CN101436928B - Parameter estimation method for chaos system - Google Patents

Abstract

The invention discloses a method for estimating a parameter for a chaotic system. The method comprises a chaotic system type determination process, a chaotic parameter rang estimation process and a chaotic parameter determination process. Under the condition of only obtaining a finite chaotic state sequence, the method can estimate the type and parameters of chaotic mapping. The method breaks the condition limitation that the prior chaotic parameter estimation method can only aim at the prior chaotic mapping and carry out estimation under the condition that only one of the system parameters is unknown. The parameter estimation method can be widely applied to security analysis based on a chaotic cipher system and a security communication system, does not need to be based on the chaotic mapping type, parameters and other conditions in the analytic process and has strong applicability and maneuverability.

Description

A kind of method for parameter estimation that is used for chaos system

Technical field

The invention belongs to information security safety analysis technology, be specifically related to a kind ofly, can be widely used in based on fields such as the cryptographic system of chaos and secret signalling safety analysis at chaos method for parameter estimation based on the cryptographic system and the secret signalling of chaos.

Background technology

Existing safety analysis technology at chaos cipher system and secret signalling mainly adopts as methods such as genetic algorithm, improvement particle swarm optimization algorithms to be estimated the parameter of chaotic maps.Yet in actual application, these safety analytical methods exist realizes that difficulty is big, and efficiency of evolution is low, problems such as low precision.Therefore said method only is in theoretical research stage, can not be as the safety analysis method of actual cryptographic system or secret signalling.

Paper " a kind of chaotic maps method for parameter estimation based on genetic algorithm " (is worn, Ma Xikui, Li Fucai, outstanding brave, Acta Physica Sinica, 2002 (51), 11, a kind of chaotic maps method for parameter estimation based on genetic algorithm has been proposed in P2459-4), this method is by suitable fitness function of structure, the parameter Estimation problem of chaotic maps is converted into the optimizing problem of a parameter, utilizes the global optimization search capability of genetic algorithm that it is found the solution then.Yet this method only can be meaningful at short notice.Along with the long-time evolution of system, the error between estimated parameter institute's characterization system and the real system will increase.In addition, this method only at the situation of a unknown parameters of chaotic maps, yet for the condition of whole system parameters the unknowns, can only the searching for of this method in a bigger scope, efficient is low, poor accuracy.

Summary of the invention

The object of the present invention is to provide a kind of method for parameter estimation that is used for chaos system, this method can be under the condition of unknown password system or the employed chaotic maps type of secret signalling, parameter to chaotic maps estimates that it is fast that it has analysis speed, the characteristics that accuracy is high.

For realizing purpose of the present invention, the method for parameter estimation that is used for chaos system provided by the invention, its step comprises:

(A1) according to chaos system type and parameter information, determine parameter group, the quantity of parameter group is l, and l is a positive integer;

(A2) use respectively every group of parameter generate one with the isometric comparison chaos sequence { C1 of chaos sequence to be analyzed { T (i) } _j(i) }, wherein, i represents the sequence number of element in the chaos sequence to be analyzed, i=1, and 2 ..., N, N represent the length of chaos sequence to be analyzed, j represents to be used to generate the sequence number of the parameter group of this comparison chaos sequence, and j=1,2 ..., l;

(A3) calculate the nonlinear characteristic amount of chaos sequence to be analyzed { T (i) }, comprise time delay τ and embed dimension d;

(A4) according to time delay τ that calculates and embedding dimension d, chaos sequence to be analyzed { T (i) } and each are compared chaos sequence { C1 _j(i) } carry out phase space reconfiguration, obtain respectively the vector sequence { T (i) } that reconstitutes and C1 ' _j(i) };

T′(i)＝{T(i)，T(i+τ)，...，T(i+(d-1)τ)}

C1′ _j(i)＝{C1 _j(i)，C1 _j(i+τ)，...，C1 _j(i+(d-1)τ)}

(A5) calculate each vector sequence that reconstitutes { T (i) } and C1 ' _j(i) } Central Moment Feature amount λ _TWith

(A6) according to vector sequence T (i), utilize statistical analysis technique, calculate the threshold epsilon that is used for critical parameter;

(A7) the Central Moment Feature amount λ of chaos sequence more to be analyzed _TWith the Central Moment Feature amount that compares chaos sequence

, obtain the comparison chaos sequence of its difference, with the system parameters of the comparison chaos sequence that obtains parameter Estimation result as chaos sequence to be analyzed smaller or equal to ε.

