Encryption system of digital signals using synchronization of chaos

ENCRYPTION SYSTEM OF DIGITAL SIGNALS

USING SYNCHRONIZATIN OF CHAOS

0003Technical Field

0004The present invention relates to an encryption system of digital signals using synchronization of chaos, and more particularly, to an encryption of digital signals using synchronization of chaos wherein a plurality of chaotic devices which generate signals with chaotic characteristics (hereinafter, referred as to "chaotic signals") are modulated by noise signals and the chaotic devices are synchronized each other.

0005Background Art

0006Recently, a number of researches have been vigorously made to apply "Chaos Theory" to various industrial fields. Since the chaotic devices evolving chaotically display sensitivity to initial conditions, when two substantially identical devices start with slightly different initial conditions, two identical quickly evolve to values with different trajectories which are vastly different and become totally uncorrelated as time evolves. This makes chaotic devices nonperiodic and unpredictable over long times. The phenomenon is due to the sensitivity to initial conditions which is called Butterfly Effect. In a chaotic system with the master chaotic device and the slave chaotic device, synchronization means that state variables of the master chaotic device become identical to state variables of the slave chaotic device to control the chaotic phenomenon. However, such chaotic devices are impossible to synchronize by conventional methods. Thus, new numerous methods have been proposed and developed to synchronize signals of nonlinear dynamical devices and also to apply the synchronized chaotic devices to secure communication.

0007Considering known prior arts, methods are described in papers by Louis M. pecora and Thomas L. Carrol entitled "Synchronization in Chaotic System" (PHYSICAL REVIEW LETTERS, Vol.4 No.8, p.821, 1990) and entitled "Synchronizing Chaotic Circuits"(IEEE TRANSACTIONS CIRCUIT AND SYSTEMS, p.453, Apr. 1991). These articles disclose a theory of synchronizing two chaotic devices and describe a circuit demonstrating such synchronization. Also, U.S. Pat. No. 5,245,660 to Pecora and Carroll discloses a system for producing synchronized signal.

0008FIG. 1 shows the synchronization concept disclosed in U.S. Pat. No. 5,245,660 to Pecora and Carroll. Referring to FIG. 1, a primary system as a master chaotic device is divided into first subsystem 2 as a drive signal generator and second subsystem 3. A new subsystem 3' identical to the subsystem 3 is linked with the primary system 1, there forming a response subsystem 1 ' as a slave chaotic device. The master and slave devices construct an overall chaotic system. The driving output signal X4 of the first subsystem 2 is transmitted to the second subsystem 3 and response subsystem 3' to synchronize the second subsystem 3' wherein the variables XI ', X2', X3' of the response subsystem 3' correspond to the variables XI, X2, X3 of the second subsystem. As a result, the variables XI ', X2', X3', X4' of the slave chaotic device and the variables XI, X2, X3, X4 of the master chaotic device are in synchronization with each other.

0009On the other hand, synchronization in chaotic system has high potentiality of practical applications in secure communication, optics, and nonlinear dynamicsw model identification. Specially, the secure communication using a synchronizing system is disclosed in U.S. Pat. No. 5,291,555 to Cuomo and Oppenheim which employs the synchronizing concept of Pecora and Carroll thereto.

0010FIG. 2 shows the communication system disclosed in U.S. Pat. No. 5,291,555 to Cuomo and Oppenheim. The communication system comprises a chaotic transmitter 10 including a drive signal generator 12 for producing a chaotic drive signal u(t) and an adder 14 for adding message signal m(t) to the drive signal u(t) to produce a transmitted signal, and a receiver 20 for receiving the transmitted signal including a drive signal regenerator 22 for reconstructing the drive signal u'(t) from the received signal u(t)+m(t), and a subtracter 24 for subtracting the reconstructed drive signal u'(t) from the received signal u(t)+m(t) to detect therefrom message signal m'(t).

