CN113259085A - Three-dimensional multi-cavity chaotic system construction method based on rotation method and pseudo-random sequence generator - Google Patents
Three-dimensional multi-cavity chaotic system construction method based on rotation method and pseudo-random sequence generator Download PDFInfo
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Abstract
The invention discloses a method for constructing a three-dimensional multi-cavity chaotic system based on a rotation method, which comprises the steps of firstly establishing a one-dimensional chaotic system of which an iteration sequence always keeps the same polarity with an initial value of the system under a specific condition, then establishing a three-dimensional hollow multi-cavity hyperchaotic system on the basis of the one-dimensional chaotic system, and finally adding filling items to two dimensions on the basis of the three-dimensional hollow multi-cavity hyperchaotic system to obtain a three-dimensional solid multi-cavity hyperchaotic system, wherein the establishment method is simple; the three-dimensional multi-cavity chaotic system has the characteristics of three-dimensional multi-cavity structure, controllable signal polarity, better performance (clear attractor phase diagram and rich dynamic characteristics) and the like. The invention also discloses a pseudo-random sequence generator, which is obtained based on the three-dimensional multi-cavity chaotic system, and the output sequence of the pseudo-random sequence generator passes all tests, has excellent randomness, is suitable for the engineering fields of encryption and the like, and has wide application.
Description
Technical Field
The invention relates to the technical field of nonlinear systems and digital signals, in particular to a three-dimensional multi-cavity chaotic system construction method based on a rotation method and a pseudo-random sequence generator.
Background
Chaos is the unpredictable behavior of a deterministic system due to extreme sensitivity to initial values and parameters. Chaotic systems include continuous systems and discrete systems. Compared with a continuous system, the discrete system is simple to realize, has low algorithm overhead and can generate a chaotic sequence with higher complexity, thereby having better practicability. Chaotic systems can be classified into one-dimensional systems and high-dimensional systems. The one-dimensional chaotic system has the advantages of simple structure, few parameters, narrow chaotic range, easy prediction of phase space orbits, difficulty in resisting attacks such as parameter estimation and the like, and thus, the safety requirement of practical application cannot be met. The high-dimensional chaotic system has a more complex structure and better nonlinear performance, is easy to generate a hyperchaotic state, and is a research hotspot in recent years. The hyperchaotic state means that the chaotic system has two or more positive Lyapunov indexes, the system phase space orbits are separated in multiple dimensions, and the dynamic behavior is more complex.
The multi-cavity chaotic system is one kind of high-dimensional discrete chaotic system, and the attractor consists of a plurality of cavities with similar structures, so that the multi-cavity chaotic system has richer topological structures and more complex dynamic behaviors compared with a common discrete chaotic system. The existing generalized construction method of the discrete multi-cavity chaotic system only has a closed-loop modulation coupling model, and the multi-cavity chaotic system constructed based on the method can observe a multi-cavity structure only when an attractor is projected to a two-dimensional phase plane. In addition, in engineering practice, there are often requirements on the polarity of the signal, and it is desirable that the signal does not exceed a zero point. However, the existing discrete multi-cavity chaotic mapping can only generate bipolar signals, which is not beneficial to engineering application. Meanwhile, the generation of the pseudorandom sequence in the encryption field is one of the core technologies of the encryption system, and the security of the encryption system is greatly influenced by the quality of the pseudorandom sequence. Based on the reasons, the invention is significant in that a novel generalized construction model for constructing the discrete multi-cavity chaotic system with controllable signal polarity is provided, and a pseudo-random sequence generator is designed on the basis of the model.