When chaos cipher system or secret signalling adopt multiple chaos system, under chaos system type condition of unknown, entering step (A1) before, according to following step the chaos sequence that chaos sequence to be analyzed and chaotic maps to be selected generate is carried out the nonlinear correlation degree relatively, determine the chaos system type that chaos cipher system to be analyzed or secret signalling are adopted when generating chaos sequence to be analyzed:

(B1) according to the design rule of chaos cipher system to be analyzed or secret signalling, the n kind chaos system type that definite random source that generates this chaos sequence to be analyzed { T (i) } can be adopted;

(B2) the n kind chaos system that utilizes above step to determine adopts classical parameter setting, generates and the identical comparison chaos sequence { C2 of chaos sequence to be analyzed { T (i) } length _k(i) }, k represents to be used to generate the sequence number of the chaos system type of this comparison chaos sequence, k=1, and 2 ..., n, n represent the quantity of chaos system type;

(B3) the nonlinear characteristic amount of calculating chaos sequence to be analyzed comprises embedding dimension d and time delay τ;

(B4) according to the embedding peacekeeping time delay of determining, to chaos sequence to be analyzed { T (i) } and comparison chaos sequence { C2 _k(i) } carry out phase space reconfiguration, obtain:

T′(i)＝{T(i)，T(i+τ)，...，T(i+(d-1)τ)}

C2′ _k(i)＝{C2 _k(i)，C2 _k(i+τ)，...，C2 _k(i+(d-1)τ)}

(B5) calculate T (i) and the C2 that reconstruct obtains _k' (i) dynamics auto-correlation factor index, that is:

$Q_{{T\_C2}_k}=\underset{\mathrm{\ϵ}\→0}{\lim }\left|\ln \frac{C_T\left(\mathrm{\ϵ}\right)}{C_{{C2}_k}\left(\mathrm{\ϵ}\right)}\right|$

Wherein:

$C_T\left(\mathrm{\&epsiv;}\right)=P\left(\right||T^{\&prime;}\left(i\right)-T^{\&prime;}\left(j\right)||<\mathrm{\&epsiv;})$

$=\frac{2}{(N-d)(N-d+1)}\&times;\underset{i=1}{\overset{N-m}{\mathrm{\&Sigma;}}}\underset{j=i+1}{\overset{N-m+1}{\mathrm{\&Sigma;}}}\mathrm{\&Theta;}(\mathrm{\&epsiv;}-||T^{\&prime;}\left(i\right)-T^{\&prime;}\left(j\right)|\left|\right)$

$C_{{C2}_k}\left(\mathrm{\&epsiv;}\right)=P\left(\right||{{C2}_k}^{\&prime;}\left(i\right)-{{C2}_k}^{\&prime;}\left(j\right)||<\mathrm{\&epsiv;})$

$=\frac{2}{(N-d)(N-d+1)}\&times;\underset{i=1}{\overset{N-m}{\mathrm{\&Sigma;}}}\underset{j=i+1}{\overset{N-m+1}{\mathrm{\&Sigma;}}}\mathrm{\&Theta;}(\mathrm{\&epsiv;}-||{{C2}_k}^{\&prime;}\left(i\right)-{{C2}_k}^{\&prime;}\left(j\right)|\left|\right)$

Θ is the Heaviside step function.

(B6) comparison chaos sequence and the chaos sequence to be analyzed with chaos sequence dynamics auto-correlation factor index value minimum to be analyzed results from identical chaos system; Draw the chaos system type that chaos cipher system to be analyzed or secret signalling are adopted thus when generating chaos sequence to be analyzed.

