A variable-structure coupled chaotic system and application thereof

Abstract

This invention discloses a structurally variable coupled chaotic system and its applications, belonging to the field of information security technology. The coupled chaotic system is formed by the coupling of multiple discrete chaotic systems. During system evolution, the coupling structure between different subsystems is dynamically changed in real time by manipulating the coupling matrix. The coupled system exhibits complex dynamic behavior and high nonlinearity, and the dynamic changes in structure cause continuous abrupt changes in the system's dynamics, resulting in non-stationary chaotic sequences. This system is resistant to attacks from statistical analysis, algebraic analysis, and other methods, exhibiting extremely high security. Furthermore, this system is applicable to various chaotic systems and complex chaotic networks of arbitrary scale. It can also generate complex dynamics based on simple discrete chaotic systems. Under the same complexity and security requirements, controlling structural changes results in lower overhead. It can be used to design high-performance chaotic cryptographic systems and as a component in network security applications.

Description

Technical Field

[0001] This invention belongs to the field of information security technology, and more specifically, relates to a structurally variable coupled chaotic system and its applications. Background Technology

[0002] Chaotic cryptography is a novel cryptographic technique characterized by its simplicity, efficiency, and security. It represents an important application of chaos theory and its related technologies.

[0003] Existing chaotic systems have simple structures, lack variability, and are vulnerable to statistical and mathematical analysis. Currently used parameter variation methods have a slow impact on system dynamics and are susceptible to time-series-based parameter identification attacks. Furthermore, when chaotic systems are implemented on devices with finite precision, they are prone to degradation, exhibiting characteristics such as short periods, low linear complexity, and strong correlations, leading to inherent security vulnerabilities. Common methods for overcoming chaotic degradation, to ensure system security, typically significantly increase the implementation difficulty and resource consumption of the chaotic system.

[0004] Therefore, how to use relatively simple chaos functions to generate chaotic systems with complex dynamic behaviors under limited resources in order to ensure the security of chaotic encryption is a problem that needs to be solved. Summary of the Invention

[0005] In view of the above-mentioned defects or improvement needs of the existing technology, the present invention provides a structurally variable coupled chaotic system and its application. The purpose is to construct a secure chaotic system of arbitrary scale, while meeting the security requirements of cryptography and reducing the implementation difficulty and resource consumption.

[0006] To achieve the above objectives, in a first aspect, the present invention provides a coupled chaotic system with variable structure, comprising: a coupling module and n discrete chaotic systems; n≥2;

[0007] The coupling module is used to start the first iteration after receiving the start command, take the input initial system state variables as the system state variables under the first iteration of the coupled chaotic system, and input the n state values ​​one by one into the n discrete chaotic systems.

[0008] Discrete chaotic systems are used to generate corresponding discrete chaotic values ​​by performing chaotic mapping after receiving state value input, and then output them to the coupling module.

[0009] The coupling module is also used in each iteration to perform coupled chaos mapping on each discrete chaotic value after obtaining the discrete chaotic values ​​generated by n discrete chaotic systems, so as to obtain a set of coupled chaotic values ​​for the current iteration and output them. At the same time, the set of coupled chaotic values ​​is used as the system state variables of the coupled chaotic system in the next iteration, and the n state values ​​are input one by one into the n discrete chaotic systems for the next iteration.

[0010] Wherein, the i-th coupled chaotic value obtained by performing coupled chaotic mapping on the i-th discrete chaotic value in the k-th iteration is:

[0011]

[0012] f is the i-th discrete chaotic value in the k-th iteration; i (·) is the chaotic mapping function of the i-th discrete chaotic system; Let A be the i-th state value of the system state variables in the k-th iteration of the coupled chaotic system; the coupling matrix A k This is a randomly generated n×n 0-1 matrix in the k-th iteration; Let be the system state variables of the coupled chaotic system in the kth iteration.

[0013] More preferably, the i-th coupled chaotic value obtained by performing coupled chaotic mapping on the i-th discrete chaotic value in the k-th iteration is:

[0014]

[0015] Where, λ k The coupling coefficient is randomly generated in the k-th iteration.

[0016] More preferably, after obtaining the coupled chaotic value through the coupled chaotic mapping, the coupling module further restricts the obtained coupled chaotic value within the attraction domain of the discrete chaotic system.

[0017] Wherein, the i-th coupled chaotic value is confined within the attraction domain of the discrete chaotic system. Where mod is the modulo operator; the region of attraction of the i-th discrete chaotic system is α. i .

[0018] More preferably, the coupling matrix A k The method for generating any element in M ​​is as follows: randomly select matrix M. k The element in the r-th row and c-th column is used as A k Any element in;

[0019] in, Matrix M kThe 0-1 matrix of size s×t is randomly generated in the k-th iteration. It is obtained by performing matrix self-operation on the initial matrix M0 in the k-th iteration. The initial matrix M0 is a 0-1 matrix of size s×t, which is randomly generated by the coupled chaotic system and then fixed.

[0020] More preferably, the coupling module is also used to stop the iteration after receiving a termination instruction.

[0021] In a second aspect, the present invention provides a coupled chaotic sequence generation and control method, applied to a coupling module in a structurally variable coupled chaotic system provided in the first aspect of the present invention, comprising:

[0022] After receiving the start command, the first iteration begins. The input initial system state variables are used as the system state variables under the first iteration of the coupled chaotic system, and the n state values ​​are input one by one into the n discrete chaotic systems.