Disclosure of Invention
The invention aims to provide a method for constructing a three-dimensional multi-cavity chaotic system based on a rotation method, which specifically comprises the following steps:
establishing a one-dimensional chaotic system, wherein a model of the chaotic system is an expression 1):
establishing a three-dimensional hollow multi-cavity hyperchaotic system, wherein the model of the three-dimensional hollow multi-cavity hyperchaotic system is expressed as an expression 2):
wherein:
is an envelope control function;
establishing a three-dimensional solid multi-cavity hyperchaotic system, which specifically comprises the following steps: on the basis of a three-dimensional hollow multi-cavity hyperchaotic systemx、yAdding a filling item into the dimension to obtain a three-dimensional solid multi-cavity hyperchaotic system;
the model of the three-dimensional multi-cavity chaotic system is expressed as expression 3):
wherein:x、y、zis a state variable;nthe number of iterations to solve for the system;cis the internal disturbance frequency;hin order to make the attractor height of the chaotic system,
；ras an attractorx-yThe projected radius of the coordinate plane.
The construction method of the three-dimensional solid multi-cavity hyperchaotic system has simple steps and is beneficial to industrial application; the three-dimensional multi-cavity chaotic system has the characteristics of three-dimensional multi-cavity structure, controllable signal polarity, better performance (clear attractor phase diagram and rich dynamic characteristics) and the like.
The invention also discloses a pseudo-random sequence generator, which is obtained according to the three-dimensional multi-cavity chaotic system obtained by the construction method;
the pseudo-random sequence is obtained by the following method: a chaotic sequence generated according to a certain dimension of the model expression 3); expanding each state value in the chaotic sequence by 106A multiplication module 256 to obtain a sequence of integers with a value varying between 0 and 255; converting each value in the integer sequence into an eight-bit binary number to obtain a pseudorandom sequence;
the pseudo-random sequence obtained by the pseudo-random sequence generator is expressed by expression 4):
wherein:PRBSin order to generate a pseudo-random series,floorin order to carry out the rounding-down operation,modin order to carry out the mould-taking operation,x i representing the selected chaotic sequenceiA state value.
In the present invention, when satisfying
The iterative sequence of the expression 1) always keeps the same polarity with the initial value of the system;call values of
。
Random sequence obtained by the method of the invention is subjected to random characteristic test, and NIST test on the sequence shows that: the output sequence of the pseudo-random sequence generator based on the three-dimensional sine filled multi-cavity hyper-chaotic system passes all tests, has excellent randomness, and can be widely applied to the engineering fields of encryption and the like.
In addition to the objects, features and advantages described above, other objects, features and advantages of the present invention are also provided. The present invention will be described in further detail below with reference to the drawings.
Drawings
The accompanying drawings, which are incorporated in and constitute a part of this application, illustrate embodiments of the invention and, together with the description, serve to explain the invention and not to limit the invention. In the drawings:
FIG. 1 is a block diagram of a three-dimensional hollow multi-cavity hyperchaotic model;
FIG. 2 is a three-dimensional sine hollow multi-cavity hyperchaotic system attractor;
FIG. 3 is a three-dimensional sine hollow multi-cavity hyperchaotic system coexisting attractor;
FIG. 4 is
The complexity of Lyapunov index spectrum, bifurcation diagram and fuzzy entropy of the time-three-dimensional sinusoidal hollow multi-cavity hyper-chaotic system is shown in a figure 4(a), a figure 4(b) and a figure 4(c) in sequence;
FIG. 5 is
The complexity of Lyapunov index spectrum, bifurcation diagram and fuzzy entropy of the time-three-dimensional sinusoidal hollow multi-cavity hyper-chaotic system is shown in a figure 5(a), a figure 5(b) and a figure 5(c) in sequence;
FIG. 6 is
The complexity of Lyapunov index spectrum, bifurcation diagram and fuzzy entropy of the time-varying three-dimensional sinusoidal hollow multi-cavity hyper-chaotic system is shown in the figure 6(a), the figure 6(b) and the figure 6(c) in sequence;
FIG. 7 is a block diagram of a three-dimensional solid multi-cavity hyperchaotic model;
FIG. 8 is a three-dimensional sine solid multi-cavity hyperchaotic system attractor;
FIG. 9 is a three-dimensional sine solid multi-cavity hyperchaotic system coexisting attractor;
FIG. 10 is
The complexity of Lyapunov index spectrum, bifurcation diagram and fuzzy entropy of the time-varying three-dimensional sinusoidal solid multi-cavity hyper-chaotic system is shown in FIG. 10(a), FIG. 10(b) and FIG. 10 (c);
FIG. 11 is
Time three-dimensional sineThe Lyapunov exponential spectrum, the bifurcation diagram and the fuzzy entropy complexity of the solid multi-cavity hyper-chaotic system are sequentially shown in a figure 11(a), a figure 11(b) and a figure 11 (c);
FIG. 12 is
The Lyapunov exponential spectrum, the bifurcation diagram and the fuzzy entropy complexity of the time-three-dimensional sinusoidal solid multi-cavity hyper-chaotic system are sequentially shown in FIG. 12(a), FIG. 12(b) and FIG. 12 (c).
