A Novel Design Architecture of Secure Communication System with Reduced Order Linear Receiver

1. International Journal of Trend in International Open Access Journal International Open Access Journal | www.ijtsrd.com International Journal of Trend in Scientific Research and Development (IJTSRD) Research and Development (IJTSRD) www.ijtsrd.com ISSN No: 2456 ISSN No: 2456 - 6470 | Volume - 3 | Issue – 1 | Nov 1 | Nov – Dec 2018 A Novel Design Architecture A Novel Design Architecture of Secure Communication System Reduced-Order Linear Receiver Secure Communication System with Reduced Yeong-Jeu Sun Professor, Department of Electrical Engineering Electrical Engineering, I-Shou University, Kaohsiung Kaohsiung, Taiwan ABSTRACT In this paper, a new concept about secure communication system is introduced and a novel secure communication design with reduced linear receiver is developed to guarantee the global exponential stability of the resulting error signals. Besides, the guaranteed exponential convergence rate of the proposed secure communication system can be correctly calculated. Finally, simulations are given to demonstrate the feasibility and effectiveness of the obtained results. Key Words: Chaotic system, secure c system, reduced-order linear receiver 1.INTRODUCTION As we know, because chaotic system sensitive to initial value, the output behaves like a random signal. Frequently, chaos in many dynamic systems is an origin of the generation of oscillation and an origin of instability. Several kinds of chaotic systems have been widely applied in various applications such as secure communication, slave chaotic systems, image encryption systems, chemical reactions, system identification and ecological systems; see, for instance the references therein. In recent years, numerous secure communications have been extensively explored; see, for example, [ 12] and the references therein. Generally speaking secure communication is composed of transm receiver and reduced-order linear receiver merits of low price and easy implementation. Therefore, searching a lower-dimensional order linear receiver for the secure chaotic communication system constitutes an important area for practical control design. In this paper, we will propose a new communication system and a communication system with receiver will be developed resulting error signals can converge to zero in some exponential convergence rate guaranteed exponential convergence rate proposed chaotic secure communication system be accurately estimated. Finally, simulations are proposed to exhibit feasibility of the main results. This paper is organized as follows. The problem formulation and main results 2. Several numerical simulations are 3 to illustrate the main result. Finally, remarks are drawn in Section paper, ℜ denotes the n-dimensional Euclidean space, = : denotes the Euclidean norm of the vector x, and a denotes the number a. 2.PROBLEM FORMULATION AND MAIN RESULTS In this paper, we develop the following communication system with simple linear receiver and its block diagram is shown in Figure 1. Transmitter: ( ) 3 2 1 1 t x t x a t x = & ( ) 2 3 1 2 2 t x a t x a t x + = & ( ) 2 1 5 4 3 t x t x a a t x + = & (1c) ( ) 2 7 1 6 t x a t x a t y + = ( ) . 0 , 1 ≥ ∀ + = t t m t x C t m m φ In this paper, a new concept about secure communication system is introduced and a novel secure communication design with reduced-order linear receiver is developed to guarantee the global exponential stability of the resulting error signals. guaranteed exponential convergence rate of the proposed secure communication system can be correctly calculated. Finally, simulations are given to demonstrate the feasibility and effectiveness of the obtained results. we will propose a new idea about secure a novel design of secure communication system with reduced-order linear to guarantee that the onverge to zero in some convergence rate. Meanwhile, the guaranteed exponential convergence rate of the proposed chaotic secure communication system can some some numerical numerical Finally, some numerical exhibit the capability and Chaotic system, secure communication This paper is organized as follows. The problem are presented in Section simulations are given in