CN113190865B - Coupling chaotic system and application thereof - Google Patents
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Abstract
The invention discloses a coupling chaotic system and application thereof, belonging to the field of trustThe field of information security, the system state variable X in the (k+1) th iteration k + 1 The method comprises the following steps: x is X k + 1 ＝Q k ×(f(X k )+Δ k ) mod 1; wherein f () is a function sequence composed of mapping functions of one or more discrete chaotic systems; q (Q) k ＝diag(V k ) The method comprises the steps of carrying out a first treatment on the surface of the X and V are n-dimensional vectors, respectively represent system state variables and system parameters of the coupled chaotic system, and subscripts represent iteration times; delta k Constructing n-dimensional coupling terms based on system state variables in the kth iteration; "mod 1" represents a decimal operation; further, the coupling item dynamically changes along with the iteration times, and the coupling item also carries information of a time-lag state variable; the system parameters are time-varying parameters. The invention effectively reduces the realization difficulty and the resource expenditure while meeting the security requirement of cryptography, and further improves the system security by realizing the hyperchaotic digital system with variable structure, variable parameters and variable time delay.
Description
Technical Field
The invention belongs to the field of information security, and particularly relates to a coupling chaotic system and application thereof.
Background
The chaotic cipher technology is one new kind of cipher technology. Typical characteristics of the chaotic system such as sensitivity of initial conditions, random-like behavior, topological transfer and mixing characteristics and characteristics of the encryption system have good correspondence, for example, sensitivity of chaos to parameters corresponds to sensitivity of the encryption system to keys, and diffusion and confusion characteristics of chaos topological transfer and mixing characteristics correspond to encryption, so that the chaotic system is very suitable for encryption.
The security of the chaotic cipher mainly depends on the chaotic system used, and the chaotic system adopted in the practical encryption application has continuous and discrete, high-dimensional and low-dimensional partitions. The high-dimensional chaotic system has good chaotic performance, can better meet the safety requirement when being applied to cryptography, but is a continuous time system in most cases, and increases the realization difficulty of the system and the cost of resources in practical application.
The low-dimensional discrete chaotic mapping system has the advantages of simple structure, easy realization of software and hardware, high running speed and the like, but also has the problems of small parameter range, small state variable, simple structure and the like, so that the low-dimensional discrete chaotic mapping system is easy to be attacked by passwords such as phase space reconstruction, parameter attack, statistical attack and the like when being used in cryptography, and has great potential safety hazard.
Generally, how to construct a discrete chaotic system with better chaotic performance and high safety, so that the discrete chaotic system can be applied to cryptography, and therefore, the realization difficulty and the resource cost are reduced while the safety requirement of the cryptography is met, and the discrete chaotic system is a problem to be solved.
Disclosure of Invention
Aiming at the defects and improvement demands of the prior art, the invention provides a coupling chaotic system and application thereof, and aims to provide a discrete chaotic system with good chaotic performance and high safety, which can reduce the realization difficulty and the resource expense while meeting the safety demands of cryptography.
To achieve the above object, according to one aspect of the present invention, there is provided a coupled chaotic system, which has a system state variable X in the (k+1) th iteration k+1 The method comprises the following steps:
X k+1 ＝Q k ×(f(X k )+Δ k )mod 1；
wherein f () is a function sequence composed of mapping functions of one or more discrete chaotic systems; q (Q) k ＝diag(V k ) The method comprises the steps of carrying out a first treatment on the surface of the X and V are n-dimensional vectors, respectively represent system state variables and system parameters of the coupled chaotic system, and subscripts represent iteration times; delta k Constructing n-dimensional coupling terms based on system state variables in the kth iteration; "mod 1" means a decimal operationAnd (3) doing so.
The coupling chaotic system provided by the invention constructs the coupling item based on one or more original discrete chaotic systems and adds the coupling item to the original discrete chaotic system, the coupling chaotic system finally obtained is a higher-dimensional discrete chaotic system, the performance of the plurality of discrete chaotic systems can be integrated through the coupling item, the complexity of the coupling chaotic system finally obtained is higher, and the chaotic performance of the coupling chaotic system are effectively improved under the conditions of lower realization difficulty and resource expense.
