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.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    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 
       [0002]    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 
       [0003]    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. 
         [0004]    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. 
         [0005]    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. 
         [0006]    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 
       [0007]    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. 
         [0008]    One embodiment of the invention provides an analog-digital mixed chaotic system, including: 
         [0009]    a digital chaotic system adapted to generate a digital chaotic sequence; 
         [0010]    a continuous chaotic system adapted to generate a continuous chaotic signal; 
         [0011]    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; 
         [0012]    a disturbance coupling module adapted to disturb the digital chaotic system according to the state of the continuous chaotic system; and 
         [0013]    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. 
         [0014]    Another embodiment of the invention provides a method for generating an analog-digital mixed chaotic signal, including: 
         [0015]    generating a digital chaotic sequence from a digital chaotic system; 
         [0016]    generating a continuous chaotic signal from a continuous chaotic system; 
         [0017]    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. 
         [0018]    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. 
         [0019]    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) 
         [0020]    The present invention will be further described in conjunction with the figures and embodiments as follows, in which: 
           [0021]      FIG. 1  is a schematic diagram of an analog-digital mixed chaotic system according to an embodiment of the present invention; 
           [0022]      FIG. 2  is a block diagram of one timing/counting module as shown in  FIG. 1  according to an embodiment of present invention; 
           [0023]      FIG. 3  is a block diagram of the other timing/counting module as shown in  FIG. 1  according to an embodiment of present invention; 
           [0024]      FIG. 4  is a schematic view of a main timing for running the analog-digital mixed chaotic system shown in  FIG. 1 ; 
           [0025]      FIG. 5  is a schematic view of a main timing for running the synchronization coupling module shown in  FIG. 1 ; 
           [0026]      FIG. 6  is a schematic view of a main timing for running the disturbance coupling module shown in  FIG. 1 ; 
           [0027]      FIG. 7  is a phase diagram of the digital chaotic system  1 ; 
           [0028]      FIG. 8  is a phase diagram of the continuous chaotic system  1 ; 
           [0029]      FIG. 9  is a schematic view showing errors between the state variables of the continuous Lorenz chaotic systems  1  and  2 ; and 
           [0030]      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 
       [0031]      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. 
         [0032]    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. 
         [0033]    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. 
         [0034]    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): 
         [0000]        X   j ( i+ 1)= G ( X   j ( i )) i=0, 1, 2, . . . ; j=1,2  (1) 
         [0000]    where, j=1, 2 denotes the digital chaotic systems  11  and  21  respectively, 
         [0035]    X j (i) denotes a m-dimensional state variable of the digital chaotic system, in which X 1 (0) and X 2 (0) represent the initial conditions of the digital chaotic systems  11  and  21 , respectively, and X 1 (0)=X 2 (0). 
         [0036]    The digital chaotic systems  11  and  21  generate an output every time after Δ 5 ′ times of iterations are completed. It is assumed that T 1  and T 2  are the respective time required for the digital chaotic systems  11  and  21  to execute one iteration. For simplicity, given that T 1 &lt;T 2  without losing universality, the transmitting and receiving parties may take T 2  as the time required for each of digital chaotic systems  11  and  21  to execute one iteration. 
         [0037]    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. 
         [0038]    The dynamics characteristics of the continuous chaotic systems  12  and  22  may be expressed by the following equation (2): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
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         [0000]    where, j=1,2 denotes the continuous chaotic systems  12  and  22  respectively, 
         [0039]    {tilde over (X)} j  (t) is a n-dimensional state variable of the continuous chaotic system, in which {tilde over (X)} 1 (0) and X 2 ( 0 ) are initial conditions of the continuous chaotic systems  12  and  22 , respectively, 
