CN112596702A - Chaotization method based on disturbance and pseudo random sequence generation method - Google Patents
Chaotization method based on disturbance and pseudo random sequence generation method Download PDFInfo
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Abstract
The invention discloses a chaotization method based on disturbance, which comprises the steps of selecting a seed system needing chaotization; selecting a disturbance function; performing external disturbance chaotization, internal disturbance chaotization and composite disturbance chaotization on the target function by adopting a disturbance function; and selecting the optimal scheme as a final chaotization result. The invention also discloses a pseudo-random sequence generation method comprising the chaotization method based on the disturbance. The invention not only realizes chaos based on disturbance for the target function by introducing a disturbance mode, but also is suitable for one-dimensional mapping and high-dimensional mapping, the proposed multiple disturbance form can be added to any state variable of any dimension, so as to construct a plurality of new enhanced discrete chaotic systems without losing generality, and meanwhile, the method of the invention also has the characteristics of high reliability, good safety, low cost and easy implementation.
Description
Technical Field
The invention belongs to the technical field of chaos, and particularly relates to a chaos method based on disturbance and a pseudo-random sequence generation method.
Background
With the development of economic technology and the improvement of living standard of people, the chaotization method is also gradually applied to the production and life of people, and brings convenience to the production and life of people.
Based on the number of system variables, the discrete chaotic mapping is divided into a one-dimensional chaotic mapping and a high-dimensional chaotic mapping. The one-dimensional chaotic mapping such as Logistic mapping, Sine mapping, Chebyshev mapping, ICMIC mapping and the like is widely applied to the fields of cryptography, watermarking, optimization and the like due to the characteristics of low cost and high implementation efficiency. However, the one-dimensional chaotic mapping parameter space is small, the chaotic interval is narrow, and the complexity of the generated chaotic sequence is low, so that the track is easy to estimate, the safety risk exists, and the application and the characteristics are limited.
In recent years, researchers have proposed a variety of different schemes to construct new or enhanced chaotic maps, including dimension expansion, cascading chaos, fractional order continuation, etc., and improve the performance of the existing chaotic maps on the basis of meeting unpredictability, ergodicity and sensitivity to parameters and initial conditions. However, these methods bring high implementation cost when the original system is high-dimensional mapping, thereby restricting the application of the technical solution.
Disclosure of Invention
One of the objectives of the present invention is to provide a chaos method based on disturbance, which has high reliability, good safety, low cost and easy implementation.
The invention also aims to provide a pseudo-random sequence generation method comprising the perturbation-based chaos method.
The chaos method based on disturbance provided by the invention comprises the following steps:
s1, acquiring a seed system to be chaotic;
s2, selecting a disturbance function;
s3, adopting the disturbance function selected in the step S2, and adopting external disturbance chaotization, internal disturbance chaotization and composite disturbance chaotization for the seed system selected in the step S1;
and S4, evaluating the external disturbance chaotization result, the internal disturbance chaotization result and the composite disturbance chaotization result obtained in the step S3, and selecting an optimal scheme as a final chaotization result.
The perturbation function in step S2 specifically includes the following functions: g (x) x, g (x) ex、g(x)＝sinx、g(x)＝cosx、g(x)＝arctanx、
And
the step S3, which is to perform the chaotization by the external disturbance specifically according to the following formula:
xn+1＝af(xn)-b(g1(xn)+g2(xn)+…+gm(xn))
in the formula xn+1Is the result after n +1 iterations; f (x)n) Mapping f (x) values of the seed after n iterations; gi(xn) As a function of disturbance gi(x) Iterating the values n times, i being a positive integer and taking the value 1, 2.. m; a is the control parameter of f (x); b is a control parameter of the disturbance function, and b is equal to 0,1]。
The internal disturbance chaotization step S3 is specifically to perform internal disturbance chaotization by adopting the following formula:
xn+1＝a(f(xn-b(g1(xn)+g2(xn)+…+gm(xn)))
in the formula xn+1Is the result after n +1 iterations; gi(xn) For a selected perturbation function gi(x) After n iterations, i is a positive integer and takes a value of 1, 2.. m; a is the control parameter of f (x); b is a control parameter of the disturbance function, and b is equal to 0,1]。
The compound disturbance chaotization of the step S3 is specifically a compound disturbance chaotization by adopting the following formula:
xn+1＝a(f(xn-b(g11(xn)+g12(xn)+…))-b(g21(xn)+g22(xn)+…)
in the formula xn+1Is the result after n +1 iterations; g1i(xn) For a selected internal perturbation function g1i(x) Value after n iterations, g2i(xn) For a selected external disturbance function g2i(x) After n iterations, i is a positive integer, and the value of i is 1, 2.. a, a is a control parameter of f (x); b is a control parameter of the disturbance function, and b is equal to 0,1]。
The step S4 evaluates the external disturbance chaotization result, the internal disturbance chaotization result and the composite disturbance chaotization result obtained in the step S3, selects an optimal scheme as a final chaotization result, specifically calculates the complexity of the external disturbance chaotization result, the complexity of the internal disturbance chaotization result and the complexity of the composite disturbance chaotization result obtained in the step S3, and selects a scheme with the highest complexity as the final chaotization result.
