Method for generating chaotization and chaotic sequence of high-dimensional power system

Abstract

The invention discloses a method for generating chaotization and a chaotic sequence of a high-dimensional power system. The method includes the following steps that an m-dimensional discrete power system is set, Sn+1=ASnmodc, A is a constant-coefficient matrix, as no non-linear term is arranged in the m-dimensional discrete power system, the jacobian matrix is A, P=An, and the characteristic values of the parameter matrix A decide Lyapunov indexes of the system; the Lyapunov index values LE1, LE2, ..., LEm are given, the characteristic values are calculated, and a m*m-dimensional non-singularmatrix q is designed; the parameter matrix A=qGammaq-1 is calculated and put in an original model, and the discrete power system is re-constructed; the initial value of a state vector serves as an initial secret key, and a random sequence is generated through a time sequence of the re-constructed discrete power system. The Lyapunov indexes can be precisely controlled, a period system with period attractors and a system with fixed point attractors are achieved, and the chaotic sequence output through the method has more complex chaos behaviors.

Description

technical field

[0001] The invention relates to the field of information security, in particular to a chaotic high-dimensional dynamic system and a method for generating chaotic sequences. Background technique

[0002] Existing general chaotic design methods mainly rely on the Chen-Lai algorithm, given the initial state x 0 , for the control system x 1 =f 0 (x 0 )+B 0 x 0 Calculate its Jacobian matrix

[0003] J 0 (x 0 )=f 0 '(x 0 )+B 0 x 0 ,

[0004] And note T 0 =J 0 (x 0 ). take B 0 x 0 =σ 0 I and choose the constant σ 0 > 0 so that the matrix [T 0 T 0 T ] is limited and diagonally dominant. For k=0,1,2,..., consider the control system

[0005] x k+1 =f k (x k )+B k x k ,

[0006] where B k x k =σ k I has been obtained from the previous step. Now do the following calculation:

[0007] Step 1. Calculate the Jacobian matrix

[0008] J k (x k )=f k '(x k )+B k x k

[0009] mark T k =J k T k-1 .

[0010] Step 2. Choose consta...

Examples

Embodiment 1

[0046] like Figure 1-2 As shown, the chaoticization of a high-dimensional dynamic system and its chaotic sequence generation method are as follows:

[0047] Let m-dimensional discrete dynamical system

[0048] S n+1 =AS n modc

[0049] where S n is the state vector (x 1 (n),x 2 (n),x 3 (n)......x m (n)) T , A is a constant coefficient matrix,

[0050]

[0051] Since there are no nonlinear terms in the m-dimensional discrete dynamical system, the Jacobian matrix is ​​A, therefore, P=A n , let the m eigenvalues ​​of matrix A be λ 0 ,λ 1 ,...,λ m , the m Lyapunov exponents in m-dimensional discrete chaos are:

[0052]

[0053] Therefore, the eigenvalues ​​of the parameter matrix A determine the Lyapunov exponent of the system; the construction method of the parameter matrix A is as follows:

[0054] (1) Given the Lyapunov exponent value LE 1 ,LE 2 ,...LE m , and calculate the eigenvalues The eigenvalue-based diagonal matrix Λ is constructed as

[0055] ...

Embodiment 2

[0060] 1. Any given 8 eigenvalues ​​greater than 1, such as 40, 41, 42, 43, 44, 45, 46, 47, and c is defined as 1. The rounded Lyapunov exponents are 3.69, 3.71, 3.74, 3.76, 3.78, 3.81, 3.83, 3.85.

[0061] 2. A nonsingular matrix q is defined as:

[0062]

[0063] The element q(i,i)=2, i=1,2,3...m, and the remaining elements are all 1. It is easy to show that q is a nonsingular matrix. When m=8, q can be defined

[0064]

[0065] and the rounded inverse matrix q -1 for

[0066]

[0067] 3. The parameter matrix is

[0068]

[0069] Reconstructing discrete dynamical systems.

[0070]

[0071] (4) The initial value of the state vector is used as the initial key, and the chaotic sequence is generated by reconstructing the discrete dynamic system. When the output sequence is greater than 0.5, the quantization is 1, and when the output sequence is less than 0.5, the quantization is 0.

Embodiment 3

[0073] The effect of the present invention can be further illustrated by the following detection results of the present embodiment:

[0074] 1. Detection method and content:

[0075] The randomness of the chaotic sequence output by the chaotic sequence generator in Embodiment 2 of the present invention is detected by using the SP800-22 random number detection standard provided by the National Institute of Standards and Technology NIST. The detection standard includes 15 detection contents, each of which is A P-value is included in the test results produced by the test. When the P value is greater than 0.01, it means that the test content has passed.

