CN108718229B - A Key Generation Method Based on Reconstructed Discrete Dynamical System - Google Patents

A Key Generation Method Based on Reconstructed Discrete Dynamical System Download PDF

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CN108718229B
CN108718229B CN201810419378.3A CN201810419378A CN108718229B CN 108718229 B CN108718229 B CN 108718229B CN 201810419378 A CN201810419378 A CN 201810419378A CN 108718229 B CN108718229 B CN 108718229B
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丁群
王传福
余龙飞
李孝友
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Heilongjiang University
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    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
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Abstract

本发明公开一种高维动力系统的混沌化及其混沌序列产生方法，方法如下：设m维离散动力系统，Sn+1＝ASnmodc，A是常系数矩阵，由于m维离散动力系统中没有非线性项，所以雅可比矩阵是A，因此，P＝An，参数矩阵A的特征值决定了系统的李雅普诺夫指数；给定李雅普诺夫指数值LE1,LE2,...LEm，并计算特征值，设计一个m×m维非奇异矩阵q；计算参数矩阵A＝qΛq‑1，并带入原模型中重构离散动力系统；将状态向量的初始值作为初始密钥，利用重构离散动力系统的时间序列产生随机序列；本发明能够实现对李雅普诺夫指数的精确控制，实现了具有周期吸引子的周期系统和具有不动点吸引子的系统，由该方法输出的混沌序列具有更复杂的混沌行为。

Figure 201810419378

The invention discloses a chaotic high-dimensional dynamic system and a method for generating a chaotic sequence. The method is as follows: Suppose an m-dimensional discrete dynamic system, S n+1 =AS n modc, and A is a constant coefficient matrix. Since the m-dimensional discrete dynamic system is There is no nonlinear term in , so the Jacobian matrix is A, therefore, P=A n , the eigenvalues of the parameter matrix A determine the Lyapunov exponent of the system; given the Lyapunov exponent values LE 1 , LE 2 , .. .LE m , and calculate the eigenvalues, design an m×m-dimensional non-singular matrix q; calculate the parameter matrix A=qΛq ‑1 , and bring it into the original model to reconstruct the discrete dynamic system; take the initial value of the state vector as the initial density The method can realize the precise control of the Lyapunov exponent, and realize the periodic system with periodic attractor and the system with fixed point attractor. The output chaotic sequence has more complex chaotic behavior.