0011However, the aforedescribed known prior art of Pecora and Carroll has the drawback that the overall chaotic system including the master and slave devices has a strong tendency to easy synchronization in spite of somewhat different parameters between the master and slave devices since the drive signal of the driving generator is without any conversion inputted into the response subsystem to synchronize two chaotic systems. Explaining it any other way, when the subsystem in satisfied with the synchronizing condition proposed by Pecora and Carroll that the Lyapunov exponents of the subsystem are all negative, the overall system may be easily synchronized even though the parameters of the circuit elements constructing the response system are variable to a certain drgree, for example, 20 percentage. Thus, the prior art of Cuomo and Oppenheim employing the synchronizing concept of Pecora and Carroll also has the drawback that the transmitted information signals may be wiretapped since it is relatively easy to reproduce the communication apparatus on account of the strong synchronization tendency.

0012On the other hand, there is a trend that all the data are processed by digital signals in accordance with the development of digital technique. Therefore, a number of researches have been actively made to encode information signals of digital signals and various systems for encoding the digital signals are developed.

0013Disclosure of the Invention

0014Accordingly, the present invention is conceived to solve the above problems in the prior art. An object of the present invention is to provide an encryption system of digital signals using synclironization of chaos wherein a plurality of chaotic devices which generate signals are modulated by noise signals and the chaotic devices are synchronized each other.

0015In order to accomplish the above object, an encryption system of digital signals using synchronization of chaos comprises a transmitter including a master chaotic device for generating first chaotic signals with chaotic characteristics modulated by noise signals, state variables of the master chaotic device being functionally interrelated, a first integer modulation part for modulating the first chaotic signals to bit signals of integer type, a first inversion and transposition part for inverting and transposing the first chaotic signals modulated by the first integer modulation part, a first code modulation part for modulating the first chaotic signals outputted from the first inversion and transposition part to ASCII code, a first logical operation part for generating encryption signals of information signals by logically operating the first chaotic signals of the ASCII code and the information signals on a bit-by-bit basis, a mixing part for mixing the encryption signals outputted from the logical operation part with the noise signals and transmitting mixed signals; and a receiver including a separation part for separating the encryption signals and noise signals from the mixed signals and outputting the encryption signals and the noise signals, a slave chaotic device identical to the master chaotic device for generating second chaotic signals corresponding to the first chaotic signals with chaotic characteristics modulated by the noise signals, state variables of the master chaotic device being functionally interrelated, a second integer modulation part for modulating the second chaotic signals to bit signals of integer type, a second inversion and transposition part for inverting and transposing the second chaotic signals modulated by the second integer modulation part, a second code modulation part for modulating the second chaotic signals outputted from the first inversion and transposition part to the ASCII code, a second logical operation part for logically operating the second chaotic signals of the ASCII code and the encryption signals outputted from the separation part on the bit-by-bit basis and decoding the information signals.

0016Brief Description of the Drawings

0017The above and other objects, features and other advantages of the present invention will be more clearly understood from the following detailed description taken in conjunction with the accompanying drawings, in which: FIG. 1 is a block diagram illustrating synclironization concept according to the prior art proposed by Pecora and Carroll;

0018FIG. 2 is a block diagram of the communication system using the synchronized chaotic system according to the prior art;

0019FIG. 3 is an operational diagram explaining synchronization concept of a synchronized chaotic system according to the present invention;

0020FIG. 4 is an operational diagram explaining an encryption apparatus of an encryption system of digital signals according to the present invention in which noise signals are feedbacked to state variables of a chaotic device;

0021FIG. 5a to FIG. 5d are graphs showing correlations between the noise signals and the chaotic signals in a logistic chaotic system, FIG. 5a is a graph showing a correlation of the noise signals itself, FIG 5b is a graph showing a correlation of the logistic map itself, FIG. 5c is a graph showing a mutual correlation of the noise signals and the chaotic signals, FIG. 5d is a graph showing a correlation of the chaotic signals themselves; FIG. 6 is an operational diagram explaining an decryption apparatus apparatus of the encryption system of digital signals according to the present invention in which noise signals are feedbacked to state variables of a chaotic device;

0022FIG. 7 are wave form charts of the master and slave chaotic devices when the master and slave chaotic devices are not in synchronization with each other, FIG.

