# Patent

## Title
System and method for generating analog-digital mixed chaotic signal, encryption communication method thereof

## Abstract

A system and method for generating analog-digital mixed chaotic signal and an encryption communication method thereof are provided. In the system and method, the complementarity between continuous chaotic systems (12, 22) and digital chaotic systems (11, 21) are reasonably utilized. In specific, the digital chaotic systems, which are separated from each other, control the local continuous chaotic systems respectively, so as to enable the continuous chaotic systems, which are also separated from each other, to stably and synchronously work for a long time. Thus, there is no need to transmit the synchronizing signal, and as a result the anti-attack capability is increased effectively. Further, the continuous chaotic systems disturb the local digital chaotic systems to prevent the digital chaotic systems from degradation. This compensates the drawbacks of digital chaotic systems.

## Background
CROSS-REFERENCE TO RELATED APPLICATIONS
This application is a continuation of International Patent Application No. PCT/CN2006/003724, filed Dec. 30, 2006, which claims priority to Chinese Patent Application No. 200510121497.3, entitled “System and Method for Generating Analog-digital Mixed Chaotic Signals”, filed Dec. 31, 2005, both of which are hereby incorporated by reference for their entirety.


FIELD OF THE INVENTION
The present invention relates to cryptography technology in information security field, and more particularly, to a system and method for generating analog-digital mixed chaotic signals as well as an encryption communication method.
BACKGROUND
With the development of computer, communication and network technology, the progress of global informationization is speeding up. Up to now, network information system has become an infrastructure for countries, industries, groups and enterprises under development. The communication security of the network information system has therefore become an important issue associated with the continuous and stable development of economy and the safety of a country.
At present, chaos has been applied in the secure communication field. The so-called chaos refers to a complex external manifestation resulted from its internal randomicity. In other words, the chaos is a kind of non-random movements that appears to be random. The chaotic signals are characterized by ergodicity, wideband, noise like, sensitive to the initial conditions, rapidly attenuated self correlation and weak cross correlation, and the like. These characteristics provide numerous mechanisms and methods to achieve secure communication. The Chaotic Encryption System in the art includes digital chaotic system and continuous (or analog) chaotic system.
The digital chaotic system is less affected by noise interference and device parameters, and thereby it is easy to run stably for a long time and may be duplicated accurately. However, because digital chaotic sequences are all generated on computers or other equipment with limited precision, any digital chaotic sequence generator may be classified into a finite automation. In this case, the generated digital chaotic sequence presents degraded characteristics, such as short period, strong correlation and small linear complexity; in other words, the dynamics characteristics of a real digital chaotic system are much different from those of ideal real-value chaotic system.
Compared with the digital chaotic system, the continuous chaotic system (i.e. analog chaotic system) has a continuous state (for example, voltage, current etc.) space, which allows the generated chaotic sequences to have a good cryptology characteristic. However, due to the effects of the deviation of device parameters and the circuit noise, it is hard to achieve a stable synchronization between two continuous chaotic systems for a long time. This severely slows the progress of practical application of the chaotic synchronization based secure communication system.
SUMMARY
Various embodiments of the invention provide a system and method for generating analog-digital mixed chaotic signal as well as an encryption communication method thereof, in which the problem of characteristic degradation is not tended to occur in the digital chaotic system, and the continuous chaotic system is capable of running stably and synchronously for a long time.
One embodiment of the invention provides an analog-digital mixed chaotic system, including:
a digital chaotic system adapted to generate a digital chaotic sequence;
a continuous chaotic system adapted to generate a continuous chaotic signal;
a synchronization coupling module adapted to generate a synchronizing impulse signal to the continuous chaotic system for controlling the continuous chaotic system, according to states of the digital chaotic system and the continuous chaotic system, and output the synchronizing impulse signal to the continuous chaotic system;
a disturbance coupling module adapted to disturb the digital chaotic system according to the state of the continuous chaotic system; and
a controlling module adapted to control the states of the digital chaotic system, the continuous chaotic system, the synchronization coupling module and the disturbance coupling module.
Another embodiment of the invention provides a method for generating an analog-digital mixed chaotic signal, including:
generating a digital chaotic sequence from a digital chaotic system;
generating a continuous chaotic signal from a continuous chaotic system;
generating a synchronizing impulse signal for controlling the continuous chaotic system according to states of the digital chaotic system and the continuous chaotic system, and outputting the synchronizing impulse signal to the continuous chaotic system; and
disturbing the digital chaotic system according to the state of the continuous chaotic system.
Yet another embodiment of the invention provides an encryption communication system, comprising a transmitting party, and a receiving party configured to communicate with the transmitting party, wherein a first analog-digital mixed chaotic system included in the transmitting party is configured to encrypt a plaintext to be transmitted by using a chaotic signal generated by itself, and outputting a resultant cipher text; and a second analog-digital mixed chaotic system included in the receiving party is configured to decrypt a received cipher text by using a chaotic signal generated by itself, and outputting a resultant plaintext; and the system is wherein: initial conditions and system parameters of the digital chaotic systems in the first and the second analog-digital mixed chaotic systems are substantially the same, and the first and the second analog-digital mixed chaotic systems according to the one embodiment of present invention.
Embodiments of the invention utilize the complementarity, in terms of the cryptography characteristics and the implementation technology, between a digital chaotic system and a chaotic synchronization based secure communication system. In these embodiments, the digital chaotic systems, which are separated from each other in location, control their own local continuous chaotic systems respectively, so that both of the separated continuous chaotic systems may run in a stable and synchronous state for a long time without transmitting synchronizing signals. This effectively enhances anti-attack ability of a chaotic system. Also in these embodiments, the continuous chaotic systems in turn disturb the respective local digital chaotic system to prevent the digital chaotic systems from characteristic degradation. Therefore, in the analog-digital mixed chaotic system, the shortcomings that the continuous chaotic systems cannot run in a stable and synchronous state for a long time and thereby the synchronizing signals need to be transmitted through a channel can be overcome, and meanwhile, the problem, i.e. the characteristic degradation of digital chaotic systems can be avoided.



BRIEF DESCRIPTION OF THE DRAWING(S)
The present invention will be further described in conjunction with the figures and embodiments as follows, in which:
FIG. 1 is a schematic diagram of an analog-digital mixed chaotic system according to an embodiment of the present invention;
FIG. 2 is a block diagram of one timing/counting module as shown in FIG. 1 according to an embodiment of present invention;
FIG. 3 is a block diagram of the other timing/counting module as shown in FIG. 1 according to an embodiment of present invention;
FIG. 4 is a schematic view of a main timing for running the analog-digital mixed chaotic system shown in FIG. 1;
FIG. 5 is a schematic view of a main timing for running the synchronization coupling module shown in FIG. 1;
FIG. 6 is a schematic view of a main timing for running the disturbance coupling module shown in FIG. 1;
FIG. 7 is a phase diagram of the digital chaotic system 1;
FIG. 8 is a phase diagram of the continuous chaotic system 1;
FIG. 9 is a schematic view showing errors between the state variables of the continuous Lorenz chaotic systems 1 and 2; and
FIG. 10 is a schematic view of the distribution of the 2,000,000 keys generated by the analog-digital mixed chaotic system.



DETAILED DESCRIPTION
FIG. 1 shows a secure communication system according to an embodiment of the present invention, in which two analog-digital mixed chaotic systems 1 and 2 are separated from each other.
The analog-digital mixed chaotic system 1 includes a digital chaotic system 11, a continuous chaotic system 12, a timing/counting module 13, a controlling module 14, a synchronization coupling module 15, a disturbance coupling module 16, an analog to digital (A/D) converting module 17, encoding modules 18 and 19, a multiplexing switch 110, data buffering modules 111 and 112, and a encrypting/decrypting module 113, and the like.
The analog-digital mixed chaotic system 2 includes a digital chaotic system 21, a continuous chaotic system 22, a timing/counting module 23, a controlling module 24, a synchronization coupling module 25, a disturbance coupling module 26, an A/D converting module 27, encoding modules 28 and 29, a multiplexing switch 210, data buffering modules 211 and 212, and a encrypting/decrypting module 213, and the like.
Here, the digital chaotic systems 11 and 21 may be represented by a difference equation of a continuous chaotic system with finite precision, or a discrete chaotic system with other finite precision. The dynamics characteristics of the digital chaotic systems 11 and 21 may be expressed as the following equation (1):

Xj(i+1)=G(Xj(i)) i=0, 1, 2, . . . ; j=1,2  (1)

where, j=1, 2 denotes the digital chaotic systems 11 and 21 respectively,

Xj(i) denotes a m-dimensional state variable of the digital chaotic system, in which X1(0) and X2(0) represent the initial conditions of the digital chaotic systems 11 and 21, respectively, and X1(0)=X2(0).
The digital chaotic systems 11 and 21 generate an output every time after Δ5′ times of iterations are completed. It is assumed that T1 and T2 are the respective time required for the digital chaotic systems 11 and 21 to execute one iteration. For simplicity, given that T1<T2 without losing universality, the transmitting and receiving parties may take T2 as the time required for each of digital chaotic systems 11 and 21 to execute one iteration.
Since the initial conditions and system parameters of the digital chaotic systems 11 and 21 are identical to each other, the digital chaotic systems 11 and 21 can be completely synchronized. Here, in order to ensure that the continuous chaotic systems 12 and 22 can be synchronized even in the case that a data transmission delay occurs and the respective time required for one iteration of the digital chaotic systems 11 and 21 is inconsistent, the operating states of the digital chaotic systems 11 and 21 may be classified into a running state (i.e. iterative computing state) and a holding state.
The dynamics characteristics of the continuous chaotic systems 12 and 22 may be expressed by the following equation (2):




  
    
      
        
          
            
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where, j=1,2 denotes the continuous chaotic systems 12 and 22 respectively,