Figure 201810419378

Description

Secret key generation method based on reconstructed discrete power system
Technical Field
The invention relates to the field of information safety, in particular to a chaotization method and a chaos sequence generation method of a high-dimensional power system.
Background
The existing general chaotic design method mainly depends on Chen-Lai algorithm, and gives an initial state x0For control system x1＝f0(x0)+B0x0Calculate its jacobian matrix
J0(x0)＝f0'(x0)+B0x0，
And remember T0＝J0(x0). Get B0x0＝σ0I and selecting a constant sigma0> 0 such that the matrix [ T0T0 T]Limited and diagonal predominance. For k 0,1,2, consider a control system
xk+1＝fk(xk)+Bkxk，
In the formula Bkxk＝σkI has been determined from the previous step. Now the following calculations are made:
step 1, calculating a Jacobian matrix
Jk(xk)＝fk'(xk)+Bkxk
Note Tk＝JkTk-1。
Step 2, selecting a constant sigmak> 0, so that the matrix
[TkTk T]-e2kcI
Limited and diagonal predominance, wherein the constant c > 0 satisfies 0 < c ≦ λi(x0)＜∞，i＝0,1,2,...,n,。
And 3, performing the following modular operation on the control system:
xk+1＝fk(xk)+Bkxk(mod1)
the first two steps of the algorithm described above make the Lyapunov (Lyapunov) exponent of the controlled system strictly positive, so that the system trajectory expands in all directions. A simple controller that satisfies steps 1 and 2 is
uk＝Bkxk＝σkIk，σk＝N+ec。
The modulo operation of the third step makes the entire orbit globally bounded.
The Chen-Lai algorithm can only ensure that the chaotic system has positive Lyapunov (Lyapunov) indexes, and cannot realize accurate control on all Lyapunov (Lyapunov) indexes.
Secondly, selecting a discrete dynamic system, and piecing together a positive Lyapunov (Lyapunov) index by using a Jacobian method:
discrete chaotic system with m dimension
Figure BDA0001650297620000021
The jacobian matrix is:
Figure BDA0001650297620000022
iterating the jacobian matrix J n timesi＝J(x1(i),x2(i)...xm(i) I ═ 0,1,2.. n). Let P be J0·J1·····JnAnd m eigenvalues of the matrix P are λ0,λ1,...,λmThen the Lyapunov (Lyapunov) index is
Figure BDA0001650297620000023
If a positive Lyapunov (Lyapunov) index exists, the discrete power system has chaotic behavior, the method utilizes parameters to piece together the positive Lyapunov (Lyapunov) index, has no clear design direction, and cannot accurately control the Lyapunov (Lyapunov) index, and the design of the high-dimensional hyperchaotic system is very difficult by utilizing the method.
Disclosure of Invention
Based on the problems, the chaotization of the high-dimensional power system and the chaos sequence generation method thereof are provided, the problems that the high-dimensional hyper-chaos system is difficult to design and the Lyapunov exponent in the chaos system is difficult to control are solved, and all the Lyapunov exponents in the designed chaos system can be well controlled by the method.
The technology adopted by the invention is as follows: a chaotization method of a high-dimensional power system and a chaos sequence generation method thereof are as follows:
discrete power system with m dimensions
Sn+1＝ASn modc
Wherein SnIs a state vector (x)1(n),x2(n),x3(n)......xm(n))TAnd A is a constant coefficient matrix,
Figure BDA0001650297620000031
since there are no non-linear terms in the m-dimensional discrete dynamical system, the jacobian matrix is a, so P ═ anLet m eigenvalues of matrix A be λ0,λ1,...,λmAnd m Lyapunov indexes in the m-dimensional discrete chaos are as follows:
Figure BDA0001650297620000032
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:
(1) given the Lyapunov index value LE1,LE2,...LEmAnd calculating the characteristic value
Figure BDA0001650297620000041
The diagonal matrix Lambda based on eigenvalues is constructed as
Figure BDA0001650297620000042
(2) Designing an m x m dimensional nonsingular matrix q;
(3) calculating parameter matrix A ═ q Λ q-1And the data are brought into the original model to reconstruct a discrete power system;
(4) and taking the initial value of the state vector as an initial key, and generating a chaotic sequence by utilizing the reconstructed discrete power system.
