A Non-symmetric Digital Image Secure Communication Scheme Based on Generalized Chaos Synchronization System

Commun. Theor. Phys. (Beijing China) 44 (2005) pp. 1115 1124 c International Academic Publishers Vol. 44 No. 6 December 15 2005 A Non-symmetric Digital Image Secure Communication Scheme Based on Generalized Chaos Synchronization System ZHANG Xiao-Hong 12 and MIN Le-Quan 1 1 Applied Science School and Information Engineering School University of Science and Technology Beijing Beijing 100083 China 2 Information Engineering School Jiangxi University of Science and Technology Ganzhou 341000 China (Received April 4 2005) Abstract Based on a generalized chaos synchronization system and a discrete Sinai map a non-symmetric true color (RGB) digital image secure communication scheme is proposed. The scheme first changes an ordinary RGB digital image with 8 bits into unrecognizable disorder codes and then transforms the disorder codes into an RGB digital image with 16 bits for transmitting. A receiver uses a non-symmetric key to verify the authentication of the received data origin and decrypts the ciphertext. The scheme can encrypt and decrypt most formatted digital RGB images recognized by computers and recover the plaintext almost without any errors. The scheme is suitable to be applied in network image communications. The analysis of the key space sensitivity of key parameters and correlation of encrypted images imply that this scheme has sound security. PACS numbers: 87.18.Sn 05.45.Xt 05.45.Vx 05.45.Pq Key words: generalized chaos synchronization Sinai map image confusion non-symmetric key data origin authentication network communication 1 Introduction Chaos is one type of complex dynamic behaviors generated by determined nonlinear dynamic systems but is similar to statistic process. A chaos signal has the characteristics of aperiodicity continuous wide spectrum similarity to noise and so on. In the past decade the study of chaos-based secure communication systems has been fascinatingly developed. [1 19] Recently the Generalized Chaos Synchronization (GCS) approaches have also been gradually developed [71119 22] which may provided new tools for constructing secure communication systems. [1122 26] In recent papers [232526] several text secure communication schemes have been proposed via GCS systems. Network communication has become a main tool for information distributions. Digital image transformations have played more and more roles. In this paper a nonsymmetric GCS-based digital true color (RGB) image encryption scheme is proposed. The scheme uses the chaotic source signals generated by the responding system in a GCS system and a discrete Sinai map to encrypt RGB images. A receiver may use his non-symmetric keys for the authentication of received data origin without communication with the sender of the data origin. Theoretical analysis experimental simulation and statistical analysis show that this scheme is extreme sensitivity to keys and has a 10 135 large key space sound security and university. Our researches imply that the digital image encryption scheme may be used in digital image network communications. 2 Digital Image Secure Communication Scheme 2.1 Some Basics Firstly let us remember the definition on GCS and a GCS theorem to be used for constructing chaotic source generators. Definition 1 be (Similar to Refs. [6] and [20]) Let systems Ẋ = F (X) (1) Ẏ = G(Y X m ) (2) X(t) = (x 1 (t) x 2 (t)... x n (t)) T R n Y R m (3) X m (t) = (x 1 (t) x 2 (t)... x m (t)) T m n (4) F (X) = (f 1 (X) f 2 (X)... f n (X)) T (5) G(Y X m ) = (g 1 (Y X m ) g 2 (Y X m )... g m (Y X m )) T. (6) System (1) is called the driving system and system (2) is said to be the responding system. The two systems The project supported by National Natural Science Foundation of China under Grant Nos. 60074034 and 70271068 the Foundation for University Key Teachers and the Research Fund for the Doctoral Program of Higher Education under Grant No. 20020008004 by the Ministry of Education of China Correspondence author E-mail: lqmin@sohu.com