{tilde over (X)}j (t) is a n-dimensional state variable of the continuous chaotic system, in which {tilde over (X)}1(0) and X2(0) are initial conditions of the continuous chaotic systems 12 and 22, respectively,
A is a constant matrix of order n, and
Φ:Rn→Rn is a function for describing the nonlinear component of the continuous chaotic system.
Also, in order to ensure that the continuous chaotic systems 12 and 22 may be synchronized even in the case that a data transmission delay occurs and the respective time required for one iteration of the digital chaotic systems 11 and 21 is inconsistent, the operating states of the continuous chaotic systems 12 and 22 may be classified into a running state and a holding state as well.
The timing/counting modules 13 and 23 of FIG. 1 have the following functions:
Firstly, the timing/counting modules 13 and 23 may record the times of iterations of the digital chaotic systems 11 and 21, respectively.
By reference to FIGS. 2 and 3, three counters are provided in each of the timing/counting modules 13 and 23, i.e. C11, C12, C13 and C21, C22 and C23. The counters C11 and C21 provide the respective local controlling modules 14 and 24 with control signals for controlling the output of the digital chaotic systems 11 and 21, respectively. The counters C12 and C22 provide the respective local controlling modules 14 and 24 with control signals for controlling the digital chaotic systems 11 and 21 to output state values to the synchronization coupling modules 15 and 25, respectively. The counters C13 and C23 provide the respective local controlling modules 14 and 24 with control signals for controlling the disturbance coupling modules 16 and 26 to disturb the digital chaotic systems 11 and 21, respectively.
Secondly, the timing/counting modules 13 and 23 may record the durations for which the continuous chaotic systems 12 and 22 stay in the continuous running state (i.e. running time of the continuous chaotic system) respectively.
By reference to FIGS. 2 and 3, two timers are provided in each of the timing/counting modules 13 and 23, i.e., timers T11, T12 and T21, T22. The timers T11 and T21 provide the respective local controlling modules 14 and 24 with control signals for controlling the synchronization coupling modules 15 and 25 to output synchronized impulses, respectively. The timers T12 and T22 provide the respective local controlling modules 14 and 24 with control signals for controlling the disturbance coupling modules 16 and 26 to sample the continuous chaotic systems. It should be clarified that: the timers may record the running time of the continuous chaotic system, that is, the timers in the timing/counting modules 13 and 23 suspend when the continuous chaotic systems stay in a holding state. Therefore, with regard to the running time, the duration for which the above-mentioned synchronizing impulse is applied to the continuous chaotic system is Δ1, and the period of the sampling control signal is Δ2.
The controlling modules 14 and 24 of FIG. 1 have the following functions:
Firstly, the controlling modules 14 and 24 may control respective local digital chaotic systems 11 and 21 to initiate each key generation process, and control respective local digital chaotic systems 11 and 21 to complete each key generation process, according to the states of the respective local data buffering modules 111, 112 or 211, 212 and the signals from the counters C11 and C21 in the timing/counting modules 13 and 23.
Secondly, the controlling modules 14 and 24 may control the output of the respective local digital chaotic systems 11 and 21 to the respective local synchronization coupling modules 15 and 25, according to the signals from the counters C12 and C22 in the respective local timing/counting modules 13 and 23.
Thirdly, the controlling modules 14 and 24 may control the time at which the synchronization coupling modules 15 and 25 output the synchronizing impulse signals, according to the signals from the timers T11 and T21 in the respective local timing/counting modules 13 and 23, so that the continuous chaotic systems 12 and 22 may realize stable chaotic synchronization, without transmitting the synchronizing impulse signals.
Fourthly, the controlling modules 14 and 24 may control the time at which the disturbance coupling modules 16 and 26 sample the respective local continuous chaotic systems 12 and 22, according to the signals from the timers T11 and T21 in the respective local timing/counting modules 13 and 23.
Fifthly, the controlling modules 14 and 24 may control the time at which the disturbance coupling modules 16 and 26 output the disturbing signals, according to the signals from the counters C13 and C23 in the respective local timing/counting modules 13 and 23, so that the digital chaotic systems 11 and 21 can overcome the problem of characteristic degradation.
It can be seen from the above, the controlling modules 14 and 24 may control the operating states of respective local digital chaotic systems 11 and 21 and the operating states of respective local continuous chaotic systems 12 and 22, according to the signals from all the counters and timers in the respective local timing/counting modules 13 and 23. Further, the controlling modules 14 and 24 may control data rate of the dataflow inputted into the data buffers, according to the states of respective local data buffers.
Under control of the controlling modules 14 and 24 of FIG. 1, the synchronization coupling modules 15 and 25 generate synchronizing impulse signals, according to the state values inputted by the respective local digital chaotic systems and current state values of the respective local continuous chaotic systems, and control the respective local continuous chaotic systems 12 and 22 synchronously according to the synchronizing impulse signal, so that a stable chaotic synchronization may be achieved between the separated continuous chaotic systems 12 and 22 without transmitting the synchronizing impulse signals.
The synchronization coupling modules 15 and 25 may be adapted to realize the synchronization between the separated continuous chaotic systems 12 and 22 according to the following mathematical model (3):

Δ{tilde over (X)}j|t=tk=−B·(W(C(Xj(nk)))−{tilde over (X)}j(tk)), k=1, 2, . . . ,  (3)

where, j=1,2 denotes the synchronization coupling modules 15 and 25 respectively;

B is a diagonal matrix of order n;
tk denotes the time at which the synchronization coupling modules 15 and 25 output the synchronizing impulse signal, and tk=k·Δ1;
{tilde over (X)}j(tk) denotes state values of the continuous chaotic systems 12 and 22 at the time t=tk;
nk denotes the times of iterations when the digital chaotic systems 11 and 21 output state values to the respective local synchronization coupling modules 15 and 25;
Xj (nk) denotes state values of the digital chaotic systems 11 and 21 at i=nk;
C(•) is a function for converting binary state values of the digital chaotic systems 11 and 21 into decimal state values (i.e. to realize digital to analog conversion), and
W(•) is function for mapping an independent variable from an m-dimension space to a n-dimension space.
At the transmitting and receiving parties, it is required to construct, according to a criterion for keeping stable synchronization between the continuous chaotic systems 12 and 22, a matrix B for adjusting the error magnitude between W(C(Xj(nk))) and {tilde over (X)}j(tk). Further, it is required to determine the time tk at which the synchronization coupling modules 15 and 25 output the synchronizing impulse signal, and to synchronously control the respective local continuous chaotic systems. In this way, a stable chaotic synchronization can be achieved between the separated continuous chaotic systems 12 and 22, without transmitting the synchronizing impulse signals therebetween.
Under control of the controlling modules 14 and 24 of FIG. 1, the disturbance coupling modules 16 and 26 respectively sample the states of the respective local continuous chaotic systems 12 and 22, and output the sampled states to the respective local digital chaotic systems 11 and 21 after a conversion of a disturbance coupling function, so as to prevent characteristic degradation of the digital chaotic systems 11 and 21, respectively.
The disturbance coupling modules 16 and 26 may prevent characteristic degradation of the digital chaotic systems 11 and 12 according to the following mathematical model (4):




  
    
      
        
          
            
              
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where, j=1,2 denotes the disturbance coupling modules 16 and 26 respectively;