[0023] In each iteration, after obtaining the discrete chaotic values ​​generated by n discrete chaotic systems, a coupled chaotic mapping is performed on each discrete chaotic value to obtain a set of coupled chaotic values ​​for the current iteration and output them. At the same time, this set of coupled chaotic values ​​is used as the system state variables for the next iteration of the coupled chaotic system, and the n state values ​​are input one by one into the n discrete chaotic systems for the next iteration.

[0024] Among them, after receiving the state value input, the discrete chaotic system performs chaotic mapping to generate the corresponding discrete chaotic value;

[0025] The i-th coupled chaotic value obtained by performing coupled chaotic mapping on the i-th discrete chaotic value in the k-th iteration is:

[0026]

[0027] f is the i-th discrete chaotic value in the k-th iteration; i (·) is the chaotic mapping function of the i-th discrete chaotic system; Let A be the i-th state value of the system state variables in the k-th iteration of the coupled chaotic system; the coupling matrix A k This is a randomly generated n×n 0-1 matrix in the k-th iteration; Let be the system state variables of the coupled chaotic system in the kth iteration.

[0028] Thirdly, the present invention also provides a computer-readable storage medium comprising a stored computer program, wherein the computer program, when executed by a processor, controls the device where the storage medium is located to execute the coupled chaotic sequence generation and control method provided in the second aspect of the present invention.

[0029] In summary, the above-described technical solutions conceived in this invention can achieve the following beneficial effects:

[0030] 1. This invention provides a structurally variable coupled chaotic system, formed by the coupling of multiple discrete chaotic systems. During system evolution, the coupling structure between different sub-discrete chaotic systems is dynamically changed in real time by manipulating the coupling matrix. The coupled system exhibits complex dynamic behavior and high nonlinearity, and the dynamic changes in structure cause the system's dynamics to undergo continuous abrupt changes, thereby generating non-stationary chaotic sequences. This system is resistant to attacks from statistical analysis, algebraic analysis, etc., and has extremely high security. Furthermore, it can generate complex dynamics based on simple discrete chaotic systems, and can control structural changes with lower overhead while maintaining the same complexity and security requirements.

[0031] 2. The variable-structure coupled chaotic system provided by this invention, based on two or more original discrete chaotic systems, constructs a continuously changing system structure based on the system state and adds it to the original discrete chaotic systems, ultimately obtaining a higher-dimensional variable-structure chaotic system. Through structural transformation, the performance of multiple chaotic systems can be integrated, and the chaoticity of the system is greatly increased, resulting in a more complex and stronger chaotic system. This effectively improves the chaotic performance and security of the chaotic system with relatively low implementation difficulty and resource overhead. Overall, the variable-structure chaotic system provided by this invention has high chaotic performance and good security performance, and can be applied to cryptography, meeting certain security requirements while reducing implementation difficulty and resource overhead.

[0032] 3. The structurally variable coupled chaotic system provided by this invention controls the system structure in the current iteration through the system state of the previous iteration. The system structure changes with the change of the system's own state, which can ensure the randomness of the system structure and realize a variable structure chaotic system with high chaotic performance and security.

[0033] 4. The chaotic system structure provided by this invention changes continuously with the system's own state during the iteration process. This invention proposes to control the system based on a matrix M, which is a 0-1 matrix with any power. The transformation of M is controlled by the state of the system in the previous iteration. This structural change makes the system structure more complex and greatly improves the chaotic performance of the system. Attached Figure Description

[0034] Figure 1 This is a structurally variable chaotic mapping state sequence diagram provided by the first aspect of the present invention;

[0035] Figure 2 The present invention provides a sequence diagram, phase diagram, autocorrelation function diagram, and frequency distribution histogram of the original Logistic chaotic mapping; wherein, (a) is a state sequence diagram of the original Logistic chaotic mapping provided by the present invention; (b) is a phase diagram of the original Logistic chaotic mapping provided by the present invention; (c) is an autocorrelation function diagram of the state sequence of the original Logistic chaotic mapping provided by the present invention; and (d) is a frequency distribution diagram of the output sequence of the original Logistic chaotic mapping provided by the present invention.

[0036] Figure 3 This is a schematic diagram of the topology of the variable structure chaotic system provided in Example 1 of the present invention;

[0037] Figure 4 The following are the sequence diagram, phase diagram, autocorrelation function diagram, and frequency distribution histogram of the structurally variable chaotic mapping provided in Example 1 of the present invention; wherein, (a) is the state sequence diagram of the structurally variable chaotic mapping provided in Example 1 of the present invention; (b) is the phase diagram of the structurally variable chaotic mapping provided in Example 1 of the present invention; (c) is the autocorrelation function diagram of the state sequence of the structurally variable chaotic mapping provided in Example 1 of the present invention; and (d) is the frequency distribution diagram of the output sequence of the structurally variable chaotic mapping provided in Example 1 of the present invention.

[0038] Figure 5 This is a schematic diagram of the topology of the variable structure chaotic system provided in Example 1 of the present invention;

[0039] Figure 6 Examples of the present invention are: a sequence diagram, a phase diagram, an autocorrelation function diagram, and a frequency distribution histogram of a structure-variable chaotic mapping provided in Example 2 of the present invention; wherein, (a) is a state sequence diagram of a structure-variable chaotic mapping provided in Example 2 of the present invention; (b) is a phase diagram of a structure-variable chaotic mapping provided in Example 2 of the present invention; (c) is an autocorrelation function diagram of the state sequence of a structure-variable chaotic mapping provided in Example 2 of the present invention; and (d) is a frequency distribution diagram of the output sequence of a structure-variable chaotic mapping provided in Example 2 of the present invention.

