zero, we get R_{1}=R_{7}=R_{9}=1kΩ, R_{4}=R_{5}=R_{11}=R_{12}=100 kΩ, and R_{3}=R_{8}=R_{10}=0.6666 kΩ, where C_{1}=C_{2}=C_{3}=100 nf. According to this circuit design, the implementation results for the time domain of x, y, z, are shown in Figure 5 (af).
4.1 The proposed cryptosystem
A new encryption method is proposed; it is based on the principle of stream segmentation. The DDDMD system is used to provide the key. It is called the Chaotic Stream Segmentation (CSS) method. The schematic diagram of the proposed CSS method is illustrated in Figure 6. This method is used to encrypt different types of inputs (text, images, voice, etc.). It shows high encryption efficiency according to the used performance analyzes (histogram, Number of pixel change rate (NPCR), unified averaged changed intensity (UACI), correlation coefficient analysis, and information entropy). The algorithm of the proposed cryptosystem is presented as follows:
Figure 6. Schematic diagram for the CSS method
The Algorithm of the proposed cryptosystem
I. Key generating part
Key generating is the main part of any encryption scheme. When the keyspace is large, it is resulted in a highly secure system to resist brute force attack. In this work, this property is achieved because we used a chaos system for generating the key
Input: the chaotic system (1)
Initial condition k={x_{10}, x_{20}, x_{30}, x_{40}, x_{50}, a_{0}, b_{0}, c_{0}, d_{0}, e_{0}}
Output: state as square arrangement, in which each pixel is 8 bits.
II. Encryption part
Input: (image P of size mxn)
Output Cipher image C
$S P=X O R(S, P)$
For i=1 to n
For j=1 to m
For k=1 to m
If $a(j, i)=S P(k, i)$
$t(j, i)=\operatorname{perm}(k, i)$
k=m+z
End
End k
End j
End i
NSP=t
$S T_i=X O R\left(S^i, S P_i\right)$
End
III. Decryption part
The decryption is the reverse of the encryption part.
ommunication system, as shown in Figure 7. It is based on secure chaotic modulation.
Figure 7. Design of secure optical communication system
5.1 Multicarrier chaotic masking modulation (MCCMM)
Single carrier modulation approaches have grown more common in communication applications. Multicarrier modulation (MCM) gains a lot of attraction to improve the signal processing techniques.
In this part, the chaos masking technique has been proposed with multicarrier modulation. This model can create c_{i} (where i=1, 2, 3, ..., N) chaotic sequences, which may be used to build masking vectors. To create masking vectors, we use the binary form of the chaotic sequences c_{i}, which may be represented as c_{1}, c_{2}, ..., c_{N}, to make the chaotic sequences more unpredictable. First, the symbols in the supplied data are divided into subsequences. Then, the masking vectors are created by assigning different beginning values to separate subsequences. Figure 8 illustrates the multimasking design.
Multicarrier approaches can improve data transmission rates. The Wavelength Division Multiplexing (WDM) technology is a key technique for achieving high data rates (increasing the transmission capacity of optical communication systems). WDM is a potential technology for exploiting the optical fiber's huge bandwidth. It is a technique for simultaneously transmitting two or more optical signals on the same fiber. WDM divides the optical transmission spectrum into a number of nonoverlapping wavelength bands. The multiplexer combines signals from many sources and feeds them into the transmission channel, an optical fiber. A demultiplexer separates the signals at the receiving side, in which the photodetectors can detect them.
Figure 8. Multicarrier chaotic masking
In this section, several performance criteria are used to assess the efficiency of the proposed cryptosystem. The obtained result is also compared with other highperformance systems. The designed system has a high ability to encrypt various image types. This can be accomplished by converting images into random pixels, which are difficult to recognize without any knowledge of the original images. The implemented encryption system for different types of images (color and grayscale images) with the histograms of both of them is shown in Figure 9.
6.1 Histogram
A histogram is used to interpret the numerical data visually. It shows the number of points within a particular range of data. Besides, it can be used to display the value of the data frequency and their distribution, which displays its outliers or gaps. The histogram of the input images and the encrypted images are shown in Figure 9. Obviously, there is a significant difference between them. As a result, we may conclude that the given algorithm is statistically resistant.