{tilde over (X)}j (τk) (j=1,2) indicates state value of each the continuous chaotic systems 12 and 22 at the time of t=τk (i.e. the sample value obtained by sampling the individual local continuous chaotic systems 12 and 22 at the time of t=τk by each of the disturbance coupling modules 16 and 26, where τk=k·Δ2);
D(•) is a function for converting a decimal sample value into a binary sample value with finite precision,
K is a predetermined positive integer, and
Δ3′ indicates a predetermined number of times of iterations, and
H(•) is a disturbance coupling function, which disturbs the digital chaotic systems every time that Δ3′ times of iterations are completed in the digital chaotic systems 11 and 21.
At the transmitting and receiving parties, it is required to construct the disturbance coupling function H(•) according to the characteristics of the continuous chaotic systems 12 and 22 and the characteristic degradation property of the digital chaotic systems 11 and 21. Further, it is required to determine the time tk (i.e. sampling clock period Δ2) at which the continuous chaotic systems is to be sampled, and the times of iterations Δ3′ for disturbing the respective local digital chaotic systems, so that the statistical characteristic of the disturbance coupling function H(•) may approach to a probability distribution that truncation error of a digital chaotic system follows, with probability 1 in a certain precision. Therefore, the characteristic degradation caused by the finite precision effect may be overcome in the digital chaotic systems affected by the disturbed signal.
The A/D converting modules 17 and 27 of FIG. 1 convert the continuous chaotic signals generated by the respective local continuous chaotic systems 12 and 22 into digital chaotic sequences.
The encoding modules 18, 19 and 28, 29 of FIG. 1 encode the respective local digital chaotic sequences.
The multiplexing switches 110 and 210 of FIG. 1 select one of the following sequences: the digital chaotic sequences generated by from the digital chaotic systems 11 and 21, the continuous (analog) chaotic signals generated by the continuous chaotic systems 12 and 22, and the digital chaotic sequences obtained by A/D converting the output of the continuous chaotic systems 12 and 22, as the output of the analog-digital mixed chaotic system. The digital chaotic sequence generated by the system may be adapted to encode the data as either a pseudorandom number sequence or a key stream. The continuous (analog) chaotic signals generated by the system can be adapted to design a chaotic synchronization based secure communication system.
The data buffering modules 111, 112 and 211, 212 of FIG. 1 are adapted to buffer the respective local plaintext and cipher text, respectively.
The encrypting/decrypting modules 113 and 213 of FIG. 1 are adapted to encrypt/decrypt the data inputted into the respective local data buffers. The encrypting/decrypting modules may encrypt/decrypt digital signals by using the encoded digital chaotic sequences, or may realize chaotic synchronization based secure communication by using the continuous (analog) chaotic signals generated by the continuous chaotic systems.
In the analog-digital mixed chaotic systems 1 and 2, the controlling modules 14 and 24 control the respective local digital chaotic systems, the continuous chaotic systems, the synchronization coupling modules, the disturbance coupling module and the data buffers, according to the various clock signals and control signals provided by the timing/counting modules 13 and 23.
Next, the main timing for running the analog-digital mixed chaotic system 1 will be illustrated by taking an example in which the system 1 encrypts a data signal.
In FIGS. 4, 5 and 6, signals I1,1, D1,1, D1,2, C1,0, C1,1, C1,2, C1,3, C1,4, C1,5, C1,6, C1,7, C1,8, C1,9 respectively indicate the following meanings:
(1) I1,1 indicates the input data stream, where each impulse indicates that one bit of data is inputted into the data buffering module 112 of the analog-digital mixed chaotic system 1.
(2) D1,1 and D1,2 indicate whether data in the two buffers B11 and B12 of the data buffering module 112 is ready, respectively. If data in a buffer is ready, a state flag of the buffer is set to “1”, otherwise, the state flag of the buffer is set to “0”. If a state flag of a buffer is “1”, new data is prohibited to be inputted into this buffer, and if the state flag is “0”, new data is allowed to be inputted into this buffer. Moreover, when data within the above buffer is encrypted/decrypted, the corresponding state flag will be set to “0”.
(3) C1,0 indicates the data of which buffers will be processed by the analog-digital mixed chaotic system 1. Data in the buffer B11 will be processed by the analog-digital mixed chaotic system 1 if C1,0 is “1”, and data in the buffer B12 will be processed if C1,0 is
(4) C1,1 indicates that the digital chaotic system 11 starts a process of generating one element of a chaotic sequence.
(5) C1,2 indicates that the digital chaotic system 11 ends a process of generating one element of the chaotic sequence. If data in the buffer is not ready, the digital chaotic system 11 suspends the process of generating the next element of the chaotic sequence, and if data in the buffer is ready, the digital chaotic system 11 starts the process of generating the next element of the chaotic sequence automatically.
(6) C1,3 indicates that all of the data in the buffer has been processed.
(7) Each impulse of C1,4 corresponds to the completion of one iteration in the digital chaotic system 11.
(8) C1,5 indicates that the digital chaotic system 11 outputs a state value to the synchronization coupling module 15 every time after Δ4′ times of iterations are completed.
(9) C1,6 indicates the output of the synchronization coupling module 15. With respect to the definition of running time, the duration for which the synchronization coupling module 15 output the synchronizing impulse is Δ1.
(10) C1,7 indicates the output of the continuous chaotic system 12.
(11) C1,8 indicates the control to the digital chaotic system 11. The digital chaotic system 11 executes Δ′3 times of iterations within two adjacent impulses.
(12) C1,9 indicates that the disturbance coupling module 16 samples the continuous chaotic system 12. The period for the disturbance coupling module 16 to sample the continuous chaotic system 12 is Δ2, with respect to the definition of running time.
The running timing of the analog-digital mixed chaotic system 1 is shown as follows:
In FIG. 4, when signal C1,0 is “1”, the analog-digital mixed chaotic system 1 processes the data in buffer B11. The input data I1,1 continually arrive at the analog-digital mixed chaotic system 1. When the data in the buffer B11 are ready, the state signal D1 changes from “0” to “1”. At the rising edge of the state signal D1, the starting signal C1,1 of the digital chaotic system 11 is generated, and thereby the digital chaotic system 11 starts to execute iteration with the signal C1,0 changing from “1” to “0”. Under the trigger of the rising edge of the state signal D1, the timers T11, T12 and the counters C11, C12, C13 in the timing/counting module 13 begin timing and counting. When the timer C11 records that the digital chaotic system 11 has completed Δ5′ times of iterations, the counter C11 generates a completion signal (shown as C1,2 in FIG. 4), and the counter C11 is reset. Under the trigger of the rising edge of the signal C1,2, the digital chaotic system 11 outputs a key for encryption. The analog-digital mixed chaotic system 1 processes the data in the buffer B11 with the encryption key generated by the digital chaotic system 11. When all of the data in the buffer B11 has been processed, a completion signal (shown as C1,3 in FIG. 4) is generated, and the state signal D1 of buffer B11 is set to 0. At this time, the buffer B11 enters a data preparation period.
When signal C1,0 is “0”, the analog-digital mixed chaotic system 1 processes the data in buffer B12. If the state signal D2 (shown as D2 in FIG. 2(a)) of the buffer B12 is “1”, the starting signal restarts the digital chaotic system 11 when the counter C11 generates the completion signal, the counter C11 resumes counting, and thereby the digital chaotic system 11 enters a process of generating the next key, meanwhile the signal C1,0 changes from “0” to “1”. The timers T11, T12 and the counters C12, C13 continues timing and counting. If the state signal D2 of the buffer B12 is “0”, the digital chaotic system 11 and the continuous chaotic system 12 stay in a holding state when the counter C11 generates a completion signal, and meanwhile, the timers T11, T12 and the counters C12, C13 stop timing and counting.
In FIG. 5, it is assumed that, in normal conditions, the time required for the digital chaotic system 11 to complete an iterative computation is T2. In this case, when the counter C12 records that the digital chaotic system 11 has completed Δ4′ times of iterations, the running time Δ1 of the continuous chaotic system recorded by the timer T11 is just T2·Δ4′ (shown as segment I in FIG. 4). At this time, the digital chaotic system 11 outputs a state value to the synchronization coupling module 15 under control of the controlling module 14, the synchronization coupling module 15 outputs and applies a synchronizing impulse to the continuous chaotic system 12 under control of the controlling module 14, and the counter C12 and the timer T11 are reset.
In actual situation, because of the effect of outer conditions such as running environment on the digital chaotic system 11, the time required for executing each iteration may be not T2. Therefore, when the counter C12 records that the digital chaotic system 11 has completed Δ4′ times of iterations, the running time Δ1 of the continuous chaotic system 12 may be less than T2·Δ4′ (shown as segment II in FIG. 5), and thus the digital chaotic system 11 is required to stay in the holding state after the Δ4′ times of iterations, until the running time Δ1 of the continuous chaotic system 12 arrives T2·Δ4′. Once the running time Δ1 of the continuous chaotic system 12 arrives T2·Δ4′, the digital chaotic system 11 leaves the holding state, the synchronization coupling module 15 outputs and applies the synchronizing impulse to the continuous chaotic system 12, and then the counter C12 and the timer T11 are reset.
When the running time Δ1 of the continuous chaotic system 12 arrives T2·Δ4′, the digital chaotic system 11 has not completed the Δ′ times of iterations (shown as segment III in FIG. 5). In this case, the continuous chaotic system 12 is required to stay in the holding state after the running time Δ1 arrives T2·Δ4′, until the digital chaotic system 11 completes the Δ4′ times of iterations. When the digital chaotic system 11 has completed Δ4′ times of iterations, the continuous chaotic system 12 leaves the holding state, the synchronization coupling module 15 outputs and applies the synchronizing impulse to the continuous chaotic system 12 to synchronously control the continuous chaotic system, and then the counter C12 and the timer T11 are reset.
In FIG. 6, it is assumed that, in normal conditions, the time required for executing one iteration is T2. When the counter C13 records that the digital chaotic system 11 has completed A′ times of iterations, the running time Δ2 of the continuous chaotic system 12 recorded by the timer T12 is just T2 Δ3′ (shown as segment I in FIG. 6). At this time, the disturbance coupling module 15 samples the continuous chaotic system 12 (shown as C1,9 in FIG. 6) and converts the sampled values, under control of the controlling module 14, and outputs a disturbing signal to the digital chaotic system 11 to disturb the digital chaotic system 11, and then the counter C13 and the timer T12 are reset.
In actual situation, because of the effect of outer conditions such as running environment on the digital chaotic system 11, the time required for executing each iteration may be not T2. Therefore, when the running time Δ2 of the continuous chaotic system 12 arrives T2·Δ3′, the digital chaotic system 11 has not completed Δ3′ times of iterations (shown as segment II in FIG. 6). In this case, continuous chaotic system 12 should stay in the holding state after the running time Δ2 arrives T2·Δ3′, until the digital chaotic system 11 completed Δ3′ times of iterations. When the digital chaotic system 11 has completed Δ3′ times of iterations, the continuous chaotic system 12 leaves the holding state, the disturbance coupling module 15 samples the continuous chaotic system 12 (shown as C1,9 in FIG. 6) and converts the sampled values under control of the controlling module 14, and outputs a disturbing signal to the digital chaotic system 11 to disturb the digital chaotic system 11, and then the counter C13 and the timer T12 are reset.
When the counter C13 records that the digital chaotic system 11 has completed Δ3′ times of iterations, the running time Δ2 of the continuous chaotic system 12 is less than T2·Δ3′ (shown as segment III in FIG. 6). In this case, the digital chaotic system 11 should stay in the holding state after completing Δ3′ times of iterations, until the running time Δ2 of the continuous chaotic system 12 arrives T2·Δ3′. When the running time Δ2 of the continuous chaotic system 12 arrives T2·Δ3′, the digital chaotic system 11 leaves the holding state, the disturbance coupling module 15 samples the continuous chaotic system 12 (shown as C1,9 in FIG. 6) and converts the sampled values under control of the controlling module 14, and outputs a disturbing signal to the digital chaotic system 11 to disturb the digital chaotic system 11, and then the counter C13 and the timer T12 are reset.
A criterion for holding a stable synchronization between the continuous chaotic systems 12 and 22 is illustrated as below.
(1) Given that B is a diagonal matrix of order n, I is an identity matrix of order n, d is the largest Eigenvalue of matrix (I+B)T·(I+B), q is the largest Eigenvalue of (A+AT), and L is a local Lipschitz constant of the nonlinear mapping Φ in equation (2). The digital chaotic systems 11 and 21 output their state values to the respective local synchronization coupling modules 15 and 25 every time that Δ4′ times of iterations are completed. When the duration Δ1 for which the synchronization coupling modules 15 and 25 output the synchronizing impulse signal meets the criterion of:

Δ1=Δ4′·T2  (5)

and

(q+2L)·Δ1≦−1n(ξ·d) ξ>1,d<1  (6a),

a stable chaotic synchronization may be achieved between the continuous chaotic systems 12 and 22 without transmission of the synchronizing impulse.

(2) The disturbance coupling modules 16 and 26 disturbs the respective local digital chaotic systems 11 and 21 every time after Δ3′ times of iterations are completed in the digital chaotic systems 11 and 21. The sampling clock period Δ2 and the times of iterations Δ3′ should meet the following relation:

Δ2=Δ3′·T2  (6b)

(3) If the digital chaotic system is a difference equation of a continuous chaotic system with finite word length, which is identical to the equation (2), the digital chaotic system may be expressed as equation (7)

Xj(i+1)=(I+τ·A)·Xj(i)+τ·Φ(Xj(i))  (7)

where, τ is a positive real number.

λ1 is assumed to be the largest Eigenvalue of (I+τ·A)T(I+τ·A).
The duration Δ1 for which the synchronization coupling modules 15 and 25 output the synchronizing impulse signals should further satisfy the equation (8a), in addition to satisfying equation (5).




  
    
      
        
          
            Δ
            1
          
          >
          
            
              -
              τ
            
            ·
            
              ⌊
              
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  d
                
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ln
                    ⁡
                    
                      (
                      
                        
                          
                            λ
                            1
                          
                        
                        +
                        
                          τ
                          ⁢
                          
                            
                          
                          ⁢
                          L
                        
                      
                      )
                    
                  
                
              
              ⌋
            
          
        
        ;
      
    
    
      
        (
        
          8
          ⁢
          a
        
        )
      
    
  




and Δ4′ should meet equation (8b), in addition to satisfying equation (6a).





