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1、精品论文大全Partial Generalized Synchronization Theorem of Differential and Discrete Systems with Applications in Encryption Scheme Jianyi Jing1,2 , Lequan Min1,2, , Geng Zhao31 Applied Science School, University of Science and Technology Beijing, 100083,2 Information and Engineer School, University of Sc
2、ience and Technology Beijing, 100083,3 Beijing Electric Science and Technology Institute, Beijing 100079jingjianyijAAbstractThis paper presents two theorems for designing controller to make two independent (chaotic) differential equation systems or two inde- pendent (chaotic) discrete systems realiz
3、e directional partial general- ized synchronization (PGS). Two numerical simulation examples are given to illustrate the effectiveness of the proposed Theorems. A non- symmetric encryption scheme is introduced based on the theorems. It can be expected that our theorems provide new tools for under- s
4、tanding and studying PGS phenomena and information encryption.Keywords: partial generalized synchronization; differential equa- tion systems, discrete systems, chaos-based encrytion scheme1 IntroductionSince the last decades, the research of chaos has got much attention (1- 12. One of the reasons fo
5、r this is that synchronization can be found in many physical, biological and engineering systems.The idea of generalized synchronization (GS) was also gradually developed(13Supported by the National Nature Science Foundation of China (Nos.70271068, 60674059), the Opening Re- search Fund for the Key
6、Lab of Information Security and Secrecy at the Beijing Electronic Science and Technology Institute (No.KYKF200605). Corresponding author- 1 -16), i.e., all state trajectories of the driven system synchronize with the driving system via a transformation.The researches on GS may provide new tools for
7、constructing secure communication systems(15, 16-22).Partial synchronization (PS) is defined as the situation where part of state tra- jectories of synchronized systems mutually asymptotically converge as time goes to infinity. Recently, the researches on PS in network have received attention(11,12,
8、 23,24).In this paper, firstly, a new Theorem on PGS for differential equation systems is presented. Secondly, a constructive theorem to PGS for two discrete systems is established. Based on the two Theorems, we can design controllers such that two indenpendent systems are in PGS with respect to a p
9、rescribed transformer. As a example, the Duffing chaotic system can be in PGS with the Lorenz chaotic system via designing a controller. As a second example, the discrete Lorenz three- dimensional chaotic map can be in PGS with the discrete Burgers map. Thirdly, a chaos encryption scheme based on ou
10、r theorems is introduced.2. PGS Theorem of Differential Equation SystemsGiven two independent chaotic systems, one is used as the driving system and an- other as the driven system. The goal of partial generalized synchronization is to design an appropriate controller U for the driven system, such th
11、at part state vari- ables of the controlled system can be in GS with the driving system.Definition 1 (4, 13) Consider two systems- 13 -X = F (X ), Y= G(Y, X ) (1.1)whereX Rn , Y Rm , F (X ) = (f1(X ), f2(X ), , fn (X )T Rn , G(Y, X ) = (g1(Y, X ), g2(Y, X ), , gm(Y, X ) Rm.If there exists a transfor
12、mation H : Rn Rm , and a subset B = Bx By Rn Rmsuch that all trajectories of (1.1) with initial conditions in B satisfieslimt+kH (X ) Y k = 0.Then the systems in (1.1) are said to be in GS with respect to the transformationH .Definition 2 Consider two independent systemsX = F (X ) (1.2)精品论文大全Y k Gk
13、(Y )whereY =Y mk=Gmk(Y )(1.3)X = (x1 , x2, . . . , xn )T Rn , Y = (y1, y2, . . . , ym)T Rm, Y k = (y1, y2, . . . , yk )T Rk ,Y mk = (yk+1, yk+2, . . . , ym)T Rmk ,F (X ) = (f1(X ), f2 (X ), . . . , fn(X )T , Gk (Y ) = (g1(Y ), g2(Y ), . . . , gk (Y )T ,Gmk (Y ) = (gk+1(Y ), gk+2(Y ), . . . , gm(Y )T
14、 .If there exists a controller Umk (X, Y ) = (uk+1(X, Y ),uk+2(X, Y), . . . , um(X, Y )T such that Y k in the controlled systemY k Gk (Y )Y =Y mk=Gmk(Y ) + Umk(1.4)and X in system (1.2) are in GS with respect to a transformation L : Rn Rk ,that is,limt+kL(X ) Y k k = 0.Then the systems (1.2) and (1.
