JP2000049771A - Synchronization device of chaos system and privacy communication system using the same - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide the synchronization device for a chaos system for synchronizing the main chaos device and subordinate chaos device of the same constitution for generating chaos signals by respectively impressing external signals provided with noise or chaos characteristics to the main chaos device and the subordinate chaos device as control signals and a privacy communication system using it. SOLUTION: This synchronization device synchronizes a main chaos device 30 and a subordinate chaos device 40, by respectively impressing the external signals provided with the noise or the chaos characteristics to the at least one state variable of the main chaos device 30 and the state variable of the subordinate chaos device 40 corresponding to it, modulating them and feeding them back to the main chaos device 30 and the subordinate chaos device 40. The mutually corresponding respective state variables of the main chaos device 30 and the subordinate chaos device 40 are synchronized by matching time spatial change with each other, and at this time, the chaos signals of the main chaos device 30 and the subordinate chaos device 40 form unique time space attraction which is completely different from chaos attraction initially provided in the chaos device.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

The present invention relates to chaos (chaos).
s) In a system composed of a plurality of chaotic devices having the same configuration for generating a signal having a characteristic (hereinafter, referred to as a “chaotic signal”) (hereinafter, referred to as a “chaotic system”), an external signal having noise or chaotic characteristics Synchronize the chaotic devices according to the “external chaotic signal” (hereinafter referred to as “external chaotic signal”), and use the synchronizing device of the chaotic system so that the chaotic devices generate chaotic signals that change in the same manner. The present invention relates to a communication system.

[0002]

2. Description of the Related Art Recently, as interest in chaos has been concentrated, research has been actively conducted to apply chaos theory to various fields of industry as much as possible. As is well known, chaos is one of the complex physical phenomena that occurs in nonlinear systems, and two chaotic systems with the same configuration are completely different over time, even if the initial conditions are very slightly different. Represents Such chaotic systems have an aperiodic, completely unpredictable characteristic over time, the "Butterfly effect".
ct) ”has the property that it is sensitive to initial conditions. In a chaotic system, the synchronization of the chaotic system is necessary to control the chaotic phenomenon. To make the state variables of a chaotic system and the state variables of other chaotic systems change or become identical.

[0003] However, synchronization of chaotic systems is not achieved by conventional general methods. Therefore, recently various methods for synchronizing nonlinear chaotic systems have been proposed and developed. In addition, researches for applying the chaotic system and the synchronization technology of the chaotic system to secret communication in particular are being actively conducted.

Conventional techniques for synchronizing chaotic systems are described by Pecora and Carol (LMPecora and TLCarroll) as “Synchronizing in Chaotic S
ystem) ”, published in Physics Review Letters, Vol.4 No.8, p821 in 1990, and published by the same person in IEEE 1991 (IEEE TRANSACTION CIRCUIT AND SUSTEM, p453 Apr). .19
91) “Synchronizing Chaotic Ci
The paper describes the theory of synchronizing two chaotic systems and a circuit that achieves synchronization. US Pat. No. 5,245, Pecola and Karol No. 660 also discloses the synchronization technique in detail.

FIG. 19 shows a technique for synchronizing pecora and carol disclosed in US Pat. No. 5,245,660. As shown in the figure, the main chaotic system 1 is divided into a first subsystem 2 and a second subsystem 3 which are drive signal generators, and a new subsystem 3 ′ is linked to the main chaotic system 1. Subsystem 3 'is the second subsystem 3
And a response subsystem is composed of the subordinate chaotic system 1 ′. At this time, the main chaotic system and the subordinate chaotic system combine to form the entire chaotic system. Drive output signal of first subsystem 2 (X4)
Is a second subsystem 3 having state variables (X1, X2, X3) and a response subsystem having state variables (X'1, X'2, X'3) corresponding to the state variables (X1, X2, X3). In order to synchronize the system 3 ', it is transmitted to the second subsystem 3 and the response subsystem 3', respectively. As a result, the state variables (X′1, X ′) of the subordinate chaotic system 1 ′
2, X'3, X4) and the state variables (X
1, X2, X3, X4) are mutually synchronized. In summary, U.S. Pat. No. 5,245,660 to Pecora and Karol describes a synchronization in which one variable of a subordinate chaotic system is replaced by one variable of a main chaotic system in order to synchronize the main and subordinate chaotic systems. Is presented.

On the other hand, the synchronization technique of the chaotic system can be applied to secret communication, optical and nonlinear dynamic modeling. In particular, in the communication field, research on secret communication for protecting communication contents from a third party by a synchronization technique of a chaotic system is in progress. Cuomo
o) and Oppenheim US Patent No. 5,2
No. 91,555 discloses in detail a secret communication system to which the synchronization technique of Pecora and Carol is applied.

FIG. 20 shows a secret communication system between Kuomo and Oppenheim disclosed in US Pat. No. 5,291,555. As shown in the figure, the communication system includes a driving signal generator 12 for generating a chaotic driving signal (u (t)) and a chaotic driving signal (u (t)) for forming a transmission signal.
And a drive signal for reproducing a drive signal (u ′ (t)) from a received signal (m (t) + u (t)). Regenerator 22
And the reproduction drive signal (u ′) for detecting the information signal (m ′ (t)) from the received signal (u (t) + m (t)).
(T)) and a receiver 20 comprising a subtractor 24 for subtracting (t)).

[0008] However, the technique for synchronizing Pecora and Karol described above synchronizes the two chaotic systems regardless of how much the parameters differ between the main chaotic system and the subordinate chaotic system. Input to the response subsystem without conversion, and tends to be easily synchronized in the entire chaotic system. In other words, if the response subsystem satisfies only the synchronization condition in which the Lyapunov Exponents presented by Pecora and Karol are all negative numbers, the parameter values of the circuit elements constituting the response subsystem are changed by, for example, about 20%. Even the whole chaotic system can be easily synchronized. Therefore, even in the conventional secret communication systems of Kuomo and Oppenheim that employ the technique of synchronizing Pecora and Karol, it is relatively easy to reconfigure the communication system due to the strong synchronization tendency, so that the information signal is simple. There is a disadvantage that the possibility of leakage is high.