This method is compared with existing method, has overcome the condition restriction that existing chaos method for parameter estimation can only only have the condition an of the unknown to estimate at known chaotic maps and system parameters.This system can type and the parameter to chaotic maps estimate under the condition that only obtains limited chaos state sequence.This method for parameter estimation can be widely used in the safety analysis based on the cryptographic system and the secret signalling of chaos, and analytic process need not to have stronger adaptability and operability based on conditions such as chaotic maps type and parameters.Compare with documents " a kind of chaotic maps method for parameter estimation based on genetic algorithm ", the present invention has following characteristics:

(1), the chaos sequence that chaos sequence to be analyzed and chaotic maps to be selected are generated carries out the nonlinear correlation degree relatively, determine the type of chaotic maps to be analyzed, thereby can under the condition of unknown chaotic maps type, analyze, strengthen the adaptability of this analytical method.

(2), compare, seek possible the value space of parameter, dwindle the parameter search scope, improve analysis efficiency, reduction overall calculation complexity by the Central Moment Feature amount.

(3), in determining parameter space, utilize possible parameter to generate chaos sequence, the chaos sequence of chaos sequence to be analyzed and generation is compared, determine whole system parameters spans of chaotic maps to be analyzed.

Description of drawings

Fig. 1 is the flow chart of chaos method for parameter estimation of the present invention;

Fig. 2 is for determining the method flow diagram of chaotic maps type;

Fig. 3 is for determining the method flow diagram of chaos parameter value scope.

Embodiment

The present invention is further detailed explanation below in conjunction with accompanying drawing and example.

The invention provides a kind of to based on the cryptographic system of chaos and the safety analysis method of secret signalling.This method is estimated the chaos system type of cryptographic system and secret signalling employing by the chaos sequence for the treatment of that analysis collects, according to the Given informations such as parameter information of cryptographic system and secret signalling the parameter that this chaos system adopts is estimated on this basis.

As shown in Figure 1, the step of the inventive method is as follows:

(1) cryptographic system or the secret signalling based on chaos might adopt following dual mode to design random source: one, single chaos system; Two, multiple chaos system.If chaos cipher system to be analyzed or secret signalling adopt mode directly to enter step (2) first, if chaos cipher system to be analyzed or secret signalling adopt mode two, then this step is carried out the nonlinear correlation degree relatively with the chaos sequence that chaos sequence to be analyzed and chaotic maps to be selected generate, and determines the chaos system type that chaos cipher system to be analyzed or secret signalling are adopted when generating chaos sequence to be analyzed.Its specific implementation process comprises as shown in Figure 2:

(1.1) design rule of chaos cipher system to be analyzed or secret signalling, the n kind chaos system type that definite random source that generates this chaos sequence to be analyzed may be adopted;

(1.2) each chaos system that utilizes above step to determine adopts classical parameter setting, generates the comparison chaos sequence { C2 that is all length N with chaos sequence to be analyzed { T (i) } _k(i) }, wherein, i represents the sequence number of element in the chaos sequence to be analyzed, i=1,2, ..., N, N represent the length of chaos sequence to be analyzed, k represents to be used to generate the sequence number of the chaos system type of this comparison chaos sequence, k=1,2 ..., n, n represent the quantity of chaos system type;

(1.3) the nonlinear characteristic amount of calculating chaos sequence to be analyzed: embed dimension d and time delay τ;

(1.4) according to the result of step (1.3), to chaos sequence to be analyzed { T (i) } and chaos sequence { C2 relatively _k(i) } carry out phase space reconfiguration, obtain:

T′(i)＝{T(i)，T(i+τ)，...，T(i+(d-1)τ)}

C2′ _k(i)＝{C2 _k(i)，C2 _k(i+τ)，...，C2 _k(i+(d-1)τ)}

(1.5) calculate T (i) and the C2 that reconstruct obtains _k' (i) dynamics auto-correlation factor index, that is:

$Q_{T\_{C2}_k}=\underset{\mathrm{\&epsiv;}\&RightArrow;0}{\lim }\left|\ln \frac{C_T\left(\mathrm{\&epsiv;}\right)}{C_{{C2}_k}\left(\mathrm{\&epsiv;}\right)}\right|$

Wherein:

$C_T\left(\mathrm{\&epsiv;}\right)=P\left(\right||T^{\&prime;}\left(i\right)-T^{\&prime;}\left(j\right)||<\mathrm{\&epsiv;})$

