The Secure Transmission of Videos Using the Karhunen-Loéve ( KL ) Decomposition and the Synchronization of the Unified Chaotic System with the Hyperchaotic Chen System

A unique secure communication scheme that can be used for the transmission of gray-scale and color videos is presented in this paper.The proposed scheme is developed by using theKarhunen-Loéve (K-L) decomposition and the synchronization of the unified chaotic system with the hyperchaotic Chen system. First, the gray-scale or color video is represented by a set ofN frames. In order to reduce the data, the K-L decomposition is used to come up with data coefficients and eigenfunctions that optimally obtain the crux of the N frames. Using only the most energetic eigenfunctions to approximate the original frames results in computational savings. The data coefficients corresponding to the most energetic eigenfunctions are encrypted and transmitted using a master system composed of a combination of the unified chaotic system and the hyperchaotic Chen system. At the receiver end, these coefficients are recovered and a controller of the sliding mode type is utilized forcing the master and slave systems to synchronize. Simulation results illustrate how the proposed control law is able to synchronize the master and the slave systems. In addition, a demonstration of the recovery of the original frames using the decrypted data coefficients along with the eigenfunctions of the frame is provided. The presented simulations indicate that the proposed scheme results in an excellent performance.


Introduction
Historically, the field of chaos synchronization has attracted many researchers from an array of disciplines  due to a number of applications which utilize chaos.Secure communication systems are among the potential applications of chaos synchronization.The concept relies on the idea of hiding the transmitted signal in the states of the transmitter's chaotic system.Retrieving the transmitted signal requires control or synchronization of the chaotic system at the transmitter end with the chaotic system at the receiver end.
Chaotic systems are extremely sensitive to initial conditions; this fact makes their dynamical behaviors unpredictable.Ott et al. [1] were the first to introduce the idea of control of chaos by forcing the behavior of chaotic dynamical system to follow a desired behavior.Pecora and Carroll [2] demonstrated that chaotic systems can be synchronized using a drive-response approach.Since then, chaotic synchronization methods have become very attractive in many fields.
In 1993, the chaotic modulation method was applied by Wu and Chua [3] in secure communication, and in 1996, chaotic parameter modulation method was introduced by Yang and Chua [4].In 2005, Lu and Cao [7] designed adaptive controllers to synchronize the states of the chaotic Lorenz system with the states of the hyperchaotic Chen system.Moreover, adaptive control laws were then designed by Park [8] to synchronize two hyperchaotic Chen systems using Lyapunov stability theory.Recently, different control designs have been investigated to synchronize hyperchaotic and chaotic systems have been examined; for example, see .
This paper proposes an unprecedented secure communication scheme developed by using the K-L decomposition and the synchronization of hyperchaotic Chen with the unified chaotic systems for transmitting gray-scale and color videos.A promising advantage of using the unified chaotic system is its ability to have different behaviors and dynamical properties for different values of system parameter [10].First, we represent each of the gray-scale and color videos by a set of N frames.Then, the K-L decomposition is utilized to generate data coefficients and eigenfunctions that optimally capture the original N frames.The data coefficients corresponding to the most energetic eigenfunctions are encrypted and transmitted using a master system composed of a combination of the unified chaotic system and the hyperchaotic Chen system.Even though the idea of using synchronization of hyperchaotic systems for secure communication is available in the literature, the novelty of this work is in the proposed scheme of decomposing the transmitted data before rather than masking the data directly as will be shown later.A sliding mode controller is then used at the receiver end to synchronize the two systems consisting of hyperchaotic Chen and unified chaotic systems to recover these data coefficients and thus reconstruct the transmitted videos.The sliding mode technique is chosen due to its popularity, robustness, finite-time convergence, and simplicity in implementation.
The rest of the paper is structured as follows: The Karhunen-Loéve decomposition is described in Section 2. Section 3 presents the unified chaotic system and the hyperchaotic Chen system.The design of the proposed controller is detailed in Section 4. Section 5 illustrates the proposed secure communication scheme.Simulation results validating the developed scheme are presented and discussed in Section 6.Finally, some concluding remarks are provided in Section 7.
First, the data is considered to be a sequence of  realvalued vectors {  }  =1 , where the dimension of   is  such that   = [  1 ,   2 , ⋅ ⋅ ⋅ ,    ]  .These vectors can represent a set of images.
Using the snapshot method [64], one can compute the symmetric and positive definite covariance matrix  as follows: In (1), ⟨⋅, ⋅⟩ denotes the usual Euclidian inner product.The eigenfunctions of the data are orthogonal and defined such that where  []    is the  ℎ component of the  ℎ eigenvector.Let  be such that In (3), Ψ  is the  ℎ eigenfunction and   ( = 1, . . ., ) are the data coefficients computed by projecting the data vector onto an eigenfunction such that The energy of the data, , is defined as follows: where   is the eigenvalue of the ℎ eigenfunction.Because the energy percentage of each eigenfunction depends on the eigenfunction's associated eigenvalue,   , it can be calculated as follows: In case that all the eigenfunctions are used, then the original data can be fully restored.Moreover, using the most energetic eigenfunctions, an approximation of the original data can be reconstructed as follows:

System Description
This section presents the hyperchaotic Chen system and the unified chaotic system needed for the development of the secure communication scheme; the hyperchaotic Chen system as well as the unified chaotic system is used to generate the master and slave systems.
The master system is obtained by combining the hyperchaotic Chen system with the unified chaotic system.The hyperchaotic Chen system is a fourth-order system of ODEs defined such as where , , , and  represent the system states and the scalars , , , , and  are the parameters of the system.This system exhibits a hyperchaotic behavior when  = 35,  = 3,  = 12,  = 7, and 0 ≤  ≤ 0.085 [10].The unified chaotic system is defined such that where   ,   , and   are the system states and  is a real parameter satisfying  ∈ [0, 1].Depending on the value of , system (9) can display different behaviors and dynamical properties.For instance, system (9) exhibits the behavior of the Lorenz chaotic system when  = 0 [55].When 0 <  < 0.8, system (9) becomes the generalized Lorenz chaotic system.System (9) will be the  ü chaotic system if  = 0.8 [6].The unified chaotic system becomes the Chen chaotic system when  = 1 [5]; and it becomes the generalized chaotic system when 0.8 <  < 1.
Therefore, the master system is defined as follows: where   ,   ,   , and   represent the states of the Chen hyperchaotic system (A) with , , , , and  being its parameters.The states of the unified chaotic system (B) are  1 ,  1 , and  1 , and  is the system parameter.Also,  = () represents the message () to be transmitted (data coefficients) with a scaling factor .The overall system is depicted in Figure 1.
At the receiver side, the slave system is taken to be a combination of the hyperchaotic Chen system (C) and the unified chaotic system (D) (see Figure 1).It should be noted that the parameters of the master and slave systems should be the same.
Hence, the description of the slave system can be written as follows: where   ,   ,   , and   are the states of the Chen hyperchaotic system (C),  2 ,  2 , and  2 are the states of the unified chaotic system (D), and the parameters , , , , and  are the same as for system (10).Also, the controllers of the slave system are  1 ,  2 ,  3 , and  4 .These controllers will be designed to synchronize the master and slave systems at the transmitter and receiver sides, respectively.It can be seen from ( 10) and ( 11) that the signal   () is used to drive the two unified chaotic systems at both sides.Also, the parameter  used in both systems is generated from   () using a generation rule that must be the same at the transmitter and receiver sides.