Further, in the iterative process, the coupling term Δ k Is dynamically changed with the number of iterations.
According to the coupling chaotic system provided by the invention, the coupling items dynamically change along with the iteration times, so that a dynamically changing coupling mode is realized, and the chaotic performance of the system can be further improved.
Further, delta k ＝σ 1 (U k-1 )；
Wherein u= (U) 0 ,u 1 ,...,u 2n-1 ) T The vector is a 2 n-dimensional vector and is used for storing a system state variable, and subscripts represent iteration times; sigma (sigma) 1 The operator is defined as follows:
Further, the coupling term Δ constructed in the kth iteration k But also carries information of the time-lapse state variables.
The chaotic coupling system provided by the invention has the advantages that the coupling items comprise the system state variables and the time-lag state variables, so that the coupling chaotic system constructed by the invention is a time-varying time-lag coupling system.
In some of the alternative embodiments of the present invention,
wherein m is a positive integer;
U 1 ＝(u 0 ,u 1 ,...,u n-1 ) T represents the first half of U, U 2 ＝(u n ,u n +1 ,...,u 2n-1 ) T The latter half of U is represented, and the subscript indicates the number of iterations; sigma (sigma) 3 The operator is used for disturbing the sequence of elements in the 2 n-dimensional vector to obtain a new 2 n-dimensional vector.
In some of the alternative embodiments of the present invention,
wherein sigma 3 The operator is used for disturbing the sequence of elements in the 2 n-dimensional vector to obtain a new 2 n-dimensional vector.
The invention constructs the vector U according to the two modes, so that the vector U can dynamically change along with the iteration number k, and the vector U stores the system state variable X of the current iteration k Also included is the time lag of the system state variable, i.e. the system state variable X of the last iteration k-1 The coupling term constructed based on the vector U is enabled to dynamically change along with the iteration times, and a dynamically changing coupling mode and a dynamically changing time lag are achieved.
Further, the system parameter V of the coupling chaotic system in the kth iteration k Is a time-varying parameter.
According to the coupled chaotic system provided by the invention, the system parameters are time-varying parameters, so that the coupled chaotic system is a discrete chaotic system with time-varying parameters, and the chaotic performance and the safety of the coupled chaotic system are further improved.
Further, V k ＝S(V k-1 )；
The operator S is used for changing the value of each element in the n-dimensional vector within a preset range to obtain a new n-dimensional vector.
The coupling chaotic system provided by the invention has the advantages that the system parameters in each iteration are obtained by changing the values of each element in the system parameters of the previous iteration, so that the randomness of the system parameters can be ensured, and the time-varying (including variable structure, variable parameters and variable time lag) hyperchaotic digital system is realized.
According to another aspect of the invention, the application of the coupled chaotic system in the fields of chaotic ciphers, chaotic digital modulations, chaotic pseudorandom number generators and the like is provided.
In general, through the above technical solutions conceived by the present invention, the following beneficial effects can be obtained:
(1) The coupling chaotic system provided by the invention constructs the coupling item based on one or more original discrete chaotic systems and adds the coupling item to the original discrete chaotic system, the coupling chaotic system finally obtained is a higher-dimensional discrete chaotic system, the performance of the plurality of discrete chaotic systems can be integrated through the coupling item, the complexity of the coupling chaotic system finally obtained is higher, and the chaotic performance and the safety of the coupling chaotic system are effectively improved under the conditions of lower realization difficulty and resource expense. In general, the coupling chaotic system provided by the invention has higher chaotic performance and better safety performance, and can be applied to cryptography, so that the realization difficulty and the resource expense are reduced while certain safety requirements are met.
(2) The coupled chaotic system provided by the invention, wherein the vector U for storing the system state variable dynamically changes along with the iteration number k, and the vector U stores the system state variable X of the current iteration k Also included is the time lag of the system state variable, i.e. the system state variable X of the last iteration k-1 The coupling term constructed based on the vector U is enabled to dynamically change along with the iteration times, a dynamically changing coupling mode and a dynamically changing time lag are achieved, and the system safety is further improved.