[0009]

SUMMARY OF THE INVENTION Accordingly, the present invention has been devised to solve the above-mentioned disadvantages, and a first object of the present invention is to use an external signal having noise or chaotic characteristics as a control signal as a main chaotic signal. It is an object of the present invention to provide a synchronizing device for a chaotic system that synchronizes a main chaos device and a subordinate chaos device having the same configuration to generate a chaotic signal by applying the chaos signal to the device and the subordinate chaos device.

A second object of the present invention is to provide a secret communication system using a synchronization device for a chaotic system.

[0011]

To achieve the first object, a synchronizing device for a chaotic system according to the present invention comprises: a main chaotic device for generating a chaotic signal whose state variables have a mutual functional relationship; In a chaotic system that is configured the same as the chaotic device and includes a subordinate chaotic device that generates a chaotic signal corresponding to the chaotic signal generated by the main chaotic device, at least one or more state variables of the main chaotic device and noise or A first synchronization unit that receives an external signal having chaotic characteristics, modulates a state variable of the main chaotic device with the external signal, and feeds back a state variable modulated by the external signal to the main chaotic device; At least one or more state variables of the subordinate chaotic device corresponding to at least one or more state variables of the main chaotic device and the at least one state variable applied to the first synchronization means; A second synchronization unit that receives the external signal, modulates the state variable of the slave chaos by the external signal, and feeds back the state variable modulated by the external signal to the slave chaos. Features.

In order to achieve the second object, a secret communication system using the chaotic system synchronization device of the present invention comprises:
A transmitter including a main chaotic device that generates a chaotic signal whose state variables have a mutual functional relationship, and a subordinate configured to be the same as the main chaotic device and generating a chaotic signal corresponding to the chaotic signal generated by the main chaotic device. In a secret communication system comprising a receiver having a chaotic device and transmitting an information signal of the transmitter to the receiver via a predetermined transmission medium, the transmitter is configured to include at least one of the main chaotic devices. A first synchronization for receiving a state variable and an external signal having noise or chaotic characteristics, modulating the state variable of the main chaotic device with the external signal, and feeding back the state variable modulated by the external signal to the main chaotic device; Encoding means, and an adder for adding an information signal to the chaotic signal of the main chaotic device to generate a cryptographic signal, wherein the receiver includes at least the main chaotic device. Receiving at least one or more state variables of the dependent chaotic device corresponding to one or more state variables and the external signal applied to the first synchronization means of the transmitter, and receiving the dependent signal by the external chaotic signal; Second synchronization means for modulating a state variable of the chaotic device and feeding back a state variable modulated by the external chaotic signal to the subordinate chaotic device, and receiving from the adder of the transmitter to recover the information signal And a subtractor for removing a chaotic signal of the slave chaotic device synchronized with the chaotic signal of the master chaotic device from the encrypted signal to be obtained.

[0013]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS Preferred embodiments of the present invention will be described below in detail with reference to the accompanying drawings.

Generally, a mathematical model of a chaotic system is divided into a system represented by a difference equation and a system represented by a differential equation. Lorentz system, Rossl (Rossl)
er) system and Duffing are widely known, and as a system of difference equations, a logistic map is widely known.

[0015] Such a chaotic system is a state variable (Stat
e Variable), and these state variables are used to implement the main circuit in a chaotic system. Thus, a hardware circuit configuration corresponding to an arbitrary chaotic system represented by a state variable is easy for those skilled in the art. For example, a well-known Lorentz system is represented by the following Equation 1, and a circuit that implements such an equation using an integrator, an adder and a subtractor using an operational amplifier is disclosed in U.S. Pat. No. 5,29 to Kuomo and Oppenheim.
No. 1,555.

[0016]

(Equation 1) Circuits corresponding to the Rosler system and the modified duffing system are disclosed in detail in U.S. Pat. No. 5,402,334 to Pecora and Karol, and encryption technology using a logistic map is disclosed in U.S. Pat. It is disclosed in Japanese Patent No. 5,048,086.

As shown in FIG. 1, the main chaotic device 30 is given as n-dimensional state variables x (t), y (t), z (t),. The chaotic device 40 is also the n-dimensional state variable x of the main chaotic device 30
Are given as n-dimensional state variables x '(t), y' (t), z '(t), ... corresponding to (t), y (t), z (t), .... According to the present invention, the same external chaotic signal (g) is assigned to at least one of the state variables of the main chaotic device and the corresponding state variable of the subordinate chaotic device.
(T)) to apply the master chaos 30 and the slave chaos 40
Is modulated and fed back to each of the master chaos 30 and the slave chaos 40 to synchronize the two chaos.

As shown in FIG. 1, the same two chaotic devices form the entire chaotic system. At this time, one chaotic device is the main chaotic device 30 and another chaotic device is the subordinate chaotic device 40. Then, a device for synchronizing the main chaos device 30 and the subordinate chaos device 40 is the first synchronization unit 5.
0 and the second synchronization unit 60.

The first synchronizer 50 synchronizes the main chaotic device 30 and the subordinate chaotic device 40 with a first scaler that scales the external chaotic signal (g (t)) by a first scaling factor (α). 52 and any one state variable value of the main chaotic device 30, for example, x, from the external chaotic signal (αg (t)) scaled by the first scaler 52.
A subtracter 54 for subtracting (t), a second scaler 56 for again scaling the output signal (αg (t) −x (t)) of the subtractor 54 by the second scaling coefficient (β), and a second scaler 56.
The output signal of the scaler 56 (f (t) = β [αg (t)
−x (t)]) and the adder 58 that adds the state variable value (x (t)) to the main chaotic device 30 and feeds back the result to the main chaotic device 30.