6.2 Correlation analysis
The correlation measure effectively specifies the link between two existent random variables. It can be described as the relationship between two neighboring pixels in an image. As a result, the proposed encryption is utilized to shatter the association between image adjacent pixels, and the correlation value after encryption relates to the encryption system's efficiency. Figure 10 depicts the results of parrot image correlation in three orientations (horizontal, vertical, and diagonal). The correlation values can be calculated using the following equations:
$C A=\frac{E(xE(x)) E(yE(y))}{\sqrt{D(x) \sqrt{D}(y)}}$ (4)
where, $E(x)=\frac{1}{N} \sum_{i=1}^N x_i$, and $D(x)=\frac{1}{N} \sum_{i=1}^N\left(x_iE(x)\right)^2$ .
Figure 9. The of simulation results of CSS: (a) Color and grayscale plain images; (b) the histogram of (a); (c) cipher images of (a); (d) the histogram of (c)
Figure 10. Pixels correlation: (a) the plain and cipher images, along with the directions of (b) horizontal, (c) vertical, and (d) diagonal
The fundamental goal of the encryption process is to break the correlation between pixels. The correlations between pixels in the original image are strong, whereas the correlations between pixels in the encrypted image are weak since these pixels are distributed randomly. As a result, the efficiency of the encryption system is indicated by how low the correlation value is. Some results for the correlation of pixels that have been used previously are shown in Table 2. Table 3 shows the comparison with some literatures results based on the correlation coefficients factor.
6.3 Information entropy
The entropy of information is a measure of disorder or unpredictability and thus of uncertainty. As a result, entropy can be used to evaluate the relevance of a ciphering approach. The effectiveness of the encryption algorithm is assessed with higher entropy. A message source's entropy H(s) is defined as:
$H(s)=\sum_{i=0}^{2 N1} P\left(s_i\right) \log _2 P\left(s_i\right)$ (5)
where, s is the source, N is the gray level number with a probability P(s_{i}).
For completely random images, the value of H(s) is recommended to be equal to 8. However, the proposed encryption algorithm is considered secured based on the entropy values presented in Table 1.
6.4 Differential attack
Another security criterion is the differential attack resistance, which is based on the sensitivity of ordinary images to change. The sensitivity to any change in the value of the pixels indicates that the encryption system is highly resistant to differential attacks. This attack is evaluated using two statistical tests: The NPCR (denotes the number of pixels change rate in the encrypted image when changing one pixel in the plain image) and the UACI (used to measure the average intensity for the differences between inputs and encrypted images). Their ideal values are close to 100% and 33.33, respectively. To evaluate the ability of the proposed encryption method, some images are chosen, in which the average values are obtained by selecting their pixels randomly. The results in Table 1 show the high security of the proposed system against a differential attack. Eqns. 6, 7 that calculate the UACI and NPCR are given by:
$\mathrm{UACI}=\frac{\sum_{i=1}^N \sum_{j=1}^MC 1(i, j)C 2(i, j)}{255 \times M \times N} \times 100 \%$ (6)
where, M, and N are the image's width and height, respectively, for an 8bit image, and C_{k}(i,j) represents the pixels of the ciphered image.
$N P C R=\frac{\sum_{i=1}^M \sum_{j=1}^{M N} D(i, j)}{M \times N} \times 100 \%$ (7)
where, $D(i j)= \begin{cases}1, & \text { if } C 1 \neq C 2 \\ 0, & \text { if } C 1=C 2\end{cases}$ , C1 and C2 are the two cipher images, M and N are the images' width and height, respectively.
The key sensitivity of the hyperchaotic system is examined by a little change in the initial values (x_{0}, y_{0}, z_{0}, w_{0}, u_{0}) or the control parameters (a, b, c, d, e). The UACI identifies the mean intensity of the difference in the two encrypted images, whereas the NPCR calculates the ratio of the different pixel counts between two encrypts images. Tables 4, and 5 show the key sensitivity based on tiny change in the system variables and control variables.
However, the deviation between the plain and encrypted image measures the encryption quality. The quality of the proposed encryption method is presented in Table 1, which shows good performance quality.
Table 1. The security tests
Image 
Image dimension 
Correlation coefficient 
Entropy 
UACI 
NPCR 
1200×857 
0.0016 
7.9583 
33.30 
99.60 