00237a is a wave form chart of one variable y_n of the master chaotic device, FIG. 7b is a wave form chart of one variable y'_n of the slave chaotic device, FIG. 7c is a wave form chart of the difference y_n-y'_n between two variables;

0024FIG. 8 are wave form charts of the master and slave chaotic devices when the master and slave chaotic devices are in synchronization with each other, FIG. 8a is a wave form chart of the variable y_n of the master chaotic device, FIG. 8b is a wave chart of the variable y'_n of the slave chaotic device, FIG. 8c a wave form chart of the difference y_n-y'_n between two variables; and

0025FIG. 9a is a shape of phase space of one variable y'_n of the slave chaotic device one variable y_n of the master chaotic device when the master and slave chaotic devices are not in synchronization with each other, FIG. 9b is a shape of phase space of one variable y'_n of the slave chaotic device one variable y_n of the master chaotic when the master and slave chaotic devices are in synchronization with each other.

0026Best Mode for Carrying Out the Invention

0027Hereinafter, a preferred embodiment of an encryption system of digital signals using synchronization of chaos according to the present invention will be described in detail with respect to the accompanying drawings. In general, mathematical models of chaotic systems often involve two types of systems, that is, the systems expressed as difference equations and the systems expressed as differential equations. The systems of the differential equations include a well known Lorenz system, a Rossler system, a Duffing system, and etc. The logistic map is well known as a system of the difference equation. The chaotic systems are functionally defined in terms of state variables which are used to construct a main electronic circuit in the chaotic system. For example, the electronic circuit corresponding to well known Lorenz system is disclosed in U.S. Pat. No. 5,291,555. and can be expressed as the following equation (1) u = σ(v — u) v = γu - v - 20uw w = 5uv - bw (1)

0028The electronic circuits corresponding to Rossler and modified Duffing systems are disclosed in U.S. Pat. No. 5,402,334, and an encryption technology using the logistic map is disclosed in U.S. Pat. 5,048,086. Those skilled in the art can easily construct electronic circuits in correspondence to arbitrary chaotic systems defined in terms of the state variables. Thus, the detail explanation of construction of the electronic circuits is not omitted in here.

0029Referring to FIG. 3, a master chaotic device 30 is given as n-dimensional state variables x, y, z,.... and a slave chaotic device 40 identical to the master chaotic device is given as n-dimensional state variables x', y', z',.... corresponding to the n-dimensional state variables x, y, z The master chaotic device 30 is synchronized with the slave chaotic device 40 by modulating at least one arbitrary variable, for example x, of the master chaotic device and at least one arbitrary variable, for example x', of the slave chaotic device corresponding to the variable of the master chaotic device by means of external signals as noise signals, chaotic signals, random numbers(ξ_n), etc. (hereinafter, referred as to "noise signals") and feedbacking the modulated signals to the master chaotic device 30 and the slave chaotic device 40. On the other hand, in order to synchronize the master chaotic device with the slave chaotic device, the parameters of the master and slave chaotic devices may be modulated by external noise signals or the noise signals may be applied to the chaotic devices as external forces.

0030Referring to FIG. 3, as mentioned above, one chaotic system comprises two chaotic devices. A first synchronizing part 50 and a second synchronizing part 60 are provided to synchronize the master chaotic device 30 and the slave chaotic device 40, respectively. The first synchronizing part 50 includes a first sealer 51 for scaling the noise signals (ξ_n) by a first scaling factor(α), a second sealer 53 for scaling one arbitrary state variable x_n of the master chaotic device 30 by a second scaling factor(β), and an adder 54 for adding the noise signals(αξ_n) scaled by the first sealer 51 to the output signals(βx_n) of the second sealer 53 and feedbacking the added signals to the master chaotic device 30. Also, the second synchronizing part 60 identical to the first synchronizing part 50 includes a first sealer 61 for scaling the noise signals(ξ_n) by the first scaling factor(α), a second sealer 63 for scaling one state variable x'_n of the slave chaotic device 40 corresponding to the state variable of the master chaotic device 30 by the second scaling factor(β), and an adder 64 for adding the noise signals(αξ_n) scaled by the first sealer 61 to the output signals(βx_n) of the second sealer 63 and feedbacking the added signals to the slave chaotic device 40.