         [0040]    A is a constant matrix of order n, and 
         [0041]    Φ:R n →R n  is a function for describing the nonlinear component of the continuous chaotic system. 
         [0042]    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. 
         [0043]    The timing/counting modules  13  and  23  of  FIG. 1  have the following functions: 
         [0044]    Firstly, the timing/counting modules  13  and  23  may record the times of iterations of the digital chaotic systems  11  and  21 , respectively. 
         [0045]    By reference to  FIGS. 2 and 3 , three counters are provided in each of the timing/counting modules  13  and  23 , i.e. C 11 , C 12 , C 13  and C 21 , C 22  and C 23 . The counters C 11  and C 21  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 C 12  and C 22  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 C 13  and C 23  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. 
         [0046]    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. 
         [0047]    By reference to  FIGS. 2 and 3 , two timers are provided in each of the timing/counting modules  13  and  23 , i.e., timers T 11 , T 12  and T 21 , T 22 . The timers T 11  and T 21  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 T 12  and T 22  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 Δ 1 , and the period of the sampling control signal is Δ 2 . 
         [0048]    The controlling modules  14  and  24  of  FIG. 1  have the following functions: 
         [0049]    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 C 11  and C 21  in the timing/counting modules  13  and  23 . 
         [0050]    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 C 12  and C 22  in the respective local timing/counting modules  13  and  23 . 
         [0051]    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 T 11  and T 21  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. 
         [0052]    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 T 11  and T 21  in the respective local timing/counting modules  13  and  23 . 
         [0053]    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 C 13  and C 23  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. 
         [0054]    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. 
         [0055]    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. 
         [0056]    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): 
         [0000]      Δ {tilde over (X)}   j | t=t     k     =−B ·( W ( C ( X   j ( n   k )))− {tilde over (X)}   j ( t   k )), k=1, 2, . . . ,  (3) 
         [0000]    where, j=1,2 denotes the synchronization coupling modules  15  and  25  respectively; 
         [0057]    B is a diagonal matrix of order n; 
         [0058]    t k  denotes the time at which the synchronization coupling modules  15  and  25  output the synchronizing impulse signal, and t k =k·Δ 1 ; 
         [0059]    {tilde over (X)} j (t k ) denotes state values of the continuous chaotic systems  12  and  22  at the time t=t k ; 
         [0060]    n k  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 ; 
         [0061]    X j  (n k ) denotes state values of the digital chaotic systems  11  and  21  at i=n k ; 
         [0062]    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 
         [0063]    W(•) is function for mapping an independent variable from an m-dimension space to a n-dimension space. 
         [0064]    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(X j (n k ))) and {tilde over (X)} j (t k ). Further, it is required to determine the time t k  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. 
         [0065]    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. 
         [0066]    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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         [0000]    where, j=1,2 denotes the disturbance coupling modules  16  and  26  respectively; 
         [0067]    {tilde over (X)} j  (τ k ) (j=1,2) indicates state value of each the continuous chaotic systems  12  and  22  at the time of t=τ k  (i.e. the sample value obtained by sampling the individual local continuous chaotic systems  12  and  22  at the time of t=τ k  by each of the disturbance coupling modules  16  and  26 , where τ k =k·Δ 2 ); 
         [0068]    D(•) is a function for converting a decimal sample value into a binary sample value with finite precision, 
         [0069]    K is a predetermined positive integer, and 
         [0070]    Δ 3 ′ indicates a predetermined number of times of iterations, and 
         [0071]    H(•) is a disturbance coupling function, which disturbs the digital chaotic systems every time that Δ 3 ′ times of iterations are completed in the digital chaotic systems  11  and  21 . 
         [0072]    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 t k  (i.e. sampling clock period Δ 2 ) at which the continuous chaotic systems is to be sampled, and the times of iterations Δ 3 ′ 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. 
         [0073]    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. 