$=\frac{2}{(N-d)(N-d+1)}\&times;\underset{i=1}{\overset{N-m}{\mathrm{\&Sigma;}}}\underset{j=i+1}{\overset{N-m+1}{\mathrm{\&Sigma;}}}\mathrm{\&Theta;}(\mathrm{\&epsiv;}-||T^{\&prime;}\left(i\right)-T^{\&prime;}\left(j\right)|\left|\right)$

$C_{{C2}_k}\left(\mathrm{\&epsiv;}\right)=P\left(\right||{{C2}_k}^{\&prime;}\left(i\right)-{{C2}_k}^{\&prime;}\left(j\right)||<\mathrm{\&epsiv;})$

$=\frac{2}{(N-d)(N-d+1)}\&times;\underset{i=1}{\overset{N-m}{\mathrm{\&Sigma;}}}\underset{j=i+1}{\overset{N-m+1}{\mathrm{\&Sigma;}}}\mathrm{\&Theta;}(\mathrm{\&epsiv;}-||{{C2}_k}^{\&prime;}\left(i\right)-{{C2}_k}^{\&prime;}\left(j\right)|\left|\right)$

Θ is the Heaviside step function.

(1.6) comparison chaos sequence and the chaos sequence to be analyzed with chaos sequence dynamics auto-correlation factor index value minimum to be analyzed results from identical chaos system.Draw the chaos system type that chaos cipher system to be analyzed or secret signalling are adopted thus when generating chaos sequence to be analyzed.

(2) can in specific scope, choose the parameter generation chaos sequence of specific parameter according to key protocol based on the cryptographic system or the secret signalling of chaos as chaos system.Therefore will in possible parameter value, compare the possible value of searching and definite parameter by the Central Moment Feature amount.Its specific implementation process comprises as shown in Figure 3:

(2.1) according to the result and the parameter information of step (1), determine possible parameter group, species number is l;

(2.2), generate and the isometric comparison chaos sequence { C1 of chaos sequence to be analyzed according to every group of possible parameter _j(i) } (j=0,1,2..., l);

(2.3) result according to step (1.3) carries out phase space reconfiguration { C1 according to the method for step (1.4) to each comparison chaos sequence _j(i) } (j=0,1,2... l), obtains the vector sequence after the reconstruct;

C1′ _j(i)＝{C1 _j(i)，C1 _j(i+τ)，...，C1 _j(i+(d-1)τ)}

(2.4) calculate vector sequence T (i) and the C1 ' that each reconstitutes _j(i) Central Moment Feature amount λ _TWith

(2.5) according to vector sequence T (i), utilize statistical analysis technique, calculate the threshold epsilon that is used for critical parameter;

(2.6) the Central Moment Feature amount λ of chaos sequence more to be analyzed _TWith the Central Moment Feature amount that compares chaos sequence

, obtain the comparison chaos sequence of its difference, with the system parameters of the comparison chaos sequence that relatively obtains estimated result as chaos sequence to be analyzed smaller or equal to ε.

Example 1:

Suppose A for the chaos cipher system, according to its design and key protocol, known following information:

(1) this cryptographic system may be chosen a random source of conduct in a plurality of chaos systems, these chaos systems comprise the Lorenz chaos system, Chen chaos system, three kinds of three-dimensional chaos systems of Lu chaos system, and the sampling interval that produces chaos sequence is 1ms, and its chaos system equation is as follows:

Lorenz chaotic maps equation:

$\left\{\begin{array}{c}\mathrm{dx}/\mathrm{dt}=a(y-x)\\ \mathrm{dy}/\mathrm{dt}=\mathrm{bx}-y-\mathrm{xz}\\ \mathrm{dz}/\mathrm{dt}=\mathrm{xy}-\mathrm{cz}\end{array}\right.$

Chen chaotic maps equation:

$\left\{\begin{array}{c}\mathrm{dx}/\mathrm{dt}=a(y-x)\\ \mathrm{dy}/\mathrm{dt}=(b-a)x+\mathrm{by}-\mathrm{xz}\\ \mathrm{dz}/\mathrm{dt}=\mathrm{xy}-\mathrm{cz}\end{array}\right.$

Lu chaotic maps equation:

$\left\{\begin{array}{c}\mathrm{dx}/\mathrm{dt}=a(y-x)\\ \mathrm{dy}/\mathrm{dt}=-\mathrm{xz}+\mathrm{cy}\\ \mathrm{dz}/\mathrm{dt}=\mathrm{xy}-\mathrm{bz}\end{array}\right.$

(2) for the Lorenz chaos system, its parameter value scope is:

a∈{x|x＝7+0.2*i，i＝0，1，...，30}

b∈{x|x＝25+0.2*i，i＝0，1，...，30}

c∈{x|x＝2/3+0.2*i，i＝0，1，...，20}

For the Chen chaos system, its parameter value scope is:

a∈{x|x＝32+0.2*i，i＝0，1，...，30}

b∈{x|x＝25+0.2*i，i＝0，1，...，30}

c∈{x|x＝2/3+0.2*i，i＝0，1，...，20}

For the Lu chaos system, its parameter value scope is:

a∈{x|x＝33+0.2*i，i＝0，1，...，30}

b∈{x|x＝17+0.2*i，i＝0，1，...，30}

c∈{x|x＝1+0.2*i，i＝0，1，...，20}

(3) chaos sequence to be analyzed comprises x, y, three dimension { T of z _x(i) }, { T _y(i) }, { T _z(i) }, wherein, i represents the sequence number of element in the chaos sequence to be analyzed, i=1, and 2 ..., 1,000,000.

By analyzing known chaos sequence { T _x(t _i), { T _y(t _i), { T _z(t _i), the type and the system parameters of the chaos system that generates this sequence are estimated that concrete implementation method is as follows:

(1) determine the chaos system type:

(1.1), learn that the chaos cipher system may adopt following 3 kinds of chaos systems: Lorenz chaos system, Chen chaos system, Lu chaos system according to Given information (1);

(1.2) chaos system of determining according to step (1.1), generation length are 1,000,000 three-dimensional comparison chaos sequence { C2 _{Lorenz_x}(i) }, { C2 _{Lorenz_y}(i) }, { C2 _{Lorenz_z}(i) }, { C2 _{Chen_x}(i) }, { C2 _{Chen_y}(i) }, { C2 _{Chen_z}(i) }, { C2 _{Lu_x}(i) }, { C2 _{Lu_y}(i) }, { C2 _{Lu_z}(i) }, wherein the parameter of chaos system is classical parameter setting, { a _Lorenz=10, b _Lorenz=28, c _Lorenz=8/3}, { a _Chen=35, b _Chen=28, c _Chen=8/3}, { a _Lu=36, b _Lu=20, c _Lu=3};

(1.3) utilize pseudo-neighbor point to calculate the embedding dimension d of chaos sequence to be analyzed _x, d _y, d _z, utilize mutual information method to calculate the time delay τ of chaos sequence to be analyzed _x, τ _y, τ _z

(1.4) according to the result of step (1.3), to chaos sequence { T to be analyzed _x(i) }, { T _y(i) }, { T _z(i) } chaos sequence { C2 and relatively _{Lorenz_x}(i) }, { C2 _{Lorenz_y}(i) }, { C2 _{Lorenz_z}(i) }; { C2 _{Chen_x}(i) }, { C2 _{Chen_y}(i) }, { C2 _{Chen_z}(i) }; { C2 _{Lu_x}(i) }, { C2 _{Lu_y}(i) }, { C2 _{Lu_z}(i) }, carry out phase space reconfiguration, obtain the sequence { T after the reconstruct _x' (i) }, { T _y' (i) }, { T _z' (i) }; { C2 _{Lorenz_x}' (i) }, { C2 _{Lorenz_y}' (i) }, { C2 _{Lorenz_z}' (i) }; { C2 _{Chen_x}' (i) }, { C2 _{Chen_y}' (i) }, { C2 _{Chen_z}' (i) }; { C2 _{Lu_x}' (i) }, { C2 _{Lu_y}' (i) }, { C2 _{Lu_z}' (i) };