1116 ZHANG Xiao-Hong and MIN Le-Quan Vol. 44 are said to be in GCS with respect to a transformation H : R m R m if there exists an open subset B = B X B Y R n R m such that any trajectory (X(t) Y (t)) of systems (1) and (2) with initial conditions (X(0) Y (0)) B has the property: lim t + X m(t) H 1 (Y (t)) = 0. Theorem 1 [20] Let X X m Y F (X) and G(Y X) be defined by Eqs. (1) (6). Suppose that H : R m R m is a C 1 diffeomorphism and X m = V (Y ) is the inverse function of H. If systems (1) and (2) are in GCS via transformation Y = H(X m ) then the function G(Y X) in Eq. (2) takes the following form: G(Y X) = [ V (Y )] 1 [F m (X) q(x m Y )] (7) V 1 V 1 y 1 y 2 V 2 V 2 y V (Y ) = 1 y 2 V m V m y 1 y 2 and the function V 1 y m V 2 y m (8) V m y m F m (X) = (f 1 (X) f 2 (X)... f m (X)) T (9) q(x m Y ) = (q 1 (X m Y ) q 2 (X m Y )... q m (X m Y )) T guarantees that the zero solution of the error equation (10) ė = Ẋm V (Y )Ẏ = q(x m Y ) (10) is asymptotically stable. In subsection 2.2 we shall first construct a Lorenz GCS system by means of Theorem 1. A non-symmetric digital RGB image secure communication scheme will be presented in subsection 2.3 via the discrete Sinai map. 2.2 GCS System Let the driving system Ẋ = F (X) be the Lorenz system ẋ 1 = f 1 (X) = σ(x 1 x 2 ) ẋ 2 = f 2 (X) = γx 1 x 2 x 1 x 3 ẋ 3 = f 3 (X) = x 1 x 2 βx 3. (11) Now construct a diffeomorphism H : R 3 R 3 as follows: H(X) = (h 1 (X) h 2 (X) h 3 (X)) T = ( 4 e x 1/20 h(x) 3 ex2/15 h(x) 5 ) ex3/18 (12) h(x) h(x) = ( e x1/20 e x2/15 + 2 e x2/15 e x3/18 + 3 e x1/20 e x3/18 ). (13) Then V (Y ) = (V 1 (Y ) V 2 (Y ) V 3 (Y )) (14) V 1 (Y ) = 20[ln(15y 1 ) ln( )] (15) V 2 (Y ) = 15[ln(20y 2 ) ln( )] (16) V 3 (Y ) = 18[ln(12y 3 ) ln( )]. (17) It follows that a 11 a 12 a 13 V (Y ) = a 21 a 22 a 23 a 31 a 32 a 33 a 11 = 160y 2 y 3 y 1 ( ) 20(5y 1 + 8y 3 ) a 12 = 20(9y 1 + 8y 2 ) a 13 = 15(5y 2 + 9y 3 ) a 21 = a 22 = 135y 1 y 3 y 2 ( ) 15(9y 1 + 8y 2 ) a 23 = 18(5y 2 + 9y 3 ) a 31 = 18(5y 1 + 8y 3 ) a 32 = a 33 = 90y 1 y 2 y 3 ( ). Consequently b 11 b 12 b 13 [ V (Y )] 1 = b 21 b 22 b 23 b 31 b 32 b 33 b 11 = 2y 1 y 2 y 3 5( ) y 1 y 2 (5y 1 + 8y 3 ) b 12 = 15( ) y 1 y 3 (9y 1 + 8y 2 ) b 13 = 18( ) y 1 y 2 (5y 2 + 9y 3 ) b 21 = 20( ) b 22 = 3y 1 y 2 y 3 5( ) y 2 y 3 (9y 1 + 8y 2 ) b 23 = 18( )

No. 6 A Non-symmetric Digital Image Secure Communication Scheme Based on Generalized Chaos Synchronization System 1117 Let y 1 y 3 (5y 2 + 9y 3 ) b 31 = 20( ) y 2 y 3 (5y 1 + 8y 3 ) b 32 = 15( ) b 33 = 5y 1 y 2 y 3 18( ). q(x Y ) = (q 1 (X Y ) q 2 (X Y ) q 3 (X Y )) T = (V 1 (Y ) x 1 V 2 (Y ) x 2 V 3 (Y ) x 3 ) G(Y X) = [ V (Y )] 1 [F (X) q(x Y )]. Then from Theorem 1 we obtain the responding system Ẏ = G(Y X). (18) Select the initial states of the systems (11) and (18) as follows: and X(0) = (x 1 (0) x 2 (0) x 3 (0)) = ( 1.4702 3.4971 3.0812) (19) Y (0) = (y 1 (0) y 2 (0) y 3 (0)) = (h 1 (X(0)) h 2 (X(0)) h 3 (X(0))). (20) The trajectories of the GCS system are shown in Fig. 1. It can be seen that the state variables X and Y are in GCS with respect to the transformation V. Fig. 1 Chaotic trajectories of the state variables. (a) x 1 x 2 and x 3; (b) y 1 y 2 and y 3; (c) the decoded trajectory; (d) the variables are in chaotic GCS via a transformation V 1. 2.3 Sinai Map Discrete Sinai map is a 2-dimensional chaotic system. The representation of Sinai map has the form x n+1 = x n + y n + α cos(2πy n ) mod (1) y n+1 = x n + 2y n mod (1) (21) α [0 1]. From the Sinai EQ. it follows that 0 x n 1 0 y n 1. Let (x(0) y(0)) = (0 0). (22) Then figures 2(a) 2(c) show the positions of the points (x n y n ) of the first 1000 iterations of Sinai maps with pa-