  
    
      
        
          Δ
          4
          ′
        
        >
        
          -
          
            ⌊
            
              
                ln
                ⁢
                
                  
                
                ⁢
                d
              
              
                2
                ⁢
                
                  
                
                ⁢
                
                  ln
                  ⁡
                  
                    (
                    
                      
                        
                          λ
                          1
                        
                      
                      +
                      
                        τ
                        ⁢
                        
                          
                        
                        ⁢
                        L
                      
                    
                    )
                  
                
              
            
            ⌋
          
        
      
    
    
      
        (
        
          8
          ⁢
          b
        
        )
      
    
  




At this time, a chaotic synchronization may be achieved between the continuous chaotic systems 12 and 22. On the other hand, the errors between the digital chaotic system and local continuous chaotic system may not approach to zero, that is, the continuous chaotic system and the local digital chaotic system may not be completely synchronized, so that the continuous chaotic system may provide a disturbance to overcome the characteristic degradation.

It should be noted that all the time in the above criterions is present by taking the running time of the continuous chaotic system as a reference time.
Embodiment 1
In this embodiment, the analog-digital mixed chaotic system is designed by using the Lorenz chaotic system. The operation precision of the digital chaotic system is 19 bits, which 7 bits are adapted to indicate the integer component of a variable in the digital chaotic system, 11 bits are adapted to indicate the decimal component of the variable in the digital chaotic system, and the remaining 1 bit is adapted to indicate the sign of the variable in the digital chaotic system.
It is assumed that the digital Lorenz chaotic systems 11 and 21 are




  
    
      
        {
        
          
            
              
                
                  
                    
                      x
                      j
                    
                    ⁡
                    
                      (
                      
                        i
                        +
                        1
                      
                      )
                    
                  
                  =
                  
                    
                      
                        (
                        
                          1
                          -
                          
                            τ
                            ·
                            σ
                          
                        
                        )
                      
                      ·
                      
                        
                          x
                          j
                        
                        ⁡
                        
                          (
                          i
                          )
                        
                      
                    
                    +
                    
                      τ
                      ·
                      σ
                      ·
                      
                        
                          y
                          j
                        
                        ⁡
                        
                          (
                          i
                          )
                        
                      
                    
                    +
                    
                      
                        η
                        1
                      
                      ·
                      
                        h
                        ⁡
                        
                          (
                          
                            i
                            ,
                            
                              
                                
                                  x
                                  ~
                                
                                j
                              
                              ⁡
                              
                                (
                                t
                                )
                              
                            
                          
                          )
                        
                      
                    
                  
                
                ⁢
                
                  
                
              
            
          
          
            
              
                
                  
                    
                      
                        
                          y
                          j
                        
                        ⁡
                        
                          (
                          
                            i
                            +
                            1
                          
                          )
                        
                      
                      =
                      
                        
                          τ
                          ·
                          r
                          ·
                          
                            
                              x
                              j
                            
                            ⁡
                            
                              (
                              i
                              )
                            
                          
                        
                        +
                        
                          
                            (
                            
                              1
                              -
                              τ
                            
                            )
                          
                          ·
                          
                            
                              y
                              j
                            
                            ⁡
                            
                              (
                              i
                              )
                            
                          
                        
                        -
                      
                    
                  
                
                
                  
                    
                      
                        
                          
                            τ
                            ·
                            
                              
                                x
                                j
                              
                              ⁡
                              
                                (
                                i
                                )
                              
                            
                            ·
                            
                              
                                z
                                j
                              
                              ⁡
                              
                                (
                                i
                                )
                              
                            
                          
                          +
                          
                            
                              
                                η
                                2
                              
                              ·
                              
                                h
                                ⁡
                                
                                  (
                                  
                                    i
                                    ,
                                    
                                      
                                        
                                          y
                                          ~
                                        
                                        j
                                      
                                      ⁡
                                      
                                        (
                                        t
                                        )
                                      
                                    
                                  
                                  )
                                
                              
                            
                            ⁢
                            
                              
                            
                            ⁢
                            i
                          
                        
                        =
                        0
                      
                      ,
                      1
                      ,
                      2
                      ,
                      ...
                    
                  
                
              
            
          
          
            
              
                
                  
                
                ⁢
                
                  
                    
                      z
                      j
                    
                    ⁡
                    
                      (
                      
                        i
                        +
                        1
                      
                      )
                    
                  
                  =
                  
                    
                      
                        (
                        
                          1
                          -
                          
                            τ
                            ·
                            β
                          
                        
                        )
                      
                      ·
                      
                        
                          z
                          j
                        
                        ⁡
                        
                          (
                          i
                          )
                        
                      
                    
                    +
                    
                      τ
                      ·
                      
                        
                          x
                          j
                        
                        ⁡
                        
                          (
                          i
                          )
                        
                      
                      ·
                      
                        
                          y
                          j
                        
                        ⁡
                        
                          (
                          i
                          )
                        
                      
                    
                    +
                    
                      
                        η
                        3
                      
                      ·
                      
                        h
                        ⁡
                        
                          (
                          
                            i
                            ,
                            
                              
                                
                                  z
                                  ~
                                
                                j
                              
                              ⁡
                              
                                (
                                t
                                )
                              
                            
                          
                          )
                        
                      
                    
                  
                
                ⁢
                
                  
                
              
            
          
        
      
    
    
      
        (
        9
        )
      
    
  




where, j=1,2 denotes the digital chaotic systems 11 and 21, respectively,

the parameters are σ=10, r=28, β=2.5, τ=5×10−4 s;
h(i,•) is the disturbance coupling function, and




  
    
      
        
          h
          ⁡
          
            (
            
              i
              ,
              x
            
            )
          
        
        =
        
          {
          
            
              
                0
              
              
                
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    mod
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Δ
                      3
                      ′
                    
                  
                  ≠
                  
                    0
                    ⁢
                    
                      
                    
                    ⁢
                    or
                    ⁢
                    
                      
                    
                    ⁢
                    i
                  
                  <
                  K
                
              
            
            
              
                
                  D
                  ⁡
                  
                    (
                    x
                    )
                  
                
              
              
                
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    mod
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Δ
                      3
                      ′
                    
                  
                  =
                  
                    
                      0
                      ⁢
                      
                        
                      
                      ⁢
                      and
                      ⁢
                      
                        
                      
                      ⁢
                      i
                    
                    ≥
                    K
                  
                
              
            
          
        
      
    
    
      
        (
        10
        )
      
    
  




where D(●) is an A/D conversion function, and In this embodiment, the output of the continuous chaotic system is converted into a digital value of 19 bits, the disturbance coupling intensities are η1=0.5, η2=0.5, η3=0.5.

It is assumed that the continuous Lorenz chaotic systems 12 and 22 are




  
    
      
        {
        
          
            
              
                
                  
                    
                      d
                      
                        d
                        t
                      
                    
                    ⁢
                    
                      
                        
                          x
                          ~
                        
                        j
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                  =
                  
                    
                      
                        -
                        σ
                      
                      ·
                      
                        
                          
                            x
                            ~
                          
                          j
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                    +
                    
                      σ
                      ·
                      
                        
                          
                            y
                            ~
                          
                          j
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                  
                
                ⁢
                
                  
                
              
            
          
          
            
              
                
                  
                    d
                    
                      d
                      t
                    
                  
                  ⁢
                  
                    
                      
                        y
                        ~
                      
                      j
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
                =
                
                  
                    r
                    ·
                    
                      
                        
                          x
                          ~
                        
                        j
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                  -
                  
                    
                      
                        y
                        ~
                      
                      j
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  -
                  
                    
                      
                        
                          x
                          ~
                        
                        j
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    ·
                    
                      
                        
                          z
                          ~
                        
                        j
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
              
            
          
          
            
              
                
                  
                    
                      d
                      
                        d
                        t
                      
                    
                    ⁢
                    
                      
                        
                          z
                          ~
                        
                        j
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                  =
                  
                    
                      
                        -
                        β
                      
                      ·
                      
                        
                          
                            z
                            ~
                          
                          j
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                    +
                    
                      
                        
                          
                            x
                            ~
                          
                          j
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ·
                      
                        
                          
                            y
                            ~
                          
                          j
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                  
                
                ⁢
                
                  
                
              
            
          
          
            
              
                
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        x
                        ~
                      
                      j
                    
                  
                  =
                  
                    
                      b
                      1
                    
                    ·
                    
                      (
                      
                        
                          
                            
                              x
                              ~
                            
                            j
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                        -
                        
                          w
                          ⁡
                          
                            (
                            
                              
                                x
                                j
                              
                              ⁡
                              
                                (
                                
                                  n
                                  k
                                
                                )
                              
                            
                            )
                          
                        
                      
                      )
                    
                  
                
                ⁢
                
                  
                
              
            
          
          
            
              
                
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        y
                        ~
                      
                      j
                    
                  
                  =
                  
                    
                      b
                      2
                    
                    ·
                    
                      (
                      
                        
                          
                            
                              y
                              ~
                            
                            j
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                        -
                        
                          w
                          ⁡
                          
                            (
                            
                              
                                y
                                j
                              
                              ⁡
                              
                                (
                                
                                  n
                                  k
                                
                                )
                              
                            
                            )
                          
                        
                      
                      )
                    
                  
                
                ⁢
                
                  
                
              
            
          
          
            
              
                
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        z
                        ~
                      
                      j
                    
                  
                  =
                  
                    
                      b
                      3
                    
                    ·
                    
                      (
                      
                        
                          
                            
                              z
                              ~
                            
                            j
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                        -
                        
                          w
                          ⁡
                          
                            (
                            
                              
                                z
                                j
                              
                              ⁡
                              
                                (
                                
                                  n
                                  k
                                
                                )
                              
                            
                            )
                          
                        
                      
                      )
                    
                  
                
                ⁢
                
                  
                
              
            
          
        
      
    
    
      
        (
        11
        )
      