Detailed Description
Embodiments of the invention will be described in detail below with reference to the drawings, but the invention can be implemented in many different ways, which are defined and covered by the claims.
Example 1:
a three-dimensional multi-cavity chaotic system based on a rotation method is established by the following steps:
the first step is to establish a one-dimensional chaotic system, and the chaotic system is modeled as an expression 1):
wherein:zis a state variable;nthe number of iterations to solve for the system; h is the attractor height of the chaotic system; c is internal disturbance frequency, and the value ranges are all
。
When it is satisfied with
And keeping the same polarity of the iteration sequence of the expression 1) and the initial value of the system all the time, and verifying the conditions as follows:
in the first case:
；
in the second case:
；
from the above, when satisfying
And in time, the iterative sequence of the expression 1) always keeps the same polarity with the initial value of the system, so that the polarity of the chaotic signal can be controlled by setting the positive and negative of the initial value.
Secondly, establishing a three-dimensional hollow multi-cavity hyperchaotic system, wherein the model is expressed as formula 2):
wherein:x、y、zis a state variable;hin order to make the attractor height of the chaotic system,ras an attractorx-yThe radius of the projection of the coordinate plane,cis the internal disturbance frequency;
the shape of the envelopes of the attractors can be controlled as desired as an envelope control function.
The block diagram of the three-dimensional hollow multi-cavity hyperchaotic model is shown in figure 1 in detail.
Setting parameters
Initial value of
And performing simulation by using Matlab to obtain an attractor of the expression 2) as shown in FIG. 2.
Is provided withParameter(s)
Two sets of initial values are respectively
And
the coexistence attractor of expression 2) is shown in fig. 3.
Therefore, it can be seen that when the condition is satisfied
Due to system
Polarity of dimension
Control so as to form a pair of
And (3) a surface-symmetrical coexisting attractor. The system dynamics were analyzed as follows:
(1) when in use
Height of attractor
Varying from 0 to 10, the Lyapunov exponential spectrum, bifurcation diagram, and fuzzy entropy complexity of the system are shown in fig. 4(a) -4 (c). The system is only on
And a narrow period window is provided, and two positive Lyapunov indexes are always provided for the values of other parameters, so that the state is a hyperchaotic state. Fuzzy entropy complexity of system
The variation fluctuates around 1.5.
(2) When in use
Radius of projection of attractor
Varying from 0 to 10, the Lyapunov exponential spectrum, bifurcation diagram, and fuzzy entropy complexity of the system are shown in fig. 5(a) -5 (c). The system always has two positive Lyapunov indexes; the system is in a hyperchaotic state in the interval. Fuzzy entropy complexity of system
The increase is increased, and the maximum value can reach 3.97.
(3) When in use
Frequency of internal disturbance
Varying from 0 to 100, the Lyapunov exponential spectrum, bifurcation map and fuzzy entropy complexity of the system are shown in fig. 6(a) -6 (c). In that
In the method, a narrow periodic window is provided, the other range systems always have two positive Lyapunov indexes, and the system is in a hyperchaotic state. Fuzzy entropy complexity of system
The variation fluctuates around 1.6.