1118 ZHANG Xiao-Hong and MIN Le-Quan Vol. 44 rameters α = 0.01 0.30 and 0.99 respectively. It can be seen that the Sinai map with parameter α = 0.01 has an orbit with uniform density. Fig. 2 The orbits of Sinai maps for the first 1000 iterations. (a) α = 0.01 (b) α = 0.30 (c) α = 0.99. 2.4 Digital Image Secure Communication Scheme Popular digital images recognized by Matlab are classified into three kinds: (i) RGB images. An RGB image can be represented as an m n 3 data matrix m n stand for the rows and the columns of the pixels in the image and 3 represents 3 color planes red green and blue each plane with 256 levels of intensity denoted as (R G B). Such an image is called an RGB 24 bitmap. Since each color has an 8-bit binary system representation there are altogether 2 24 kinds of colors. (ii) Index images. An index image includes a data matrix X and a color map matrix P. P is an m 3 data matrix and 3 represents 3 color planes red green and blue with double accuracy in [0 1] of intensity. (iii) Intensity images. An intensity image is a gray scale image consisting of one gray scale intensity m n matrix. The gray scale intensity at each pixel (i j) is a double accuracy value in [0 1]. Now let us describe the image communication scheme for the case that the image to be sent is an RGB image. In the cases of other kind images the scheme is still effective after slightly amending. Assume that sender A has the systems (11) and (18) and receiver B holds the system (11) non-symmetric key K n = (V 1 V 2 ) defined by Eqs. (15) and (16). Both A and B have three Sinai maps (21) with system parameter α = 0.01 0.08 and 0.99. A and B share a symmetric keys K s = {k1 k 2 k 3 k 4 k 5 k 6 k 7 k 8 k 9 }. Here {k 1 k 2 k 3 } = {σ γ β} which are the parameters of the Lorenz equation; {k 4 k 5 k 6 } = {k x1(0) k x2(0) k x3(0)} which are real numbers to be used to encrypt the initial X(0) conditions of the Lorenz equation in our cryptographic algorithm; {k 7 k 8 k 9 } = {α x(0) y(0)} α x(0) and y(0) are the parameter and the initial condition of the Sinai map (21) respectively. The sender A and the receiver B hold a meeting before the remote transmission start and agree on iterate times 2N for the Lorenz equation and N for the Sinai map N = m n is the number of all the pixels of the image (m rows and n columns) received by B. Now sender A wants to transmit an RGB image M to receiver B. Denote the level of intensity of the 3 color planes as g(r) g(g) and g(b) respectively. The following procedure describes our secure communication scheme: (i) Sender A first uses the GCS systems (11) and (18) with initial conditions (19) and (20) to generate a chaotic trajectory (X(t) Y (t)) as shown in Fig. 1. Then the sender A uses the chaotic signal y 1 (t) to diffuse the intensity of the 3 color planes red green and blue by c 1 = 1000 + g(r) + round(kkk 1 reshap(y 1 (m n + 1 : 2m n) m n)) + round(kk 1 reshap(ys 1 m n)) + round(200 reshap(x 1 (m n + 1 : 2m n) m n) reshap(xs 1 m n)) + round(400 reshap(y 1 (m n + 1 : 2m n) m n) reshap(ys 2 + xs 3 m n)) c 2 = 500 + g(g) + round(kkk 2 reshap(y 1 (m n + 1 : 2m n) m n))