Section chaotic system is highly the output behaves like a , chaos in many dynamic to illustrate the main result. Finally, conclusion in Section 4. Throughout this dimensional Euclidean space, the Euclidean norm of the column n systems is an origin of the generation of oscillation T⋅ x x x everal kinds of chaotic absolute value of a real have been widely applied in various applications such as secure communication, master- image encryption, biological ROBLEM FORMULATION AND MAIN system identification, instance, [1-3] and the following new secure communication system with simple reduced-order and its block diagram is shown in , numerous secure communications have been extensively explored; see, for example, [4- erally speaking, a secure communication is composed of transmitter and order linear receiver has the s of low price and easy implementation. ( ) ( ), ( ) (1a) (1b) ( ), ( ) ( ), ( ), ( ) dimensional reduced- secure chaotic constitutes an important area ( ) (1d) (1e) ( ) @ IJTSRD | Available Online @ www.ijtsrd.com www.ijtsrd.com | Volume – 3 | Issue – 1 | Nov-Dec 2018 Dec 2018 Page: 1154

2. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 in Scientific Research and Development (IJTSRD) ISSN: 2456 in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 Receiver: ( ) 1 t z =   a a ( ) t ( ) t ( ) t   3 = + + − 2 7 a x a x a z2 1 a  a 2 1 3 2 3 ( ) t ( ), t a  (2a) − + 7 z y 6 2 a  a [ ] 6 a 6 ( ) t ( ) t − + 2 a x a x 6 1 7 2 a a a ( ) t ( ) t ( ), t     t (2b) = − − + 2 a 7 2 z & a z y 6 2 3 2 a     a a a a 6 6 , 0 ≥ ( ) t ( ) t       3 = − + + − − 2 7 2 7 a x a z ( ) t ( ) t ( ) (2c) = φ − ∀ , 2 3 2 m where ( ) of transmitter, ( ) 1 : = z t z ( ) 1 ℜ ∈ t m ( ) 2 ℜ ∈ t m N q∈ . It is noted that the chaotic Sprott B the special case 1 5 3 3 2 1 = − = − = = = a a a a a same parameters of the chaotic Sprott B 0 7 6 > a a . Apparently, a good secure communication system means that we can recover the message in the receiver system; i.e., the error vector ( ) m t m t e 1 2 : − = can converge to zero in some sense. Before presenting the main result, let us introduce a definition which will be used in the main theorem Definition 1: The system (1) with (2) is called secure communication system with exponential convergence type if there are positive numbers k and ( ) exp : 1 2 − ≤ − = k t m t m t e α In this case, the positive number α exponential convergence rate. Now we present the main results communication system of (1) with (2). Theorem 1: The system (1) with (2) is a secure communication system with exponential convergence type. Besides, the convergence rate is given by C z t a a  2 m m 6 6   a a [ ] ( ) t ( ) t      [ ] = − − − ( ) t ( ) t ∈ 2 7 a x z is the partial is the output of transmitter is the state vector is the information vector, is the signal recovered from haotic Sprott B system is special case of the . In the sequel, we adopt the chaotic Sprott B system with , a good secure communication system means that we can recover the message in the receiver system; i.e., the error vector can converge to zero in some sense. partial state vector of transmitter, T = ∈ ℜ 2 : x t x x 3 2 2 a 1 2 6 ( ) ℜ t ℜ ∈ y    a a a a ( ) t ( ) t       3 [ ] = − + + − − ( ) t × q ( ) t 2 7 2 7 a x a z is the state vector of receiver, T 2 z 2 3 2 a a  2 6 6 , and , with × ∈ ℜ 1 2 q C   a a m ( ) t      = − − 2 7 a w ( ) t from × 1 q m1 3 2 a 6  a ( ). t     = − + 7 1 w the of . In the sequel, we adopt the the system system (1) (1) with with 2 a 6 This implies that     a ( ) t ( ) 0 . (4)     = ⋅ − + 7 exp 1 w w t   2 2 a     ( ) t 6 m1 From (1)-(4), it is easy to see that ( ) z t x t w 1 1 1 − =  − = a a 6 6 ), it is easy to see that ( ) 1 ( ) t ( ) ( ) t  a ( ) t ( ) t 7 y x     2 let us introduce a   which will be used in the main theorem. 1 a ( ) t ( ) t − − + 7 z y   2 a a  6 [ 6 The system (1) with (2) is called secure communication system with exponential convergence a ] ( ) t ( ) t = − − 7 x z 2 2 a 6 and α such that a ( ) t ( ) ( ) ( ) = − 7 w . ∀ ≥ , 0 t t 2 a 6 ( ) 0     a w a     . (5) = − ⋅ − + 7 2 7 is called the exp 1 t   a a     6 6 Hence, from (3)-(5), it results ( ) 1 w t w t w + = ( ) ( ) t s for secure 2 2 2 + 2 6 2 7 a a ( ) 0 ≤ ⋅ w 2 2 6 a tem (1) with (2) is a secure     communication system with exponential convergence a (6)     ⋅ − + ∀ ≥ 7 exp 1 , . 