[0040] Figure 7 A topological diagram of a variable-structure chaotic system provided in Example 1 of this invention;

[0041] Figure 8Examples 3 of this invention provide a sequence diagram, phase diagram, autocorrelation function diagram, and frequency distribution histogram of a structurally variable chaotic mapping; wherein, (a) is a state sequence diagram of a structurally variable chaotic mapping provided in Example 3 of this invention; (b) is a phase diagram of a structurally variable chaotic mapping provided in Example 3 of this invention; (c) is an autocorrelation function diagram of the state sequence of a structurally variable chaotic mapping provided in Example 3 of this invention; and (d) is a frequency distribution diagram of the output sequence of a structurally variable chaotic mapping provided in Example 1 of this invention. Detailed Implementation

[0042] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0043] To provide a chaotic system with good chaotic performance and high security, while meeting cryptographic security requirements and reducing implementation difficulty and resource overhead, this invention provides a variable-structure chaotic system and its application. The overall idea is as follows: based on two or more original discrete chaotic systems, a variable-structure chaotic system structure is constructed to integrate the performance of multiple discrete chaotic systems. The structure of the chaotic system is continuously changed, resulting in a more complex chaotic system. This effectively improves the chaotic performance of the chaotic system with lower implementation difficulty and resource overhead. Furthermore, the 0-1 matrix of the system structure is determined by the system's state at the previous iteration, making the dynamic changes of the system structure determined by the system's own state, further improving the chaotic performance of the discrete chaotic system and thus further enhancing system security, enabling its application in high-security cryptographic systems.

[0044] The variable structure chaotic system provided by this invention is actually a chaotic system that can be directly applied to cryptography.

[0045] Specifically, in the first aspect, such as Figure 1 As shown, this invention provides a structurally variable coupled chaotic system, comprising: a coupling module and n discrete chaotic systems; n≥2;

[0046] The coupling module is used to start the first iteration after receiving the start command, take the input initial system state variables as the system state variables under the first iteration of the coupled chaotic system, and input the n state values ​​one by one into the n discrete chaotic systems.

[0047] Discrete chaotic systems are used to generate corresponding discrete chaotic values ​​by performing chaotic mapping after receiving state value input, and then output them to the coupling module.

[0048] The coupling module is also used in each iteration to perform coupled chaos mapping on each discrete chaotic value after obtaining the discrete chaotic values ​​generated by n discrete chaotic systems, so as to obtain a set of coupled chaotic values ​​for the current iteration and output them. At the same time, the set of coupled chaotic values ​​is used as the system state variables of the coupled chaotic system in the next iteration, and the n state values ​​are input one by one into the n discrete chaotic systems for the next iteration.

[0049] Wherein, the i-th coupled chaotic value obtained by performing coupled chaotic mapping on the i-th discrete chaotic value in the k-th iteration is:

[0050]

[0051] f is the i-th discrete chaotic value in the k-th iteration; i (·) is the chaotic mapping function of the i-th discrete chaotic system; Let A be the i-th state value of the system state variables in the k-th iteration of the coupled chaotic system; the coupling matrix A k This is a randomly generated n×n 0-1 matrix in the k-th iteration; Let be the system state variables of the coupled chaotic system in the kth iteration.

[0052] The present invention constructs a variable structure chaotic system in the manner described above. The structure of this chaotic system changes with the state of the system itself. Therefore, the chaotic system constructed by the present invention is a variable structure chaotic system based on its own state, and its chaotic performance and security are further improved.

[0053] It should be noted that the discrete chaotic systems can be the same or different. Specifically, discrete chaotic systems can be Logistic chaotic systems, Tent chaotic systems, Henon chaotic systems, piecewise linear mappings, etc.

[0054] Furthermore, to ensure that the degree of mutual influence between any two chaotic systems is random, preferably, the i-th coupled chaotic value obtained by performing a coupled chaotic mapping on the i-th discrete chaotic value in the k-th iteration is:

[0055]

[0056] Where, λ k Let be the coupling coefficient randomly generated in the k-th iteration. Specifically, the coupling coefficient of the coupled chaotic system is related to the iteration number k and the coupling coefficient λ between every two discrete chaotic systems, denoted as λ. k =I(λ,k), indicating that the coupling coefficient changes with the iteration process.

[0057] Furthermore, in order to prevent the values ​​generated by the coupled chaotic system from increasing indefinitely during the iteration process, preferably, after obtaining the coupled chaotic value through the coupled chaotic mapping, the coupling module also restricts the obtained coupled chaotic value to the attraction domain of the discrete chaotic system.

[0058] Wherein, the i-th coupled chaotic value is confined within the attraction domain of the discrete chaotic system. Where mod is the modulo operator; the region of attraction of the i-th discrete chaotic system is α. i .

[0059] It should be noted that a coupling matrix is ​​randomly generated in each iteration to describe the coupling structure of the chaotic system. One approach is to prepare a matrix set encompassing all n×n 0-1 matrices, and then randomly select a coupling matrix from this set in each iteration. However, this method requires significant memory to store the matrix set. To address this issue, preferably, the coupling matrix A... k The method for generating any element in M ​​is as follows: randomly select matrix M. k The element in the r-th row and c-th column is used as A k any element in; where, Matrix M k The 0-1 matrix of size s×t is randomly generated in the k-th iteration. It is obtained by performing matrix self-operation on the initial matrix M0 in the k-th iteration. The initial matrix M0 is a 0-1 matrix of size s×t, which is randomly generated by the coupled chaotic system and then fixed.