Controller Design
A sliding mode control law is proposed in this section to synchronize the master and slave systems described in (10) and (11), respectively.This design is validated and proven to synchronize the two systems.The proof is divided into two parts.The first part of the proof shows that the signal   () will synchronize the unified chaotic system (D) at the receiver end with the one at the transmitter side.The second part of the proof demonstrates that the proposed control law is able to synchronize the system at the receiver side with the one at the transmitter side.
The unified chaotic systems at both the transmitter and the receiver sides are described, respectively, as follows: and Notice that the signal   () is driving the two systems.The errors between the two unified chaotic systems are defined such that By differentiating these errors and using ( 12) and ( 13), we obtain the following: Theorem 1.If the parameter  is designed to be in the interval [0, 1/29), the signal   () will synchronize the unified chaotic system ( ) at the receiver with the unified chaotic system ( ) at the transmitter (i.e., lim Proof.Let the Lyapanov function candidate  1 be such that [10] where  = [  ,   ,   ]  and  is such that Clearly, the matrix  is positive definite when  is in the interval [0, 1/29).This implies that  1 > 0 when  ∈ [0, 1/29).
The derivative of  1 with respect to time along the trajectories of the error system (15) is where  is defined as When  ∈ [0, 1/29), the matrix  is positive definite which implies that V 1 is negative definite.Therefore, it is guaranteed that the errors   ,   , and   asymptotically converge to zero when  ∈ [0, 1/29).In conclusion, the signal   () will force the slave system (13) to be synchronized with the master system (12).
Hence, after a time   , the unified chaotic systems will be synchronized, and the states  1 ,  1 , and  1 can be retrieved at the receiver side.These states are then used to unmask the states    ,    , and    at the receiver side to obtain the states of the Chen system at the transmitter side as follows: Let the synchronization errors between the two Chen systems be such that By differentiating these errors with respect to time and using (10) and (11), we obtain the following error system: Theorem 2. Let the controllers be designed such that If the above controllers are applied to the slave system ( ) at the receiver, the asymptotic convergence of the synchronization errors to zero is guaranteed.
Proof.Consider the following Lyapunov function candidate, Using the proposed controllers in (23), the derivative of  2 with respect to time along the trajectories in ( 22) can be written as follows: It is evident from (25) that V 2 < 0 for  2 > | 2 |.Therefore, the asymptotic convergence of   ,   ,   , and   is guaranteed.Furthermore, if  ∈ [0, 1/29), it is ensured that   ,   , and   will asymptotically converge to zero as well according to (1).Thus, the proposed controllers force the synchronization errors to converge to zero asymptotically.
As a result, the proposed controllers in (23) guarantee that the slave system is synchronized with the master system; a noisy version of the transmitted message can be recuperated according to the equation m() =   −   .
Remark .The signal   () is used to generate the parameter () to make sure that () ∈ [0, 1/29].One of the possible generation rules that can be used is () = |  |/29(|  | + 1).Note that the  generation rule must be the same at both the transmitter and receiver sides.

The Proposed Secure Communication Scheme
A unique secure communication scheme developed by using the K-L decomposition and the synchronization of a combined hyperchaotic Chen system and the unified chaotic system is examined in this section.The developed scheme is utilized to transmit gray-scale and color videos through the encryption of the transmitted data.Figure 1 depicts a block diagram of the proposed secure communication scheme.The description of the scheme is as follows.
Consider a video (we will call it data) that requires secure transmission.We apply a separation rule to the data such that (i) If the data is a gray-scale video, then the data is decomposed into  frames.The gray-scale frames are represented by an  ×  data matrix containing the gray levels of the pixels.
(ii) If the data is a color video, then it is decomposed into  frames.The color frames (RGB frame) are represented using an  ×  × 3 data matrix which specifies the levels of red, green, and blue color components for each pixel.
Using the Karhunen-Loéve decomposition on the  frames, one can generate the data coefficients and the eigenfunctions of the data.The eigenfunctions are transmitted without encryption through a public channel; this is the case because it is impossible to recreate the original data using only the eigenfunctions.To encrypt the data coefficients, we add them to one of the states in the master system.This is done by transforming the data coefficients into binary format to form a sequence of pulses () where the scalar  represents the amplitude of the pulses.The sequence of pulses is appended to the state   of the master system to be encrypted.The states of the new system are then transmitted to the receiver through a noise-free public channel.
It is well known that hyperchaotic systems are extremely sensitive to initial conditions; hence it will not be possible to recover the original data using the transmitted states even if these states are intercepted.The reason for that is that the eigenfunctions of the original data are not known to the intruder who successfully intercepts the master system even in the case when the data coefficients are successfully retrieved.Hence, the proposed scheme for data transmission is very secure.
In Figure 1, the master system at the transmitter is considered to be a combination of the hyperchaotic Chen system (System A) and transmitted directly through the public channel.The states of the unified chaotic system (System B) at the transmitter ( 1 ,  1 , and  1 ) are used to mask the   ,   , and   states of the hyperchaotic Chen system (System A) producing the masked states    ,    , and    , respectively.This stage is considered to increase the complexity of the proposed encryption to ensure that intruders are unable to retrieve any information from the transmitted data.The masked states are then transmitted to the receiver through the public channel.It should be noted that the masking rules should be the same at the transmitter and receiver sides.
The received   state is utilized to drive the unified chaotic system (System D) and is used to generate the parameter () with the same generation rule that is used at the transmitter side.After the synchronization between the two unified chaotic systems, the states  1 ,  1 , and  1 are retrieved at the receiver side.These states are then utilized to unmask the received masked states    ,    , and    to obtain the states   ,   , and   .Using these generated states, a controller which is based on the sliding mode technique is used to synchronize the hyperchaotic Chen system (System C) at the receiver with the one at the transmitter.After the synchronization, a noisy version of the transmitted message   () (data coefficients) can be retrieved and filtered in order to reconstruct the original message m().
Once we obtain the retrieved message m(), we can recover the transmitted binary data coefficients and convert them back to real values.The obtained real-valued data coefficients along with the received eigenfunctions are used to recreate the original transmitted video by using (7).