The invention also provides a pseudo-random sequence generation method comprising the chaotization method based on the disturbance, which specifically comprises the following steps:
A. for the existing chaotic system, carrying out chaotization according to the chaotization method based on disturbance to generate a chaotic sequence;
B. converting the chaotic sequence into a 01 sequence by adopting a binary quantitative optimization algorithm;
C. the generated pseudo-random sequence is subjected to performance evaluation through a NIST test, thereby generating a final pseudo-random sequence.
The chaotization method based on disturbance and the pseudo random sequence generation method provided by the invention not only realize the chaotization based on the disturbance aiming at the target function by introducing the disturbance mode, but also are suitable for one-dimensional mapping and high-dimensional mapping, the proposed multiple disturbance form can be added to any state variable of any dimension, so that a plurality of new enhanced discrete chaos systems can be constructed without losing the generality of the systems, and meanwhile, the method also has the characteristics of high reliability, good safety, low cost and easy implementation.
Drawings
FIG. 1 is a schematic diagram of the method flow of the chaos method of the present invention.
FIG. 2 is a model diagram of an external perturbation scheme of the chaos method of the present invention.
FIG. 3 is a model diagram of an internal perturbation scheme of the chaos method of the present invention.
FIG. 4 is a model diagram of a complex perturbation scheme of the chaos method of the present invention.
Fig. 5 is a schematic diagram of a chaotic attractor when a is 3.4, which is a characteristic of the sinusoidal chaotic map of the chaotic method of the present invention.
Fig. 6 is a bifurcation diagram varying with a of the characteristics of the sinusoidal chaotic map of the chaotic method according to the present invention.
Fig. 7 is a Lyapunov exponential spectrum diagram of the sinusoidal chaos mapping characteristic of the chaos method of the present invention.
FIG. 8 is a schematic diagram of the balance point analysis of the external disturbance system mapped by the sinusoidal chaos in the chaos method of the present invention.
Fig. 9 is a schematic diagram of a chaotic attractor of an external perturbation system when a is 3.5 and b is 0.6 in the chaotization method of the invention.
FIG. 10 is a schematic diagram of fuzzy complexity of an external perturbation system with a chaotization method varying with a and b according to the present invention.
FIG. 11 is a schematic diagram of the bifurcation of the external perturbation system with the chaotization method varying with a and b according to the invention.
FIG. 12 is a schematic spectrum of Lyapunov exponent of an externally perturbed system of the chaotization method varying with a and b.
FIG. 13 is a schematic diagram of the balance point analysis of the internal perturbation system of the sinusoidal chaotic map of the chaotization method of the invention.
Fig. 14 is a schematic diagram of the chaotic attractor of the internal perturbation system when a is 3.5 and b is 0.6 in the chaos transformation method of the present invention.
FIG. 15 is a schematic diagram of fuzzy yEn complexity chaos of an internal perturbation system with a chaotization method varying with a and b according to the present invention.
FIG. 16 is a schematic diagram of the bifurcation of the internal perturbation system with the variation of the chaotization method according to the invention as a and b.
FIG. 17 is a schematic spectrum of Lyapunov exponent of an internal perturbation system of the chaotization method varying with a and b.