0031Generally, since the chaotic devices display sensitivity to initial conditions, when the initial conditions of the two identical chaotic devices 30, 40 are not identical to each other, the trajectory of the state variable x_n of the master chaotic device 30 is completely different from the trajectory of the state variable x'_n of the slave chaotic device 40 as time evolves. Therefore, the master chaotic device 30 and the slave chaotic device 40 will have different trajectories as time evolves without the first synchronizing part 50 and the second synchronizing part 60. On the other hand, the master chaotic device 30 and the slave chaotic device 40 will be in synchronization with each other and have identical trajectories by means of the first synchronizing part 50 and the second synchronizing part 60 and the external noise signals. In other words, the corresponding variables x_n and x'_n , y_n and y'_n, z_n and z'_n,.... Of the master and slave chaotic devices 30 and 40 will have identical trajectories.

0032As described above, when the master chaotic device 30 is synchronized with the slave chaotic device 40 by the first synchronizing part 50 and the second synchronizing part 60, resulting in x_n = x'_n, y_n = y'_n, z_n = z'_n That is, the values of the state variables of the master chaotic device 30 becomes identical to those of the corresponding state variables of the slave chaotic device 40. In order to express the synchronizing method according to the present invention as mathematical equations, the synchronized state variable in the master chaotic device 30 is given as x_n and the synchronized state variable in the slaver chaotic device 40 is given as x'_n. Also,' The external noise signal is given as ξ_n. Then, the signal feedbacked to the master chaotic device 30 is x_n=αξ_n+βx_n and the signal feedbacked to the slave chaotic device 40 is x'_n=αξ_n+βx'_n, Thus, the feedbacked signals are substituted with each state variable of the master and slave chaotic devices 30,40.

0033According to the present invention, the information signals are added to the signals from the master chaotic device and the mixed signals are transmitted together with the noise signals to the slave chaotic device. Then, the noise signals are used to synchronize the slave chaotic device with the master chaotic device, the difference between the signals of the master chaotic device and the signals of slave chaotic device becomes the information signals. As a result, the two master and slave chaotic devices can be employed as a encryption system, At this time, when the exponents of the state variables and the parameters of the master chaotic device are different those of the slave chaotic device, the master and slave chaotic devices are in synclironization with each other. Therefore, the exponents of the state variables and the parameters may be used as keys for encryption. Since the values of the keys may be real numbers and the number of the keys is numerous.

0034Next, the synchronization of the two chaotic devices using the logistic map will be described. The logistic map is given as the following equation (2). x_n+ι = λx_n(l-x_n) (2)

0035In this equation (2), the chaos is determined in accordance with values of λ. For example, when λ=3.9, The device will show chaotic characteristics. When the slave chaotic device is given as the following equation (3), the synclironization of two chaotic devices which are synchronized by random numbers will be described. x'_n= λx'_n(l-x'_n) (3)

0036A random number γ_n is feedbacked to the master and slave chaotic devices. Then, the equations (2) and (3) becomes the following equations (4) and (5), respectively.

0037X_n+ι = λx_n(l-X_n) γ_n (4) x'_n+ι = λx'_n(l-x'_n) +γ_n (5)

0038When the random number ξ_n is a value between 0 and 1, and the random number γ_n is a value between 0 and β, the random number feedbacked to the two chaotic devices γ_n =βξ_n and the equations (4) and (5) are in synchronization with each other. To show the synchronization of the equations (4) and (5), when x_n+- - γ_n_ i =αy_n and x_n+ι - γ_n - y_n+ι by the Homeomorphism, since x_n+1 = γ_n_ι +αy_n, x_n+1 = y_n._x +«*y_n, The equations (4) and (5) becomes the following equation (6). αy_n+ι + γ_n = λ(αy_n + γ_n- (l-(αy_n + γ_n- ) + γ_n αy'n+i + Yn = λ(αy'„ + γ_n.ι) (l-(αy'_n+ γ„-ι)) + Yn (6)