On the other hand, an external chaotic signal (g (t)) applied to the first synchronization unit 50 is also applied to the second synchronization unit 60, and the second synchronization unit 60 is External chaotic signal (g (t)) to synchronize chaotic device 40
Is the same as the first scaling factor of the first synchronization unit 50.
A first scaler 62 that scales by a scaling factor (α), and a main chaotic device 30 from the external chaotic signal (αg (t)) scaled by the first scaler 62 determines a subordinate corresponding to the state variable value (x (t)). A subtractor 64 for subtracting the state variable value (x ′ (t)) of the chaotic device 40 and an output signal (αg (t) −x ′ (t)) of the subtractor 64 are again converted to a second signal.
A second scaler 66 for scaling by a scaling factor (β), and an output signal (f ′) of the second scaler 66
(T) = β [αg (t) −x ′ (t)]) and the adder 68 that adds the state variable value (x ′ (t)) and feeds it back to the subordinate chaotic device 40.

In general, since the chaotic device is very sensitive to the initial conditions, if the two chaotic devices 30 and 40 operate without being synchronized with each other, the state variables generated by the main chaotic device 30 with the passage of time will be described. The trajectory and the trajectory of the state variable generated by the subordinate chaos device 40 are completely different. That is, if the first and second synchronization units 50 and 60 are not provided in the master chaos device 30 and the slave chaos device 40 shown in FIG. 1, all the corresponding state variables of the two chaos devices 30 and 40 are different. You will move on the trajectory. Therefore, when the first and second synchronizers 50 and 60 having the above-described configurations are connected to the main chaotic device 30 and the subordinate chaotic device 40, respectively, and the two chaotic devices 30 and 40 are synchronized, two chaotic devices 30 and 40 are obtained. The devices 30, 40 will be synchronized to have the same trajectory. Thereby, the corresponding state variables x (t) and x ′ (t), y (t) and y ′ of the main chaotic device 30 and the subordinate chaotic device 40 are obtained.
(T), z (t) and z '(t), ... move along the same trajectory.

As described above, the first synchronization unit 5 according to the present invention
When the master chaos device 30 and the slave chaos device 40 are synchronized by the zero and second synchronization unit 60, x (t) = x ′ (t), y (t) = y ′ (t), and z (T) = z ′
(T), ... That is, all the state variable values of the main chaotic device 30 and all the state variable values of the subordinate chaotic device 40 become the same. In order to express such a chaotic system synchronization technique by a mathematical expression, the state variable to be synchronized is x (t) in the main chaotic device 30, and the state variable corresponding to the variable of the main chaotic device in the subordinate chaotic device 40. X '
Assuming that (t) and the external chaotic signal acting on this is g (t), the signal fed back to the main chaotic device 30 is f (t) = β (αg (t) −x (t)) The signal fed back to the chaos device 40 is f ′ (t) =
β (αg (t) −x ′ (t)). When the feedback signal is applied to each state variable of the two chaotic devices, the result is x (t) = x (t) + f (t) in the main chaotic device 30.
In the subordinate chaotic device 40, x '(t) = x'
(T) + f ′ (t). Synchronizing the two chaotic devices with each other in this manner is the synchronization technology of the chaotic system according to the present invention.

In order to implement such a synchronizer of the present invention in an actual device, the preferred embodiment of the present invention uses the Lorentz equation. As shown in FIG. 2, a circuit that embodies the Lorentz equation is easily realized by an operational amplifier (Op-Amp) and an analog multiplier.

In the first operational amplifier 301, a signal v (t) is input to its inverting terminal via a resistor R1, and a signal u (t) is input to its non-inverting terminal via a resistor R2. The signals v (t) and u (t) are converted and amplified by a gain value determined by a resistance value connected to the operational amplifier. The output of the first operational amplifier 301 is connected to the second operational amplifier 302 connected to the capacitor C1 via the variable resistor R3.
Is integrated by

The first analog multiplier 311 combines the output signals u (t) and w (t) of the second operational amplifier 302, and the third operational amplifier 303 outputs the signal u input to its inverting terminal.
(T) is non-inverted and amplified by the ratio of the resistors R4 and R5 and applied to the non-inverted terminal of the fourth operational amplifier 304.

The fourth operational amplifier 304 has a combined signal u (t) .w (t) at its inverting terminal via a resistor R6 and an output signal of the third operational amplifier 303 at its non-inverting terminal at a resistor R7. , And converts and amplifies these input signals with a gain value determined by a resistance value connected to the operational amplifier.

The output signal of the fourth operational amplifier 304 is input to the inverting terminal of the fifth operational amplifier 305 via the resistor R10, and is inverted and amplified by the ratio of the resistors R10 and R11. The output of the fifth operational amplifier 305 is integrated by the sixth operational amplifier 306 connected to the capacitor C2 to output a signal v (t).

The second analog multiplier 312 outputs the signal u (t)
And v (t) are combined, and the seventh operational amplifier 307 has the combined signal u (t) · v (t) connected to its inverting terminal via the resistor R14 and w (t) connected to the non-inverting terminal via the resistor R17. These input signals are converted and amplified by a gain value determined by a resistance value connected to the operational amplifier. The output of the seventh operational amplifier 307 is the resistor R16
And a signal w (t) is integrated by an eighth operational amplifier 308 connected to the capacitor C3.

As described above, the main chaotic device 30 and the subordinate chaotic device 40 constitute a circuit diagram designed as shown in FIG. At this time, according to the present invention, two chaotic devices 30, 40
Are added to the main chaotic devices 30 and 40 in order to synchronize the signals. The chaotic signal added from the outside is, for example, a chaotic circuit of a Rossler system, a chaotic circuit of a duffing system, Navier-Stokes (Navi Stokes).
er-Stokes) The chaotic signal generated by the chaotic circuit of the system can be applicable. These circuit configurations can be easily manufactured by those skilled in the art having ordinary knowledge, and a detailed description thereof will be omitted below. Meanwhile, an arbitrary noise signal may be filtered and adjusted to have an appropriate frequency band, and then used as an external synchronization signal.