(3) The system parameter of the coupled chaotic system provided by the invention is a time-varying parameter, so that the coupled chaotic system is a discrete chaotic system with a time-varying parameter, and the chaotic performance of the coupled chaotic system is further improved; in the preferred scheme, the system parameters in each iteration are obtained by changing the values of each element in the system parameters of the last iteration, so that the randomness of the system parameters can be ensured, a time-varying (including variable structure, variable parameters and variable time lag) hyperchaotic digital system is realized, the system safety is further improved, and the high safety requirement of cryptography is met.
Drawings
FIG. 1 is a bifurcation, phase and frequency map of an original logistic chaotic map; wherein, (a) is a bifurcation diagram, (b) is a phase diagram, and (c) is a frequency distribution diagram;
fig. 2 is a schematic diagram of a topology structure of a coupling chaotic system provided in embodiment 1 of the present invention;
fig. 3 is a bifurcation diagram of each element in a system state variable of the coupled chaotic system provided in embodiment 1 of the present invention; wherein (a) to (c) are three elements x in the system state variables of the coupled chaotic system provided in embodiment 1 of the present invention 0 、x 1 And x 2 Is a bifurcation diagram of (1);
fig. 4 is a phase diagram of each element in a system state variable of the coupled chaotic system provided in embodiment 1 of the present invention; wherein (a) - (c) are three elements x in the system state variables of the coupled chaotic system provided in embodiment 1 of the present invention 0 、x 1 And x 2 Is a phase diagram of (2);
fig. 5 is a frequency distribution histogram of each element in a system state variable of the coupled chaotic system provided in embodiment 1 of the present invention; wherein (a) to (c) are three elements x in the system state variables of the coupled chaotic system provided in embodiment 1 of the present invention 0 、x 1 And x 2 Is a frequency distribution histogram of (1);
fig. 6 is a phase diagram of each element in a system state variable of the coupled chaotic system provided in embodiment 2 of the present invention; wherein (a) to (c) are respectively the coupled chaotic system provided in embodiment 2 of the present inventionThree elements x in a system state variable of (2) 0 、x 1 And x 2 Is a phase diagram of (2);
fig. 7 is a frequency distribution histogram of each element in a system state variable of the coupled chaotic system provided in embodiment 2 of the present invention; wherein (a) to (c) are three elements x in the system state variables of the coupled chaotic system provided in embodiment 2 of the present invention 0 、x 1 And x 2 Is a frequency distribution histogram of (1);
fig. 8 is a schematic diagram of a time-varying hyperchaotic digital system with variable structure, variable parameters and variable time lag according to embodiment 2 of the invention.
Detailed Description
The present invention will be described in further detail with reference to the drawings and examples, in order to make the objects, technical solutions and advantages of the present invention more apparent. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not intended to limit the scope of the invention. In addition, the technical features of the embodiments of the present invention described below may be combined with each other as long as they do not collide with each other.
In the present invention, the terms "first," "second," and the like in the description and in the drawings, if any, are used for distinguishing between similar objects and not necessarily for describing a particular sequential or chronological order.
In order to provide a discrete chaotic system with good chaotic performance and high safety, and reduce the implementation difficulty and resource expenditure while meeting the safety requirement of cryptography, the invention provides a coupling chaotic system and application thereof, and the whole thought is as follows: based on one or more original discrete chaotic systems, a coupling item is constructed based on a system state variable and added into the original discrete chaotic systems, so that the performances of the plurality of discrete chaotic systems are integrated, the complexity of the finally obtained coupling chaotic system is higher, and the chaotic performance of the coupling chaotic system is effectively improved under lower realization difficulty and resource expense; on the basis, the system adopts a dynamic coupling mode and a dynamic system parameter changing mode to further improve the chaotic performance of the discrete chaotic system, so that the system safety is further improved, and the system can be applied to a high-safety password system.
The coupling chaotic system provided by the invention is actually a chaotic system which can be directly applied to passwords.