15、4) are said to be in PGS with respect to the trans- formation Y k = L(X ).Theorem 1 Let two independent differential (chaotic) system be defined by eqution(1.2) and (1.3), and L : Rn Rk be a transformation given byL(X ) = (l1 (X ), l2(X ), , lk (X)T = Y kSuppose that the function qi(Xe , Ye ), (i =
16、1, 2) ensures that the zero solution of the following error equation (1.5) is zero solution asymptotically stable with respect to e = Ye Xee = qi (Xe , Ye ) (1.5)If vector Y mk can be solved from the equation li Gk (Y k , Y mk ) xjF (X ) = q1 (L(X ), Y k ) (1.6)knvia function Y mk = M (X, Y k ) = (M
17、1(X, Y k ),M2(X, Y k ), . . . , Mmk (X, Y k )T . Then we can design a controller Umk viadM (X, Y k )Umk (X, Y ) =dt+q2 (M (X, Y k ), Y mk )Gmk (Y ). (1.7)wheredM (X, Y k ) =dt M1M1 M1M1 x1xny1yk Mmkmkmkmkx1Mn xMy1M k y F (X ) !,Gk (Y )such that (1.2) and (1.4) are in PGS with respect to Y k = L(X ).
18、Proof: To be submitted elsewhere.3. PGS Theorem of Discrete SystemsDefinition 3 letX (i + 1) = F (X (i), (1.8)Y (i + 1) = Yk (i + 1)Ymk (i + 1)i = 1, 2, Gk (Y (i)=Gmk (Y (i)(1.9)be two discrete systems, whereX Rn , Y Rm , Yk Rk , Ymk Rmk .If there exists a transform L : Rn Rk , k m, such thatlimi+kL
19、(X (i) Y k (i)k = 0.then the systems (1.8) and (1.9) are said to be in PGS with respect to the transformY k (i) = L(X (i)Theorem 2 Given two discrete systems defined by (1.8) and (1.9). Let L : Rn Rkbe a transformation given byL(X ) = (l1 (X ), l2(X ), , lk (X)T = Y kSuppose that the function qj (Xe
20、 (i), Ye (i), (j = 1, 2) ensures that the zero solution of the following error equation (1.10) is zero solution asymptotically stable with respect to e(i + 1) = Ye (i + 1) Xe (i + 1)e(i + 1) = qj (Xe (i), Ye (i), j = 1, 2. (1.10) If vector Y mk (i) can be solved from the equationGk (Y k (i), Y mk (i
21、) L(F (X (i) = q1 (L(X (i), Y k (i) (1.11)via functionY mk (i) = M (X (i), Y k (i)= (M1 (X (i), Y k (i), M2(X (i), Y k (i), . . . , Mmk (X (i), Y k (i)T .Then we can design a controller Umk (i) viaUmk (i) = M (F (X (i), Gk (Y (i)+q2(Y mk (i), M (X (i), Y k (i)Gmk (Y (i)= (uk+1(i), uk+2(i), , um(i)Ts
22、uch that the systems (1.12) and (1.13)X (i + 1) = F (X (i), (1.12)Y (i + 1) = Yk (i + 1)Ymk (i + 1) Gk (Y (i)=Gmk (Y (i) + Umk (i)(1.13)are in PGS with respect to Y k (i + 1) = L(X (i + 1).Proof: To be submitted elsewhere.4. Numerical Simulation4.1 Example for Theorem 1Let two differential chaotic s
23、ystems be the Lorenz system (as driving system)x 1 = x1 + x2x 2 = x1 x2 x1 x3x 3 = x1 x2 x3(1.14)精品论文大全where = 10, = 28, =8, and the Duffing system3y1 = y4 y2 = y33 2y = y1 y4 = y4 + y1 y3 + y2(1.15)where = 0.25, = 0.2. If initial conditions of systems (1.14) and (1.15) is selected as (3.3432, 1.987
24、3, 3.5872)T and (1.8641, 25.9282, 13.1516, 4.0240)T , then the systems (1.14) and (1.15) are chaotic. Figure 1 shows their chaotic trajectories. We choose qi(Xe , Ye ) = Xe Ye (i = 1, 2), i,eq1 (L(X ), Y k ) = L(X ) Y kq2 (M (X, Yk ), Y mk ) = M (X, Yk ) Y mkWe want systems (1.14) and (1.15) be in P
25、GS via the transformation:L(X ) = (l1(X ), l2 (X )wherel1(X ) = k11 x1 + k12x2 + k13 x3 + b1l2(X ) = arctan x2 .where k11 = 2, k12 = 3, k13 = 4, b1 = 5. Then from the new theorem, we can getU42 (X, Y ) = (u1(X, Y ), u2(X, Y ),whereu3=3X k1i (xi + 2x i + xi) + b1i=112y4 y1 (y4 + y1 y3 + y2)u4 = arcta
26、n x2 +2x 2 + x221 + x22x2 x 222(1 + x2 )22y3 y2 (y2).where x i, xi , (i = 1, 2, 3) can be obtained from system(1.14). The result of PGS of variables X and Y 2 are shown in Figure 2. The errors go to zero. The numerical simulation verify our theoretical analysis.4.2 Example for Theorem 2Let two discr
27、ete chaotic systems be the Burgers map system(as driving system)( x1 (i + 1) = ax1 (i) x2 (i)2x2 (i + 1) = bx2 (i) + x1(i)x2 (i)(1.16)where a = 0.75, b = 1.75, and the Lorenz three-dimensional chaotic map system y1(i + 1) = y3(i)y2(i + 1) = y1(i)y3(i + 1) = y1(i)y3(i) y2(i)(1.17)If initial condition