[0076] 2. Test results:

[0077] Referring to Example 2, make it generate 100 groups of 10,000,000 random sequences, and use the SP800-22 random number detection standard provided by the National Institute of Standards and Technology (NIST) for detection. One group of results is shown in Table 1-8:

[0078] Table 1x 1 (n) Out...

technical field

[0001] The invention relates to the field of information security, in particular to a chaotic high-dimensional dynamic system and a method for generating chaotic sequences. Background technique

[0002] Existing general chaotic design methods mainly rely on the Chen-Lai algorithm, given the initial state x 0 , for the control system x 1 =f 0 (x 0 )+B 0 x 0 Calculate its Jacobian matrix

[0003] J 0 (x 0 )=f 0 '(x 0 )+B 0 x 0 ,

[0004] And note T 0 =J 0 (x 0 ). take B 0 x 0 =σ 0 I and choose the constant σ 0 > 0 so that the matrix [T 0 T 0 T ] is limited and diagonally dominant. For k=0,1,2,..., consider the control system

[0005] x k+1 =f k (x k )+B k x k ,

[0006] where B k x k =σ k I has been obtained from the previous step. Now do the following calculation:

[0007] Step 1. Calculate the Jacobian matrix

[0008] J k (x k )=f k '(x k )+B k x k

[0009] mark T k =J k T k-1 .

[0010] Step 2. Choose consta...

Examples

Embodiment 1

[0046] like Figure 1-2 As shown, the chaoticization of a high-dimensional dynamic system and its chaotic sequence generation method are as follows:

[0047] Let m-dimensional discrete dynamical system

[0048] S n+1 =AS n modc

[0049] where S n is the state vector (x 1 (n),x 2 (n),x 3 (n)......x m (n)) T , A is a constant coefficient matrix,

[0050]

[0051] Since there are no nonlinear terms in the m-dimensional discrete dynamical system, the Jacobian matrix is ​​A, therefore, P=A n , let the m eigenvalues ​​of matrix A be λ 0 ,λ 1 ,...,λ m , the m Lyapunov exponents in m-dimensional discrete chaos are:

[0052]

[0053] Therefore, the eigenvalues ​​of the parameter matrix A determine the Lyapunov exponent of the system; the construction method of the parameter matrix A is as follows:

[0054] (1) Given the Lyapunov exponent value LE 1 ,LE 2 ,...LE m , and calculate the eigenvalues The eigenvalue-based diagonal matrix Λ is constructed as

[0055] ...

Embodiment 2

[0060] 1. Any given 8 eigenvalues ​​greater than 1, such as 40, 41, 42, 43, 44, 45, 46, 47, and c is defined as 1. The rounded Lyapunov exponents are 3.69, 3.71, 3.74, 3.76, 3.78, 3.81, 3.83, 3.85.

[0061] 2. A nonsingular matrix q is defined as:

[0062]

[0063] The element q(i,i)=2, i=1,2,3...m, and the remaining elements are all 1. It is easy to show that q is a nonsingular matrix. When m=8, q can be defined

[0064]

[0065] and the rounded inverse matrix q -1 for

[0066]

[0067] 3. The parameter matrix is

[0068]

[0069] Reconstructing discrete dynamical systems.

[0070]

[0071] (4) The initial value of the state vector is used as the initial key, and the chaotic sequence is generated by reconstructing the discrete dynamic system. When the output sequence is greater than 0.5, the quantization is 1, and when the output sequence is less than 0.5, the quantization is 0.

Embodiment 3

[0073] The effect of the present invention can be further illustrated by the following detection results of the present embodiment:

[0074] 1. Detection method and content:

[0075] The randomness of the chaotic sequence output by the chaotic sequence generator in Embodiment 2 of the present invention is detected by using the SP800-22 random number detection standard provided by the National Institute of Standards and Technology NIST. The detection standard includes 15 detection contents, each of which is A P-value is included in the test results produced by the test. When the P value is greater than 0.01, it means that the test content has passed.

[0076] 2. Test results:

[0077] Referring to Example 2, make it generate 100 groups of 10,000,000 random sequences, and use the SP800-22 random number detection standard provided by the National Institute of Standards and Technology (NIST) for detection. One group of results is shown in Table 1-8:

[0078] Table 1x 1 (n) Out...

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