(1.5) calculate { T _xAnd { C2 (i) } _{Lorenz_x}(i) }; { T _xAnd { C2 (i) } _{Chen_x}(i) }; { T _xAnd { C2 (i) } _{Lu_x}(i) }; { T _yAnd { C2 (i) } _{Lorenz_y}(i) }; { T _yAnd { C2 (i) } _{Chen_y}(i) }; { T _yAnd { C2 (i) } _{Lu_y}(i) }; { T _zAnd { C2 (i) } _{Lorenz_z}(i) }; { T _zAnd { C2 (i) } _{Chen_z}(i) }; { T _zAnd { C2 (i) } _{Lu_z}Dynamics auto-correlation factor index (ii) }:

$Q_{T_x\_C{2}_{\mathrm{Lorenz}\_x}},Q_{T_x\_C{2}_{\mathrm{Chen}\_x}},Q_{T_x\_C{2}_{\mathrm{Lu}\_x}}$

$Q_{T_y\_C{2}_{\mathrm{Lorenz}\_y}},Q_{T_y\_C{2}_{\mathrm{Chen}\_y}},Q_{T_y\_C{2}_{\mathrm{Lu}\_y}}$

$Q_{T_z\_C{2}_{\mathrm{Lorenz}\_z}},Q_{T_z\_C{2}_{\mathrm{Chen}\_z}},Q_{T_z\_C{2}_{\mathrm{Lu}\_z}}$

(1.6) the dynamics auto-correlation factor index that calculates by comparison step (1.5), the result points out:

$\min (Q_{T_x\_C{2}_{\mathrm{Lorenz}\_x}},Q_{T_x\_C{2}_{\mathrm{Chen}\_x}},Q_{T_x\_C{2}_{\mathrm{Lu}\_x}})=Q_{T_x\_{C2}_{\mathrm{Chen}\_x}}$

$\min (Q_{T_y\_C{2}_{\mathrm{Lorenz}\_y}},Q_{T_y\_C{2}_{\mathrm{Chen}\_y}},Q_{T_y\_C{2}_{\mathrm{Lu}\_y}})=Q_{T_y\_{C2}_{\mathrm{Chen}\_y}}$

$\min (Q_{T_z\_C{2}_{\mathrm{Lorenz}\_z}},Q_{T_z\_C{2}_{\mathrm{Chen}\_z}},Q_{T_z\_C{2}_{\mathrm{Lu}\_z}})=Q_{T_z\_{C2}_{\mathrm{Chen}\_z}}$

Be the dynamics auto-correlation factor index minimum of the comparative sequences of chaos sequence to be analyzed and Chen chaos system generation, illustrate that chaos sequence to be analyzed is to have the Chen chaos system to generate;

(2) determine the possible value of the parameter of chaos system:

(2.1) chaos system of Chen as a result and the Given information of determining according to step (1) (2) determined parameter range;

a∈{x|x＝32+0.2*i，i＝0，1，..，30}

b∈{x|x＝25+0.2*i，i＝0，1，...，30}

c∈{x|x＝2/3+0.2*i，i＝0，1，...，20}

(2.2) the parameter value scope of determining according to step (2.1) is utilized possible parameter group, and generation length is 1,000,000 comparison chaos sequence $\left\{{C1}_{x_j}\left(i\right)\right\},\left\{{C1}_{y_j}\left(i\right)\right\},\left\{{C1}_{z_j}\left(i\right)\right\},$ Wherein, i represents the sequence number of element in the chaos sequence to be analyzed, i=1, and 2 ..., 1,000,000, j represents to be used to generate the sequence number of the parameter group of this comparison chaos sequence, j=1,2 ..., 20181;

(2.3) result according to step (1.3) compares chaos sequence to each

Carry out phase space reconfiguration, obtain the vector sequence after the reconstruct

(2.4) calculate the vector sequence { T that each reconstitutes _x' (i) }, { T _y' (i) }, { T _z' (i) } and