[0060] More preferably, the coupling module is also used to stop the iteration after receiving a termination instruction.

[0061] Correspondingly, the present invention also provides a coupled chaotic sequence generation and control method, applied to the coupling module in the structurally variable coupled chaotic system provided in the first aspect of the present invention, comprising:

[0062] After receiving the start command, the first iteration begins. The input initial system state variables are used as the system state variables under the first iteration of the coupled chaotic system, and the n state values ​​are input one by one into the n discrete chaotic systems.

[0063] In each iteration, after obtaining the discrete chaotic values ​​generated by n discrete chaotic systems, a coupled chaotic mapping is performed on each discrete chaotic value to obtain a set of coupled chaotic values ​​for the current iteration and output them. At the same time, this set of coupled chaotic values ​​is used as the system state variables for the next iteration of the coupled chaotic system, and the n state values ​​are input one by one into the n discrete chaotic systems for the next iteration.

[0064] Among them, after receiving the state value input, the discrete chaotic system performs chaotic mapping to generate the corresponding discrete chaotic value;

[0065] The i-th coupled chaotic value obtained by performing coupled chaotic mapping on the i-th discrete chaotic value in the k-th iteration is:

[0066]

[0067] f is the i-th discrete chaotic value in the k-th iteration; i (·) is the chaotic mapping function of the i-th discrete chaotic system; Let A be the i-th state value of the system state variables in the k-th iteration of the coupled chaotic system; the coupling matrix A k This is a randomly generated n×n 0-1 matrix in the k-th iteration; Let be the system state variables of the coupled chaotic system in the kth iteration.

[0068] The related technical solutions are the same as those provided in the first aspect of this invention for the structurally variable coupled chaotic system, and will not be described in detail here.

[0069] It should be noted that this invention can construct new variable-structure chaotic systems based on any low-dimensional chaotic system. Without loss of generality, unless otherwise specified, the following embodiments are all based on the classic one-dimensional Logistic chaotic system. It should be understood that the descriptions of these embodiments are for the purpose of aiding understanding of this invention, but do not constitute a limitation thereof.

[0070] Before explaining the technical solution of this invention in detail, the original Logistic chaotic system will be briefly introduced as follows:

[0071] The mapping equation for a Logistic chaotic system is as follows:

[0072] χ n+1 =μχ n (1-χ n )

[0073] Where, χ n The subscript represents the system state variable, and the subscript indicates the iteration number; μ represents the system parameter.

[0074] The sequence diagram, phase diagram, autocorrelation function diagram, and frequency distribution histogram of the original Logistic chaotic map are shown below. Figure 2 As shown in (a), (b), (c), and (d) in the figure.

[0075] The following is an example.

[0076] Example 1:

[0077] A 3D Logistic variable structure chaotic system, wherein the i-th state value of the system state variable in the (k+1)-th iteration is... for:

[0078]

[0079] Where f(·) is a mapping function of a chaotic system; λ k A is the coupling coefficient in the k-th iteration of the variable structure chaotic system; k It is an n×n 0-1 matrix randomly generated in the k-th iteration, representing the coupling structure of the chaotic system; The system state variable represents the coupled chaotic system. The superscript indicates different chaotic systems, and the subscript indicates the number of iterations. "mod1" indicates that the state of the chaotic system is restricted to the attraction domain 1.

[0080] In this embodiment, n = 3, and correspondingly, the system state variable in the k-th iteration can be expressed as: Different iterations are distinguished by subscripts.

[0081] f(·) represents a Logistic chaotic system, i.e.:

[0082]

[0083] In this example, each system variable is calculated using a 64-bit length, with 64 decimal places, meaning the data precision is 64 bits. A 3D chaotic system is used for coupling to observe the chaotic performance of a small-scale chaotic system.

[0084] Furthermore, A k Let M represent the structure of the chaotic system during the k-th iteration. k It is an arbitrary s×t matrix (s,t≥n), where the subscripts indicate the iteration number;

[0085] In this example, Where P is a permutation matrix of order non-zero, U is a strictly upper triangular matrix, and each row of (UE) contains exactly one element 1, J r,t Let m be an r×t matrix with all elements equal to 1. When m is odd, t = (n-1) / 2; when m is even, t = n / 2-1 or t = n / 2. It can be rigorously proven that any power of the 0-1 matrix M is still a 0-1 matrix.

[0086] Extensive experiments have shown that matrix P is the most dynamically changing part of matrix M, and the values ​​of matrix M exhibit periodic changes with increasing exponent, with the period being positively correlated with the size of matrix P. Therefore, when selecting matrix M, the dimension of matrix P should be maximized; thus, m is chosen to be an even number, and U is set to a 1×1 matrix. In this embodiment, n = 3, so the number of rows and columns of matrix P is greater than or equal to 3. Here, P is set to a 5×5 square matrix. To minimize the space occupied by the matrix, m = 2×5 = 10. Therefore, the initial matrix of matrix M is... Among them, U 1,1 =(0), E 1,5 = (10000), matrix P 5,5 In the middle, P 1,3 =1,P 2,4 =1,P 3,1 =1,P 4,5 =1,P 5,2 =1, and all other elements are 0.

[0087] After each iteration, a 3×3 matrix A is selected from matrix M to construct the chaotic system structure. The selection method is as follows:

[0088] For any a ij ∈A k (0≤i,j≤n) satisfies:

[0089] {i1,i2,…,i n}∈{1,2,…,s}

[0090] {j1,j2,…,j n}∈{1,2,…,t}

[0091] Where i = G i (s,k)mod s,j=G j (t,k)mod t, G(·) is a function of s(t) and k, and i and j are calculated based on the value of k in each iteration.