FIG. 3 is a circuit diagram of the main chaotic device 30 and the first synchronization unit 50 embodied by the Lorentz equation.
3 shows a circuit diagram of the subordinate chaotic device 40 and the second synchronization unit 60 similarly realized by the Lorentz equation.

As shown in FIG. 3, first, the main chaotic device 30 has the same configuration as the chaotic signal generation circuit shown in FIG. First
The synchronizer 50 corresponds to the variable resistor R21 corresponding to the first scaler 52 in FIG.
9 and a ninth operational amplifier 501 including the peripheral circuit, the variable resistor 25 corresponding to the second scaler 56, and the adder 5
And a tenth operational amplifier 502 corresponding to the operational amplifier U10 and its peripheral circuits, and an eleventh operational amplifier 503 composed of the operational amplifier U11 and its peripheral circuits and inverting and amplifying the output of the tenth operational amplifier 502. Is done.

On the other hand, as shown in FIG.
0 has the same configuration as the chaotic signal generation circuit shown in FIG. The second synchronizing unit 60 corresponds to the variable resistor R31 corresponding to the first scaler 62, corresponds to the subtractor 64, the twelfth operational amplifier 601 including the operational amplifier U12 and its peripheral circuit, and the variable resistor corresponding to the second scaler 66. The operational amplifier U13 corresponds to the resistor R35 and the adder 68.
And a thirteenth operational amplifier 602 comprising an operational amplifier U14 and peripheral circuits thereof, and a fourteenth operational amplifier 6 comprising an operational amplifier U14 and peripheral circuits thereof and inverting and amplifying the output of the thirteenth operational amplifier 602.
03.

The operation of the present invention having the above-described configuration will now be described.

The well-known Lorentz equation is applied to the present invention, and the following equation (2) expresses the non-linear terms xy and xz
2 shows a three-dimensional Lorentz system including:

[0035]

(Equation 2) In Equation 2, x, y, and z are state variables,
σ, R and b are parameters. Here, σ = 10, R =
When 8/3, b = 8, this chaotic system generates a chaotic signal. When this chaotic system is used as the main chaotic device, if the circuit configuration is the same as that of the main chaotic device and only the trajectories of its variables are different, the subordinate chaotic device can be expressed by the following mathematical formula 3.

[0036]

(Equation 3) The parameters of the subordinate chaotic device are the same as those of the main chaotic device.
At this time, if the initial conditions of x, y, z and x ', y', z 'are set to be different from each other, the two chaotic devices have independently different trajectories. For example, when examining the x 'value corresponding to the state variable x in the main chaotic device and the state variable x in the subordinate chaotic device, two trajectories are different, and different temporal characteristics as shown in FIGS. 5A and 5B are obtained. Represent. Here, FIG.
A shows the waveform of the state variable in the main chaotic device, and FIG. 5B shows the waveform of the state variable in the subordinate chaotic device.

However, the first synchronizing unit and the second synchronizing unit are connected to the master chaos device and the slave chaos device, respectively, and the external chaos signal is applied to the mutually corresponding state variables x and x 'of the two chaos devices. When applied to synchronize these two chaotic devices,
The main chaos device and the subordinate chaos device are expressed by the following Equations 4 and 5, respectively.

[0038]

(Equation 4)

(Equation 5) Here, the meaning of being synchronized means that the spatio-temporal trajectories of state variables of the same nonlinear system have the same trajectory between corresponding state variables. That is,
This means that x = x ′, y = y ′, and z = z ′. Such synchronization can be easily confirmed by computer simulation. The external chaotic signal applied to the main chaotic device and the subordinate chaotic device has a value of 0 to 1 obtained by a random number generator, one random number is defined as the amplitude A of the external chaotic signal, and the other random number is defined as ω. The scaled external chaotic signal becomes αAsin (ωt) and is used as a synchronization signal. FIG. 6A to FIG. 6 show waveform diagrams when two chaotic devices are synchronized according to an external chaotic signal.
D. FIG. 6A is a waveform diagram of an external chaotic signal used for synchronization of a main chaotic device and a subordinate chaotic device, and FIG. 6B is a waveform diagram showing a trajectory of state variables of the main chaotic device.
D is a waveform diagram showing the trajectory of the state variable of the subordinate chaotic device corresponding to the state variable of the main chaotic device, and FIG. 6C is a waveform diagram showing the difference between the state variables of the two chaotic devices.

As shown in FIGS. 6A to 6D, when synchronization is established between the two chaotic devices, the chaotic signals of the two chaotic devices immediately match, and the state variable (eg, x) of the main chaotic device and The state variable (x ') of the subordinate chaotic device becomes the same.
To confirm this, FIG. 6C shows the difference between the two state variables as 1
I tried to enlarge it by 0000 times, and the difference was '0' immediately.
Can be confirmed.

In order to examine the difference between the two chaotic signals before and after synchronization, the phase space in the yz plane before and after synchronization is examined. Attraction of Two Chaotic Devices in a Chaotic Attractor
Find out.

FIGS. 7 and 8 show the shape of the chaotic attraction of the two chaotic devices in the yz plane before and after synchronization, respectively. FIG. 7 shows that the attraction of the chaotic device before being synchronized in response to the external chaotic signal maintains the attraction of the normal Lorentz chaotic system. However, FIG. 8 does not have the general chaotic attracting fractal shape after being synchronized in response to an external chaotic signal, but shows chaotic attracting similar to very complex noise. Therefore, after being synchronized, the altered chaotic attraction as shown in FIG. 8 cannot find the shape of the original normal chaotic attraction.