         [0074]    The encoding modules  18 ,  19  and  28 ,  29  of  FIG. 1  encode the respective local digital chaotic sequences. 
         [0075]    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. 
         [0076]    The data buffering modules  111 ,  112  and  211 ,  212  of  FIG. 1  are adapted to buffer the respective local plaintext and cipher text, respectively. 
         [0077]    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. 
         [0078]    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 . 
         [0079]    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. 
         [0080]    In  FIGS. 4 ,  5  and  6 , signals I 1,1 , D 1,1 , D 1,2 , C 1,0 , C 1,1 , C 1,2 , C 1,3 , C 1,4 , C 1,5 , C 1,6 , C 1,7 , C 1,8 , C 1,9  respectively indicate the following meanings: 
         [0081]    (1) I 1,1  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 . 
         [0082]    (2) D 1,1  and D 1,2  indicate whether data in the two buffers B 11  and B 12  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”. 
         [0083]    (3) C 1,0  indicates the data of which buffers will be processed by the analog-digital mixed chaotic system  1 . Data in the buffer B 11  will be processed by the analog-digital mixed chaotic system  1  if C 1,0  is “1”, and data in the buffer B 12  will be processed if C 1,0  is 
         [0084]    (4) C 1,1  indicates that the digital chaotic system  11  starts a process of generating one element of a chaotic sequence. 
         [0085]    (5) C 1,2  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. 
         [0086]    (6) C 1,3  indicates that all of the data in the buffer has been processed. 
         [0087]    (7) Each impulse of C 1,4  corresponds to the completion of one iteration in the digital chaotic system  11 . 
         [0088]    (8) C 1,5  indicates that the digital chaotic system  11  outputs a state value to the synchronization coupling module  15  every time after Δ 4 ′ times of iterations are completed. 
         [0089]    (9) C 1,6  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 Δ 1 . 
         [0090]    (10) C 1,7  indicates the output of the continuous chaotic system  12 . 
         [0091]    (11) C 1,8  indicates the control to the digital chaotic system  11 . The digital chaotic system  11  executes Δ′ 3  times of iterations within two adjacent impulses. 
         [0092]    (12) C 1,9  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 Δ 2 , with respect to the definition of running time. 
         [0093]    The running timing of the analog-digital mixed chaotic system  1  is shown as follows: 
         [0094]    In  FIG. 4 , when signal C 1,0  is “1”, the analog-digital mixed chaotic system  1  processes the data in buffer B 11 . The input data I 1,1  continually arrive at the analog-digital mixed chaotic system  1 . When the data in the buffer B 11  are ready, the state signal D 1  changes from “0” to “1”. At the rising edge of the state signal D 1 , the starting signal C 1,1  of the digital chaotic system  11  is generated, and thereby the digital chaotic system  11  starts to execute iteration with the signal C 1,0  changing from “1” to “0”. Under the trigger of the rising edge of the state signal D 1 , the timers T 11 , T 12  and the counters C 11 , C 12 , C 13  in the timing/counting module  13  begin timing and counting. When the timer C 11  records that the digital chaotic system  11  has completed Δ 5 ′ times of iterations, the counter C 11  generates a completion signal (shown as C 1,2  in  FIG. 4 ), and the counter C 11  is reset. Under the trigger of the rising edge of the signal C 1,2 , the digital chaotic system  11  outputs a key for encryption. The analog-digital mixed chaotic system  1  processes the data in the buffer B 11  with the encryption key generated by the digital chaotic system  11 . When all of the data in the buffer B 11  has been processed, a completion signal (shown as C 1,3  in  FIG. 4 ) is generated, and the state signal D 1  of buffer B 11  is set to 0. At this time, the buffer B 11  enters a data preparation period. 
         [0095]    When signal C 1,0  is “0”, the analog-digital mixed chaotic system  1  processes the data in buffer B 12 . If the state signal D 2  (shown as D 2  in  FIG. 2(   a )) of the buffer B 12  is “1”, the starting signal restarts the digital chaotic system  11  when the counter C 11  generates the completion signal, the counter C 11  resumes counting, and thereby the digital chaotic system  11  enters a process of generating the next key, meanwhile the signal C 1,0  changes from “0” to “1”. The timers T 11 , T 12  and the counters C 12 , C 13  continues timing and counting. If the state signal D 2  of the buffer B 12  is “0”, the digital chaotic system  11  and the continuous chaotic system  12  stay in a holding state when the counter C 11  generates a completion signal, and meanwhile, the timers T 11 , T 12  and the counters C 12 , C 13  stop timing and counting. 