[0092] In this example, from matrix P 5,5 The method for selecting a 3×3 matrix A is as follows:

[0093] {i1,i2,i3}∈{1,2,…,5}

[0094] {j1,j2,j3}∈{1,2,…,5}

[0095] Where i = G i (t,k)mod(5-3+1),j=G j (t,k) mod 3, G(·) is a function of t,k: G i (t,k)=t+k+i,Gj (t,k)=t 2k+j In each iteration, i and j are calculated based on the value of k.

[0096] Furthermore, M k =H(M,k), where H(M,k) represents the H(·) iterative mapping of matrix M based on the value of k in each iteration (the self-operation of the matrix, such as self-multiplication, inversion, and transpose when M is a square matrix).

[0097] In this example, In each iteration, the number of times matrix M is raised to the power of k is controlled, thereby controlling the values ​​of matrices P and A, so as to achieve the purpose of the system structure changing with the state of the system itself. The subscript indicates the number of iterations.

[0098] In this example, the initial values ​​are set to X0 = (0.75, 0.5, 0.375), λ k ≡3.75, μ 1 =μ 2 =μ 3 =3.75.

[0099] Figure 3 A topological diagram of a variable-structure chaotic system provided as an example of the invention;

[0100] In this embodiment, the chaotic mapping state sequence diagrams are as follows: Figure 4 x in (a) of the figure 1 ,x 2 ,x 3 As shown, according to Figure 4 As shown in Figure (a), the results demonstrate that the elements in the system state variables exhibit good chaotic properties in this example.

[0101] In this embodiment, the phase diagrams of the chaotic mapping are as follows: Figure 4 χ in Figure (b) 1 ,χ 2 ,χ 3 As shown, according to Figure 4 As shown in Figure (b), in this embodiment, the phase space orbits of each element in the system state variables fill the entire interval. (Comparison) Figure 2 As shown in Figure (b), this embodiment effectively expands the scope of chaos, and the phase diagram structure is more complex. It can not only effectively resist attacks by methods such as phase space reconstruction, but also has good confusion characteristics.

[0102] In this embodiment, the autocorrelation function graphs of the chaotic mapping state sequence are as follows: Figure 4 χ in Figure (c) 1 ,χ 2 ,χ 3As shown, according to Figure 4 As shown in Figure (c), there is no strong correlation among the elements of the system state variables in this example.

[0103] In this embodiment, the frequency distribution diagrams of the chaotic mapping output sequence are as follows: Figure 4 χ in (d) of the figure 1 ,χ 2 ,χ 3 As shown, according to Figure 4 As shown in Figure (d), in this embodiment, the frequencies of each element in the system state variables are uniformly distributed; compared with... Figure 2 As shown in Figure (d), this embodiment can improve the distribution characteristics of the original chaotic system, making the system output more random.

[0104] The variable-structure chaotic system provided in this embodiment modifies the system's structure based on its state, coupling three one-dimensional Logistic chaotic systems into a three-dimensional chaotic system. This chaotic system can integrate the performance of multiple chaotic systems and increases the complexity of the chaotic system through structural changes. With relatively low implementation difficulty and resource overhead, it effectively improves the chaotic performance and security performance of the chaotic system. Overall, the variable-structure chaotic system provided by this invention exhibits high chaotic performance and good security performance, making it applicable to cryptography and meeting cryptographic security requirements while reducing implementation difficulty and resource overhead.

[0105] Example 2:

[0106] A 5-dimensional logistic variable structure chaotic system, wherein the i-th state value of the system state variable in the (k+1)-th iteration is... for:

[0107]

[0108] Where f(·) is a mapping function of a discrete chaotic system; λ k A is the coupling coefficient in the k-th iteration of the variable structure chaotic system; k It is an n×n 0-1 matrix randomly generated in the k-th iteration, representing the coupling structure of the chaotic system; This represents the system state variable of the coupled chaotic system in the k-th iteration. The superscript indicates different discrete chaotic systems, and the subscript indicates the iteration number. "mod1" indicates that the state of the chaotic system is restricted to the attraction domain 1.

[0109] In this embodiment, n = 5, and correspondingly, the system state variable in the k-th iteration can be expressed as: Different iterations are distinguished by subscripts.

[0110] f(·) represents a Logistic chaotic system, i.e.:

[0111]

[0112] Furthermore, in this example, each system variable uses a 64-bit length during computation, with 64 bits for the decimal places, meaning the data precision is 64 bits. Unlike Example 1, this example uses five coupled Logistic systems to improve the system's chaotic performance by increasing M.

[0113] Furthermore, A k Let M represent the structure of the chaotic system during the k-th iteration. k It is an arbitrary s×t, (s,t≥n) 0-1 matrix, where the subscripts indicate the iteration number;

[0114] In this example, Where P is a permutation matrix of order non-zero, U is a strictly upper triangular matrix, and each row of (UE) contains exactly one element 1, J r,t Let m be an r×t matrix with all elements equal to 1. When m is odd, t = (n-1) / 2; when m is even, t = n / 2-1 or t = n / 2. It can be rigorously proven that any power of the 0-1 matrix M is still a 0-1 matrix.