0039In the equation (6), all the γ_n are removed, and both sides are divided by α. Also, γ_n is substituted with βξ_n and λ/αis substituted with μ. Then, the absolute value is adopted to protect divergence, and the modulus 1 is adopted to maintain the value under 1. Thus, the equation (6) becomes the following equation (7) y_n+ι = I μ (αy_π+ βξ_n- (l-(αy_n+ βξ_n- ) | modi y'_n+ι = I μ ( y'_n+ βξ_n-ι) (l-(αy'_n+ βξ_n- ) I modi (7)

0040To confirm the synchronization of the equation (7), when the difference of the variables of two equations of the equation (7) y_n- y'_n = z_nthe difference of the two equations is given as the following equation (8). z_n+1 = μα (l-βξ_n - 2αy_n) z_n +μα²z²_π (8)

0041This equation shows a nonlinear difference equation which defines a new chaotic system. In the equation (8), there are values which are modulated by y_n and ξ_n as the parameters of z_n. Since the methods for modulating the nonlinear systems by the noise signals or the chaotic signals are well lαiown, the detailed descriptions thereof are omitted in here.

0042Analyzing the phenomena of the nonlinear system in which the parameters are inverted in terms of the chaotic signals or noise signals, the system assumes very complicated phenomena. The inverted system irregularly oscillates from the chaotic signals to the value very close to zero, or converges to zero, or appears chaotic. The oscillation form the chaos to the value very close to zero is defined as on-off intermittency. There is a critical value condition that the system including the variable differences of a master chaotic device and a slave chaotic device generates infinite period of laminar phase which is connected with on-off intermittency. At a value above the critical value α_c, the new chaotic device immediately converges to zero. Accordingly, the new chaotic device is synchronized with the master chaotic device since the differences between two identical variables become zero. That is, when α > α_c, the devices generate infinite period of laminar phase and the master chaotic device and the slave chaotic device are in synchronization with each other Comparing the results according to the present invention with the result of

0043Pecora-Carroll synchronization, it is easily understood what the features of the present invention is. In the synchronizing method according to Pecora-Carroll, although the synchronization is carried out by the synchronization condition that the

0044Lyapunov exponents of the slave chaotic device are all negative, this is not the synchronization phenomena which carried out by the critical value condition that the system including the variable differences of a master chaotic device and a slave chaotic device generates infinite period of laminar phase which is connected with on-off intermittency.

0045The encryption system using the abovementioned synchronization method according to the present invention includes an encryption apparatus shown in FIG. 4, and a decryption apparatus shown FIG. 6

0046In order to explain the encryption apparatus, referring to FIG. 4, the state variables of the master chaotic device 30 is functionally interrelated. The noise signals ξ_n are inputted into the master chaotic device 30 and the master chaotic device 30 generates first chaotic signals. A first integer modulation part 70 is connected to the master chaotic device 30 to modulate the first chaotic signals to bit signals of integer type and a first inversion and transposition part 80 is connected to the first integer modulation part 70 to invert and transpose the first chaotic signals. A first code modulation part 90 is connected to the first inversion and transposition part 80 to modulate the first chaotic signals outputted from the first inversion and transposition part 80 to ASCII code. A first exclusive OR gate 100 is connected to the first code modulation part 90 to generate encryption signals of information signals. . The first exclusive OR gate 100 performs an exclusive OR operation to logically operate the first chaotic signals of the ASCII code and the information signals on a bit-by-bit basis. A mixing part 110 is connected to the first exclusive OR gate 100 to mix the encryption signals outputted from the first exclusive OR gate 100 with the noise signals and transmit the mixed signals. The mixing part 110 is an exclusive OR gate.