On the other hand, when z and z 'are used as the feedback state variables of the master chaos and the slave chaos, respectively, FIG. 9 shows z (t) -z (t) when there is no noise.
+ Τ) shows the shape of the chaotic attraction in the time-delayed phase space, and FIG. 10 shows the time-delayed phase space of z (t) -z (t + τ) when the main chaos and the subordinate chaos are synchronized by noise. Shows the shape of the chaotic attraction. 9 and 10 show the case where τ is 0.5. FIG. 10 and FIG.
It has no fractal shape of general chaotic attraction, and exhibits chaotic attraction similar to very complex noise, and the altered chaotic attraction cannot find the shape of the original normal chaotic attraction.

Next, the principle of synchronizing two chaotic devices using a logistic map will be examined. At this time, the logistic map is expressed by the following equation (6).

[0044]

(Equation 6) In Equation 6, the chaos is determined by the α value. For example, when α is 3.9, the device represents chaos and characteristics, and the subordinate chaotic device is given as x ′ _{n + 1} = αx ′ _n (1−x ′ _n ). If the two chaotic devices are synchronized by a random number, and the random number has a value between 0 and 1, the feedback value of the master chaotic device is f _n = β (γ _n −x _n ), and the feedback value of the slave device is f ' _n = β (γ _n -x' _n ). Then, the two chaotic devices, the main chaotic device and the subordinate chaotic device, are given by the following equations 7 and 8, respectively.

[0045]

(Equation 7)

(Equation 8) In Equations 7 and 8, the difference between the two state variables is x _n −
If x ' _n = y _n , the difference between the two state variables is
Given as

[0046]

(Equation 9) Equation 9 is a nonlinear difference equation defining a new chaotic system. In Equation 9, parametric y _n is modulated at a variable main chaotic system x _n and γ _n, y ² _n, it is seen that the modulated variable main chaotic system. in this way,
Since a method of modulating a nonlinear system with a noise signal or a chaotic signal is widely known, a detailed description thereof will be omitted below.

When the parameters of a nonlinear system are modulated with a chaotic signal or a noise signal, and the phenomenon is analyzed, the chaotic system has a very complicated appearance. The converted chaotic system may randomly fluctuate between the chaotic signal and the value very close to the value of 0 depending on the condition of each parameter, sometimes converge to the value of 0, and sometimes chaos. May be represented. Such chaos and fluctuations very close to the value of 0 are called on-off intermittentness. When such intermittentness occurs, the average laminar length becomes infinitely long, and a critical value condition occurs in which the difference between the two variables converges to a value of zero. Above this critical value α _c , a new chaotic device formed by the difference between the state variables of two identical chaotic devices will immediately converge to zero. Therefore,
When the difference between the variables becomes 0, there is no difference in the trajectory between the two chaotic devices, so that the trajectories of the two chaotic devices become the same and are immediately synchronized. That is, α> α _c
, The condition of the critical value of the on-off intermittent condition is the condition of α where the average laminar length becomes infinitely long and the difference between the two variables converges to 0. The condition to be synchronized. Equation 9 is simply converged to 0 and synchronized by the value of β multiplied by the difference between the two variables, and such a synchronization method feeds back the state variables of the main chaotic device to the subordinate chaotic device. It has the same synchronization conditions as the method.

By comparing the synchronization method according to the present invention with the synchronization method using Pecora and Karol, it is easy to see what the features of the present invention are.

That is, in the synchronization method by Pecora and Karol, the sub-Lyapunov of the subordinate chaotic system
The synchronization is performed under the condition that the exponent is always a negative value, but this is not a synchronization phenomenon due to the critical value condition in which the length of the on-off intermittent average laminar becomes infinitely long. However, in the present invention, the average laminar length of the intermittent on-off is infinitely long and the two variables are not synchronized under a critical value condition in which the difference between the two variables converges to zero.

When the aspect of the synchronization is examined based on the difference equation, the main difference equation and the dependent difference equation can be defined by the following equations (10) and (11), respectively.

[0051]

(Equation 10)

[Equation 11] In Equations 10 and 11, a is 0.89 and b is 0.
9

In the above-mentioned difference equation, when the respective synchronization equations are put in x _n and x ′ _n as in a logistic map, the main chaotic device and the subordinate chaotic device can be expressed by the following equation (12).
And can be expressed by Equation 13.

[0053]

(Equation 12) In Expression 12, if ζ is a noise signal having a value between 0 and 1, f _n = 0.4 (ζ _n −x _n ) and g _n =
0.4 (ζ _n −y _n ).

[0054]

(Equation 13) In Equation 13, when ζ is a noise signal having a value between 0 and 1, f ′ _n = 0.4 (ζ _n −x ′ _n ), and
g ′ _n = 0.4 (ζ _n −y ′ _n ).

Further, in Expressions 10 to 13,
'mod1' is 0 and 1 for both master and slave chaos
In order to have only values in between, if it is greater than 1, only the value after the decimal point is taken, and if it is negative, a positive integer is added so as to have only the value after the decimal point.

FIGS. 11A to 11C show the state variables and the differences between the variables y _n and y ′ _n of the chaotic device when they are not synchronized in consideration of the above-described difference equation. 11A is variable y _n of the main chaotic system, FIG. 11B 'difference _n, FIG. 11C is variable y dependent chaotic system' y _n and y indicates the _n. Examination of FIG. 11B, it can be seen that the state variable y _'n of state variables y _n and dependent chaotic apparatus main chaotic system has a time-independent value.

FIGS. 12A to 12C show the state variables and the differences between the variables y _n and y ′ _n of the two chaotic devices when synchronized in consideration of the above-described difference equation. 12A is the variable y _n of the main chaotic device, and FIG.
Times enlarged 'difference _n, FIG. 12C is variable y dependent chaotic system' y _n and y respectively the _n. Examination of FIG. 12B, when the synchronization between the two chaotic system is performed, the variable y _'n of the variable y _n and dependent chaotic apparatus main chaotic system is seen to have a time identical value. That is, looking at FIG. 12B, since the two chaotic devices have different initial conditions, the variables of the two chaotic devices move independently of each other initially, but how much time has elapsed when the synchronization started. Later, it can be seen that the two chaotic devices coincide with each other and have the same trajectory. Therefore, after the two chaotic devices are synchronized, the difference between the corresponding variables of the two chaotic devices is zero.