[0115] In this embodiment, n = 5, so the number of rows and columns of matrix P is >= 5. Here, we let P be a 10×10 square matrix. To minimize the space occupied by the matrix, we let m = 2×10 = 20. Therefore, the initial matrix of matrix M is chosen as follows. Select U 1,1 =(0), E 1,10 = (1 0 0 0 0 0 0 0 0 0), in matrix P 10,10 In the middle, P 1,3 =1,P 2,5 =1,P 3,7 =1,P 4,1 =1,P 5,9 =1,P 6,4 =1,P 7,6 =1,P 8,10 =1,P 9,2 =1,P 10,8 =1, and all other elements are 0.

[0116] After each iteration, an n×n matrix A is selected from matrix M to construct the chaotic system structure. The selection method is as follows:

[0117] For any a ij ∈A k (0≤i,j≤n) satisfies:

[0118] {i1,i2,…,i n}∈{1,2,…,s}

[0119] {j1,j2,…,j n}∈{1,2,…,t}

[0120] Where i = G i (s,k)mod s,j=G j (t,k)mod t, G(·) is a function of s(t) and k, and i and j are calculated based on the value of k in each iteration.

[0121] In this example, from matrix P 10,10 The method for selecting a 5×5 matrix A is as follows:

[0122] {i1,i2,…,i5}∈{1,2,…,10}

[0123] {j1,j2,…,j5}∈{1,2,…,10}

[0124] Where i = G i (t,k)mod(10-5+1),j=G j (t,k)mod6, G(·) is a function of t,k: G i (t,k)=t+k+i,G j (t,k)=t 2k+j In each iteration, i and j are calculated based on the value of k.

[0125] Furthermore, M k =H(M,k), where H(M,k) represents the H(·) iterative mapping of matrix M based on the value of k in each iteration (the self-operation of the matrix, such as self-multiplication, inversion, and transpose when M is a square matrix).

[0126] In this example, In each iteration, the number of times matrix M is raised to the power of k is controlled according to the value of k, thereby controlling the values ​​of P and A, so as to achieve the purpose of the system structure changing with the state of the system itself. The subscript indicates the number of iterations.

[0127] In this example, the initial values ​​are set to X0 = (0.75, 0.5, 0.375, 0.875, 0.25), λ k ≡3.75, μ 1 =μ 2 =μ 3 =μ 4 =μ 5 =3.75.

[0128] Figure 5 This is a schematic diagram of the topology of the variable structure chaotic system provided in this embodiment;

[0129] In this embodiment, the chaotic mapping state sequence diagrams are as follows: Figure 6 x in (a) of the figure 1 ,x 2 ,x 3 ,x 4 ,x 5 As shown, according to Figure 6 As shown in Figure (a), the results demonstrate that the elements in the system state variables exhibit good chaotic properties in this example.

[0130] In this embodiment, the phase diagrams of the chaotic mapping are as follows: Figure 6 x in (b) 1 ,x 2 ,x 3 ,x 4 ,x 5 As shown, according to Figure 6 As shown in Figure (b), in this embodiment, the phase space orbits of each element in the system state variables fill the entire interval. (Comparison) Figure 2 As shown in Figure (b), this embodiment effectively expands the scope of chaos, and the phase diagram structure is more complex. It can not only effectively resist attacks by methods such as phase space reconstruction, but also has good confusion characteristics.

[0131] In this embodiment, the autocorrelation function graphs of the chaotic mapping state sequence are as follows: Figure 6 x in (c) of the figure 1 ,x 2 ,x 3 ,x 4 ,x 5 As shown, according to Figure 6 As shown in Figure (c), there is no strong correlation among the elements of the system state variables in this example.

[0132] In this embodiment, the frequency distribution diagrams of the chaotic mapping output sequence are as follows: Figure 6 x in (d) of the figure 1 ,x 2 ,x 3 ,x 4 ,x 5 As shown, according to Figure 6 As shown in Figure (d), in this embodiment, the frequencies of each element in the system state variables are uniformly distributed; compared with... Figure 2 As shown in Figure (d), this embodiment can improve the distribution characteristics of the original chaotic system, making the system output more random.

[0133] The variable-structure chaotic system provided in this embodiment changes the system's structure based on its state, coupling five one-dimensional Logistic chaotic systems into a five-dimensional chaotic system. This chaotic system can integrate the performance of multiple chaotic systems and increases the complexity of the chaotic system through structural changes. With relatively low implementation difficulty and resource overhead, it effectively improves the chaotic performance and security performance of the chaotic system. Overall, the variable-structure chaotic system provided by this invention has high chaotic performance and good security performance. Compared with Example 1, it offers higher security for cryptographic applications, but it also increases implementation difficulty and resource overhead. In practical applications, the appropriate system should be selected based on actual needs.

[0134] Example 3:

[0135] A 5-dimensional Logistic variable structure chaotic system, unlike Example 2, has a precision of only 8 bits.

[0136] The i-th state value of the system state variables in the (k+1)-th iteration of the chaotic system. for:

[0137]

[0138] Where f(·) is a mapping function of a chaotic system; λ k A is the coupling coefficient in the k-th iteration of the variable structure chaotic system; k It is an n×n 0-1 matrix randomly generated in the k-th iteration, representing the coupling structure of the chaotic system; The system state variable represents the coupled chaotic system. The superscript indicates different chaotic systems, and the subscript indicates the number of iterations. "mod1" indicates that the state of the chaotic system is restricted to the attraction domain 1.

[0139] In this embodiment, n = 5, and correspondingly, the system state variable in the k-th iteration can be expressed as: Different iterations are distinguished by subscripts.