0 t t   guaranteed exponential a     6 a Thus, it can be readily obtained that ( ) ( ) ( ) t w C m ⋅ ≤ t can be readily obtained that . α = + 7 1 ( ) t ( ) z a = − e t m t m t 6 2 1 Proof. Define ( ) [ 1 = w t w Thus, from (1)-(3), one has ( ) z t x t w 2 2 2 & & & − = ( ) t ( ) t ( ( ) t = φ − − φ + C C x ] [ ] m m m m .(3) T T = − − ∈ ℜ 2 w x z x z 2 1 1 2 2 ( ) ( ) t + a 2 6 2 7 a a ( ) 0 ≤ ⋅ ⋅ w C 2 m 2 6   a a ( ) t ( ) t ( ) t     = + + − 2 7 a x a x a z 2 1 3 2 3 12     a a 6     ⋅ − + ∀ ≥ 7 exp 1 , , 0 t t   a a     ( ) t − 6 2 y a in view of (1), (2), and (6). This This completes the proof. 6 @ IJTSRD | Available Online @ www.ijtsrd.com www.ijtsrd.com | Volume – 3 | Issue – 1 | Nov-Dec 2018 Dec 2018 Page: 1155

3. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 in Scientific Research and Development (IJTSRD) ISSN: 2456 in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 Remark 1: It should be emphasized that receiver of (2) is linear and with lower dimens than that of the transmitter. Consequently proposed receiver of (2) has the superiorities price and easy implementation by electronic circuit 3.NUMERICAL SIMULATIONS Consider the novel secure communication system of (1)-(2) with 1 7 6 = = a a and C the synchronization of signals proposed secure communication (1) achieved with guaranteed convergence The real message ( ) t m1 , the recovered message and the error signal are depicted in Figure 2 respectively, which clearly indicates message ( ) t m1 is recovered after 3 seconds. 4.CONCLUSION In this paper, a new concept about secure communication system has been introduced and a novel secure communication design with order linear receiver has been developed to guarantee the global exponential stability of the resulting error signals. Meanwhile, the guaranteed exponential convergence rate of communication system can be correctly calculated. Finally, some numerical simulations have been offered to show the feasibility and effectiveness of the obtained results. ACKNOWLEDGEMENT The author thanks the Ministry of Science and Technology of Republic of China for supporting work under grants MOST 106-2221 MOST 106-2813-C-214-025-E, and MOST E-214-030. Besides, the author is grateful to Professor Jer-Guang Hsieh for the useful for the useful comments. It should be emphasized that the proposed 6.5 lower dimensions Consequently, the superiorities of low by electronic circuit. 6 5.5 m1(t) 5 novel secure communication system of [ m1 ] 1 and and . By Theorem 1, . By Theorem 1, ( ) t m2 = 1 − ( ) t 4.5 m for the proposed secure communication (1)-(2) can be 4 rate of message ure 2-Figure 4, that the real seconds. . , α m2 = 2 ( ) t 3.5 0 5 10 15 20 25 30 35 40 45 50 t (sec) t (sec) ( ) t Figure 2: Real message of described in the m1 transmitter of (1) transmitter of (1). 6.5 In this paper, a new concept about secure communication system has been introduced and a cation design with reduced- has been developed to guarantee the global exponential stability of the resulting error signals. Meanwhile, the guaranteed exponential convergence rate of 6 5.5 m2(t) 5 the the proposed proposed ectly calculated. secure secure 4.5 Finally, some numerical simulations have been offered to show the feasibility and effectiveness of the 4 3.5 0 5 10 15 20 25 30 35 40 45 50 Ministry of Science and t (sec) t (sec) ( ) t Figure 3: Recoverd message of Recoverd message of described in the of Republic of China for supporting this m2 2221-E-214-007, , and MOST 107-2221- grateful to Chair receiver of (2) receiver of (2). 0.25 0.2 ( ) ( ) − 1 1 System a d ( ) t ( ) t x y Transmitter Transmitter 0.15 m2(t)-m1(t) C m + ( ) t m1 + ( ) t φ 0.1 m channel channel 0.05 + − ( ) t m2 Systems (2a)-(2b) Receiver 0 ( ) t 0 5 10 15 z C m t (sec) t (sec) ( ) t Figure 1: Secure-communication scheme information vector and scheme ( is the m1 ( ) t ( ) t Figure 4: Error signal of Error signal of . − m m 2 1 ( ) t is the recovered vector) is the recovered vector). m2 @ IJTSRD | Available Online @ www.ijtsrd.com www.ijtsrd.com | Volume – 3 | Issue – 1 | Nov-Dec 2018 Dec 2018 Page: 1156

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