0047Next, the operation of the enciyption apparatus as described above is explained

0048When the noise signals ξ_n are inputted into the master chaotic device 30 and the one variable x_n of the master chaotic device 30 shows very unsteady signals. However, the variable x_n is unrelated to the noise signals ξ_n. Examining the correlation between the noise signals and the variable signal, FIG. 5a to FIG. 5d are graphs showing correlations between the noise signals and the chaotic signals in a logistic chaotic system. . FIG. 5a is a graph showing a correlation of the noise signals itself, FIG 5b is a graph showing a correlation of the logistic map itself, FIG. 5 c is a graph showing a mutual correlation of the noise signals and the chaotic signals, FIG. 5d is a graph showing a correlation of the chaotic signals themselves. From FIG. 5a to FIG. 5d, it can be seen that there is hardly correlation between the noise signals and the chaotic signals.

0049According to the present invention, the equation of the master chaotic device 30 when the noise signals ξ_n are inputted into the master chaotic device 30 is given as the following (9). x_n+1 = λ[αξ_n+ βx' J (l-[λ[αξ_n+ βx' J ] (9)

0050The variable signal x_n of the equation (9) is modulated to the bit signal of integer type by the first integer modulation part 70, the modulated bit signal is inverted and transposed by the inversion and transposition part 80.

0051The first code modulation part 90 modulates the first chaotic signals outputted from the first inversion and transposition part 80 to ASCII code and operates the chaotic signals on the bit-by-bit-basis. The exclusive OR gate 100 operates the first chaotic signals of ASCII code and the information signals. At this time, the operated information signals are changed to bit signals as digital signals. Namely, the chaotic signals are changed to bit signals and the information signals are encrypted by the first exclusive OR gate 100. The encrypted information signals are mixed with the noise signals by the mixing part 110 and the mixed signals are transmitted to a receiver. The transmitted encryption signals are given as the following equation (10).

0052Yn= Xn Θ S _n (10) Here, Θ means the exclusive OR gate.

0053Next, the operation for decrypting the encryption information signals will be described with reference to FIG. 6.

0054Referring to FIG. 6, a separation part 200 separates the encryption signals and the noise signals ξ_n from the mixed signals which are received from the encryption apparatus of FIG. 4. The slave chaotic device 40 is connected to the separation part 200. The state variables of the slave chaotic device 40 is also functionally interrelated. The noise signals ξ_n are inputted into the slave chaotic device 40 and the slave chaotic device 40 generates second chaotic signals. A second integer modulation part 210 is connected to the slave chaotic device 40 to modulate the second chaotic signals to bit signals of integer type and a second inversion and transposition part 220 is connected to the second integer modulation part 210 to invert and transpose the second chaotic signals. A second code modulation part 230 is connected to the second inversion and transposition part 220 to modulate the second chaotic signals outputted from the second inversion and transposition part 220 to ASCII code. A second exclusive OR gate 240 is connected to the second code modulation part 230 to decrypt the information signals from the encryption signals. The second exclusive OR gate 240 performs an exclusive OR operation to logically operate the second chaotic signals of the ASCII code and the information signals on a bit-by-bit basis. Next, the operation of the decryption apparatus as described above is explained.

0055The separation part 200 receives the signals transmitted from the encryption apparatus of the transmitter and separates the noise signals and the encryption signals from the received signals. The noise signals ξ_n are inputted into the slave chaotic device 40 and the enciyption signals are inputted into the second exclusive OR gate 240. Thus, the slave chaotic device 40 of the receiver is synchronized with the master chaotic device 30 of the transmitter by the noise signals ξ_n. The equation of the slave chaotic device 40 to which the noise signals ξ_n are applied is given as the following equation (11). x i = λ[x' _n + α(ξ_n-x' „)] (l-[ x'_n +(αξ_n - x' _n)]) (11)

0056The second chaotic signals x'_n of the slave chaotic device 40 of the receiver are different from the first chaotic signals x_n of the master chaotic device 30 of the transmitter at initial stage. However, the slave chaotic device 40 is immediately synchronized with the master chaotic device 30, resulting in x'=x_n. When the slave chaotic device 40 is synchronized with the master chaotic device 30 by the noise signals x_n, the second exclusive OR gate 240 operates the chaotic signals outputted from the second integer modulation part 210, the second inversion and transposition part 220 and the second code modulation part 230 and the encryption signals outputted from the separation part 200. This procedure is expressed as the following the equation (12). s„ = γ„ 0 ^χ' „ (12)

0057Here, s_n is decrypted information signal, γ_n is the encryption signal transmitted from the encryption apparatus, and x'_n is the chaotic signal synchronized in the slave chaotic device 40. The explanations as mentioned above are limited to the logistic map.