The invention has the following advantages and beneficial effects: the method can realize the accurate control of the Lyapunov exponent, and because the Lyapunov exponent is accurately controlled, a periodic system with a periodic attractor and a system with a stationary point attractor are realized, and the chaotic sequence output by the method has more complex chaotic behaviors.
Drawings
FIG. 1 is a flow chart of a power system for controlling the Lyapunov exponent configuration;
fig. 2 is a diagram of a chaotic sequence generator constructed by using a positive lyapunov exponent.
Detailed Description
The invention is further illustrated by way of example in the accompanying drawings of the specification:
example 1
As shown in fig. 1-2, a chaos method of a high-dimensional power system and a chaos sequence generation method thereof are as follows:
discrete power system with m dimensions
Sn+1＝ASn modc
Wherein SnIs a state vector (x)1(n),x2(n),x3(n)......xm(n))TAnd A is a constant coefficient matrix,
Figure BDA0001650297620000051
since there are no non-linear terms in the m-dimensional discrete dynamical system, the jacobian matrix is a, so P ═ anLet m eigenvalues of matrix A be λ0,λ1,...,λmAnd m Lyapunov (Lyapunov) indexes in the m-dimensional discrete chaos are as follows:
Figure BDA0001650297620000052
thus, the eigenvalues of the parameter matrix a determine the Lyapunov (Lyapunov) exponent of the system; the construction method of the parameter matrix A is as follows:
(1) given a Lyapunov (Lyapunov) index value LE1,LE2,...LEmAnd calculating the characteristic value
Figure BDA0001650297620000053
The diagonal matrix Lambda based on eigenvalues is constructed as
Figure BDA0001650297620000054
(2) Designing an m x m dimensional nonsingular matrix q;
(3) calculating parameter matrix A ═ q Λ q-1And the data are brought into the original model to reconstruct a discrete power system;
(4) and taking the initial value of the state vector as an initial key, and generating a chaotic sequence by utilizing the reconstructed discrete power system.
Example 2
1. Any given 8 eigenvalues greater than 1, such as 40,41,42,43,44,45,46,47, and c is defined as 1. Rounded Lyapunov (Lyapunov) indices are 3.69,3.71,3.74, 3.76,3.78,3.81,3.83,3.85.
2. A non-singular matrix q is defined as:
Figure BDA0001650297620000061
where the element q (i, i) is 2, i is 1,2,3.. m, and the remaining elements are all 1. It is easy to prove that q is a non-singular matrix. When m is 8, q may be defined
Figure BDA0001650297620000062
And the rounded inverse matrix q-1Is composed of
Figure BDA0001650297620000063
3. The parameter matrix is
Figure BDA0001650297620000071
And reconstructing the discrete power system.
Figure BDA0001650297620000072
(4) And taking the initial value of the state vector as an initial key, and generating a chaotic sequence by utilizing the reconstructed discrete power system. The quantization is 1 when the output sequence is greater than 0.5 and 0 when the output sequence is less than 0.5.
Example 3
The effect of the present invention can be further illustrated by the following detection results of this embodiment:
1. the detection method and the content are as follows:
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 national institute of standards and technology, NIST, wherein the detection standard comprises 15 detection contents, and a detection result generated by each detection contains a P value. When the P value is larger than 0.01, the detection content is passed.
2. And (3) detection results:
referring to example 2, 100 sets of 10000000 random sequences were generated and tested using the SP800-22 random number test standard provided by the national institute of standards and technology NIST, wherein one set of results are shown in tables 1-8:
TABLE 1 x1(n) output sequence testing
Figure BDA0001650297620000081
TABLE 2 x2(n) output sequence testing
Figure BDA0001650297620000082
TABLE 3 x3(n) output sequence testing
Figure BDA0001650297620000091
TABLE 4 x4(n) output sequence testing
Figure BDA0001650297620000092
TABLE 5 x5(n) output sequence testing
Figure BDA0001650297620000101
TABLE 6 x6(n) output sequence testing
Figure BDA0001650297620000102
TABLE 7 x7(n) output sequence testing
Figure BDA0001650297620000111
TABLE 8 x8(n) output sequence testing
Figure BDA0001650297620000112
The 100 groups of data are tested, and the passing rate value is not lower than 0.96.