         [0096]    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 T 2 . In this case, when the counter C 12  records that the digital chaotic system  11  has completed Δ 4 ′ times of iterations, the running time Δ 1  of the continuous chaotic system recorded by the timer T 11  is just T 2 ·Δ 4 ′ (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 C 12  and the timer T 11  are reset. 
         [0097]    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 T 2 . Therefore, when the counter C 12  records that the digital chaotic system  11  has completed Δ 4 ′ times of iterations, the running time Δ 1  of the continuous chaotic system  12  may be less than T 2 ·Δ 4 ′ (shown as segment II in  FIG. 5 ), and thus the digital chaotic system  11  is required to stay in the holding state after the Δ 4 ′ times of iterations, until the running time Δ 1  of the continuous chaotic system  12  arrives T 2 ·Δ 4 ′. Once the running time Δ 1  of the continuous chaotic system  12  arrives T 2 ·Δ 4 ′, 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 C 12  and the timer T 11  are reset. 
         [0098]    When the running time Δ 1  of the continuous chaotic system  12  arrives T 2 ·Δ 4 ′, 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 Δ 1  arrives T 2 ·Δ 4 ′, until the digital chaotic system  11  completes the Δ 4 ′ times of iterations. When the digital chaotic system  11  has completed Δ 4 ′ 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 C 12  and the timer T 11  are reset. 
         [0099]    In  FIG. 6 , it is assumed that, in normal conditions, the time required for executing one iteration is T 2 . When the counter C 13  records that the digital chaotic system  11  has completed A′ times of iterations, the running time Δ 2  of the continuous chaotic system  12  recorded by the timer T 12  is just T 2  Δ 3 ′ (shown as segment I in  FIG. 6 ). At this time, the disturbance coupling module  15  samples the continuous chaotic system  12  (shown as C 1,9  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 C 13  and the timer T 12  are reset. 
         [0100]    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 T 2 . Therefore, when the running time Δ 2  of the continuous chaotic system  12  arrives T 2 ·Δ 3 ′, the digital chaotic system  11  has not completed Δ 3 ′ 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 Δ 2  arrives T 2 ·Δ 3 ′, until the digital chaotic system  11  completed Δ 3 ′ times of iterations. When the digital chaotic system  11  has completed Δ 3 ′ 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 C 1,9  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 C 13  and the timer T 12  are reset. 
         [0101]    When the counter C 13  records that the digital chaotic system  11  has completed Δ 3 ′ times of iterations, the running time Δ 2  of the continuous chaotic system  12  is less than T 2 ·Δ 3 ′ (shown as segment III in  FIG. 6 ). In this case, the digital chaotic system  11  should stay in the holding state after completing Δ 3 ′ times of iterations, until the running time Δ 2  of the continuous chaotic system  12  arrives T 2 ·Δ 3 ′. When the running time Δ 2  of the continuous chaotic system  12  arrives T 2 ·Δ 3 ′, the digital chaotic system  11  leaves the holding state, the disturbance coupling module  15  samples the continuous chaotic system  12  (shown as C 1,9  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 C 13  and the timer T 12  are reset. 
         [0102]    A criterion for holding a stable synchronization between the continuous chaotic systems  12  and  22  is illustrated as below. 
         [0103]    (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+A T ), 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 Δ 4 ′ times of iterations are completed. When the duration Δ 1  for which the synchronization coupling modules  15  and  25  output the synchronizing impulse signal meets the criterion of: 
         [0000]      Δ 1 =Δ 4   ′·T   2   (5) 
         [0000]      and 
         [0000]      ( q+ 2 L )·Δ 1 ≦−1 n (ξ· d ) ξ&gt;1,d&lt;1  (6a), 
         [0000]    a stable chaotic synchronization may be achieved between the continuous chaotic systems  12  and  22  without transmission of the synchronizing impulse. 
         [0104]    (2) The disturbance coupling modules  16  and  26  disturbs the respective local digital chaotic systems  11  and  21  every time after Δ 3 ′ times of iterations are completed in the digital chaotic systems  11  and  21 . The sampling clock period Δ 2  and the times of iterations Δ 3 ′ should meet the following relation: 
         [0000]      Δ 2 =Δ 3   ′·T   2   (6b) 
         [0105]    (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) 
         [0000]        X   j ( i+ 1)=( I+τ·A )· X   j ( i )+τ·Φ( X   j ( i ))  (7) 
         [0000]    where, τ is a positive real number. 
         [0106]    λ 1  is assumed to be the largest Eigenvalue of (I+τ·A) T (I+τ·A). 
         [0000]    The duration Δ 1  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). 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Δ 
                       1 
                     