For ease of description, the following operators are defined:
wherein delta is 0 ＝u 0 +u n ,Δ 1 ＝u 1 +u n+1 ,...,Δ n-1 ＝u n-1 +u 2n-1 The method comprises the steps of carrying out a first treatment on the surface of the Through sigma 1 An operator maps the vector U of 2n dimensions to the vector delta of n dimensions, each element delta in the vector delta i Comprising two elements U in a vector U i And u i+n ，i＝0,1,2,…n-1；
Wherein y= (Y) 0 ,y 1 ,...,y n-1 ) T ,X＝(x 0 ,x 1 ,...,x n-1 ) T Q=diag (V), f (X) is an n-dimensional discrete chaotic system, v= (μ) 0 ,μ 1 ,...,μ n-1 ) T ,Δ＝(Δ 0 ,Δ 1 ,...,Δ n-1 ) T The method comprises the steps of carrying out a first treatment on the surface of the Through sigma 2 An operator can add n-dimensional coupling items into the original n-dimensional chaotic discrete chaotic system to obtain a higher-dimensional discrete chaotic system;
wherein,,
Y 1 ＝(y 0 ,y 1 ,...,y n-1 ) T ，Y 2 ＝(y n ,y n+1 ,...,y 2n-1 ) T ，U 1 ＝(u 0 ,u 1 ,...,u n-1 ) T ，U 2 ＝(u n ,u n+1 ,...,u 2n-1 ) T ；
by an operator sigma 3 The new 2n vector can be obtained by disturbing the arrangement sequence of the elements in the vector U through element replacement;
the value of each element in the n-dimensional vector V can be changed within a preset range through an S operator, so that a new n-dimensional vector W is obtained.
Based on the defined operator, at the initial value
The coupled chaotic system constructed by the invention can be expressed as an iterative system as follows:
X k+1 ＝σ 2 (X k ；V k ,Δ k )；
wherein X, V and Δ represent system state variables, system parameters, and coupling terms, respectively, and subscripts represent the number of iterations;
the invention can construct a new coupling chaotic system on the basis of any low-dimensional discrete chaotic system, and constructs a three-dimensional coupling chaotic system on the basis of a classical one-dimensional logistic chaotic system in the following embodiments without losing generality.
Before explaining the technical scheme of the invention in detail, the following brief description is made on an original logistic chaotic system:
the mapping equation of the logistic chaotic system is as follows:
x k+1 ＝f(x k )＝μx k (1-x k )；
wherein x represents a system state variable and the subscript represents the number of iterations; μ represents a system parameter;
the bifurcation diagram, the phase diagram and the frequency distribution histogram of the original logistic chaotic system are respectively shown as (a), (b) and (c) in fig. 1; according to the method shown in fig. 1 (a), the system is in a chaotic state when the system parameter mu epsilon (3.569,4), the change range of the system parameter of the original logistic chaotic system is small, according to the method shown in fig. 1 (b), the range of the attractor occupied by the original logistic chaotic system is small, the phase diagram is simple in structure, and according to the method shown in fig. 1 (c), the frequency distribution of the original logistic chaotic system is in non-uniform distribution with low middle and high two sides.
The following are examples.
Example 1:
a coupled chaotic system comprises a system state variable X in the (k+1) th iteration k+1 The method comprises the following steps:
X k+1 ＝Q k ×(f(X k )+Δ k )mod 1；
wherein f () is a function sequence composed of mapping functions of one or more discrete chaotic systems; q (Q) k ＝diag(V k ) The method comprises the steps of carrying out a first treatment on the surface of the X and V are n-dimensional vectors, respectively represent system state variables and system parameters of the coupled chaotic system, and subscripts represent iteration times; delta k Constructing n-dimensional coupling terms based on system state variables in the kth iteration; "mod 1" represents a decimal operation;
in this embodiment, n=3, and accordingly, the system state variable can be expressed as x= (X) 0 ,x 1 ,x 2 ) In different iterations, distinguishing through subscripts; f () represents a logistic chaotic system;
optionally, in this embodiment, the n-dimensional coupling term Δ constructed based on the system state variables in the kth iteration k Each element delta i From system state variables X k-1 Obtained by adding other elements than the i-th element, i.e.
At this time, the coupled chaotic system provided in this embodiment has a fixed topology structure, and the parameters are fixed, as shown in fig. 2, the system state variables in the (k+1) th iteration
The method comprises the following steps:
wherein the superscript indicates the element number.