Step three, establishing a three-dimensional solid multi-cavity hyperchaotic system, which specifically comprises the following steps: on the basis of a three-dimensional hollow multi-cavity hyperchaotic systemx、yAdding a filling item into the dimensionality to obtain a three-dimensional solid multi-cavity hyperchaotic system, wherein the model of the three-dimensional multi-cavity hyperchaotic system is an expression 3):
wherein:x、y、zis a state variable;hin order to make the attractor height of the chaotic system,ras an attractorx-yThe radius of the projection of the coordinate plane,cis the internal disturbance frequency.
The block diagram of the three-dimensional solid multi-cavity hyperchaotic model is shown in detail in FIG. 7.
Setting parameters
Initial value of
Simulation was performed using Matlab, and the attractor of expression 3) was obtained as shown in fig. 8.
Setting parameters
Two sets of initial values are respectively
And
the coexistence attractor of expression 3) is shown in fig. 9.
Thus, it can be seen that: when the condition is satisfied
Due to system
Polarity of dimension
Control so as to form a pair ofx-yAnd (3) a surface-symmetrical coexisting attractor. The system dynamics were analyzed as follows:
(1) when in use
Height of attractor
Varying from 0 to 10, the Lyapunov exponential spectrum, bifurcation diagram, and fuzzy entropy complexity of the system are shown in fig. 10(a) -10 (c). The system always has three positive Lyapunov indexes which are in a hyperchaotic state. Fuzzy entropy complexity of systemhVarying to fluctuate around 1.
(2) When in use
Radius of projection of attractor
Varying from 0 to 10, the Lyapunov exponential spectrum, bifurcation map and fuzzy entropy complexity of the system are shown in fig. 11(a) -11 (c). The system always has three positive and dependent parameters
Increasing and decreasing the Lyapunov exponent; the system is in a hyperchaotic state in the interval. Fuzzy entropy complexity of system
The size is increased and can reach 3.04 at most.
(3) When in use
Frequency of internal disturbance
Varying from 0 to 100, the Lyapunov exponential spectrum, bifurcation map and fuzzy entropy complexity of the system are shown in fig. 12(a) -12 (c). In that
There is a narrow window, and the rest of the range system always has three positive and dependent parameterscThe Lyapunov exponent is increased by increasing,the system is in a hyperchaotic state. Fuzzy entropy complexity of systemcVarying to fluctuate around 1.
It can be seen that the invention makes attractors parallel to the model by changing the envelope control function of the modelx-yCross-sectional radius of face as a function of cross-sectional heightzAnd varied to control the shape of the attractor. When the envelope control function is determined, the radius of the section is dependent onzAnd determining the relation of the changes. Chaotic behavior of system is composed ofzChaotic behavior driving of dimensions and passingx,yThe dimension strengthens the chaotic behavior of the system. The system model can also be used for constructing other discrete multi-cavity chaotic systems, and has strong practicability; by modifying the envelope control function, a system with clear attractor phase diagram and rich dynamic characteristics can be obtained.
Example 2:
a pseudo-random sequence generator is obtained based on a three-dimensional multi-cavity chaotic system obtained by the construction method disclosed by embodiment 1.
The method for obtaining the pseudorandom sequence in the embodiment is as follows: a chaotic sequence generated according to a certain dimension of the model expression 3); expanding each state value in the chaotic sequence by 106A multiplication module 256 to obtain a sequence of integers with a value varying between 0 and 255; converting each value in the integer sequence into an eight-bit binary number to obtain a pseudorandom sequence;
the pseudo-random sequence obtained by the pseudo-random sequence generator is expressed by expression 4):
wherein:PRBSin order to generate a pseudo-random series,floorin order to carry out the rounding-down operation,modin order to carry out the mould-taking operation,x i is a chaotic sequence.