                     &gt; 
                     
                       
                         - 
                         τ 
                       
                       · 
                       
                         ⌊ 
                         
                           
                             ln 
                              
                             
                                 
                             
                              
                             d 
                           
                           
                             2 
                              
                             
                                 
                             
                              
                             
                               ln 
                                
                               
                                 ( 
                                 
                                   
                                     
                                       λ 
                                       1 
                                     
                                   
                                   + 
                                   
                                     τ 
                                      
                                     
                                         
                                     
                                      
                                     L 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ⌋ 
                       
                     
                   
                   ; 
                 
               
               
                 
                   ( 
                   
                     8 
                      
                     a 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    and Δ 4 ′ should meet equation (8b), in addition to satisfying equation (6a). 
         [0000]    
       
         
           
             
               
                 
                   
                     Δ 
                     4 
                     ′ 
                   
                   &gt; 
                   
                     - 
                     
                       ⌊ 
                       
                         
                           ln 
                            
                           
                               
                           
                            
                           d 
                         
                         
                           2 
                            
                           
                               
                           
                            
                           
                             ln 
                              
                             
                               ( 
                               
                                 
                                   
                                     λ 
                                     1 
                                   
                                 
                                 + 
                                 
                                   τ 
                                    
                                   
                                       
                                   
                                    
                                   L 
                                 
                               
                               ) 
                             
                           
                         
                       
                       ⌋ 
                     
                   
                 
               
               
                 
                   ( 
                   
                     8 
                      
                     b 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    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. 
         [0107]    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 
       [0108]    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. 
         [0109]    It is assumed that the digital Lorenz chaotic systems  11  and  21  are 
         [0000]    
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                 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 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where, j=1,2 denotes the digital chaotic systems  11  and  21 , respectively, 
         [0110]    the parameters are σ=10, r=28, β=2.5, τ=5×10 −4  s; 
         [0111]    h(i,•) is the disturbance coupling function, and 
         [0000]    
       
         
           
             
               
                 
                   
                     h 
                      
                     
                       ( 
                       
                         i 
                         , 
                         x 
                       
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           0 
                         
                         
                           
                             
                               i 
                                
                               
                                   
                               
                                
                               mod 
                                
                               
                                   
                               
                                
                               
                                 Δ 
                                 3 
                                 ′ 
                               
                             
                             ≠ 
                             
                               0 
                                
                               
                                   
                               
                                
                               or 
                                
                               
                                   
                               
                                
                               i 
                             
                             &lt; 
                             K 
                           
                         
                       
                       
                         
                           
                             D 
                              
                             
                               ( 
                               x 
                               ) 
                             
                           
                         
                         
                           
                             
                               i 
                                
                               
                                   
                               
                                
                               mod 
                                
                               
                                   
                               
                                
                               
                                 Δ 
                                 3 
                                 ′ 
                               
                             
                             = 
                             
                               
                                 0 
                                  
                                 
                                     
                                 
                                  
                                 and 
                                  
                                 
                                     
                                 
                                  
                                 i 
                               
                               ≥ 
                               K 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0000]    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 η 1 =0.5, η 2 =0.5, η 3 =0.5. 
         [0112]    It is assumed that the continuous Lorenz chaotic systems  12  and  22  are 
         [0000]    
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                  
                                 
                                    
                                   t 
                                 
                               
                                
                               
                                 
                                   
                                     x 
                                     ~ 
                                   
                                   j 
                                 
                                  
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                             = 
                             
                               
                                 
                                   - 
                                   σ 
                                 
                                 · 
                                 
                                   
                                     
                                       x 
                                       ~ 
                                     
                                     j 
                                   
                                    
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                               
                               + 
                               
                                 σ 
                                 · 
                                 
                                   
                                     
                                       y 
                                       ~ 
                                     
                                     j 
                                   
                                    
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                               
                             
                           
                            
                           
                               
                           
                         
                       
                     
                     
                       
                         
                           
                             
                                
                               
                                  
                                 t 
                               
                             
                              
                             
                               
                                 
                                   y 
                                   ~ 
                                 
                                 j 
                               
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                           = 
                           
                             
                               r 
                               · 
                               
                                 
                                   
                                     x 
                                     ~ 
                                   
                                   j 
                                 
                                  
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                             - 
                             
                               
                                 
                                   y 
                                   ~ 
                                 
                                 j 
                               
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             - 
                             
                               
                                 
                                   
                                     x 
                                     ~ 
                                   
                                   j 
                                 
                                  
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                               · 
                               
                                 
                                   