Simulation Studies
Two simulation cases of the proposed communication scheme are presented in this section.The first case deals with the transmission of gray-scale videos while the second case deals with the transmission of color videos.In each of the two cases, the original video is decomposed into  = 49 frames; each frame is represented by a 120 × 160 data matrix for each of the gray-scale frames and for the color frames.These two cases are presented in the following two subsections.
. .Transmission of a Gray-Scale Video.In this subsection, we present the case of transmitting a gray-scale video using the proposed secure communication scheme.We consider a video from the Matlab toolbox of a traffic camera that observes vehicular behavior on a road for about 3.25 seconds.This video has 49 frames with a dimension of 120×160 pixels, and these frames are shown in Figure 2.
First, the K-L decomposition is applied on these 49 frames to obtain 49 eigenfunctions.Figure 3 depicts the most 16 energetic eigenfunctions.The generated 16 most energetic eigenfunctions are transmitted using a public channel.The energy associated with each eigenfunctions is shown in Figure 4. Figure 4(a) depicts the energy of all eigenfunctions while Figure 4(b) illustrates the energy associated with the remaining eigenfunctions not including the first one.It is noted from this figure that the first eigenfunction captures most of the energy (about 97.13 %).
The data coefficients are converted into binary numbers, and a sequence of pulses with a period   = 1 is formed to represent the data coefficients.That is, the formed message is such that The message () is added to the master system at the transmitter in the   state.Additionally, we define the generation rule of the  parameter to be such that This choice of the  generation rule ensures that  ∈ [0, 1/29).
Then the produced states of the unified chaotic system  1 ,  1 , and  1 are used to mask the states   ,   , and   .The masking rules are chosen in the following recursive manner: Figure 2: The frames of the original gray-scale video to be transmitted using the proposed scheme.After adding the message to the master system, masking the states, and transmitting them, the proposed controller forces the slave system to synchronize with the master system.This fact can be clearly seen from Figures 5 and 6; Figure 5 depicts the synchronization errors of the Chen systems while Figure 6 depicts the synchronization errors of the unified chaotic systems.It is evident that all errors converge to zero except   which contains a noisy version of the transmitted data.Therefore, the original data is recovered from   after filtering it.Part of the recovered message is shown in Figure 7.
Using the recovered message, the data coefficients are then recovered and utilized alongside the received eigenfunctions in order to reconstruct the original vectors according to (7) which are reshaped to 120 × 160 arrays to reconstruct the original frames.The reconstructed frames using all eigenfunctions are presented in Figure 8.This clearly verifies the efficiency of the proposed scheme.
To demonstrate that the developed communication scheme has another attribute besides being secure, the transmitted video can be reconstructed using some of the Mathematical Problems in Engineering     eigenfunctions (see Figure 9).Figure 9 depicts the reconstructed video using the most 35 energetic eigenfunctions along with their corresponding data coefficients.Furthermore, the Peak Signal-to-Noise Ratio (PSNR) [65], which is a known video quality metric, was used to compare the quality of the reconstructed videos for different cases.The calculated PSNR values are 51.6 dB, 40.3 dB, 36.8 dB, and 27.5 dB when using 49, 35, 30, and 15 most energetic eigenfunctions, respectively.It should be noted that PSNR was calculated by comparing each frame with its reference and then taking the average for the whole video.It can be seen from these results that a better quality can be obtained by using more eigenfunctions, and the best case is by using all eigenfunctions.
. .Transmission of a Color Video.In this subsection, we consider the transmission of a color video using the developed communication scheme.The color version of the traffic video used in Section 6.1 is utilized here.Figure 10 presents the frames of the color video which are represented by a 120 × 160 × 3 data matrices.
First, K-L decomposition is applied on these 49 frames to obtain 49 eigenfunctions.Figure 11 depicts the most 16 energetic eigenfunctions.The generated 16 most energetic eigenfunctions are transmitted using a public channel.Figure 12 depicts the energy associated with each eigenfunctions.Figure 12   with the rest of the eigenfunctions not including the first one.
Next, we transform the obtained data coefficients into binary to form a sequence of pulses as described in Section 6.1.This sequence of pulses is added to the   state of the master system and all the states of the system are masked and transmitted.The sliding mode control law at the receiver end is used to synchronize the master and slave systems; see Figures 13 and 14.A noisy version of the transmitted data is then recovered by using   where a part of that message is shown in Figure 15.
Once the sent message is recovered, it is transformed back from binary to obtain the real-valued data coefficients.The recovered data coefficients are used in conjunction with    the received eigenfunctions to reconstruct the data vectors using (7).Finally, the data vectors are reshaped to retrieve the transmitted color frames.The retrieved frames using all eigenfunctions which proves the efficiency of the developed communication scheme are presented in Figure 16.
The reconstructed frames when only the most 35 energetic eigenfunctions are used are shown in Figure 17.Moreover, the PSNR was calculated as a quality metric by considering different cases using 49, 35, 30, and 15 most energetic eigenfunctions for transmission and reconstruction.The corresponding PSNR values are 56.7 dB, 41.8 dB, 37.4 dB, and 27.5 dB, respectively.It is obvious that the reduction of transmitted eigenfunctions and coefficients comes at the expense of the reconstructed video quality which is a tradeoff that can be made depending the desired requirements.