28、s of systems (1.16) and (1.17) are selected as (0.1, 0.1)T and (0.5, 1,0.5)T , then the systems (1.16) and (1.17) are chaotic. Figure 3 shows their chaotic trajectories.2We choose qj (Xe , Ye ) = 1 (Xe Ye )(j = 1, 2), and want systems (1.16) and (1.17)to be in PGS via the transformation:L(X ) = l1 (
29、X ) = k1x1 + k2x2 + B.where k1 = 2, k2 = 3, B = 4. Then from the new theorem, we can getU31 (i) = (u2(i), u3(i)Twhereu2(i) = 01u3(i) = L(F (F (X (i) L(F (X (i) + 4 L(X (i)1 4 y1(i) + y3(i) y1(i)y3(i) + y2(i).The result of PGS of variables X and Y 1 are shown in Figure 4. The error goes to zero. The
30、numerical simulation verify our theoretical analysis.(a) Lorenz(b) Duffing5040x330204010 200 040 20y302040510020 100 10 x12020 40 x20y2 5 10 y1Figure 1. Chaotic tra jectories of (a) Lorenz system and (b) Duffing system.(a)40(b)520 002040y2 21 1 22 2 23 3 2k x k x k x b6080y atan(x )5321015201000 10
31、20 3040t250 10203040tFigure 2 Partial generalized sychronization errors vs. time: (c) e1 = y1 - l1 (X) (d) e2 = y2 l2 (X).5. PGS Based Encryption SchemeBase on the PGS system, one can establishes non-symmetric encryption scheme. The purposes of the scheme are Alice needs her subordinate Bob to provi
32、de secrete information from public chan-nels. However Alice does not hope Bob to know her encryption and decryption systems. The scheme has datum authentication function. The scheme has one-pad function.Assume that Alice has the system (1.2) and knows Bobs system (1.3) but Bob does not know Alices s
33、ystem (1.2).Alice and Bob share two invertible transforms T1 : Rk Rk and T2 : R Rwhere T2 (0) = 0 for embedding and extracting signal M in chaos stream ciphers.Alice designs a transform L : Rn Rk , functions q1 (X , Y ) and a controllerUmk (Y ) via formulas (1.7).Alice gives the controller Umk (Y )
34、to Bob.Now the secure communication scheme is described as follows.1. Bob informs Alice the length N of the secure message to be transmitted.2. Alice uses, at each time, different initial condition X (0) to generate chaos streamSx = X (i) : i = 1, 2, . . . , N . And then transmit L(X (0) and Sx to B
35、ob.(a)2(b)21 1x2y20 01 132 10202x1y12 2Figure 3. Chaotic tra jectories of (a) Burgers map system and (b) Lorenz three-dimensional chaotic map system.(a)64x , y212x ,1011 12 22050100050100ii416x 1015y k x k x B10505(b)Figure 3 (a) Tra jectories of the variables x1 (solid line), x2 (dashed line), and
36、y1 (dotted line). (b) Partial generalized sychronization errors vs. time: e = y1 k1 yx1 k2 x2 B.3. Bob uses the initial conditionY (0) = (Y k (0), Y mk (0)= (L(X (0), M (X (0), L(X (0),the chaos stream Sx and (1.4) to generate the chaos streamY k= (y1(i), y2(i), . . . , yk (i) : i = 1, 2, . . . , N.
37、4. Bob uses the transform T2 to embed his secure message plaintextM = m(i) : i = 1, 2, . . . , N in to Y k by means ofYk= (y1(i) + T2 (m(i), y2(i) + T2 (m(i), y3(i),. . . , yk (i) : i = 1, 2, . . . , N 5. Bob uses transform T1 to generate a ciphertext C .C = T1 (Y k ).6. Bob sends C to Alice.7. Alic
38、e firstly decrypts the ciphertext C byT 11 C L(X )= m 1 (i), m 2(i), . . . , m k (i) :i = 1, 2, . . . , N8. Ifm 1 (i) m 2(i),m j (i) 0,j = 2, 3, . . . , k; i = 1, 2, . . . , N.Then the ciphertext C is valid and the plaintext can be decrypted via2M= T 1(m 1(i) : i = 1, 2, . . . , N Otherwise, at leas
39、t one among L(X ), S and C has certainly altered by some intruder.Remark. Since each time Alice uses different initial condition X (0) to generate chaos stream, the pseudo-randomness of chaos trajectories of systems (1.2), (1.3) can guarantee the pseudo-randomness of the key stream Y k .Since Bob is
40、 difficult to conjecture, from the controller Umk (see (1.7) , thetransform L and Alices system (1.2), the security of the communication system is higher than that of identity synchronization or known each other encryption systems.Step 8) makes Alice be able to confirm the authentication of the data. Intruders are difficult to figure out the Alice and Bobs system via the key streams X andY k .Since Bob usesY (0) = (Y k (