                                     z 
                                     ~ 
                                   
                                   j 
                                 
                                  
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                  
                                 
                                    
                                   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 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where, j=1,2 denotes the continuous Lorenz chaotic systems  12  and  22 , respectively; 
         [0113]    the parameters are σ=10, r=28, β=2.66, b 1 =−0.2, b 2 =−0.6, b 3 =−0.6; 
         [0114]    Lipschiz index L of the continuous Lorenz chaotic system is about 50. 
         [0115]    The workflow of the analog-digital mixed chaotic cipher system is shown as follows. 
         [0116]    (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: 
         [0117]    the time T 2 =0.35 ms for the digital chaotic systems  11  and  21  to execute one iteration; 
         [0118]    the period Δ 4 ′=10 for the digital chaotic systems  11  and  21  to output signals to the individual synchronization coupling modules  15  and  25 ; 
         [0119]    the duration Δ 1 =3.5 ms for which the synchronization coupling modules  15  and  25  output synchronization impulses; 
         [0120]    the period Δ 3 ′=1000 for the disturbance coupling modules  16  and  26  to output disturbance signals to the individual digital chaotic systems  11  and  21 ; 
         [0121]    the predetermined constant K=1000 for the disturbance coupling modules, and 
         [0122]    the period Δ 2 =0.35 s for the disturbance coupling modules  16  and  26  to sample the respective continuous chaotic systems  12  and  22 . 
         [0123]    (2) The two communication parties further appoint that: 
         [0124]    the initial conditions of the digital chaotic systems  11  and  21  are x 1 (0)=x 2 (0)=0.454, y 1 (0)=y 2 (0)=0.5236, z 1 (0)=z 2 (0)=0.55878; 
         [0125]    the times of iterations when the digital chaotic systems output the encryption/decryption key is Δ 1 ′=1, and 
         [0126]    the output signal selected by the multiplexing switch is the digital chaotic sequence generated by the digital chaotic systems  11  and  21 . 
         [0127]    (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. 
         [0128]    (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 e x (t)={tilde over (x)} 1 (t)−{tilde over (x)} 2 (t), e y (t)={tilde over (y)} 1 (t)−{tilde over (y)} 2 (t), and e z (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. 
         [0129]    (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. 
         [0130]    (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 v j (2·i)=500·( x   j (i+1000)+25), v j (2·i+1)=500·(y j (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 v 1 (i) and v 2 (i)(i=0, 1, 2, . . . ) are converted with the nonlinear conversion F(•) (shown as Table 1), and key sequences k 1 (i) and k 2 (i)(i=0, 1, 2, . . . ) are obtained.  FIG. 10  is the statistical distribution diagram for the 2,000,000 keys k 1 (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). 
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 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 
       [0131]    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. 
         [0132]    It is assumed that the digital Chen chaotic systems  11  and  21  are 
         [0000]    
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               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 
                   ) 
                 
               
             
           
         
       
     
         [0133]    where, j=1,2 denotes the digital chaotic systems  11  and  21 , respectively; 
         [0134]    the parameters are a=35, b=3, c=28, τ=5×10 −4  S; 
         [0135]    h(i,•) is a disturbance coupling function, and 
         [0000]    
       
         
           
             
               
                 
                   
                     h 
                      
                     
                       ( 
                       
                         i 
                         , 
                         x 
                       
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           0 
                         
                         
                           
                             
                               i 
                                
                               
                                   
                               
                                
                               mod 
                                
                               
                                   
                               
                                
                               
                                 Δ 
                                 3 
                                 ′ 
                               
                             
                             ≠ 
                             
                               0 
                                
                               
                                   
                               
                                
                               or 
                                
                               
                                   
                               
                                
                               i 
                             
                             &lt; 
                             K 
                           
                         
                       
                       
                         
                           
                             D 
                              
                             
                               ( 
                               x 
                               ) 
                             
                           
                         
                         
                           
                             
                               i 
                                
                               
                                   
                               
                                
                               mod 
                                
                               
                                   
                               
                                
                               
                                 Δ 
                                 3 
                                 ′ 
                               
                             
                             = 
                             
                               
                                 0 
                                  
                                 
                                     
                                 
                                  
                                 and 
                                  
                                 
                                     
                                 
                                  
                                 i 
                               
                               ≥ 
                               K 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
         [0136]    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; 
         [0137]    the disturbance coupling intensities are η 1 =0.01, η 2 =0.01, η 3 =0.01. 
         [0138]    It is assumed that the continuous Chen chaotic systems  12  and  22  are 
         [0000]    
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                  
                                 
                                    
                                   t 
                                 
                               
                                
                               
                                 
                                   
                                     x 
                                     ~ 
                                   
                                   j 
                                 
                                  
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                             = 
                             
                               
                                 
                                   - 
                                   a 
                                 
                                 · 
                                 
                                   
                                     
                                       x 
                                       ~ 
                                     
                                     j 
                                   
                                    
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                               
                               + 
                               
                                 a 
                                 · 
                                 
                                   
                                     