No. 6 A Non-symmetric Digital Image Secure Communication Scheme Based on Generalized Chaos Synchronization System 1119 + round(kk 2 reshap(ys 2 m n)) + round(150 reshap(x 2 (m n + 1 : 2m n) m n) reshap(xs 2 m n)) + round(300 reshap(y 1 (m n + 1 : 2m n) m n) reshap(ys 3 + xs 1 m n)) c 3 = 1000 + g(b) + round(kkk 3 reshap(y 1 (m n + 1 : 2m n) m n)) + round(kk 3 reshap(ys 3 m n)) + round(250 reshap(x 3 (m n + 1 : 2m n) m n) reshap(xs 3 m n)) + round(500 reshap(y 1 (m n + 1 : 2m n) m n) reshap(ys 1 + xs 2 m n)) (x 1 x 2 x 3 ) are the trajectory of Eq. (11) y 1 is the component of the trajectory of Eq. (18) and (xs i ys i ) (i = 1 2 3) are the trajectories of the Sinai maps with parameters α = 0.01 0.08 0.99 respectively. A B represents the cross product of two matrices A and B. kk 1 = 4.5e + 4 kkk 1 = 3e + 4 kk 2 = 4.8e + 4 kkk 2 = 2e + 4 kk 3 = 5e + 4 kkk 3 = 1.5e + 4. (23) Then sender A transforms these array data into a 16-bit RGB ciphertext image: (c 1 c 2 c 3 ) C = (g (R) g (G) g (B)). (ii) Sender A transmits the diffused initial condition X(0) + (k x1(0) k x2(0) k x3(0)) the ciphertext C and a tag σ = y 2 + C to receiver B. (iii) Receiver B uses the keys {k 4 k 5 k 6 } = {k x1(0) k x2(0) k x3(0)} to figure out the initial condition X(0) and then solves the system (11) and obtains the trajectory X. (iv) Receiver B uses the key K n and X to obtain y 1 y 2. (v) Receiver B calculates = σ y 2. If C then B stops decrypting otherwise goes to step (vi). (vi) Receiver B uses the inverse procedures of the steps 2 and 1 to decrypt the ciphertext and obtains the plaintext M. Remarks This scheme uses two chaotic systems to encrypt the plaintext. Therefore ciphertext has stronger ability against the attack than one chaos system based encryption scheme. Since the GCS system has been used the scheme has a data origin authentication function which is difficult to be realized in the chaos synchronization based scheme. Since a receiver only knows partially the diffeomorphism function V = (V 1 V 2 V 3 ) he cannot reconstruct the GCS responding system (18). Consequently as tools of encryption GCS systems have higher security than chaos synchronization systems. 3 Experiment Simulation Our scheme can be realized by using Matlab 6.1. We select a plain-image M: flowers.tif stored in \matlab6p1\ toolbox\images\imdemos. The picture M is a 8-bit RGB image. We use the scheme described in Sec. 2.4. Here N = m n = 362 500. Our simulation results are shown in Fig. 3. It can be seen that the original image flowers.tif (Fig. 3(a)) has been successfully changed into un-recognizable disorder codes (Fig. 3(b)). The decrypted image is shown in Fig. 3(c) which is almost the same as the original image that is the maximum error and the mean error between the levels of intensity of the color planes {g(r) g(g) g(b)} of the original image Fig. 3(a) and the decrypted image (see Fig. 3(c)) are 1 and 7.1823 10 5 respectively. We have also used the scheme to encrypt and decrypt some other kinds of digital images which can be recognized by computer (For examples:.bmp.jpg.jpeg.tif.tag.gif.hdf.png.xwd.pcd.ico.cur). The similar results are

1120 ZHANG Xiao-Hong and MIN Le-Quan Vol. 44 obtained. This implies that our GCS-based encryption scheme has universality. Fig. 3 (a) Original image; (b) Encrypted image; (c) Decrypted image. 4 Security Analysis A good encryption scheme should resist all kinds of known attacks. In the following subsections key space analysis and statistical analysis are given. The results show that our scheme has satisfactory security. 