In order to know how much the difference between the chaotic signals before and after synchronization is different, the phase space in the x _n -y _n plane before and after synchronization is known. Examining the chaotic attraction of these two chaotic devices is as follows.

FIGS. 13 and 14 show the shape of the chaotic attraction of the two chaotic devices before and after synchronization, respectively.
As shown in FIG. 13, before synchronization, the attraction of the chaotic device maintains the normal attraction of the coupling map. However, as shown in FIG. 14, after being synchronized in response to an external chaotic signal, the attraction of the chaotic device does not have the fractal shape of general chaotic attraction, but rather creates a chaotic attraction similar to very complex noise. You can confirm that you have. That is, when analyzing the chaotic attraction as shown in FIG. 14, since the actual chaotic device has an irreversible characteristic, the chaotic attraction after synchronization cannot find the shape of the original chaotic attraction.

An embodiment in which the above-described chaotic system synchronization technique according to the present invention is applied to a secret communication system will now be described.

In a secret communication system, the transmitter comprises a main chaotic device that uses an external chaotic signal as a synchronization signal, and at least one state variable of the main chaotic device is a cryptographic signal that hides an information signal (eg, a voice signal). Use as
Thus, the information signal is transmitted on the chaotic signal, and the information signal has a power spectrum much smaller than the power spectrum of the chaotic signal of the main chaotic device.
The receiver has a subordinate chaos device that uses an external chaotic signal as a synchronization signal as well as the main chaotic device of the transmitter, and the main chaos that synchronizes the subordinate chaos device with the main chaos device and is used as the encryption signal of the main chaos device Subtract device state variables. Therefore, the original information signal is restored as it is by the receiver. This is done due to the condition that, in the absence of an information signal, the two chaos will be exactly the same when the master and slave chaos are synchronized. At this time,
Since the chaotic signal coming out of the chaotic device after synchronization is very disorderly, it is not known what formula was used to form such a chaotic device, and the signal generated from the chaotic device has various chaotic predictions. The secret communication system according to the present invention has an advantage that the degree of secret maintenance is very high because it cannot be found even by a method such as a method.

FIG. 15 is a conceptual diagram for explaining a secret communication system using the chaotic system synchronization technique according to the present invention. The adder 70 adds the chaotic system synchronization apparatus according to the present invention shown in FIG. Encryption unit and subtractor 8
And a decoding unit which is set to 0.

As shown in FIG. 15, an adder 70 installed in the transmitter adds an arbitrary signal (for example, an information signal (m (t)) to y (t) in the main chaotic device 30 here. The subtractor 80 installed in the receiver converts the received signal (y (t) + m (t)) into the dependent chaotic device 40 corresponding to the variable (y (t)) of the main chaotic device 30. Of the information signal (m (t) = y
(T) + m (t) -y '(t)).

FIG. 16 shows an embodiment of the main chaotic device 30, the first synchronization unit 50 and the encryption unit 70 of FIG.
Shows an embodiment of the subordinate chaotic device 40, the second synchronization unit 60, and the decoding unit 80 in FIG.

As shown in FIG. 16, the configurations of the main chaotic device 30 and the first synchronizer 50 are the same as those in FIG. The adder 70 serving as an encryption unit includes the operational amplifier U15 and its peripheral circuits, and outputs the output signal (w (t)) of the eighth operational amplifier 308.
And the information signal (m (t)) is added and then inverted.
It comprises a fifth operational amplifier 701 and a sixteenth operational amplifier 702 which comprises an operational amplifier U16 and its peripheral circuits and inverts and amplifies the output signal of the fifteenth operational amplifier 701.

As shown in FIG. 17, the configurations of the subordinate chaotic device 40 and the second synchronizer 60 are the same as those in FIG. A subtractor 80 serving as a decoding unit includes an operational amplifier U17 and its peripheral circuits. The subtractor 80 converts the output signal of the sixteenth operational amplifier 702 shown in FIG. 16 into the output signal of the eighth operational amplifier 308 (w (t)).
From the seventeenth operational amplifier 801 that subtracts

The operation of the secret communication system according to the present invention shown in FIGS. 16 and 17 will now be described.

Using the external chaotic signal (g (t)), the first
The main chaotic device 3 is controlled by the synchronization unit 50 and the second synchronization unit 60.
0 and the subordinate chaotic device 40 are synchronized. Thereafter, the transmitter transmits the information signal (m (t)) such as an audio signal to the main chaotic device 3.
0 (y (t)) is combined with the encrypted signal (y
(T) + m (t)) to the receiver. The receiver subtracts the state variable (y ′ (t)) of the subordinate chaotic device 40 from the encrypted signal (y (t) + m (t)) to obtain the original information signal (m).
(T)) is restored.

FIGS. 18A to 18D correspond to FIGS.
FIG. 8 is a waveform diagram for explaining a secret communication process by the secret communication system shown in FIG. 7. FIG. 18A shows an external chaotic signal (g (t)) for synchronization, and FIG. 18B shows an information signal (m (t)) which is a sine wave function as a state variable (y (t)) of the main chaotic device. FIG. 18C shows an information signal which is a sine wave function restored by the receiver, and FIG.
8D is the state variable of the synchronized subordinate chaos (y '
(T)) shows the respective waveforms. As shown in FIG. 18C, it can be seen that the information signal of the weak sine wave function has been restored by the receiver. Comparing this with FIG. 6C, when the information signal was not carried, the variable difference between the two chaotic devices after synchronization was 0, but as shown in FIG. 18C, when the information signal was carried, It can be seen that the variable difference between the two synchronized chaotic devices corresponds to the information signal carried.