Conclusion
This article considers a unique secure communication scheme developed by using the Karhunen-Loéve decomposition and the synchronization of master with slave systems where each system consists of the Chen hyperchaotic system and the unified chaotic system is considered.The Karhunen-Loéve decomposition is employed as a data reduction method to successfully generate data coefficients and optimal eigenfunctions that capture the original set of video frames.Using only the most energetic eigenfunctions to approximate the original frames leads to computational savings.The obtained data coefficients are encrypted before being transmitted using a master system and received using a slave system through the secure communication scheme.A sliding mode control design is presented to synchronize the master and slave systems.Furthermore, computer simulations are executed to verify the performance of the proposed scheme, and the results of these simulations are presented and discussed.
The real-time implementation of the proposed method is a challenging task and will be the subject of future research studies.

Figure 1 :
Figure 1: A depiction of the proposed communication scheme.

Figure 4 :
Figure 4: (a) Energy associated with all eigenfunctions.(b) Energy associated with all the eigenfunctions except the first one (gray-scale video).

Figure 5 :
Figure 5: The errors of the Chen systems versus time for 50 seconds (gray-scale video).

Figure 6 :
Figure 6: The errors of the unified chaotic systems versus time for 50 seconds (gray-scale video).

Figure 8 :
Figure 8: The recovered gray-scale video using all eigenfunctions.
(a)  shows the energy of all eigenfunctions while Figure12(b) depicts the energy associated

Figure 9 :
Figure 9: The recovered gray-scale video using the most 35 energetic eigenfunctions.

Figure 10 :
Figure 10: The frames of the original color video to be transmitted using the proposed scheme.

Figure 14 :
Figure 14: The errors of the unified chaotic systems versus time for 50 seconds (color video).

Figure 16 :
Figure 16: The recovered video using all eigenfunctions (color video).

Figure 17 :
Figure 17: The recovered video using the most 35 energetic eigenfunctions (color video).