4.1 Key Space and Sensitivity Analysis Key space The symmetric key used in the scheme is double precision numbers. We take that K s = {k 1 k 2 k 3 k 4 k 5 k 6 k 7 k 8 k 9 } to be decimals with 16 characters long. Consequently our symmetric key space size is 10 135 > 2 448. Additionally the space size of our non-symmetric key K n = (V 1 V 2 ) is uncountable. Therefore the key space of our scheme should be large enough for the requirements of secure communications. Key sensitivity test The key sensitivity test is proceeded as follows. (i) The image flowers.tif is encrypted by using the key K s as above. (ii) The key K s is slightly perturbed in nine fashions: K s ( i ) = K s + [0 0... δ }{{} i 0... 0] (24) i i = 1 2... 9 δ i = 10 15. (iii) Using the four of the nine perturbed keys encrypts the image flowers.tif respectively. (iv) Compare the five encrypted images each other. Table 1 lists the percentages of numbers of pixels with different RGB values in the images encrypted via two keys and the means of different RGB values in the images. (v) Compare the image flowers.tif with the decrypted flowers.tif image G ij which is encrypted by key K( i ) but decrypted via key K( j ). The comparison results are given in Table 2 in which the two values at position (K s ( i ) K s ( j )) stand for (a) The percentages of the numbers of the pixels with different RGB values in the flowers.tif and G ij ; (b) The mean of the different RGB values in the two images. Table 1 Comparisons between two images encrypted by keys K s( i) and K s( j). %/mean K s K s( 1 ) K s( 4 ) K s( 7 ) K s( 8 ) K s 0%/0 99.98%/2417 99.98%/2257 99.98%/15996 99.99%/16003 K s( 1 ) 99.98%/2417 0%/0 99.98%/2407 99.99%/16214 99.99%/16219 K s( 4 ) 99.98%/2257 99.99%/2407 0%/0 99.99%/16195 99.99%/16219 K s( 7 ) 99.98%/15996 99.99%/16214 99.99%/16195 0%/0 99.99%/15995 Table 2 Comparisons between flowers.tif and G ij. %/mean K s K s( 1 ) K s( 4 ) K s( 7 ) K s( 8 ) K s 0%/0 99.26%/122.52 99.26%/124.10 99.24%/126.98 99.24%/127.07 K s( 1 ) 99.26%/127.21 0%/0 99.29%/124.32 99.26%/127.06 99.26%/127.21 K s( 4 ) 99.15%/122.48 99.19%/122.74 0%/0 99.25%/127.04 99.25%/127.14 K s( 7 ) 99.25%/127.07 99.26%/127.08 99.26%/127.14 0%/0 99.25%/127.06 K s( 8 ) 99.25%/127.01 99.26%/126.93 99.26%/127.00 99.24%/127.03 0%/0

No. 6 A Non-symmetric Digital Image Secure Communication Scheme Based on Generalized Chaos Synchronization System Discussions (i) Table 1 shows that encrypted images are very sensitive to perturbations of the keys k1 k4 k7 k8. In fact our computer simulation shows that encrypted images are also very sensitive to perturbations of the other 5 keys. (ii) Table 2 shows that decrypted images are very sensitive to perturbations of keys k1 k4 k7 k8. Any attempt that wants to use sightly modified keys to decrypt the encrypted images completely fails. The similar results can also be obtained for the perturbations of other 5 keys. (iii) Figures 4(a) 4(c) are the images that are encrypted by the same key Ks but decrypted via keys 1121 Ks ( 1 ) Ks ( 4 ) Ks ( 7 ) Ks ( 8 ). Table 2 shows that in these images the colors of about 80 /00 pixels are recovered correctly only. Figure 4 illustrates graphically the recovered colors of 80 /00 pixels are useless for recognizing the original image flowers.tif. (iv) We have also found that almost the same results can be obtained if using lager modified keys (10 1 perturbation) decrypts the images encrypted by unmodified one. The results imply that an intruder cannot judge from his decrypted images which one of his two approximate keys is closer to the real key even though one key is different from the real key by 10 1 and another by 10 15 only. Fig. 4 Images generated via different encrypting keys and decrypting keys. (a) K( 0 ) and K( 1 ); (b) K( 0 ) and K( 4 ); (c) K( 0 ) and K( 7 ); (d) K( 0 ) and K( 8 ). In summary the sensitivity of key for decrypted images makes intruders difficult to decrypt ciphertexts. The results imply that intruders may not recognize two encrypted images to be the same one if sender uses slightly different initial values (X(0) + 10 15 ). Consequently our scheme may have the task similar to one-time pad scheme. 4.2 Statistical Analysis First let us compare the histograms between the plain-image (original image) and the encrypted image. Select the original color image flowers.tif shown in Fig. 3(a). The histograms of the intensity level of its three color planes are shown in the first row in Fig. 5. The graphs given in the second row in Fig. 5 are the histograms of the intensity level