However, the power spectrum of the information signal must be hidden in the power spectrum of the chaotic signal in order for the information signal to be distinguished from the encryption signal and not exposed. In fact, since the magnitude of the information signal is very small, a large difference cannot be found between the waveform in which the information signal is carried on the chaotic signal of the main chaotic device and the waveform in which the information signal is not carried. That is, FIG. 18B is a waveform in which an information signal is placed on the state variable of the main chaotic device, and FIG. 18D is a waveform of the state variable of the subordinate chaotic device synchronized by the external chaotic signal. I can't get it,
When the waveform signal of FIG. 18D is subtracted in response to the waveform signal of B, the information signal as shown in FIG. 18C can be restored. Therefore, using the external chaotic signal as a synchronization signal according to the synchronization technique of the present invention, and synchronizing the master chaos device and the slave chaos device by the first synchronization unit of the transmitter and the second synchronization unit of the receiver. Thus, secret communication with a high degree of security maintenance can be executed.

On the other hand, the synchronization technique according to the present invention can be directly applied to a secret communication system using a difference equation.
The difference equation uses the above-described connection map as it is.
In a secret communication system using the difference equation, an analog electronic circuit, a computer, or a digital circuit can be used. Since a person having ordinary skill in the art can easily realize such a circuit, a detailed description thereof will be omitted, and an encryption technique using a computer will be briefly described below.

When the main and subordinate computers, which are different computers, are performing the same operation,
The information signal is carried on the chaos signal of the main chaos device of the main computer generated by random numbers or other chaotic signals, and transmitted together with the random numbers or other chaotic signals. Then, the slave computer synchronizes the chaos signal of the slave chaos with the chaos signal of the master chaos using the random number or another chaos signal as a signal for synchronizing the master chaos and the slave chaos. Thereafter, when the chaotic signal of the subordinate chaotic device is subtracted from the encryption signal obtained by combining the information signal with the chaotic signal of the main chaotic device, the desired information signal is restored. In this case, when transmitting a random number or a chaotic signal in which another chaotic signal and an information signal are combined, if an existing encryption technique for secret communication is used, the security is further improved. On the other hand, when a problem occurs in security due to a random number or another chaotic signal, security can be maximized by using a random number or another chaotic signal as an initial value as a key. This method uses the previously known chaos by providing not only the basic information signal transmission of the Kuomo and Oppenheim systems, but also a high encryption speed for current communication methods requiring considerable high speed. It offers much higher encryption speed and security than secret communication methods.

On the other hand, in the above-described embodiment, only the Lorentz chaotic system has been described in the form of a differential equation. However, the present invention can be applied to all chaotic systems given in the form of a differential equation in addition to the Lorentz chaotic system. In addition, the present invention can be easily applied not only to the difference equation described here but also to all chaotic systems given in other difference equation forms. Further, a secret communication system can be realized by using this.

[0074]

As described above, the present invention can be applied to the case where there are two chaotic systems having the same configuration for generating a chaotic signal, the same external chaotic signal and one state variable or many states of the main chaotic device. One variable or many state variables of the subordinate chaos corresponding to each variable of the variable and the main chaos are modulated and fed back by the synchronization unit added to each of the main chaos and the subordinate chaos, and the main chaos is modulated. The device and the subordinate chaotic device are synchronized so that the spatio-temporal changes of the corresponding variables are mutually matched. At this time, the signal of each chaotic device forms a unique spatiotemporal attraction that is completely different from the chaotic attraction that the chaotic device originally has,
The original chaotic signal cannot be found, and when the secret communication system is applied, the degree of maintaining the secret of the information signal can be maximized.

[Brief description of the drawings]

FIG. 1 is a block diagram showing an overall configuration of a chaotic system and a synchronization device according to the present invention.

FIG. 2 is a circuit diagram of a chaotic signal generation circuit embodied by Lorentz equation.

FIG. 3 is a detailed circuit diagram illustrating the synchronization device of FIG. 1 implemented by the chaotic signal generation circuit of FIG. 2;

FIG. 4 is a detailed circuit diagram of the synchronization device of FIG. 1 implemented by the chaotic signal generation circuit of FIG. 2;

FIG. 5 is a waveform diagram of state variables of two chaotic devices that appear when the two chaotic devices are not synchronized.

6A is a waveform diagram of an external chaotic signal for synchronization, FIGS. 6B and 6D are waveform diagrams of state variables of two chaotic devices, and FIG. 6C is a waveform illustrating a state variable difference of two chaotic devices. FIG.

FIG. 7 shows the shape of chaotic attraction in the yz phase space before being synchronized.

FIG. 8 shows the shape of chaotic attraction in the yz phase space after synchronization.

FIG. 9: When using z as a feedback signal,
FIG. 7 is a diagram showing a shape of chaotic attraction in a time delay phase space of z (t) -z (t + τ) when there is no noise.

FIG. 10 shows z (t) −z (t + τ) when synchronized with noise when using z as a feedback signal.
FIG. 3 is a diagram showing a shape of chaotic attraction in a time delay phase space.

FIG. 11 is a waveform diagram illustrating a state variable of two chaotic devices and a difference between the two state variables that appear when synchronization is not performed in the connection map.

FIG. 12 is a waveform diagram showing a state variable of two chaotic devices and a difference between the two state variables, which appear when synchronization is performed using a connection map.

13 is a diagram illustrating a chaos attracted shape in x _n -y _n phase space before synchronization occurs by binding map.

14 is a diagram illustrating a chaos attracted shape in x _n -y _n phase space after synchronization caused by binding map.

FIG. 15 is a block diagram showing an overall configuration of a secret communication system using the chaotic system synchronization device according to the present invention.

16 is a detailed circuit diagram of the secret communication system of FIG. 15 implemented by the chaotic signal generation circuit of FIG. 2;

17 is a detailed circuit diagram of the secret communication system of FIG. 15 implemented by the chaotic signal generation circuit of FIG. 2;

18 is a waveform diagram illustrating a process of restoring an original information signal in the receiver of the secret communication system of FIG.