Claims (1)

1.一种基于重构离散动力系统的密钥产生方法，其特征在于，方法如下：1. a key generation method based on reconstructed discrete dynamic system, is characterized in that, method is as follows: 设m-维离散混沌系统Let m-dimensional discrete chaotic system Sn+1＝ASnmodcS n+1 =AS n modc 其中Sn是状态向量(x1(n),x2(n),x3(n)......xm(n))T，A是参数矩阵，where S n is the state vector (x 1 (n),x 2 (n),x 3 (n)......x m (n)) T , A is the parameter matrix,
Figure FDA0002950225840000011
Figure FDA0002950225840000011
 由于m-维离散混沌系统中没有非线性项，所以雅可比矩阵是A，因此，P＝An，设参数矩阵A的m个特征值为λ0,λ1,...,λm，m-维离散混沌系统中m个李雅普诺夫指数为：Since there is no nonlinear term in the m-dimensional discrete chaotic system, the Jacobian matrix is A, therefore, P=A n , suppose the m eigenvalues of the parameter matrix A are λ 0 , λ 1 ,...,λ m , The m Lyapunov exponents in the m-dimensional discrete chaotic system are:
Figure FDA0002950225840000012
Figure FDA0002950225840000012
 参数矩阵A的特征值决定了系统的李雅普诺夫指数；The eigenvalues of the parameter matrix A determine the Lyapunov exponent of the system; 其中，参数矩阵A的构造方法如下步骤所示：Among them, the construction method of parameter matrix A is as follows: (1)给定李雅普诺夫指数值LE1,LE2,...LEm，并计算特征值
Figure FDA0002950225840000013
基于特征值的对角矩阵Λ构造为：(1) Given the Lyapunov exponent values LE 1 , LE 2 ,...LE m , and calculate the eigenvalues
Figure FDA0002950225840000013
The eigenvalue-based diagonal matrix Λ is constructed as:
Figure FDA0002950225840000021
Figure FDA0002950225840000021
 (2)设计一个m×m维非奇异矩阵q；(2) Design an m×m-dimensional non-singular matrix q; (3)计算参数矩阵A＝qΛq-1，并带入m-维离散混沌系统Sn+1＝ASnmodc中重构离散动力系统；(3) Calculate the parameter matrix A=qΛq -1 and bring it into the m-dimensional discrete chaotic system Sn +1 =AS n modc to reconstruct the discrete dynamic system; (4)将状态向量的初始值作为初始密钥，利用重构离散动力系统产生随机序列。(4) The initial value of the state vector is used as the initial key, and the random sequence is generated by reconstructing the discrete dynamic system.
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Citations (6)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN101282266A (en) * 2008-03-05 2008-10-08 中科院嘉兴中心微系统所分中心 Intelligent instruction-preventing microwave radar wireless sensor network
CN103117848A (en) * 2013-01-17 2013-05-22 王少夫 Seven-dimensional chaotic system
CN103634099A (en) * 2013-12-19 2014-03-12 哈尔滨理工大学 Five-dimensional chaotic system and chaotic signal generator based on five-dimensional chaotic system
CN104749957A (en) * 2015-03-25 2015-07-01 山东科技大学 Method for accurately configuring all Lyapunov indexes of constant discrete linear system
WO2017029600A1 (en) * 2015-08-14 2017-02-23 King Abdullah University Of Science And Technology Robust lyapunov controller for uncertain systems
CN206542421U (en) * 2017-03-21 2017-10-03 哈尔滨理工大学 A kind of secondary hyperchaos analog circuit of octuple

Patent Citations (6)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN101282266A (en) * 2008-03-05 2008-10-08 中科院嘉兴中心微系统所分中心 Intelligent instruction-preventing microwave radar wireless sensor network
CN103117848A (en) * 2013-01-17 2013-05-22 王少夫 Seven-dimensional chaotic system
CN103634099A (en) * 2013-12-19 2014-03-12 哈尔滨理工大学 Five-dimensional chaotic system and chaotic signal generator based on five-dimensional chaotic system
CN104749957A (en) * 2015-03-25 2015-07-01 山东科技大学 Method for accurately configuring all Lyapunov indexes of constant discrete linear system
WO2017029600A1 (en) * 2015-08-14 2017-02-23 King Abdullah University Of Science And Technology Robust lyapunov controller for uncertain systems
CN206542421U (en) * 2017-03-21 2017-10-03 哈尔滨理工大学 A kind of secondary hyperchaos analog circuit of octuple

Non-Patent Citations (2)

* Cited by examiner, † Cited by third party
Title
Designing Hyperchaotic Systems With;Chaowen Shen et-al;《IEEE TRANSACTIONS ON CTRCUITS AND SYSTEMS-I REGULAR PAPERS》;20140831;第61卷(第8期);第2380-2389页 *
基于混沌系统的SM4密钥扩展算法;王传福等;《物理学报》;20161220;第22卷(第2期);第1-9页 *

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