To further examine the random nature of the pseudorandom sequence, the sequence was subjected to NIST test. The test is divided into 15 sub-items, wherein some items are averaged by multiple tests, and the specific test items and test results are shown in table 1. According to the test results in table 1, it can be seen that: the output sequence of the pseudo-random sequence generator based on the three-dimensional sine filled multi-cavity hyper-chaotic system passes all tests, has excellent randomness, is suitable for the engineering fields of encryption and the like, and is widely applied.
TABLE 1 NIST test results
| Test index | Number of times | P-values | Passing rate | Test results |
| |
1 | 0.4750 | 1 | By passing |
| |
1 | 0.8978 | 1 | By passing |
| Accumulation and |
2 | 0.7476 | 1 | By passing |
| |
1 | 0.8677 | 0.98 | By passing |
| |
1 | 0.6163 | 1 | By passing |
| Binary |
1 | 0.7981 | 1 | By passing |
| |
1 | 0.1453 | 0.99 | By passing |
| Non-overlapping word match test | 148 | 0.5204 | 0.99 | By passing |
| Overlapping |
1 | 0.2622 | 0.99 | By passing |
| General |
1 | 0.0487 | 0.99 | By passing |
| |
1 | 0.9114 | 0.99 | By passing |
| Random run test | 8 | 0.4176 | 0.99 | By passing |
| Random run variable test | 18 | 0.3594 | 0.99 | By passing |
| Series of |
2 | 0.4708 | 0.99 | By passing |
| |
1 | 0.4190 | 1 | By passing |
Note: the test comprises multiple tests, listed as the average of P-values and passage rate in the test results
The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention, and various modifications and changes may be made by those skilled in the art. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.
Claims (4)
1. A method for constructing a three-dimensional multi-cavity chaotic system based on a rotation method is characterized by comprising the following steps of:
establishing a one-dimensional chaotic system, wherein a model of the chaotic system is an expression 1):
wherein:zis a state variable;nthe number of iterations to solve for the system;his the chaotic system attractor height;cis the internal disturbance frequency;
establishing a three-dimensional hollow multi-cavity hyperchaotic system, wherein the model of the three-dimensional hollow multi-cavity hyperchaotic system is expressed as an expression 2):
wherein:x、y、zis a state variable;nthe number of iterations to solve for the system;cis the internal disturbance frequency;hin order to make the attractor height of the chaotic system,
；ras an attractorx-yThe projected radius of the coordinate plane;
is an envelope control function;
establishing a three-dimensional solid multi-cavity hyperchaotic system, which specifically comprises the following steps: on the basis of a three-dimensional hollow multi-cavity hyperchaotic systemx、yAdding a filling item into the dimensionality to obtain a three-dimensional solid multi-cavity hyperchaotic system, wherein the model of the three-dimensional multi-cavity hyperchaotic system is an expression 3):
wherein:x、y、zis a state variable;nthe number of iterations to solve for the system;cis the internal disturbance frequency;hin order to make the attractor height of the chaotic system,
；ras an attractorx-yThe projected radius of the coordinate plane.
2. The construction method according to claim 1, wherein the iterative sequence of expression 1) always keeps the same polarity with the system initial value.
3. The construction method according to claim 2,call values of
。
4. A pseudo-random sequence generator, which is characterized in that the pseudo-random sequence generator is obtained based on the three-dimensional multi-cavity chaotic system obtained by the construction method of any one of claims 1 to 3;
the pseudo-random sequence is obtained by the following method: a chaotic sequence generated according to a certain dimension of the model expression 3); expanding each state value in the chaotic sequence by 106Multiplication back model 256 to obtain a value of 0 to 2A sequence of integers varying between 55; converting each value in the integer sequence into an eight-bit binary number to obtain a pseudorandom sequence;
the pseudo-random sequence obtained by the pseudo-random sequence generator is expressed by expression 4):
wherein:PRBSin order to generate a pseudo-random series,floorin order to carry out the rounding-down operation,modin order to carry out the mould-taking operation,x i representing the selected chaotic sequenceiA state value.
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