    
  




where, j=1,2 denotes the continuous Lorenz chaotic systems 12 and 22, respectively;

the parameters are σ=10, r=28, β=2.66, b1=−0.2, b2=−0.6, b3=−0.6;
Lipschiz index L of the continuous Lorenz chaotic system is about 50.
The workflow of the analog-digital mixed chaotic cipher system is shown as follows.
(1) Two communication parties appoint the following parameters according to equation (9) for the digital chaotic system, equation (11) for the continuous chaotic system and equations (5), (6a), (6b), (8a) and (8b) for the synchronization conditions:
the time T2=0.35 ms for the digital chaotic systems 11 and 21 to execute one iteration;
the period Δ4′=10 for the digital chaotic systems 11 and 21 to output signals to the individual synchronization coupling modules 15 and 25;
the duration Δ1=3.5 ms for which the synchronization coupling modules 15 and 25 output synchronization impulses;
the period Δ3′=1000 for the disturbance coupling modules 16 and 26 to output disturbance signals to the individual digital chaotic systems 11 and 21;
the predetermined constant K=1000 for the disturbance coupling modules, and
the period Δ2=0.35 s for the disturbance coupling modules 16 and 26 to sample the respective continuous chaotic systems 12 and 22.
(2) The two communication parties further appoint that:
the initial conditions of the digital chaotic systems 11 and 21 are x1(0)=x2(0)=0.454, y1(0)=y2(0)=0.5236, z1(0)=z2(0)=0.55878;
the times of iterations when the digital chaotic systems output the encryption/decryption key is Δ1′=1, and
the output signal selected by the multiplexing switch is the digital chaotic sequence generated by the digital chaotic systems 11 and 21.
(3) The digital chaotic systems 11 and 21 are initialized by using the appointed initial conditions, and the initial states of the continuous chaotic systems 12 and 22 are randomly generated.
(4) Under control of the controlling modules 14 and 24, the synchronization coupling modules 15 and 25 output the synchronizing impulse signals for applying to the respective local continuous chaotic systems 12 and 22. FIG. 7 is a phase diagram of the digital chaotic system 11, FIG. 8 is a phase diagram of the continuous Lorenz chaotic system 12, and both of them illustrate that the attractors of the analog-digital mixed chaotic system still have the same level complexity as those of the Lorenz chaotic system. FIG. 9 shows the errors between the state variables of the continuous Lorenz chaotic systems 12 and 22, where ex(t)={tilde over (x)}1(t)−{tilde over (x)}2(t), ey(t)={tilde over (y)}1(t)−{tilde over (y)}2(t), and ez(t)={tilde over (z)}1(t)−{tilde over (z)}2(t). It can be seen from FIG. 9 that errors between the state variables of the two continuous chaotic systems converge to zero, that is, the continuous chaotic system 1 and the continuous chaotic system 2 are synchronized.
(5) The disturbance coupling modules 16 and 26 sample, quantize and convert the respective local continuous chaotic systems 12 and 22 under control of the controlling modules 14 and 24, and obtain the disturbance signal for disturbing the digital chaotic systems.
(6) After the digital chaotic system has completed 1000 times of initial iterations, the analog-digital mixed chaotic system begins to output a key. A digital chaotic sequence generated by the digital chaotic systems 11 and 21 is first amplitude converted in the encoding modules 19 and 29 to get vj(2·i)=500·(xj(i+1000)+25), vj(2·i+1)=500·(yj(i+1000)+35), i=0, 1, 2, . . . , j=1,2. The amplitude converted sequence is encoded with an encoding function of S(v)=v mod 256. Finally, a key stream of the analog-digital mixed systems 1, 2 is obtained after a nonlinear conversion on the encoded sequence. In this embodiment, the encoded sequences v1(i) and v2(i)(i=0, 1, 2, . . . ) are converted with the nonlinear conversion F(•) (shown as Table 1), and key sequences k1(i) and k2(i)(i=0, 1, 2, . . . ) are obtained. FIG. 10 is the statistical distribution diagram for the 2,000,000 keys k1(i) (i=0, 1, 2, . . . , 2,000,000) generated by the analog-digital mixed chaotic system 1. As shown in FIG. 10, it is illustrated that the distribution of the key sequences generated by an analog-digital mixed chaotic system is relatively uniform. At last, a cipher text (or a plaintext) can be obtained by XOR of the key and a plaintext (or a cipher text).







TABLE 1





Nonlinear conversion F(·)































ν
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15


F(ν)
169
133
214
211
84
29
172
37
93
67
24
30
81
252
202
99


ν
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31


F(ν)
40
68
32
157
224
226
200
23
165
143
3
123
187
19
210
238


ν
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47


F(ν)
112
140
63
168
50
221
246
116
236
149
11
87
92
91
189
1


ν
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63


F(ν)
36
28
115
152
16
204
242
217
44
231
114
131
155
209
134
201


ν
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79


F(ν)
96
80
163
235
215
182
158
79
183
90
198
120
166
18
175
213


ν
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95


F(ν)
97
195
180
65
82
125
141
8
31
153
0
25
4
83
247
225


ν
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111


F(ν)
253
118
47
39
176
139
14
171
162
110
147
77
105
124
9
10


ν
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127


F(ν)
191
239
243
197
135
20
254
100
222
46
75
26
6
33
107
102


ν
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143


F(ν)
2
245
146
138
12
179
126
208
122
71
150
229
38
128
173
223


ν
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159


F(ν)
161
48
55
174
54
21
34
56
244
167
69
76
129
233
132
151


ν
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175


F(ν)
53
203
206
60
113
17
199
137
117
251
218
248
148
89
130
196


ν
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191


F(ν)
255
73
57
103
192
207
13
184
15
142
66
35
145
108
219
164


ν
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207


F(ν)
52
241
72
194
111
61
45
64
190
62
188
193
170
186
78
85


ν
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223


F(ν)
59
220
104
127
156
216
74
86
119
160
237
70
181
43
101
250


ν
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239


F(ν)
227
185
177
159
94
249
230
178
49
234
109
95
228
240
205
136


ν
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255


F(ν)
22
58
88
212
98
41
7
51
232
27
5
121
144
106
42
154









Embodiment 2
In this embodiment, the analog-digital mixed chaotic system is designed by using the Chen chaotic system. The operation precision of the digital chaotic system is 19 bits, which 7 bits are adapted to indicate the integer component of a variable in the digital chaotic system, 11 bits are adapted to indicate the decimal component of the variable in the digital chaotic system, and the remaining 1 bit is adapted to indicate the sign of the variable value in the digital chaotic system.
It is assumed that the digital Chen chaotic systems 11 and 21 are




  
    
      
        {
        
          
            
              
                
                  
                    x
                    j
                  
                  ⁡
                  
                    (
                    
                      i
                      +
                      1
                    
                    )
                  
                
                =
                
                  
                    
                      (
                      
                        1
                        -
                        
                          τ
                          ·
                          a
                        
                      
                      )
                    
                    ·
                    
                      
                        x
                        j
                      
                      ⁡
                      
                        (
                        i
                        )
                      
                    
                  
                  +
                  
                    τ
                    ·
                    a
                    ·
                    
                      
                        y
                        j
                      
                      ⁡
                      
                        (
                        i
                        )
                      
                    
                  
                  +
                  
                    
                      η
                      1
                    
                    ·
                    
                      h
                      ⁡
                      
                        (
                        
                          i
                          ,
                          
                            
                              
                                x
                                ~
                              
                              j
                            
                            ⁡
                            
                              (
                              t
                              )
                            
                          
                        
                        )
                      
                    
                  
                
              
            
          
          
            
              
                
                  
                    
                      y
                      j
                    
                    ⁡
                    
                      (
                      
                        i
                        +
                        1
                      
                      )
                    
                  
                  =
                  
                    
                      τ
                      ·
                      
                        (
                        
                          c
                          -
                          a
                        
                        )
                      
                      ·
                      
                        
                          x
                          j
                        
                        ⁡
                        
                          (
                          i
                          )
                        
                      
                    
                    +
                    
                      
                        (
                        
                          1
                          +
                          
                            τ
                            ·
                            c
                          
                        
                        )
                      
                      ·
                      
                        
                          y
                          j
                        
                        ⁡
                        
                          (
                          i
                          )
                        
                      
                    
                    -
                  
                
                ⁢
                
                  
                
              
            
          
          
            
              
                
                  
                    
                      
                        τ
                        ·
                        
                          
                            x
                            j
                          
                          ⁡
                          
                            (
                            i
                            )
                          
                        
                        ·
                        
                          
                            z
                            j
                          
                          ⁡
                          
                            (
                            i
                            )
                          
                        
                      
                      +
                      
                        
                          
                            η
                            2
                          
                          ·
                          
                            h
                            ⁡
                            
                              (
                              
                                i
                                ,
                                
                                  
                                    
                                      y
                                      ~
                                    
                                    j
                                  
                                  ⁡
                                  
                                    (
                                    t
                                    )
                                  
                                
                              
                              )
                            
                          
                        
                        ⁢
                        
                          
                        
                        ⁢
                        i
                      
                    
                    =
                    0
                  
                  ,
                  1
                  ,
                  2
                  ,
                  ...
                
                ⁢
                
                  
                
              
            
          
          
            
              
                
                  
                    z
                    j
                  
                  ⁡
                  
                    (
                    
                      i
                      +
                      1
                    
                    )
                  
                
                =
                
                  
                    
                      (
                      
                        1
                        -
                        
                          τ
                          ·
                          b
                        
                      
                      )
                    
                    ·
                    
                      
                        z
                        j
                      
                      ⁡
                      
                        (
                        i
                        )
                      
                    
                  
                  +
                  
                    τ
                    ·
                    
                      
                        x
                        j
                      
                      ⁡
                      
                        (
                        i
                        )
                      
                    
                    ·
                    
                      
                        y
                        j
                      
                      ⁡
                      
                        (
                        i
                        )
                      
                    
                  
                  +
                  
                    
                      η
                      3
                    
                    ·
                    
                      h
                      ⁡
                      
                        (
                        
                          i
                          ,
                          
                            
                              
                                z
                                ~
                              
                              j
                            
                            ⁡
                            
                              (
                              t
                              )
                            
                          
                        
                        )
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        12
        )
      
    
  




where, j=1,2 denotes the digital chaotic systems 11 and 21, respectively;
the parameters are a=35, b=3, c=28, τ=5×10−4 S;
h(i,•) is a disturbance coupling function, and




  
    
      
        
          h
          ⁡
          
            (
            
              i
              ,
              x
            
            )
          
        
        =
        
          {
          
            
              
                0
              
              
                
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    mod
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Δ
                      3
                      ′
                    
                  
                  ≠
                  
                    0
                    ⁢
                    
                      
                    
                    ⁢
                    or
                    ⁢
                    
                      
                    