FIG. 18 is a schematic diagram of the balance point analysis of the sinusoidal chaotic mapping composite perturbation system of the chaotization method of the present invention.
Fig. 19 is a schematic diagram of a chaotic attractor of a complex perturbation system when a is 3.5 and b is 0.6 in the chaotization method of the invention.
FIG. 20 is a schematic diagram of fuzzy En complexity chaos of a complex perturbation system with a chaotization method varying with a and b according to the present invention.
FIG. 21 is a bifurcation diagram of the complex perturbation system with the chaotization method varying with a and b according to the invention.
FIG. 22 is a schematic spectrum of Lyapunov exponent of a complex perturbation system of the chaotization method varying with a and b.
FIG. 23 is a schematic diagram of a comparison of complexity distributions of multiple discrete systems according to the chaotization method of the present invention.
Fig. 24 is a schematic method flow diagram of a pseudo-random sequence generating method of the present invention.
Detailed Description
FIG. 1 is a schematic flow chart of the method of the present invention: the chaos method based on disturbance provided by the invention comprises the following steps:
s1, acquiring a seed system to be chaotic;
s2, selecting a disturbance function; table 1 below classifies the preferred perturbation functions into three different categories, linear or nonlinear functions, periodic or non-periodic functions, and bounded or unbounded functions. Wherein a linear function x and an exponential function exTends to infinity, which is defined as unbounded, while the trigonometric function is bounded;
TABLE 1 disturbance function class schematic
S3, adopting the disturbance function selected in the step S2, and adopting external disturbance chaotization, internal disturbance chaotization and composite disturbance chaotization for the seed system selected in the step S1;
in specific implementation, the following formula is adopted to carry out external disturbance chaotization (a one-dimensional chaotic map f (x) is taken as a seed map, and a cascade disturbance function g (x) is adopted to carry out disturbance on the seed map):
xn+1＝af(xn)-b(g1(xn)+g2(xn)+…+gm(xn))
in the formula xn+1Is the result after n +1 iterations; f (x)n) Mapping f (x) values of the seed after n iterations; gi(xn) For a selected perturbation function gi(x) Iterating the values n times, i being a positive integer and taking the value 1, 2.. m; a is the control parameter of f (x); b is a control parameter of the disturbance function, and b is equal to 0,1](ii) a As shown in fig. 2, D is unit delay, F is seed map F (x), G is a conditioning system that provides perturbation term G (x);
the following formula is adopted to carry out internal disturbance chaotization (the function g (x) is used as an internal disturbance term to disturb the one-dimensional mapping f (x)):
xn+1＝a(f(xn-b(g1(xn)+g2(xn)+…+gm(xn)))
in the formula xn+1Is the result after n +1 iterations; gi(xn) For a selected perturbation function gi(x) Iterating the values n times, i being a positive integer and taking the value 1, 2.. m; a is the control parameter of f (x); b is a control parameter of the disturbance function, and b is equal to 0,1](ii) a As shown in fig. 3, D is unit delay, F is seed map F (x), G is a conditioning system that provides perturbation term G (x);
performing compound disturbance chaotization by adopting the following formula (simultaneously acting internal disturbance and external disturbance on the one-dimensional mapping f (x)):
xn+1＝a(f(xn-b(g11(xn)+g12(xn)+…))-b(g21(xn)+g22(xn)+…)
in the formula xn+1Is the result after n +1 iterations; x is the number ofn+1Is the result after n +1 iterations; g1i(xn) For selected internal functions g1i(x) Value after n iterations, g2i(xn) For a selected perturbation function g2i(x) After n iterations, i is a positive integer, and the value of i is 1, 2.. a, a is a control parameter of f (x); b is a control parameter of the disturbance function, and b is equal to 0,1](ii) a As shown in fig. 4, D is unit delay, F is seed map F (x), G is a conditioning system that provides perturbation term G (x);
s4, evaluating the external disturbance chaotization result, the internal disturbance chaotization result and the composite disturbance chaotization result obtained in the step S3, and selecting an optimal scheme as a final chaotization result; specifically, the complexity of the external disturbance chaotization result obtained in the step S3, the complexity of the internal disturbance chaotization result obtained in the step S3 and the complexity of the composite disturbance chaotization result obtained in the step S3 are calculated, and the scheme with the highest complexity is selected as the final chaotization result.