In this embodiment, the bifurcation diagrams of the elements in the system state variables are shown in (a), (b) and (c) in fig. 3, and according to the results shown in fig. 3, it can be seen that in this embodiment, the system parameters change within (0.5,10), and the system all presents a chaotic state, and comparing fig. 3 and (a) in fig. 1, it can be seen that in this embodiment, after the coupling term is added on the basis of the original Logistic chaotic system, the system parameters have a larger change range, so that the information such as the system parameters can be better covered, and the application flexibility is larger.
In this embodiment, the phase diagrams of the elements in the system state variables are shown in (a), (b) and (c) in fig. 4, respectively, and according to the results shown in fig. 4, it can be seen that in this embodiment, the phase space tracks of the elements in the system state variables are distributed over the whole interval; as can be seen from comparison of fig. 4 and (b) in fig. 1, the chaotic range is effectively enlarged, the phase diagram structure is more complex, and the method not only can effectively resist attack of methods such as phase space reconstruction, but also has good confusion characteristics.
In this embodiment, the frequency distribution histograms of the elements in the system state variable are shown in (a), (b) and (c) in fig. 5, respectively, and it can be seen from the results shown in fig. 5 that in this embodiment, the frequencies of the elements in the system state variable are uniformly distributed; as can be seen from comparing fig. 5 and fig. 1 (c), the present embodiment can improve the distribution characteristics of the original chaotic system, so that the output of the system is more random.
In this embodiment, the dynamics performance of each element in the system state variables is shown in table 1; from the results shown in table 1, it can be seen that the complexity of the system is high in this embodiment.
TABLE 1 kinetic index for each System State variable element in example 1
| x 0 | x 1 | x 2 | |
| Approximate entropy | 0.6965 | 0.6946 | 0.6964 |
| Permutation entropy | 0.9997 | 0.9997 | 0.9997 |
The coupling chaotic system provided by the embodiment constructs the coupling item based on the system state variable and adds the coupling item to the original discrete chaotic system, the finally obtained coupling chaotic system is a higher-dimensional discrete chaotic system, the performance of a plurality of discrete chaotic systems can be integrated through the coupling item, the complexity of the finally obtained coupling chaotic system is higher, and the chaotic performance and the safety performance of the coupling chaotic system are effectively improved under the conditions of lower implementation difficulty and resource expense. In general, the coupling chaotic system provided by the invention has higher chaotic performance and better safety performance, and can be applied to cryptography, so that the realization difficulty and the resource expense are reduced while the safety requirement of the cryptography is met.
Example 2:
a coupled chaotic system comprises a system state variable X in the (k+1) th iteration k+1 The method comprises the following steps:
X k+1 ＝Q k ×(f(X k )+Δ k )mod 1；
wherein f () is a function sequence composed of mapping functions of one or more discrete chaotic systems; q (Q) k ＝diag(V k ) The method comprises the steps of carrying out a first treatment on the surface of the X and V are n-dimensional vectors, respectively represent system state variables and system parameters of the coupled chaotic system, and subscripts represent iteration times; delta k Constructing n-dimensional coupling terms based on system state variables in the kth iteration; "mod 1" represents a decimal operation;
in this embodiment, n=3, and accordingly, the system state variable can be expressed as x= (X) 0 ，x 1 ，x 2 ) In different iterations, distinguishing through subscripts; f () represents a logistic chaotic system;
in the coupled chaotic system provided by the embodiment, in the iterative process, the coupling term delta k The dynamic change occurs along with the iteration times, so that a coupling mode of dynamic change can be realized, and the chaotic performance and the safety of the system are further improved;
in this embodiment, delta k ＝σ 1 (U k-1 )；
Wherein u= (U) 0 ,u 1 ,...,u 2n-1 ) T The vector is a 2 n-dimensional vector and is used for storing a system state variable, and subscripts represent iteration times;
in order to make the coupling term have better dynamic variation characteristics, in this embodiment, the vector U dynamically varies with the number of iterations k, and the vector U stores the system state variable X of the current iteration k The time lag of the system state variable is also included, and the specific change modes are as follows:
wherein m is a positive integer;
U 1 ＝(u 0 ,u 1 ,...,u n-1 ) T represents the first half of U, U 2 ＝(u n ,u n +1 ,...,u 2n-1 ) T The latter half of U is represented, and the subscript indicates the number of iterations;