                                       y 
                                       ~ 
                                     
                                     j 
                                   
                                    
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                               
                             
                           
                            
                           
                               
                           
                         
                       
                     
                     
                       
                         
                           
                             
                                
                               
                                  
                                 t 
                               
                             
                              
                             
                               
                                 
                                   y 
                                   ~ 
                                 
                                 j 
                               
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                           = 
                           
                             
                               
                                 ( 
                                 
                                   c 
                                   - 
                                   a 
                                 
                                 ) 
                               
                               · 
                               
                                 
                                   
                                     x 
                                     ~ 
                                   
                                   j 
                                 
                                  
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                             + 
                             
                               c 
                               · 
                               
                                 
                                   
                                     y 
                                     ~ 
                                   
                                   j 
                                 
                                  
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                             - 
                             
                               
                                 
                                   
                                     x 
                                     ~ 
                                   
                                   j 
                                 
                                  
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                               · 
                               
                                 
                                   
                                     z 
                                     ~ 
                                   
                                   j 
                                 
                                  
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                  
                                 
                                    
                                   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 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where, j=1,2 denotes the continuous Chen chaotic systems  12  and  22  respectively, 
         [0139]    the parameters are a=35, b=3, c=28, b 1 =−0.8, b 2 =−0.8, b 3 =−0.8; 
         [0140]    Lipschiz index L of the Chen system is about 85. 
         [0141]    The workflow of the analog-digital mixed chaotic cipher system is as follows. 
         [0142]    (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: 
         [0143]    the time T 2 =0.33 ms for the digital chaotic systems  11  and  21  to execute one iteration; 
         [0144]    the period Δ 4 ′=30 for the digital chaotic systems  11  and  21  to output signals to the respective synchronization coupling modules  15  and  25 ; 
         [0145]    the duration Δ 1 =10 ms of the synchronization coupling modules  15  and  25  outputting synchronization impulses; 
         [0146]    the period Δ 3 ′=1000 for the disturbance coupling modules  16  and  26  to output disturbance signals to the respective digital chaotic systems  11  and  21 ; 
         [0147]    the predetermined constant K=3000 for the disturbance coupling modules, and 
         [0148]    the period Δ 2 =0.33 s for the disturbance coupling modules  16  and  26  to sample the respective continuous chaotic systems  12  and  22 . 
         [0149]    (2) The two communication parties appoint further that: 
         [0150]    the initial conditions of the digital chaotic systems  11  and  21  are x 1 (0)=x 2 (0)=0.454, y 1 (0)=y 2 (0)=0.5236, z 1 (0)=z 2 (0)=0.55878, 
         [0151]    the times of iterations when the digital chaotic systems to output the encryption/decryption key is Δ 5 ′=1, and 
         [0152]    the output signal selected by the multiplexing switch is the digital chaotic sequence generated by the digital chaotic systems  11  and  21 . 
         [0153]    (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 . 
         [0154]    (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 . 
         [0155]    (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. 
         [0156]    (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 v 1 (i) and v 2 (i)(i=0, 1, 2, . . . ) with the nonlinear conversion F i mod 3 (•) so as to get the key sequences of k 1 (i) and k 2 (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). 
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 2(a) 
               
               
                   
               
               
                 Nonlinear conversion F 0 (·) 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 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 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 2(b) 
               
               
                   
               
               
                 Nonlinear conversion F 1 (·) 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 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 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 2(c) 
               
               
                   
               
               
                 Nonlinear conversion F 2 (·) 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 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 
       [0157]    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. 
         [0158]    It is assumed that the digital Logistic chaotic systems  11  and  21  are 
         [0000]        x   j ( i+ 1)=4· x   j ( i )·(1 −x   j ( i ))+η· h ( i,{tilde over (x)}   j ( t ))  (15) 
         [0000]    where, j=1,2 denotes the digital chaotic systems  11  and  21  respectively, h(i,•) is a disturbance coupling function, and 
         [0000]    
       
         
           
             
               
                 
                   
                     h 
                      
                     
                       ( 
                       
                         i 
                         , 
                         x 
                       
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           0 
                         
                         
                           
                             
                               i 
                                
                               
                                   
                               
                                
                               mod 
                                
                               
                                   
                               
                                
                               
                                 Δ 
                                 3 
                                 ′ 
                               
                             
                             ≠ 
                             
                               0 
                                
                               
                                   