The chaos method of the present invention is further described below with reference to specific embodiments:
1. characteristics of sinusoidal chaotic mapping
Selecting Sine mapping of a Sine chaotic system as an original system, wherein the equation is as follows:
xn+1＝asin(πxn)
wherein x is a state variable, a is an amplitude and a>0, pi is the angular frequency; setting an initial value x0And (4) performing numerical simulation by using Matlab under different value parameters, wherein n is 0.1 and n is 200000.
Fig. 5 is a schematic drawing of the attractor when a is 3.4; FIG. 6 is a schematic diagram of bifurcation as a function of parameter a; FIG. 7 is a schematic representation of the Lyapunov exponential spectrum as a function of parameter a. The bifurcation graph corresponds to a Lyapunov exponential spectrum, which shows that the sinusoidal chaotic system presents a periodic state when a <1 and a are near half integers.
2. Construction of disturbance system of sine chaotic mapping
And selecting a single disturbance function to act on the sinusoidal chaotic system in an internal disturbance mode and an external disturbance mode respectively, and comparing the action effects of different disturbance functions. Through comparison, two groups of disturbance functions g are selected according to the Sine mapping1(x)＝exAnd
(1) construction of external disturbance model SEPM
Introducing a perturbation function g1(x) And g2(x) The external disturbance model equation of the sinusoidal system is as follows:
(2) construction of internal disturbance model SIPM
Introducing a perturbation function g1(x) And g2(x) The internal disturbance model equation of the sinusoidal system is:
(3) construction of composite disturbance model SBPM
Introducing a perturbation function g1(x) And g2(x) The complex disturbance model equation of the sinusoidal system is as follows:
3. characteristics of perturbed systems of sinusoidal chaotic mapping
Balance points, an attractor phase diagram, complexity, a bifurcation diagram and a Lyapunov index are important indexes for kinetic characteristic analysis, and the kinetic characteristic of the system is analyzed when system parameters change by adopting a FuzzyEn complexity algorithm. The meaning, initial condition and initial value of the parameters in the equation are kept unchanged, and Matlab is used for numerical simulation.
(1) Characteristics of external disturbance model
FIG. 8 is a diagram of equilibrium point analysis of an externally perturbed system of sinusoidal chaotic mapping, equating the existence of equilibrium points of the system to y1X-asin (π x) and
the existence of an intersection. The sinusoidally chaotic mapped externally perturbed system has infinite multi-balance points.
Fig. 9 is an attractor phase diagram of the external perturbation system when a is 3.5 and b is 0.6, and the parameter space and the ergodicity are remarkably improved. FIG. 10 is a fuzzy graph of complexity of fuzzy En of the system according to a and b. Fig. 11 is a bifurcation diagram corresponding to each two pairs, and fig. 12 is a Lyapunov exponential spectrum corresponding to each two pairs, the number of periodic windows is reduced, and the chaotic state is enhanced. After external disturbance, the performance of the sinusoidal chaotic system is improved when a <1 and a are near a half integer, the chaotic range is enlarged, and the complexity is improved.
(2) Characteristics of internal disturbance model
FIG. 13 is a diagram of equilibrium point analysis for an internal perturbation system based on sinusoidal chaotic mapping, where the existence of the equilibrium point of the system is equivalent to y1X and
the existence of an intersection. The internal perturbed system of the sinusoidal chaotic map has infinite number of equilibrium points.
Fig. 14 is an attractor phase diagram for an internal perturbation system with a-3.5 and b-0.6, with clear boundaries and high density near zero. FIG. 15 is a fuzzy graph of complexity of fuzzy En of the system according to a and b. Fig. 16 is a bifurcation diagram corresponding to each other, fig. 17 is a Lyapunov exponent spectrum corresponding to each other, the number of low-complexity regions in a parameter plane is reduced by the superimposed disturbance, the number of periodic windows is reduced, and the chaos performance is enhanced.
(3) Characteristics of the composite disturbance model
FIG. 18 is a diagram of equilibrium point analysis of a sinusoidal chaotic mapping complex perturbation system, equating the existence of system equilibrium points to y1X and
the existence of an intersection. The complex perturbation system of the sinusoidal chaotic map has infinite multi-balance points.