in order to further improve the chaotic performance of the system, in this embodiment, the system parameter V of the chaotic system in the kth iteration is coupled k Is a time-varying parameter, in particular V k ＝S(V k-1 )；
Wherein the parameters are controlled
Is of the type [3.5, 4]]The implementation mode of the variable random number and the definition operator S is as follows: parameter interval [3.5, 4]]Divided into 256 parts, i.e.)>
Wherein r is k =0, 1,. -%, 255; generating 3 belonging to [0,255]Is assigned to +.>
R in (2) k . Coupling term (delta) 0 ,Δ 1 ,Δ 2 ) Also dynamically changing, an integer α εΩ is defined, where Ω= {0,1,..2 n-1}, and α is a random number satisfying statistically uniform distribution, likewise yielding [0,2n-1 ]]Assigning the random number in the memory to alpha; operator sigma 3 The arrangement order of elements in the 2 n-dimensional vector is disturbed in such a way that
j=0, 1,..2 n-1, wherein +.>
It should be noted thatHere, however, the operator sigma 3 The specific manner in which the order of elements in a 2 n-dimensional vector is disturbed is merely illustrative and should not be construed as limiting the invention in any way, and in other embodiments of the invention other manners of disturbing may be used and will not be listed here.
Finally, the coupled chaotic system provided in the embodiment is a time-varying coupled chaotic system, and the system state variable of the coupled chaotic system in the (k+1) th iteration is the same as the system state variable of the coupled chaotic system in the (k+1) th iteration
Is that
In this embodiment, the phase diagrams of the elements in the system state variables are shown in (a), (b) and (c) in fig. 6, respectively, and according to the results shown in fig. 6, it can be seen that in this embodiment, the phase space tracks of the elements in the system state variables are distributed over the whole interval; as can be seen from comparing fig. 6 and fig. 1 (b), the present embodiment effectively enlarges the chaotic range, and the phase diagram structure is more complex, so that the present embodiment not only can effectively resist attack of methods such as phase space reconstruction, but also has good confusion characteristics.
In this embodiment, the frequency distribution histograms of the elements in the system state variable are shown in (a), (b) and (c) in fig. 7, and according to the results shown in fig. 7, it can be seen that in this embodiment, the frequencies of the elements in the system state variable are uniformly distributed, and have good statistical properties; as can be seen from comparing fig. 7 and fig. 1 (c), the present embodiment can improve the distribution characteristics of the original chaotic system, so that the output of the system is more random.
In this embodiment, the dynamics performance of each element in the system state variables is shown in table 2; as can be seen from the results shown in table 2, in this embodiment, the dynamics and complexity of the system are further improved; as can be seen from comparing the results shown in tables 1 and 2, since the dynamically changing coupling mode is adopted in the present embodiment and the system parameter is a time-varying parameter, the system complexity of the present embodiment is higher than that of the above-described embodiment 1.
TABLE 2 kinetic index for each System State variable element in example 2
| x 0 | x 1 | x 2 | |
| Approximate entropy | 1.8731 | 1.8719 | 1.8717 |
| Permutation entropy | 0.9999 | 0.9999 | 0.9999 |
A time-varying system refers to a system in which one or more parameter values describing the system change over time, so that the overall characteristic also changes over time. Small changes in the chaotic system at critical parameters can result in the system exhibiting completely different dynamics. For existing time-varying time-lag systems, as the time-lag parameter values in the system change over time, the characteristics of the overall system also change over time. Time-varying time-lag systems themselves have complexity, multiple, and overlapping properties. From the point of view of theoretical analysis, in the continuous domain, the time-varying time-lag system is an infinitely-dimensional system, and the characteristic equation is an overrun equation with an infinite number of characteristic roots. Whereas in the discrete domain, the dimension of the time-varying time-lag system grows geometrically with increasing time lags. At present, for the parameter identification problem of the existing time-varying time-lag system, a great amount of blank and difficult problems are not related to and solved. A variable structure system refers to a system in which system structural parameters change over time. For a time-varying structure system, the difficulty is high, and particularly the problem of identifiability, observability, controllability and the like of the inverse problem (time-varying parameter identification problem) exist in theory, so that the problem is avoided by the conventional time-varying structure system parameter identification method.