1122 ZHANG Xiao-Hong and MIN Le-Quan Vol. 44 of the encrypted image shown in Fig. 3(b). It can be seen that the histograms of the ciphered images are fairly uniform and are significantly different from those of the plain-image. Fig. 5 RGB histograms. The first line is the RGB histograms of the original images; The second line is the RGB histograms of the encrypted images by the key K s. Secondly similar to the study given in Ref. [16] we discuss the correlations of two adjacent pixels in the original image flowers.tif and encrypted images. To test the correlation between two vertically adjacent pixels in a ciphered image the following procedure is carried out. First select all pairs of two adjacent pixels from the image flowers.tif. Then calculate the correlation coefficient of each pair by using the following formulas (25) (32). Our statistical analysis implies that the encryption algorithm posed here has superior confusion and diffusion properties which can strongly resist statistical attacks. Let P be an image with dimension m n the image grayscales of the color planes (red green and blue) at the position (i j) are denoted by x 1 ij x 2 ij x 3 ij respectively. The corresponding gray matrices are written as g x (R) g x (G) g x (B). Denote y 1 ij (= x 1 ij+1 ) y 2 ij (= x 2 ij+1 ) y 3 ij (= x 3 ij+1 ) as the gray-scales of the three color planes of the pixel which are on the right side of the pixel at the position (i j). Denote E(X k ) = E(Y k ) = D(X k ) = 1 m (n 1) 1 m (n 1) 1 m (n 1) m n 1 x k ij (25) i=1 j=1 m n 1 y k ij (26) i=1 j=1 m n 1 (x k ij E(X k )) 2 (27) i=1 j=1

No. 6 A Non-symmetric Digital Image Secure Communication Scheme Based on Generalized Chaos Synchronization System 1123 D(Y k ) = cov(x k Y k ) = 1 m (n 1) m n 1 (y k ij E(Y k )) 2 (28) i=1 j=1 1 m (n 1) m n 1 (x k ij E(X k )) i=1 j=1 (y k ij E(Y k )) k = 1 2 3. (29) Then the correlation coefficients of the gray matrices g x (R) g x (G) g x (B) can be defined respectively by r R = cov(x 1 Y 1 ) D(X1 )) D(Y 1 ) (30) r G = r B = cov(x 2 Y 2 ) D(X2 )) D(Y 2 ) (31) cov(x 3 Y 3 ) D(X3 )) D(Y 3 ). (32) Figure 6 shows the correlation distribution of two vertically adjacent pixels in the image flowers.tif and that in the cipherimage for three color planes. The correlation coefficients are 9.3704 10 6.7135 10 3 9.2080 10 1 7.2084 10 3 9.3449 10 1 and 7.6896 10 3 respectively which are remarkably different from each other. Fig. 6 Correlations of two vertically adjacent pixels in the original images (a) (b) (c) and in the encrypted images (d) (e) and (f). 5 Concluding Remarks GCS system provides a new tool for secure communication. Based on a recent GCS theorem 1 [20] and the well-known Lorenz equation this paper constructs a GCS system. Combing the GCS system and the Sinai map a non-symmetric digital image encryption scheme is proposed. The scheme has a data origin authentication function and large key space. The sensitivity of the keys and the uniformity of the histograms of encrypted images imply that the new secure communication scheme is a promising candidates for images transmission. This scheme may be suitable for network image encryption.