Fig. 19 is an attractor phase diagram of an internal perturbation system with a being 3.5 and b being 0.6, the phase points are more densely distributed compared with external perturbation, and the key space is larger compared with internal perturbation. FIG. 20 is a fuzzy graph of complexity of fuzzy En of the system according to a and b. Fig. 21 is a bifurcation diagram corresponding to each other, and fig. 22 is a Lyapunov exponential spectrum corresponding to each other, and compared with the former two cases, the composite disturbance has some local periodic states, but the overall complexity is at the top of the three disturbance models.
(4) Complexity comparison with other discrete systems
And (3) representing the complexity of the system by using a fuzzy En algorithm, and comparing the three disturbed systems with other discrete mappings. Set m to 2, r to 0.15, N to 2000, and initial value to 0.1. FIG. 23(a) shows that the complexity of the external perturbation model has a slow rising trend when a ∈ [0.1,1 ]. The complexity of the internal disturbance model rises faster. The complexity of the composite disturbance model is reduced first and then increased. Although the complexity of the three perturbation systems is different, the performance of the three perturbation systems is obviously superior to that of the original Sine mapping, 2D-Logistic mapping, Henon mapping and 2D-SLMM mapping. FIG. 23(b) shows that the complex perturbation model SBPM has the highest complexity among SEPM, SIPM, SBPM, Sine mapping, Logistic mapping, 2D-SIMM mapping, and 2D-CMC mapping when a > 1. In addition, the complexity of the disturbed one-dimensional mapping SBPM and SIPM is higher than that of the 2D-SIMM mapping and the 2D-CMC mapping. Although the performance of the SEPM is inferior to that of the other two perturbation models when a >1, the overall complexity is still obviously superior to that of the Sine mapping and the Logistic mapping of the same dimension. Therefore, the system performance can be effectively improved by introducing disturbance on the premise of not changing the system dimension.
Fig. 24 shows a method for generating a pseudorandom sequence according to the present invention: the invention also provides a pseudo-random sequence generation method comprising the chaotization method based on the disturbance, which specifically comprises the following steps:
A. for the existing chaotic system, carrying out chaotization according to the chaotization method based on disturbance to generate a chaotic sequence;
B. converting the chaotic sequence into a 01 sequence by adopting a binary quantitative optimization algorithm;
in particular, the quantization function is
Obtaining a pseudorandom sequence with the length of 8N after N iterations;
C. performing performance evaluation on the generated pseudo-random sequence through NIST test, thereby generating a final pseudo-random sequence; in specific implementation, when the NIST test is passed, the finally generated pseudo-random sequence is directly obtained; and when the NIST test is not passed, adjusting parameters of the algorithm (such as parameters in a chaotization method based on disturbance, parameters of a quantization function and the like), and repeating the steps A to C until the NIST test is passed to obtain a finally generated pseudo-random sequence.
The following describes, in conjunction with specific embodiments, the application of the pseudo-random sequence generator based on the chaos method of perturbation further:
selecting an external disturbance model of sine chaos mapping to generate a chaos sequence, and calculating the precision d to be 2-32The environment of (2) for analysis. The NIST SP800-22 test contains 15 test indexes, and each index has two judgment bases of passing rate and P-value. Significance level α is 0.01, test sequence set number β is 100, sequence length is 106. If the test sequence satisfies two conditions: 1) the passing rates of the test results are all within the confidence interval (
Internal; 2) if the P-value is greater than 0.0001, the sequence is considered to pass the NIST test. The test results are shown in table 2, and it can be seen that the pseudo-random sequence generated by the external perturbation model of the sinusoidal chaotic map successfully passes all NIST tests.