[0140] f(·) represents a Logistic chaotic system, i.e.:

[0141]

[0142] In addition, in this example, in order to study the chaotic performance of the chaotic system under finite precision, while ensuring that the system dimension and structure matrix M remain unchanged, the precision of each system variable in the calculation process is 8 bits.

[0143] Furthermore, A k Let M represent the structure of the chaotic system during the k-th iteration.k It is an arbitrary s×t matrix (s,t≥n), where the subscripts indicate the iteration number;

[0144] In this example, Where P is a permutation matrix of order non-zero, U is a strictly upper triangular matrix, and each row of (UE) contains exactly one element 1, J r,t Let m be an r×t matrix with all elements equal to 1. When m is odd, t = (n-1) / 2; when m is even, t = n / 2-1 or t = n / 2. It can be rigorously proven that any power of the 0-1 matrix M is still a 0-1 matrix.

[0145] Similar to Example 2, the initial matrix of matrix M in this example is chosen as follows: Select U 1,1 =(0), E 1,10 = (1000000000), in matrix P 10,10 In the middle, P 1,3 =1,P 2,5 =1,P 3,7 =1,P 4,1 =1,P 5,9 =1,P 6,4 =1,P 7,6 =1,P 8,10 =1,P 9,2 =1,P 10,8 =1, and all other elements are 0.

[0146] After each iteration, an n×n matrix A is selected from matrix M to construct the chaotic system structure. The selection method is as follows:

[0147] For any a ij ∈A k (0≤i,j≤n) satisfies:

[0148] {i1,i2,…,i n}∈{1,2,…,s}

[0149] {j1,j2,…,j n}∈{1,2,…,t}

[0150] Where i = G i (s,k)mod s,j=G j (t,k)mod t, G(·) is a function of s(t) and k, and i and j are calculated based on the value of k in each iteration.

[0151] In this example, from matrix P 10,10 The method for selecting a 5×5 matrix A is as follows:

[0152] {i1,i2,…,i5}∈{1,2,…,10}

[0153] {j1,j2,…,j5}∈}1,2,…,10}

[0154] Where i = G i (t,k)mod(10-5+1),j=G j (t,k) mod 6, G(·) is a function of t,k: G i (t,k)=t+k+i,G j (t,k)=t 2k+j In each iteration, i and j are calculated based on the value of k.

[0155] Furthermore, M k =H(M,k), where H(M,k) represents the H(·) iterative mapping of matrix M based on the value of k in each iteration (the self-operation of the matrix, such as self-multiplication, inversion, and transpose when M is a square matrix).

[0156] In this example, In each iteration, the number of times matrix M is raised to the power of k is controlled according to the value of k, thereby controlling the values ​​of P and A, so as to achieve the purpose of the system structure changing with the state of the system itself. The subscript indicates the number of iterations.

[0157] In this example, the initial values ​​are set to X0 = (0.75, 0.5, 0.375, 0.875, 0.25), λ k ≡3.75, μ 1 =μ 2 =μ 3 =μ 4 =μ 5 =3.75.

[0158] Figure 7 This is a schematic diagram of the topology of the variable structure chaotic system provided in this example;

[0159] In this embodiment, the chaotic mapping state sequence diagrams are as follows: Figure 8 x in (a) of the figure 1 ,x 2 ,x 3 ,x 4 ,x 5 As shown, according to Figure 8 As shown in Figure (a), the results demonstrate that the elements in the system state variables exhibit good chaotic properties in this example.

[0160] In this embodiment, the phase diagrams of the chaotic mapping are as follows: Figure 8 x in (b) 1 ,x 2,x 3 ,x 4 ,x 5 As shown, according to Figure 8 As shown in Figure (b), in this embodiment, the phase space orbits of each element in the system state variables fill the entire interval. (Comparison) Figure 2 As shown in Figure (b), this embodiment effectively expands the range of chaos, resulting in a more complex phase diagram structure. This not only effectively resists attacks from methods such as phase space reconstruction but also exhibits good confusion characteristics. Meanwhile, compared to… Figure 6 In Figure (b), the phase space orbits of each element are more densely packed in the central region than in the surrounding area, which indicates that the reduction in precision causes a slight clustering of the data.

[0161] In this embodiment, the autocorrelation function graphs of the chaotic mapping state sequence are as follows: Figure 8 x in (c) of the figure 1 ,x 2 ,x 3 ,x 4 ,x 5 As shown, according to Figure 8 The results shown in Figure (c) indicate that, in this example, there is no strong correlation among the elements of the system state variables; in contrast... Figure 6 In Figure (c), the correlation remained essentially unchanged, indicating that the significant decrease in precision did not lead to an increase in correlation.

[0162] In this embodiment, the frequency distribution diagrams of the chaotic mapping output sequence are as follows: Figure 8 x in (d) of the figure 1 ,x 2 ,x 3 ,x 4 ,x 5 As shown, according to Figure 8 As shown in Figure (d), in this embodiment, the frequencies of each element in the system state variables are uniformly distributed; compared with... Figure 2 As shown in Figure (d), this embodiment can improve the distribution characteristics of the original chaotic system, making the system output more random; at the same time, compared with Figure 6 In the (d) plot, the histogram is relatively sparse, which indicates that the reduced precision reduces the number of possible values ​​between 0 and 1.

[0163] The finite-precision variable-structure chaotic system provided in this embodiment, based on changing the system's structure according to its state, couples five one-dimensional Logistic chaotic systems into a five-dimensional chaotic system, integrating the performance of multiple chaotic systems. By altering the system structure, the complexity of the chaotic system is increased, and excellent chaotic performance is achieved with finite precision of 8 bits, significantly reducing resource utilization. Overall, the finite-precision variable-structure chaotic system provided by this invention is easy to implement, has low resource overhead, and exhibits high chaotic performance and good security, meeting the requirements of cryptographic security and possessing high application value.