0058However, the present invention can be applied to many complex systems. Thus, the encryption becomes more complex. When the chaotic devices have a number of variables and is complex, the equation is given as the following equation (13).

0059X n+l ^— IRX n. X ID J X n) X²n+ 1 = f2(x¹n, X²n. , X ø)

0060<img file="WO0217536A1_D0001.tif" />

0061The chaotic device according to the equation (13) has / variables. In equation (13), some variables to which different scaling factors are applied are selected and the noise signals are inputted into the chaotic device. Thus, all the variables are different from the noise signals and one variable is also different from the other variables. The number of encryption bits corresponds the number of the selected variables times the number of the bits of one variable. For example, when the number of the one variable is 128 and 15 variables are selected, the number of encryption bits which is processed at a time is 1920. This number of encryption bits may be changed without limit. According to the present invention, the encryption is very complex and it is impossible to decrypt the information signals.

0062FIG. 7 are wave form charts of the two chaotic devices modulated by the noise signals are in synchronization with each other in the coupled map. FIG. 7a is a wave form chart of an arbitrary variable of the master chaotic device shown in FIG. 4, FIG 7b is a wave form chart of an arbitrary of the slave chaotic device, and FIG. 7c is a wave form chart of the difference between the variable of the master chaotic device and the variable of the slave chaotic device. FIG. 8 are wave form charts of the master and slave chaotic devices when the master and slave chaotic devices are in synchronization with each other, FIG. 8a is a wave form chart of the variable y_n of the master chaotic device, FIG. 8b is a wave chart of the variable y'_n of the slave chaotic device, FIG. 8c is a wave form chart of the difference y_n-y'_n between two variables. Here, FIG. 8c shows 10000 times amplification of a real wave form chart. Comparing the wave form chart with the noise signals, since the wave form chart is very different from the noise signals, it can be seen that the wave form chart may be hardly analyzed.

0063FIG. 9a is a shape of phase space of one variable y'_n of the slave chaotic device one variable y_n of the master chaotic device when the master and slave chaotic devices are not in synchronization with each other, FIG. 9b is a shape of phase space of one variable y'_n of the slave chaotic device one variable y_n of the master chaotic when the master and slave chaotic devices are in synchronization with each other. From FIG. 9a and FIG. 9b, the original shape of the chaotic device which is synchronized by the random numbers may be hardly discriminated by analysis of the phase space.

0064Industrial Applicability

0065On the other hand, the synchronizing system according to the present invention can be applied to the secret communication system using the difference equation. The difference equation can use the combined map as it stands. The secret communication system using the difference equation can employ analog or digital electronic circuits or computers. Since those ordinarily skilled in the art will readily construct the circuits, further explanation is omitted and the only the encryption method using the computers is shortly described hereinafter. When a master computer and a slave computer are carrying out an identical operation, the information signals are mixed with the chaotic signals of the master chaotic device of the master computer generated by random numbers or another chaotic signals and are transmitted to the slave computer. Then, the slave chaotic device of the slave computer synchronizes the chaotic signal thereof with the chaotic signal of the master chaotic device of the master computer by using the random numbers or another chaotic signals as synchronizing signals. Thereafter, the chaotic signals of the slave device are subtracted from the transmitted mixed signals to retrieve the information signals. At this time, when the random numbers or another chaotic signals and the mixed signals with information signals and the chaotic signals are transmitted to the slave chaotic device of the slave computer, the conventional encryption techniques for secret communication enables the present invention to provide more excellent security. Also, the key signals which are used as the random numbers or another signals maximize the security.

0066On the other hand, the present invention can applicable not only to the Lorenz chaotic system which is described above as a preferred embodiment, but to all the chaotic systems given by differential equation forms. Also, the present invention can be applicable not only to the difference equation which is described above as a preferred embodiment, but to all the chaotic systems given by difference equation forms.