                    ⁢
                    i
                  
                  <
                  K
                
              
            
            
              
                
                  D
                  ⁡
                  
                    (
                    x
                    )
                  
                
              
              
                
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    mod
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Δ
                      3
                      ′
                    
                  
                  =
                  
                    
                      0
                      ⁢
                      
                        
                      
                      ⁢
                      and
                      ⁢
                      
                        
                      
                      ⁢
                      i
                    
                    ≥
                    K
                  
                
              
            
          
        
      
    
    
      
        (
        13
        )
      
    
  




where D(●) is an A/D conversion function, and In this embodiment, the output of the continuous chaotic system is converted into a digital amount of 19 bits;
the disturbance coupling intensities are η1=0.01, η2=0.01, η3=0.01.
It is assumed that the continuous Chen chaotic systems 12 and 22 are




  
    
      
        {
        
          
            
              
                
                  
                    
                      d
                      
                        d
                        t
                      
                    
                    ⁢
                    
                      
                        
                          x
                          ~
                        
                        j
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                  =
                  
                    
                      
                        -
                        a
                      
                      ·
                      
                        
                          
                            x
                            ~
                          
                          j
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                    +
                    
                      a
                      ·
                      
                        
                          
                            y
                            ~
                          
                          j
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                  
                
                ⁢
                
                  
                
              
            
          
          
            
              
                
                  
                    d
                    
                      d
                      t
                    
                  
                  ⁢
                  
                    
                      
                        y
                        ~
                      
                      j
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
                =
                
                  
                    
                      (
                      
                        c
                        -
                        a
                      
                      )
                    
                    ·
                    
                      
                        
                          x
                          ~
                        
                        j
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                  +
                  
                    c
                    ·
                    
                      
                        
                          y
                          ~
                        
                        j
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                  -
                  
                    
                      
                        
                          x
                          ~
                        
                        j
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    ·
                    
                      
                        
                          z
                          ~
                        
                        j
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
              
            
          
          
            
              
                
                  
                    
                      d
                      
                        d
                        t
                      
                    
                    ⁢
                    
                      
                        
                          z
                          ~
                        
                        j
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                  =
                  
                    
                      
                        -
                        b
                      
                      ·
                      
                        
                          
                            z
                            ~
                          
                          j
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                    +
                    
                      
                        
                          
                            x
                            ~
                          
                          j
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ·
                      
                        
                          
                            y
                            ~
                          
                          j
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                  
                
                ⁢
                
                  
                
              
            
          
          
            
              
                
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        x
                        ~
                      
                      j
                    
                  
                  =
                  
                    
                      b
                      1
                    
                    ·
                    
                      (
                      
                        
                          
                            
                              x
                              ~
                            
                            j
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                        -
                        
                          w
                          ⁡
                          
                            (
                            
                              
                                x
                                j
                              
                              ⁡
                              
                                (
                                
                                  n
                                  k
                                
                                )
                              
                            
                            )
                          
                        
                      
                      )
                    
                  
                
                ⁢
                
                  
                
              
            
          
          
            
              
                
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        y
                        ~
                      
                      j
                    
                  
                  =
                  
                    
                      b
                      2
                    
                    ·
                    
                      (
                      
                        
                          
                            
                              y
                              ~
                            
                            j
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                        -
                        
                          w
                          ⁡
                          
                            (
                            
                              
                                y
                                j
                              
                              ⁡
                              
                                (
                                
                                  n
                                  k
                                
                                )
                              
                            
                            )
                          
                        
                      
                      )
                    
                  
                
                ⁢
                
                  
                
              
            
          
          
            
              
                
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        z
                        ~
                      
                      j
                    
                  
                  =
                  
                    
                      b
                      3
                    
                    ·
                    
                      (
                      
                        
                          
                            
                              z
                              ~
                            
                            j
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                        -
                        
                          w
                          ⁡
                          
                            (
                            
                              
                                z
                                j
                              
                              ⁡
                              
                                (
                                
                                  n
                                  k
                                
                                )
                              
                            
                            )
                          
                        
                      
                      )
                    
                  
                
                ⁢
                
                  
                
              
            
          
        
      
    
    
      
        (
        14
        )
      
    
  




where, j=1,2 denotes the continuous Chen chaotic systems 12 and 22 respectively,

the parameters are a=35, b=3, c=28, b1=−0.8, b2=−0.8, b3=−0.8;
Lipschiz index L of the Chen system is about 85.
The workflow of the analog-digital mixed chaotic cipher system is as follows.
(1) Two communication parties appoint the following parameters according to equation (9) for the digital chaotic system, equation (11) for the continuous chaotic system and equations (5), (6a), (6b), (8a) and (8b) for the synchronization conditions:
the time T2=0.33 ms for the digital chaotic systems 11 and 21 to execute one iteration;
the period Δ4′=30 for the digital chaotic systems 11 and 21 to output signals to the respective synchronization coupling modules 15 and 25;
the duration Δ1=10 ms of the synchronization coupling modules 15 and 25 outputting synchronization impulses;
the period Δ3′=1000 for the disturbance coupling modules 16 and 26 to output disturbance signals to the respective digital chaotic systems 11 and 21;
the predetermined constant K=3000 for the disturbance coupling modules, and
the period Δ2=0.33 s for the disturbance coupling modules 16 and 26 to sample the respective continuous chaotic systems 12 and 22.
(2) The two communication parties appoint further that:
the initial conditions of the digital chaotic systems 11 and 21 are x1(0)=x2(0)=0.454, y1(0)=y2(0)=0.5236, z1(0)=z2(0)=0.55878,
the times of iterations when the digital chaotic systems to output the encryption/decryption key is Δ5′=1, and
the output signal selected by the multiplexing switch is the digital chaotic sequence generated by the digital chaotic systems 11 and 21.
(3) The two communication parties initiate the digital chaotic systems 11 and 21 by using the appointed initial conditions, and randomly generate the initial states of the continuous chaotic systems 12 and 22.
(4) Under control of the controlling modules 14 and 24, the synchronization coupling modules 15 and 25 output the synchronizing impulse signals to be applied to the respective local continuous chaotic systems 12 and 22.
(5) Also under control of the controlling modules 14 and 24, the disturbance coupling modules 16 and 26 sample, quantize and convert the respective local continuous chaotic systems 12 and 22, and obtain the disturbance signal for disturbing the digital chaotic systems.
(6) The digital chaotic sequence generated by the digital chaotic systems 11 and 21 is amplitude converted in the encoding modules 19 and 29 to get vj(3·i)=1000 (xj(i+3000)+30), vj(3·i+1)=1000·(yj(i+3000)+40), vj(3·i+2)=500·zj(i+3000), i=0, 1, 2, . . . , j=1,2. The amplitude converted sequence is encoded with the encoding function of S(v)=v mod 256. Finally, a key stream generated by the analog-digital mixed systems 1, 2 is obtained after a nonlinear conversion on the encoded sequence. In this embodiment, the chaotic sequences generated by the analog-digital mixed chaotic system are converted with three nonlinear conversions F(•)(j=0,1,2) (shown as Table 2(a), 2(b) and 2(c)), and the conversion rule is to convert the encoded sequences v1(i) and v2(i)(i=0, 1, 2, . . . ) with the nonlinear conversion Fi mod 3(•) so as to get the key sequences of k1(i) and k2(i)(i=0, 1, 2, . . . ). At last, a cipher text (or a plaintext) can be obtained by XOR of the key and a plaintext (or a cipher text).







TABLE 2(a)





Nonlinear conversion F0(·)































ν
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15


F(ν)
169
133
214
211
84
29
172
37
93
67
24
30
81
252
202
99


ν
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31


F(ν)
40
68
32
157
224
226
200
23
165
143
3
123
187
19
210
238


ν
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47


F(ν)
112
140
63
168
50
221
246
116
236
149
11
87
92
91
189
1


ν
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63


F(ν)
36
28
115
152
16
204
242
217
44
231
114
131
155
209
134
201


ν
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79


F(ν)
96
80
163
235
215
182
158
79
183
90
198
120
166
18
175
213


ν
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95


F(ν)
97
195
180
65
82
125
141
8
31
153
0
25
4
83
247
225


ν
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111


F(ν)
253
118
47
39
176
139
14
171
162
110
147
77
105
124
9
10


ν
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127


F(ν)
191
239
243
197
135
20
254
100
222
46
75
26
6
33
107
102


ν
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143


F(ν)
2
245
146
138
12
179
126
208
122
71
150
229
38
128
173
223


ν
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159


F(ν)
161
48
55
174
54
21
34
56
244
167
69
76
129
233
132
151


ν
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175


F(ν)
53
203
206
60
113
17
199
137
117
251
218
248
148
89
130
196


ν
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191


F(ν)
255
73
57
103
192
207
13
184
15
142
66
35
145
108
219
164


ν
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207


F(ν)
52
241
72
194
111
61
45
64
190
62
188
193
170
186
78
85


ν
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223


F(ν)
59
220
104
127
156
216
74
86
119
160
237
70
181
43
101
250


ν
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239


F(ν)
227
185
177
159
94
249
230
178
49
234
109
95
228
240
205
136


ν
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255


F(ν)
22
58
88
212
98
41
7
51
232
27
5
121
144
106
42
154
















TABLE 2(b)





Nonlinear conversion F1(·)































ν
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15


F(ν)
56
232
45
166
207
222
179
184
175
96
85
199
68
111
107
91


ν
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31


F(ν)
195
98
51
181
41
160
226
167
211
145
17
6
28
188
54
75


ν
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47


F(ν)
239
136
108
168
23
196
22
244
194
69
225
214
63
61
142
152


ν
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63


F(ν)
40
78
246
62
165
249
13
223
216
43
102
122
39
47
241
114


ν
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79


F(ν)
66
212
65
192
115
103
172
139
247
173
128
31
202
44
170
52


ν
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95


F(ν)
210
11
238
233
93
148
24
248
87
174
8
197
19
205
134
185


ν
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111


F(ν)
255
125
193
49
245
138
106
177
209
32
215
2
34
4
104
113


ν
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127


F(ν)
7
219
157
153
97
190
230
89
221
81
144
220
154
163
171
208


ν
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143


F(ν)
129
15
71
26
227
236
141
191
150
123
92
162
161
99
35
77


ν
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159


F(ν)
200
158
156
58
12
46
186
110
159
90
242
146
243
73
120
204


ν
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175


F(ν)
21
251
112
117
127
53
16
3
100
109
198
116
213
180
234
9


ν
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191


F(ν)
118
25
254
64
18
224
189
5
250
1
240
42
94
169
86
67


ν
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207


F(ν)
133
20
137
155
176
229
72
121
151
252
30
130
33
140
27
95


ν
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223


F(ν)
119
84
178
29
37
79
0
70
237
88
82
235
126
218
201
253


ν
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239


F(ν)
48
149
101
60
182
228
187
124
14
80
57
38
50
132
105
147


ν
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255


F(ν)
55
231
36
164
203
83
10
135
217
76
131
143
206
59
74
183
















TABLE 2(c)