[0164] In summary, the structurally variable coupled chaotic system provided by this invention is formed by the coupling of multiple discrete chaotic systems. During system evolution, the coupling structure between different subsystems is dynamically changed in real time by manipulating the coupling matrix. The coupled system exhibits complex dynamic behavior and high nonlinearity, and the dynamic changes in structure cause continuous abrupt changes in the system's dynamics, resulting in non-stationary chaotic sequences. This system is resistant to attacks from statistical analysis, algebraic analysis, and other methods, exhibiting extremely high security. Furthermore, the system can effectively overcome the finite precision effect. The variable-structure chaotic system is applicable to various chaotic systems and complex chaotic networks of arbitrary size. Complex dynamics can also be generated based on simple chaotic systems, and controlling structural changes results in lower overhead under the same complexity and security requirements. This system can be used to design high-performance chaotic cryptosystems and can also be used as a component in network security applications.

[0165] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A coupled chaotic system with variable structure, characterized in that, To implement chaotic encryption, it includes: a coupling module and n discrete chaotic systems; ; The coupling module is used to start the first iteration after receiving the start command, take the input initial system state variables as the system state variables under the first iteration of the coupled chaotic system, and input the n state values ​​therein into the n discrete chaotic systems one by one. The discrete chaotic system is used to generate corresponding discrete chaotic values ​​by performing chaotic mapping after receiving state value input, and output them to the coupling module. The coupling module is also used to perform coupled chaos mapping on each discrete chaotic value after obtaining n discrete chaotic values ​​generated by the discrete chaotic system in each iteration, to obtain a set of coupled chaotic values ​​for the current iteration and output them. At the same time, the set of coupled chaotic values ​​is used as the system state variable for the next iteration of the coupled chaotic system, and the n state values ​​are input one-to-one into the n discrete chaotic systems for the next iteration. Among them, the k In the nth iteration, the th i The first discrete chaotic value is obtained by coupling chaotic mapping. i The value of the coupled chaotic value is: This represents the i-th discrete chaotic value in the k-th iteration. Let i be the chaotic mapping function of the i-th discrete chaotic system; Let be the i-th state value of the system state variable in the k-th iteration of the coupled chaotic system; the coupling matrix. Randomly generated in the k-th iteration A 0-1 matrix of size; Let be the system state variable of the coupled chaotic system in the kth iteration.

2. The structurally variable coupled chaotic system according to claim 1, characterized in that, The i-th coupled chaotic value obtained by performing coupled chaotic mapping on the i-th discrete chaotic value in the k-th iteration is: in, For the first k The coupling coefficients are randomly generated in each iteration.

3. The structurally variable coupled chaotic system according to claim 1 or 2, characterized in that, After obtaining the coupled chaotic value through the coupled chaotic mapping, the coupling module also restricts the obtained coupled chaotic value within the attraction domain of the discrete chaotic system. Wherein, the first, confined within the attraction domain of the discrete chaotic system i A coupled chaotic value ;in, The modulo operator is used; the region of attraction for the i-th discrete chaotic system is... .

4. The structurally variable coupled chaotic system according to claim 1, characterized in that, The coupling matrix The method for generating any element in the matrix is ​​as follows: randomly select a matrix. The element in the r-th row and c-th column is used as Any element in; in, The matrix Randomly generated in the k-th iteration A 0-1 matrix of size [size missing] is obtained by: in the k-th iteration, by [method missing] the initial matrix [value missing]. The initial matrix is ​​obtained by randomly performing matrix self-operation; for A 0-1 matrix of size is randomly generated by the coupled chaotic system and then fixed.

5. The structurally variable coupled chaotic system according to claim 1, characterized in that, The coupling module is also used to stop the iteration upon receiving a termination command.

6. A method for generating and controlling coupled chaotic sequences, characterized in that, A coupling module applied to the structurally variable coupled chaotic system according to any one of claims 1-5, comprising: After receiving the start command, the first iteration begins. The input initial system state variables are used as the system state variables of the coupled chaotic system in the first iteration, and the n state values ​​are input one by one into the n discrete chaotic systems. In each iteration, after obtaining n discrete chaotic values ​​generated by the discrete chaotic system, a coupled chaotic mapping is performed on each discrete chaotic value to obtain a set of coupled chaotic values ​​for the current iteration and output them. At the same time, the set of coupled chaotic values ​​is used as the system state variable for the next iteration of the coupled chaotic system, and the n state values ​​are input one by one into the n discrete chaotic systems for the next iteration. The discrete chaotic system, upon receiving a state value input, performs a chaotic mapping to generate a corresponding discrete chaotic value. No. k In the nth iteration, the th i The first discrete chaotic value is obtained by coupling chaotic mapping. i The value of the coupled chaotic value is: This represents the i-th discrete chaotic value in the k-th iteration. Let i be the chaotic mapping function of the i-th discrete chaotic system; Let be the i-th state value of the system state variable in the k-th iteration of the coupled chaotic system; the coupling matrix. Randomly generated in the k-th iteration A 0-1 matrix of size; Let be the system state variable of the coupled chaotic system in the kth iteration.

7. A computer-readable storage medium, characterized in that, The computer-readable storage medium includes a stored computer program, wherein when the computer program is run by a processor, it controls the device where the storage medium is located to execute the coupled chaotic sequence generation and control method of claim 6.

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