251×201 
0.0041 
7.9566 
33.78 
99.43 

600×400 
0.0018 
7.9589 
33.73 
99.74 

241×181 
0.0079 
7.9546 
33.82 
99.43 

317×340 
0.0047 
7.9573 
33.46 
99.38 
Table 2. The correlations of some images in three directions (horizontal, vertical, and diagonal)
Name 
Plain image 
Cipher image 


Horizontal 
Vertical 
Diagonal 
Horizontal 
Vertical 
Diagonal 
tree 
0.986980 
0.990941 
0.975238 
0.00588 
0.005920 
0.005483 
Winnie 
0.903037 
0.994729 
0.959772 
0.002021 
0.001798 
0.002173 
ship 
0.960820 
0.949095 
0.927689 
0.006270 
0.005713 
0.005263 
Table 3. The comparison with some other litretur based on correlation coefficients factor
Direct 
Plain image 
Our proposal 
Ref. [24] 
Ref. [25] 
Ref. [26] 
Horizontal 
0.98423 
0.002182 
0.0065 
0.0230 
0.0030 
Vertical 
0.99315 
0.002131 
0.0035 
0.0019 
0.0024 
Diagonal 
0.97691 
0.002331 
0.0036 
0.0034 
0.0034 
Table 4. Key sensitivity based on tiny change in the system variables
Images 
X_{0} 
Y_{0} 
Z_{0} 
W_{0} 
U_{0} 

NPCR 
UACI 
NPCR 
UACI 
NPCR 
UACI 
NPCR 
UACI 
NPCR 
UACI 

Rose 
99.6034 
33.463 
99.60934 
33.4633 
99.60933 
33.46354 
99.60937 
33.46354 
99.60933 
33.4634 
Lena 
99.60934 
33.4636 
99.60935 
33.4635 
99.60937 
33.46354 
99.60934 
33.46354 
99.60934 
33.46354 
Tree 
99.60934 
33.4636 
99.60937 
33.4635 
99.60933 
33.46354 
99.60935 
33.46354 
99.60932 
33.4635 
Table 5. Based on tiny change in the system variables
a 
b 
c 
d 
e 

NPCR 
UACI 
NPCR 
UACI 

UACI 
NPCR 
UACI 
NPCR 
UACI 
99.60934 
0.3346 
99.6034 
33.4636 
99.60935 
33.4636 
99.60934 
33.4636 
99.60937 
33.4636 
99.60934 
33.46354 
99.60934 
33.4636 
99.6034 
33.4635 
99.60937 
33.4635 
99.60933 
33.4637 
99.60935 
33.46351 
99.60934 
33.4635 
99.60934 
33.4635 
99.60934 
33.4633 
99.60934 
33.4636 
This paper proposes a new chaotic system based on the Lorenz design. The dynamical properties of this system are demonstrated through some wellknown performance tests such as Lyapunov, which shows that the proposed system of equation is hyperchaotic system, sample entropy, and phase plain, which demonstrate the complex behavior of the system (1).
This system is a key generation for multicarrier modulation, which is newly designed to enhance the security of optical communication channels. This design helps significant saving of energy and high spectral efficiency. This also helps speed the transition process and sends only reference signals of many parallel bits. The experimental and analytical findings show that the suggested method is capable of performing encryption operations and has a high level of robustness against the majority of known attacks.
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