Nonlinear conversion F2(·)































ν
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15


F(ν)
99
124
119
123
242
107
111
197
48
1
103
43
254
215
171
118


ν
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31


F(ν)
202
130
201
125
250
89
71
240
173
212
162
175
156
164
114
192


ν
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47


F(ν)
183
253
147
38
54
63
247
204
52
165
229
241
113
216
49
21


ν
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63


F(ν)
4
199
35
195
24
150
5
154
7
18
128
226
235
39
178
117


ν
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79


F(ν)
9
131
44
26
27
110
90
160
82
59
214
179
41
227
47
132


ν
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95


F(ν)
83
209
0
237
32
252
177
91
106
203
190
57
74
76
88
207


ν
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111


F(ν)
208
239
170
251
67
77
51
133
69
249
2
127
80
60
159
168


ν
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127


F(ν)
81
163
64
143
146
157
56
245
188
182
218
33
16
255
243
210


ν
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143


F(ν)
205
12
19
236
95
151
68
23
196
167
126
61
100
93
25
115


ν
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159


F(ν)
96
129
79
220
34
42
144
136
70
238
184
20
222
94
11
219


ν
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175


F(ν)
224
50
58
10
73
6
36
92
194
211
172
98
145
149
228
121


ν
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191


F(ν)
231
200
55
109
141
213
78
169
108
86
244
234
101
122
174
8


ν
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207


F(ν)
186
120
37
46
28
166
180
198
232
221
116
31
75
189
139
138


ν
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223


F(ν)
112
62
181
102
72
3
246
14
97
53
87
185
134
193
29
158


ν
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239


F(ν)
225
248
152
17
105
217
142
148
155
30
135
233
206
85
40
223


ν
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255


F(ν)
140
161
137
13
191
230
66
104
65
153
45
15
176
84
187
22









Embodiment 3
In this embodiment, the analog-digital mixed chaotic system is designed by using a one-dimensional Logistic chaotic system and a Chen chaotic system as the digital chaotic system and the continuous chaotic system respectively. The operation precision of the digital chaotic system is 19 bits, which 18 bits are adapted to indicate the decimal component of a variable in the digital chaotic system, and the remaining 1 bit is adapted to indicate the sign of the variable in the digital chaotic system. Although the transmitting and receiving parties have appointed the time required for the digital chaotic systems 11 and 21 to perform one iteration, the actual time that the digital chaotic systems 11 and 21 take to execute each operation will be a bit different. Therefore, in this embodiment, the respective local digital chaotic system and continuous chaotic system are controlled by the controlling modules 14 and 24 in the analog-digital mixed chaotic systems 1 and 2, so that a stable synchronization between the continuous chaotic systems that are separated with each other in location can be achieved and thereby the characteristic degradation may be overcome in the digital chaotic system.
It is assumed that the digital Logistic chaotic systems 11 and 21 are

xj(i+1)=4·xj(i)·(1−xj(i))+η·h(i,{tilde over (x)}j(t))  (15)

where, j=1,2 denotes the digital chaotic systems 11 and 21 respectively,

h(i,•) is a disturbance coupling function, and




  
    
      
        
          h
          ⁡
          
            (
            
              i
              ,
              x
            
            )
          
        
        =
        
          {
          
            
              
                0
              
              
                
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    mod
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Δ
                      3
                      ′
                    
                  
                  ≠
                  
                    0
                    ⁢
                    
                      
                    
                    ⁢
                    or
                    ⁢
                    
                      
                    
                    ⁢
                    i
                  
                  <
                  K
                
              
            
            
              
                
                  D
                  ⁡
                  
                    (
                    x
                    )
                  
                
              
              
                
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    mod
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Δ
                      3
                      ′
                    
                  
                  =
                  
                    
                      0
                      ⁢
                      
                        
                      
                      ⁢
                      and
                      ⁢
                      
                        
                      
                      ⁢
                      i
                    
                    ≥
                    K
                  
                
              
            
          
        
      
    
    
      
        (
        16
        )
      
    
  




where D(●) is an analog to digital conversion function, and in this embodiment, the output of the continuous chaotic system is converted into a digital amount of 19 bits, and the disturbance coupling intensity is η=0.01.
It is assumed that the continuous Chen chaotic systems 12 and 22 are




  
    
      
        {
        
          
            
              
                
                  
                    
                      d
                      
                        d
                        t
                      
                    
                    ⁢
                    
                      
                        
                          x
                          ~
                        
                        j
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                  =
                  
                    
                      
                        -
                        a
                      
                      ·
                      
                        
                          
                            x
                            ~
                          
                          j
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                    +
                    
                      a
                      ·
                      
                        
                          
                            y
                            ~
                          
                          j
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                  
                
                ⁢
                
                  
                
              
            
          
          
            
              
                
                  
                    d
                    
                      d
                      t
                    
                  
                  ⁢
                  
                    
                      
                        y
                        ~
                      
                      j
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
                =
                
                  
                    
                      (
                      
                        c
                        -
                        a
                      
                      )
                    
                    ·
                    
                      
                        
                          x
                          ~
                        
                        j
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                  +
                  
                    c
                    ·
                    
                      
                        
                          y
                          ~
                        
                        j
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                  -
                  
                    
                      
                        
                          x
                          ~
                        
                        j
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    ·
                    
                      
                        
                          z
                          ~
                        
                        j
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
              
            
          
          
            
              
                
                  
                    
                      d
                      
                        d
                        t
                      
                    
                    ⁢
                    
                      
                        
                          z
                          ~
                        
                        j
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                  =
                  
                    
                      
                        -
                        b
                      
                      ·
                      
                        
                          
                            z
                            ~
                          
                          j
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                    +
                    
                      
                        
                          
                            x
                            ~
                          
                          j
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ·
                      
                        
                          
                            y
                            ~
                          
                          j
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                  
                
                ⁢
                
                  
                
              
            
          
          
            
              
                
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        y
                        ~
                      
                      j
                    
                  
                  =
                  
                    
                      b
                      2
                    
                    ·
                    
                      (
                      
                        
                          
                            
                              y
                              ~
                            
                            j
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                        -
                        
                          w
                          ⁡
                          
                            (
                            
                              
                                x
                                j
                              
                              ⁡
                              
                                (
                                
                                  n
                                  k
                                
                                )
                              
                            
                            )
                          
                        
                      
                      )
                    
                  
                
                ⁢
                
                  
                
              
            
          
        
      
    
    
      
        (
        17
        )
      
    
  




where, j=1,2 denotes the continuous Chen chaotic systems 12 and 22 respectively;

the parameters are a=35, b=3, c=28, b2=−0.8;
Lipschiz index L of the Chen system is about 85.
The workflow of the analog-digital mixed chaotic cipher system is shown as follows.
(1) Two communication parties appoint the following parameters according to equation (9) for the digital chaotic system, equation (11) for the continuous chaotic system and equations (5), (6a), (6b), (8a) and (8b) for the synchronization conditions:
the time T2=1 ms for the digital chaotic systems 11 and 21 to execute one iterative;
the period Δ4′=10 for the digital chaotic systems 11 and 21 to output signals to the respective synchronization coupling modules 15 and 25;
the duration Δ1=10 ms for which the synchronization coupling modules 15 and 25 output synchronizing impulses;
the period Δ3′=10 for the disturbance coupling modules 16 and 26 to output disturbance signals to the respective digital chaotic systems 11 and 21;
the predetermined constant K=5000 for the disturbance coupling modules; and
the period Δ2=10 ms for the disturbance coupling modules 16 and 26 to sample the respective continuous Chen chaotic systems 12 and 22.
(2) The two communication parties further appoint that:
the initial conditions of the digital chaotic systems 11 and 21 are x1(0)=x2(0)=0.333;
the times of iterations when the digital chaotic systems to output the encryption/decryption key is Δ5′=1, and
the output signal selected by the multiplexing switch is the digital chaotic sequence generated by the digital chaotic systems 11 and 21.
(3) The digital chaotic systems 11 and 21 are initialized by using the appointed initial conditions, and the initial states of the continuous chaotic systems 12 and 22 are randomly generated.
(4) Under control of the controlling modules 14 and 24, the synchronization coupling modules 15 and 25 output the synchronizing impulse signals to be applied to the respective local continuous chaotic systems 12 and 22.
(5) Under control of the controlling modules 14 and 24, the disturbance coupling modules 16 and 26 sample, quantize and convert the respective local continuous chaotic systems 12 and 22, and obtain the disturbance signal for disturbing the digital chaotic systems.
(6) After the digital chaotic system has completed 5000 times of initial iterations, the analog-digital mixed chaotic system begins to output a key. The digital chaotic sequence generated by the digital chaotic systems 11 and 21 is amplitude converted in the encoding modules 19 and 29 to get vj(i)=xj(i+5000)(i=0, 1, 2, . . . j=1,2). The amplitude converted sequence is encoded with the encoding function of S(v)=[256(1−ar cos v/π)], and finally a key stream of the analog-digital mixed systems 1, 2 is obtained after a nonlinear conversion. In this embodiment, the encoded sequences v1(i) and v2(i)(i=0, 1, 2, . . . ) are converted with the nonlinear conversion F(•) (shown as Table 3), and key sequences k1(i) and k2(i)(i=0, 1, 2, . . . ) are obtained. At last a cipher text (or a plaintext) can be obtained by XOR of the key and a plaintext (or a cipher text).