TABLE 2 NIST test results
| Test index | Number of times | P-values | Passing rate | Test results |
| |
1 | 0.383827 | 0.96 | By passing |
| |
1 | 0.616305 | 0.99 | By passing |
| Cumulative Sums* | 2 | 0.129620 | 0.96 | By passing |
| |
1 | 0.574903 | 0.99 | By passing |
| |
1 | 0.574903 | 0.99 | By passing |
| |
1 | 0.494392 | 0.98 | By passing |
| |
1 | 0.657933 | 0.98 | By passing |
| NonOverlapping Template* | 148 | 0.020548 | 0.96 | By passing |
| |
1 | 0.055361 | 0.97 | By passing |
| |
1 | 0.739918 | 0.98 | By passing |
| |
1 | 0.000233 | 0.97 | By passing |
| Random Excursions* | 8 | 0.275709 | 1 | By passing |
| Random Excursions Variant* | 18 | 0.414525 | 1 | By passing |
| Serial* | 2 | 0.012650 | 0.97 | By passing |
| |
1 | 0.350485 | 0.99 | By passing |
Wherein, the test in the table comprises a plurality of tests, and the worst results of P-values and passing rates in the test results are listed.
Claims (7)
1. A chaotization method based on disturbance comprises the following steps:
s1, acquiring a seed system to be chaotic;
s2, selecting a disturbance function;
s3, adopting the disturbance function selected in the step S2, and adopting external disturbance chaotization, internal disturbance chaotization and composite disturbance chaotization for the seed system selected in the step S1;
and S4, evaluating the external disturbance chaotization result, the internal disturbance chaotization result and the composite disturbance chaotization result obtained in the step S3, and selecting an optimal scheme as a final chaotization result.
2. The perturbation-based chaos method according to claim 1, wherein the perturbation function of step S2 specifically comprises the following functions: g (x) x, g (x) ex、g(x)＝sinx、g(x)＝cosx、g(x)＝arctanx、
And
3. the perturbation-based chaos method according to claim 2, wherein the step S3 is configured to perform the chaos based on the external perturbation specifically by using the following equation:
xn+1＝af(xn)-b(g1(xn)+g2(xn)+…+gm(xn))
in the formula xn+1Is the result after n +1 iterations; f (x)n) Mapping f (x) the value of the seed after n iterations; gi(xn) For a selected perturbation function gi(x) The value after n iterations, i is a positive integer and takes the value of 1, 2.. m; a is the control parameter of f (x); b is a control parameter of the disturbance function, and b is equal to 0,1]。
4. The chaos method based on disturbance according to claim 3, wherein the chaos method based on internal disturbance in step S3 specifically adopts the following formula to perform chaos method based on internal disturbance:
xn+1＝a(f(xn-b(g1(xn)+g2(xn)+…+gm(xn)))
in the formula xn+1Is the result after n +1 iterations; gi(xn) For a selected perturbation function gi(x) The value after n iterations, i is a positive integer and takes the value of 1, 2.. m; a is the control parameter of f (x); b is a control parameter of the disturbance function, and b is equal to 0,1]。
5. The perturbation-based chaos method according to claim 3, wherein the complex perturbation chaos in step S3 is specifically performed by using the following equation:
xn+1＝a(f(xn-b(g11(xn)+g12(xn)+…))-b(g21(xn)+g22(xn)+…)
in the formula xn+1Is the result after n +1 iterations; g1i(xn) For a selected internal perturbation function g1i(x) Value after n iterations, g2i(xn) For a selected external disturbance function g2i(x) After n iterations, i is a positive integer, and the value of i is 1, 2.. a, a is a control parameter of f (x); b is a control parameter of the disturbance function, and b is equal to 0,1]。
6. The perturbation-based chaos method according to any one of claims 2-5, wherein the step S4 evaluates the external perturbation chaos result, the internal perturbation chaos result and the composite perturbation chaos result obtained in the step S3, selects an optimal scheme as a final chaos result, specifically calculates the complexity of the external perturbation chaos result, the complexity of the internal perturbation chaos result and the complexity of the composite perturbation chaos result obtained in the step S3, and selects a scheme with the highest complexity as the final chaos result.
7. A pseudo-random sequence generation method comprising the perturbation-based chaos method according to any one of claims 1-6, comprising the following steps:
A. for the existing chaotic system, carrying out chaotization according to the chaotization method based on disturbance to generate a chaotic sequence;
B. converting the chaotic sequence into a 01 sequence by adopting a binary quantitative optimization algorithm;
C. the generated pseudo-random sequence is subjected to performance evaluation through a NIST test, thereby generating a final pseudo-random sequence.
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