Central Moment Feature amount λ _x, λ _y, λ _zWith

(2.5) choose 100 respectively smaller or equal to 1,000,000 positive integer: 1,000,000,1,000,000-500,1,000,000-500 * 2 ..., 1,000,000-500 * 99, computational length is for being 1,000,000,1,000 respectively, 000-500,1,000,000-500 * 2 ..., 1,000,000-500 * 99, vector sequence { T _x' (t _i), { T _y' (t _i), { T _z' (t _i) the Central Moment Feature amount $\{{\mathrm{\&lambda;}}_{x_1},{\mathrm{\&lambda;}}_{x_2},\&CenterDot;\&CenterDot;\&CenterDot;,{\mathrm{\&lambda;}}_{x_{100}}\},\{{\mathrm{\&lambda;}}_{y_1},$ ${\mathrm{\&lambda;}}_{y_2},\&CenterDot;\&CenterDot;\&CenterDot;,{\mathrm{\&lambda;}}_{y_{100}}\},\{{\mathrm{\&lambda;}}_{z_1},{\mathrm{\&lambda;}}_{z_2},\&CenterDot;\&CenterDot;\&CenterDot;,{\mathrm{\&lambda;}}_{z_{100}}\}.$ Calculate

Variance ε _x, calculate

Variance ε _y, calculate

Variance ε _z

(2.6) the Central Moment Feature amount λ of chaos sequence more to be analyzed _x, λ _y, λ _zWith the Central Moment Feature amount that compares chaos sequence

λ _C1z, obtain its difference smaller or equal to ε _x, ε _y, ε _yThe comparison chaos sequence, with the system parameters of the comparison chaos sequence that relatively obtains estimated result, from 20181 groups of possible parameter values, finally determined 25 groups of possible parameter values thus as chaos sequence to be analyzed.

Example 2:

Suppose B for the chaos cipher system, according to its design and parameter information, known following information:

(1) this cryptographic system utilizes the Lorenz chaos system as random source, and Lorenz chaotic maps equation is:

$\left\{\begin{array}{c}\mathrm{dx}/\mathrm{dt}=a(y-x)\\ \mathrm{dy}/\mathrm{dt}=\mathrm{bx}-y-\mathrm{xz}\\ \mathrm{dz}/\mathrm{dt}=\mathrm{xy}-\mathrm{cz}\end{array}\right.$

(2) the parameter value scope of Lorenz chaos system is:

a∈{x|x＝7+0.2*i，i＝0，1，...，30}

b∈{x|x＝25+0.2*i，i＝0，1，...，30}

c∈{x|x＝2/3+0.2*i，i＝0，1，...，20}

(3) chaos sequence to be analyzed comprises x, y, three dimension { T of z _x(i) }, { T _y(i) }, { T _z(i) }, the length of sequence is 1,000,000;

By analyzing known chaos sequence, the type and the system parameters of the chaos system that generates this sequence are estimated that concrete implementation method is as follows:

(1) owing to known definite chaos system type, skips steps 1 is directly estimated the parameter of chaos system;

(2) determine the possible value of the parameter of chaos system:

(2.1) determine possible parameter group according to Given information (1) and (2), have 20181 groups;

(2.2) the parameter value scope of determining according to step (2.1) is utilized possible parameter group, and generation length is 1,000,000 comparison chaos sequence $\left\{{C1}_{x_j}\left(i\right)\right\},\left\{{C1}_{y_j}\left(i\right)\right\},\left\{{C1}_{z_j}\left(i\right)\right\},$ Wherein, i represents the sequence number of element in the chaos sequence to be analyzed, i=1, and 2 ..., 1,000,000, j represents to be used to generate the sequence number of the parameter group of this comparison chaos sequence, j=1,2 ..., 20181;

(2.3) utilize pseudo-neighbor point to calculate the embedding dimension d of chaos sequence to be analyzed _x, d _y, d _z, utilize mutual information method to calculate the time delay τ of chaos sequence to be analyzed _x, τ _y, τ _z

(2.4) according to the result of step (1.3) to chaos sequence { T to be analyzed _x(i) }, { T _y(i) }, { T _zAnd each chaos sequence relatively (i) } $\left\{{C1}_{x_j}\left(i\right)\right\},\left\{{C1}_{y_j}\left(i\right)\right\},\left\{{C1}_{z_j}\left(i\right)\right\}$ Carry out phase space reconfiguration, obtain the vector sequence { T after the reconstruct _x' (i) }, { T _y' (i) }, { T _z' (i) }; With $\left\{{{C1}^{\&prime;}}_{x_j}\left(i\right)\right\},\left\{{{C1}^{\&prime;}}_{y_j}\left(i\right)\right\},\left\{{{C1}^{\&prime;}}_{z_j}\left(i\right)\right\};$