TABLE 3





Nonlinear conversion F(·)































ν
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15


F(ν)
169
133
214
211
84
29
172
37
93
67
24
30
81
252
202
99


ν
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31


F(ν)
40
68
32
157
224
226
200
23
165
143
3
123
187
19
210
238


ν
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47


F(ν)
112
140
63
168
50
221
246
116
236
149
11
87
92
91
189
1


ν
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63


F(ν)
36
28
115
152
16
204
242
217
44
231
114
131
155
209
134
201


ν
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79


F(ν)
96
80
163
235
215
182
158
79
183
90
198
120
166
18
175
213


ν
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95


F(ν)
97
195
180
65
82
125
141
8
31
153
0
25
4
83
247
225


ν
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111


F(ν)
253
118
47
39
176
139
14
171
162
110
147
77
105
124
9
10


ν
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127


F(ν)
191
239
243
197
135
20
254
100
222
46
75
26
6
33
107
102


ν
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143


F(ν)
2
245
146
138
12
179
126
208
122
71
150
229
38
128
173
223


ν
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159


F(ν)
161
48
55
174
54
21
34
56
244
167
69
76
129
233
132
151


ν
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175


F(ν)
53
203
206
60
113
17
199
137
117
251
218
248
148
89
130
196


ν
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191


F(ν)
255
73
57
103
192
207
13
184
15
142
66
35
145
108
219
164


ν
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207


F(ν)
52
241
72
194
111
61
45
64
190
62
188
193
170
186
78
85


ν
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223


F(ν)
59
220
104
127
156
216
74
86
119
160
237
70
181
43
101
250


ν
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239


F(ν)
227
185
177
159
94
249
230
178
49
234
109
95
228
240
205
136


ν
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255


F(ν)
22
58
88
212
98
41
7
51
232
27
5
121
144
106
42
154

## Claims

1. An analog-digital mixed chaotic system, comprising:
a digital chaotic system adapted to generate a digital chaotic sequence;
a continuous chaotic system adapted to generate a continuous chaotic signal;
a synchronization coupling module adapted to generate a synchronizing impulse signal for controlling the continuous chaotic system according to states of the digital chaotic system and the continuous chaotic system, and output the synchronizing impulse signal to the continuous chaotic system;
a disturbance coupling module adapted to disturb the digital chaotic system according to the state of the continuous chaotic system; and
a controlling module adapted to control operating states of the digital chaotic system and the continuous chaotic system, and operating states of the synchronization coupling module and the disturbance coupling module.

2. The analog-digital mixed chaotic system according to claim 1, further comprising a timing and counting module adapted to record the operating states of the digital chaotic system and the continuous chaotic system, and output a corresponding signal to the controlling module according to the recoded result; and
wherein the controlling module controls the operating states of the digital chaotic system, the continuous chaotic system, the synchronization coupling module and the disturbance coupling module according to the corresponding signal from the timing and counting module.

3. The analog-digital mixed chaotic system according to claim 2, wherein the timing and counting module comprises a first counter, a second counter and a third counter for counting times of iterations of the digital chaotic system, wherein the first counter provides the controlling module with a first control signal for controlling the digital chaotic system to output, the second counter provides the controlling module with a second control signal for controlling the digital chaotic system to output a state value to the synchronization coupling module, and the third counter provides the controlling module with a third control signal for controlling the disturbance coupling module to disturb the digital chaotic system; and
the timing and counting module comprises a first timer and a second timer for timing duration of the continuous chaotic system staying in a running state, wherein the first timer provides the controlling module with a fourth control signal for controlling the synchronization coupling module to output a synchronizing impulse, and the second timer provides the controlling module with a fifth control signal for controlling the disturbance coupling module to sample the continuous chaotic system.

4. The analog-digital mixed chaotic system according to claim 1, further comprising an A/D converting module for converting the continuous chaotic signal generated by the continuous chaotic system into a digital chaotic sequence.

5. The analog-digital mixed chaotic system according to claim 4, further comprising a multiplexing switch for selecting one of three received inputs as an output, the three received inputs are:
the digital chaotic sequence from the digital chaotic system;
the continuous chaotic signal from the continuous chaotic system; and
the digital chaotic sequence from the A/D converting module.

6. The analog-digital mixed chaotic system according to claim 5, further comprising a first encoding module and a second encoding module, and
wherein, the digital chaotic sequence from the digital chaotic system is outputted to the multiplexing switch after being processed by the first encoding module, the digital chaotic sequence from the A/D converting module is outputted to the multiplexing switch after being processed by the second encoding module, the multiplexing switch selects one of the three inputs from the first encoding module, the second encoding module and the continuous chaotic system as the output of the analog-digital mixed chaotic system.

7. The analog-digital mixed chaotic system according to claim 6, further comprising an encrypting/decrypting module connected to the multiplexing switch, wherein the encrypting/decrypting module configured to encrypt an inputted plaintext and output an encrypted one or configured to decrypt an inputted cipher text and output a decrypted one, according to the output of the multiplexing switch.

8. The analog-digital mixed chaotic system according to claim 7, further comprising;
a first and second data buffers connected to the encrypting/decrypting module and the controlling module, wherein the first data buffer is configured to transmit or receive the cipher text under the control of the controlling module, and the second data buffer is configured to transmit or receive the plaintext under the control of the controlling module, and
wherein the encrypting/decrypting module, according to the output of the multiplexing switch, encrypts the plaintext in the second data buffer and output the encrypted one to the first data buffer, or decrypts the cipher text in the first data buffer and outputs the decrypted one to the second data buffer.

9. The analog-digital mixed chaotic system according to claim 8, wherein, the encrypting/decrypting module encrypts the plaintext or decrypts the cipher text according to the encoded digital chaotic sequence generated by the multiplexing switch, or realizes a chaotic synchronization based secure communication according to the continuous chaotic signals from the multiplexing switch.

10. The analog-digital mixed chaotic system according to claim 3, wherein, the synchronization coupling module generates and outputs, according to the state value from the digital chaotic system and the state of the continuous chaotic system, a synchronizing impulse signal to the continuous chaotic module under control of the controlling module, the disturbance coupling module samples the state of the continuous chaotic system under control of the controlling module and the sampled value is output to the digital chaotic system after being converted by a disturbance coupling function.

11. The analog-digital mixed chaotic system according to claim 10, wherein, the operating states of the digital chaotic system are classified into a running state and a holding state, and operating states of the continuous chaotic system are classified into a running state and a holding state; the digital chaotic system output a state value to the synchronization coupling module every time after a predetermined number of times of iterations are completed; the disturbance coupling module disturbs the digital chaotic system every time after the digital chaotic system completes a predetermined number of times of iterations.

12. A method for generating an analog-digital mixed chaotic signal, comprising:
generating a digital chaotic sequence from a digital chaotic system;
generating continuous chaotic signals from a continuous chaotic system;
generating a synchronizing impulse signal for controlling the continuous chaotic system according to states of the digital chaotic system and the continuous chaotic system, and outputting the synchronizing impulse signal to the continuous chaotic system; and
disturbing the digital chaotic system according to the state of the continuous chaotic system.

13. The method according to claim 12, further comprising:
recording the operating states of the digital chaotic system and the continuous chaotic system, and outputting a corresponding signal to a controlling module according to the recoding result; and
wherein, the generating the digital chaotic sequence from the digital chaotic system and the generating the continuous chaotic signal from the continuous chaotic system are performed under control of the controlling module based on the corresponding signal;
the generating the synchronizing impulse signal for controlling the continuous chaotic system is performed by the synchronization coupling module, under control of the controlling module based on the corresponding signal; and
the disturbing the digital chaotic system is performed under control of the controlling module based on the corresponding signal.

14. The method according to claim 13, wherein, the recording the operating states of the digital chaotic system and the continuous chaotic system, and outputting a corresponding signal to the controlling module according to the recorded result comprises:
counting times of iterations of the digital chaotic system, wherein a first counter provides the controlling module with a first control signal for controlling output of the digital chaotic system, a second counter provides the controlling module with a second control signal for controlling the digital chaotic system to output a state value to the synchronization coupling module, and a third counter provides the controlling module with a third control signal for controlling the disturbance coupling module to disturb the digital chaotic system; and
timing duration of the continuous chaotic system staying in a running state, wherein a first timer provides the controlling module with a fourth control signal for controlling the synchronization coupling module to output a synchronizing impulse, and a second timer provides the controlling module with a fifth control signal for controlling the disturbance coupling module to sample the continuous chaotic system.

15. The method according to claim 13, further comprising:
converting the continuous chaotic signal generated by the continuous chaotic system into a digital chaotic sequence; and
selecting one of the digital chaotic sequence from the digital chaotic system, the continuous chaotic signal from the continuous chaotic system, and the digital chaotic sequence converted from the continuous chaotic signal, as an output.

16. The method according to claim 13, further comprising:
encoding the digital chaotic sequence generated by the digital chaotic system;
converting the continuous chaotic signal generated by the continuous chaotic system into a digital chaotic sequence, and encoding the digital chaotic sequence converted from the continuous chaotic signal; and
selecting one of the encoded digital chaotic sequence from the digital chaotic system, the continuous chaotic signal from the continuous chaotic system, and the encoded digital chaotic sequence converted from the continuous chaotic signal, as an output.

17. The method according to any one of claim 12, wherein, the synchronizing impulse signal in generating a synchronizing impulse signal for controlling the continuous chaotic system is generated according to state value from the digital chaotic system and the state of the continuous chaotic system; and
the disturbing the digital chaotic system comprises sampling the state of the continuous chaotic system, and outputting, after a conversion using a disturbance function, the sampled value to the digital chaotic system so as to disturb the digital chaotic system.

18. An encryption communication system, comprising a transmitting party, and a receiving party configured to communicate with the transmitting party, wherein a first analog-digital mixed chaotic system included in the transmitting party is configured to encrypt a plaintext to bc transmitted by using a chaotic signal generated by itself, and outputting a resultant cipher text; and a second analog-digital mixed chaotic system included in the receiving party is configured to decrypt a received cipher text by using a chaotic signal generated by itself, and outputting a resultant plaintext; and
wherein, initial conditions and system parameters of the digital chaotic systems in the first and the second analog-digital mixed chaotic systems are substantially the same, and
each of the first and the second analog-digital mixed chaotic systems respectively comprises:
a digital chaotic system adapted to generate a digital chaotic sequence;
a continuous chaotic system adapted to generate a continuous chaotic signal;
a synchronization coupling module adapted to generate a synchronizing impulse signal for controlling the continuous chaotic system according to states of the digital chaotic system and the continuous chaotic system, and output the synchronizing impulse signal to the continuous chaotic system;
a disturbance coupling module adapted to disturb the digital chaotic system according to the state of the continuous chaotic system; and
a controlling module adapted to control operating states of the digital chaotic system and the continuous chaotic system, and operating states of the synchronization coupling module and the disturbance coupling module.