                               
                                
                               or 
                                
                               
                                   
                               
                                
                               i 
                             
                             &lt; 
                             K 
                           
                         
                       
                       
                         
                           
                             D 
                              
                             
                               ( 
                               x 
                               ) 
                             
                           
                         
                         
                           
                             
                               i 
                                
                               
                                   
                               
                                
                               mod 
                                
                               
                                   
                               
                                
                               
                                 Δ 
                                 3 
                                 ′ 
                               
                             
                             = 
                             
                               
                                 0 
                                  
                                 
                                     
                                 
                                  
                                 and 
                                  
                                 
                                     
                                 
                                  
                                 i 
                               
                               ≥ 
                               K 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
         [0000]    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 q=0.01. 
         [0159]    It is assumed that the continuous Chen chaotic systems  12  and  22  are 
         [0000]    
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                  
                                 
                                    
                                   t 
                                 
                               
                                
                               
                                 
                                   
                                     x 
                                     ~ 
                                   
                                   j 
                                 
                                  
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                             = 
                             
                               
                                 
                                   - 
                                   a 
                                 
                                 · 
                                 
                                   
                                     
                                       x 
                                       ~ 
                                     
                                     j 
                                   
                                    
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                               
                               + 
                               
                                 a 
                                 · 
                                 
                                   
                                     
                                       y 
                                       ~ 
                                     
                                     j 
                                   
                                    
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                               
                             
                           
                            
                           
                               
                           
                         
                       
                     
                     
                       
                         
                           
                             
                                
                               
                                  
                                 t 
                               
                             
                              
                             
                               
                                 
                                   y 
                                   ~ 
                                 
                                 j 
                               
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                           = 
                           
                             
                               
                                 ( 
                                 
                                   c 
                                   - 
                                   a 
                                 
                                 ) 
                               
                               · 
                               
                                 
                                   
                                     x 
                                     ~ 
                                   
                                   j 
                                 
                                  
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                             + 
                             
                               c 
                               · 
                               
                                 
                                   
                                     y 
                                     ~ 
                                   
                                   j 
                                 
                                  
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                             - 
                             
                               
                                 
                                   
                                     x 
                                     ~ 
                                   
                                   j 
                                 
                                  
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                               · 
                               
                                 
                                   
                                     z 
                                     ~ 
                                   
                                   j 
                                 
                                  
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                  
                                 
                                    
                                   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 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where, j=1,2 denotes the continuous Chen chaotic systems  12  and  22  respectively; 
         [0160]    the parameters are a=35, b=3, c=28, b 2 =−0.8; 
         [0161]    Lipschiz index L of the Chen system is about 85. 
         [0162]    The workflow of the analog-digital mixed chaotic cipher system is shown as follows. 
         [0163]    (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: 
         [0164]    the time T 2 =1 ms for the digital chaotic systems  11  and  21  to execute one iterative; 
         [0165]    the period Δ 4 ′=10 for the digital chaotic systems  11  and  21  to output signals to the respective synchronization coupling modules  15  and  25 ; 
         [0166]    the duration Δ 1 =10 ms for which the synchronization coupling modules  15  and  25  output synchronizing impulses; 
         [0167]    the period Δ 3 ′=10 for the disturbance coupling modules  16  and  26  to output disturbance signals to the respective digital chaotic systems  11  and  21 ; 
         [0168]    the predetermined constant K=5000 for the disturbance coupling modules; and 
         [0169]    the period Δ 2 =10 ms for the disturbance coupling modules  16  and  26  to sample the respective continuous Chen chaotic systems  12  and  22 . 
         [0170]    (2) The two communication parties further appoint that: 
         [0171]    the initial conditions of the digital chaotic systems  11  and  21  are x 1 (0)=x 2 (0)=0.333; 
         [0172]    the times of iterations when the digital chaotic systems to output the encryption/decryption key is Δ 5 ′=1, and 
         [0173]    the output signal selected by the multiplexing switch is the digital chaotic sequence generated by the digital chaotic systems  11  and  21 . 
         [0174]    (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. 
         [0175]    (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 . 
         [0176]    (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. 
         [0177]    (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 v j (i)=x j (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 v 1 (i) and v 2 (i)(i=0, 1, 2, . . . ) are converted with the nonlinear conversion F(•) (shown as Table 3), and key sequences k 1 (i) and k 2 (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). 
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 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