1124 ZHANG Xiao-Hong and MIN Le-Quan Vol. 44 Acknowledgments The authors would like to express sincere gratitude to Prof. Guan-Rong Chen at the City University in Hong Kong for providing them references on chaos-based secure communications. References [1] L.M. Pecora and T.L. Carroll Phys. Rev. Lett. 64 (1990) 821. [2] L.J. Kocarev K.S. Halle K. Eckert L.O. Chua and U. Parliz Int. J. Bifurcation Chaos 2 (1992) 709. [3] U. Parliz L.O. Chua L.J. Kocarev K.S. Halle and A. Shang Int. J. Bifurcation Chaos 2 (1992) 973. [4] L.M. Cuomo and A.V. Oppenheim Phys. Rev. Lett. 71 (1993) 65. [5] C.W. Wu and L.O. Chua Int. J. Bifurcation and Chaos 3 (1993) 1619. [6] L. Kocarev and U. Parlitz Phys. Rev. Lett. 74 (1995) 5028. [7] L. Kocarev and U. Parlitz Phys. Rev. Lett. 76 (1996) 1816. [8] U. Feldmann M. Halser and W. Schwarz Int. J. Circuit Theory Appl. 24 (1996) 551. [9] T. Yang and L.O. Chua Int. J. Bifurcation Chaos 6 (1996) 2653. [10] T. Yang and L.O. Chua Int. J. Bifurcation Chaos 7 (1997) 645. [11] K. Murali and M. Lakshmanan Phys. Lett. A 241 (1998) 302. [12] J. Kawata Y. Nishio H. Dedieu and A. Ushida IEICE Trans. Fund. E 82-A (1999) 1322. [13] L. Kocarev IEEE Circuits and Syst. I: Fundam. Theory Appl. 48 (2001) 1498. [14] T.L. Carroll Phys. Rev. E 67 (2003) 026207. [15] F.C.M. Lau and C.K. Tse Chaos-Based Digital Communication Systems Springer (2003). [16] G. Chen Y. Mao and C.K. Chui Chaos Solitons and Fractals 21 (2004) 749. [17] W. Ye Q. Dai S. Wang et al. Phys. Lett. A 330 (2004) 75. [18] D. Zhao G. Chen and W. Liu Chaos Solitons and Fractals 22 (2004) 47. [19] T. Yang and L.O. Chua Int. J. Bifurcation Chaos 6 (1996) 215. [20] X. Zhang and L. Min J. Univ. of Sci. and Tech. Beijing 7 (2000) 224. [21] R. John and D. Gregory Chaos Solitons and Fractals 12 (2001) 145. [22] J. Lu and Y. Xi Chaos Solitons and Fractals 17 (2003) 825. [23] X. Zhang L. Zhang and L. Min Chin. Phys. Lett. 20 (2003) 211. [24] L. Min X. Zhang and M. Yang J. Beijing Univ. of Sci. and Tech. 10 (2003) 75. [25] L. Min G. Chen X. Zhang et al. Commun. Theor. Phys. (Beijing China) 41 (2004) 632. [26] J. Xu L. Min and G. Chen Chin. Phys. Lett. 21 (2004) 1445.