FIG. 19 is a block diagram showing a concept of synchronizing a chaotic system by a conventional method proposed by Pecora and Karol.

FIG. 20 is a block diagram of a communication system implemented using a conventional synchronization concept.

[Explanation of symbols]

DESCRIPTION OF SYMBOLS 1 Main chaotic system 1 'Subordinate chaotic system 2 1st subsystem 3 2nd subsystem 3' subsystem 10 Transmitter 12 Drive signal generator 14 Adder 20 Receiver 22 Drive signal regenerator 24 Subtractor 30 Main chaotic device 40 Dependent chaos device 50 First synchronizer 52 First scaler 54 Subtractor 56 Second scaler 58 Adder 60 Second synchronizer 62 First scaler 64 Subtractor 66 Second scaler 68 Adder 70 Adder 80 Subtractor 301 , 302, ..., 308 1st operational amplifier, 2nd operational amplifier, ..., 8th operational amplifier 311 1st analog multiplier 312 2nd analog multiplier 501 10th operational amplifier 502 11th operational amplifier 503 12 operational amplifier

 ──────────────────────────────────────────────────続 き Continued on the front page F term (reference) 5K013 FA04 JA03 5K047 AA11 EE00 GG33 GG57 MM03 MM53

Claims (8)

[Claims] 1. A main chaotic device (30) for generating a chaotic signal whose state variables have a mutual functional relationship, and a chaotic signal configured identically to the main chaotic device and corresponding to the chaotic signal generated by the main chaotic device. A chaotic system comprising a subordinate chaotic device (40) generating at least one of the main chaotic devices and an external chaotic signal, and modulating the state variables of the main chaotic device by the external chaotic signal. A first synchronization means (50) for feeding back a state variable modulated by the external chaotic signal to the main chaotic device, and a first synchronizing means (50) corresponding to at least one state variable of the main chaotic device. Receiving at least one or more state variables and the external chaotic signal applied to the first synchronization means, and receiving the external chaotic signal to change a state variable of the subordinate chaotic device Modulated, the synchronization apparatus of chaotic system characterized by being configured from a second synchronizing means (60) for feeding back the status variables modulated by the external chaotic signal to the slave chaotic system. 2. The first synchronizing means includes: a first scaler for scaling the external chaotic signal by a first scaling factor; and at least one of the main chaotic devices based on the external chaotic signal scaled by the first scaler. A subtracter for subtracting one or more state variables; a second scaler for scaling an output signal of the subtractor by a second scaling factor; and an output signal of the second scaler and the at least one of the main chaotic devices. The synchronizing device for a chaotic system according to claim 1, further comprising an adder that adds a state variable and feeds back to the main chaotic device. 3. The second synchronizing means includes: a first scaler for scaling the external chaotic signal by a first scaling factor; and at least one of the slave chaotic devices based on the external chaotic signal scaled by the first scaler. A subtractor for subtracting one or more state variables; a second scaler for scaling an output signal of the subtractor by a second scaling factor; and an output signal of the second scaler and the at least one of the dependent chaotic devices. The synchronizing device for a chaotic system according to claim 1, further comprising an adder for adding a state variable and feeding back to the subordinate chaotic device. 4. The main chaotic device and the subordinate chaos under the condition that the average laminar length of the on-off intermittent is infinitely long and the difference between the chaotic signals of the main chaotic device and the subordinate chaotic device converges to zero. The synchronization device of a chaotic system according to claim 1, wherein the devices are synchronized. 5. A transmitter comprising a main chaotic device that generates a chaotic signal whose state variables have a mutual functional relationship, and a chaos configured identically to the main chaotic device and corresponding to a chaotic signal generated by the main chaotic device. A secret communication system comprising a receiver having a slave chaos device that generates a signal, and transmitting an information signal of the transmitter to the receiver via a predetermined transmission medium. Receiving at least one or more state variables and an external chaotic signal, modulating a state variable of the main chaotic device with the external chaotic signal, and feeding back a state variable modulated by the external chaotic signal to the main chaotic device. (1) a synchronizing means, and an adder for adding an information signal to a chaotic signal of the main chaotic device to generate an encryption signal, wherein the receiver has a small number of the main chaotic device. Receiving at least one or more state variables of the slave chaotic device corresponding to one or more state variables and the external signal applied to the first synchronization means of the transmitter, and A second synchronization unit that modulates a state variable of the slave chaos and feeds back a state variable modulated by the external chaos signal to the slave chaos, and an adder of the transmitter to restore the information signal. A secret communication system, comprising: a subtractor for removing a chaotic signal of the slave chaotic device synchronized with a chaotic signal of the master chaotic device from a received encrypted signal. 6. The first synchronization means includes: a first scaler for scaling the external chaotic signal by a first scaling factor; and at least one of the main chaotic devices from the external chaotic signal scaled by the first scaler. A subtracter for subtracting one or more state variables; a second scaler for scaling an output signal of the subtractor by a second scaling factor; and an output signal of the second scaler and the at least one of the main chaotic devices. 6. The secret communication system according to claim 5, further comprising an adder for adding a state variable and feeding back to the main chaotic device. 7. The second synchronizing means includes: a first scaler that scales the external chaotic signal by a first scaling factor; and at least one of the subordinate chaotic devices from the external chaotic signal scaled by the first scaler. A subtractor for subtracting one or more state variables; a second scaler for scaling an output signal of the subtractor by a second scaling factor; and an output signal of the second scaler and the at least one of the dependent chaotic devices. 6. The secret communication system according to claim 5, further comprising an adder that adds a state variable and feeds back to the subordinate chaotic device. 8. The main chaotic device and the subordinate chaos under the condition that the average laminar length of the intermittent on-off is infinite and the difference between the chaotic signals of the main chaotic device and the subordinate chaotic device converges to zero. The secret communication system according to claim 5, wherein the devices are synchronized.