In general, as shown in fig. 8, the embodiment realizes a dynamically-changing coupling mode and a dynamically-changing time lag, ensures the randomness of system parameters, realizes a time-varying hyperchaotic digital system with variable structure, variable parameters and variable time lag, has good chaotic performance, and can effectively improve the system safety when applied to the field of cryptography.
Example 3:
a coupled chaotic system, the present embodiment is similar to embodiment 2 described above, except that, in the present embodiment,
and operator sigma 3 The mode of disturbing the arrangement sequence of elements in the 2 n-dimensional vector is correspondingly changed so as to ensure that the coupling structure of the coupling chaotic system is irregular.
Example 4:
the application of the coupling chaotic system provided in any one of the embodiments 1 to 3 in the fields of chaotic ciphers, chaotic digital modulations, chaotic pseudorandom number generators and the like;
the specific application mode is similar to that of a conventional chaotic system, for example, a sequence output by the coupled chaotic system, namely a system state variable, can be subjected to some nonlinear transformation to generate a binary random number, and the generated binary random number is used as a sequence password (generally, an exclusive-or operation is performed to realize encryption and decryption).
It will be readily appreciated by those skilled in the art that the foregoing description is merely a preferred embodiment of the invention and is not intended to limit the invention, but any modifications, equivalents, improvements or alternatives falling within the spirit and principles of the invention are intended to be included within the scope of the invention.
Claims (5)
1. The chaotic encryption and decryption method based on the coupled chaotic system is characterized by comprising the following steps of: the sequence output by the coupled chaotic system is subjected to nonlinear transformation to generate binary random numbers, the generated binary random numbers are used as sequence passwords, and encryption and decryption are realized through exclusive-or operation; system state variable X of the coupled chaotic system in k+1th iteration k+1 The method comprises the following steps:
X k+1 ＝Q k ×(f(X k )+Δ k )mod 1；
wherein f () is a function sequence composed of mapping functions of one or more discrete chaotic systems; q (Q) k ＝diag(V k ) The method comprises the steps of carrying out a first treatment on the surface of the X and V are n-dimensional vectors, respectively represent system state variables and system parameters of the coupled chaotic system, and subscripts represent iteration times; delta k For the n-dimensional coupling term, delta, constructed based on the system state variables in the kth iteration k ＝σ 1 (U k-1 ) The method comprises the steps of carrying out a first treatment on the surface of the "mod 1" represents a decimal operation;
U＝(u 0 ,u 1 ,...,u 2n-1 ) T the vector is a 2 n-dimensional vector and is used for storing a system state variable, and subscripts represent iteration times;
or (F)>
Wherein m is a positive integer;
U 1 ＝(u 0 ,u 1 ,...,u n-1 ) T representing the front of UHalf, U 2 ＝(u n ,u n +1 ,...,u 2n-1 ) T The latter half of U is represented, and the subscript indicates the number of iterations; sigma (sigma) 3 The operator is used for disturbing the sequence of elements in the 2 n-dimensional vector to obtain a new 2 n-dimensional vector;
σ 1 the operator is defined as follows:
2. The chaotic encryption and decryption method based on the coupled chaotic system of claim 1, wherein in an iterative process, the coupling term delta k Dynamic changes occur with iteration number.
3. The chaotic encryption and decryption method based on the coupled chaotic system as claimed in claim 2, wherein the coupling term delta constructed in the kth iteration k But also carries information of the time-lapse state variables.
4. The chaotic encryption and decryption method based on the coupled chaotic system as claimed in any one of claims 1 to 3, wherein a system parameter V of the coupled chaotic system in a kth iteration k Is a time-varying parameter.
5. The chaotic encryption and decryption method based on the coupled chaotic system of claim 4, wherein,
V k ＝S(V k-1 )；
the operator S is used for changing the value of each element in the n-dimensional vector within a preset range to obtain a new n-dimensional vector.
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