(2.5) choose 200 respectively smaller or equal to 1,000,000 natural number: 1,000,000,1,000,000-100,1,000,000-100 * 2 ..., 1,000,000-100 * 199, computational length is 1,000,000,1,000 respectively, 000-100,1,000,000-100 * 2 ..., 1,000,000-100 * 199, vector sequence { T _x' (t _i, { T _y' (t _i), { T _z' (t _i) the Central Moment Feature amount $\{{\mathrm{\&lambda;}}_{x_1},{\mathrm{\&lambda;}}_{x_2},\&CenterDot;\&CenterDot;\&CenterDot;,{\mathrm{\&lambda;}}_{x_{200}}\},\{{\mathrm{\&lambda;}}_{y_1},$ ${\mathrm{\&lambda;}}_{y_2},\&CenterDot;\&CenterDot;\&CenterDot;,{\mathrm{\&lambda;}}_{y_{100}}\},\{{\mathrm{\&lambda;}}_{z_1},{\mathrm{\&lambda;}}_{z_2},\&CenterDot;\&CenterDot;\&CenterDot;,{\mathrm{\&lambda;}}_{z_{200}}\}.$ Calculate

Average E _x, calculate

Average E _y, calculate

Average E _z, calculate

And E _xThe average ε of difference _x, calculate

And E _yThe average ε of difference _y, calculate

And E _zThe average ε of difference _z

(2.6) the Central Moment Feature amount λ of chaos sequence more to be analyzed _x, λ _y, λ _zWith the Central Moment Feature amount that compares chaos sequence

λ _C1z, obtain its difference smaller or equal to ε _x, ε _y, ε _zThe comparison chaos sequence, with the system parameters of the comparison chaos sequence that relatively obtains estimated result, from 20181 groups of possible parameter values, finally determined 25 groups of possible parameter values thus as chaos sequence to be analyzed.

The above is preferred embodiment of the present invention, but the present invention should not be confined to the disclosed content of this embodiment and accompanying drawing.So everyly do not break away from the equivalence of finishing under the spirit disclosed in this invention or revise, all fall into the scope of protection of the invention.

Claims (1)

1. one kind at the chaos method for parameter estimation based on the cryptographic system and the secret signalling of chaos, and its step comprises:

(A1) according to chaos system type and key protocol, determine parameter group, the quantity of parameter group is l, lBe positive integer;

(A2) use respectively every group of parameter generate one with the isometric comparison chaos sequence { C1 of chaos sequence to be analyzed { T (i) } _j(i) }, wherein, i represents the sequence number of element in the chaos sequence to be analyzed, i=1, and 2 ..., N, N represent the length of chaos sequence to be analyzed, j represents to be used to generate the sequence number of the parameter group of this comparison chaos sequence, and j=1,2 ..., l

(A3) calculate the nonlinear characteristic amount of chaos sequence to be analyzed { T (i) }, comprise time delay τ and embed dimension d;

(A4) according to time delay τ that calculates and embedding dimension d, chaos sequence to be analyzed { T (i) } and each are compared chaos sequence { C1 _j(i) } carry out phase space reconfiguration, the vector sequence that obtains respectively reconstituting T ' (i) } and C1 ' _j(i) };

T′(i)＝{T(i)，T(i+τ)，...，T(i+(d-1)τ)}

C1′ _j(i)＝{C1 _j(i)，C1 _j(i+τ)，...，C1 _j(i+(d-1)τ)}

(A5) calculate each vector sequence that reconstitutes { T ' (i) } and C1 ' _j(i) } Central Moment Feature amount λ _TWith

(A6) according to vector sequence T ' (i), utilize statistical analysis technique, calculate the threshold epsilon that is used for critical parameter;

(A7) Correlation Centre moment characteristics amount λ _TWith the Central Moment Feature amount that compares chaos sequence

Obtain the comparison chaos sequence of its difference, with the system parameters of the comparison chaos sequence that obtains parameter Estimation result as chaos sequence to be analyzed smaller or equal to ε.

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