Chaotic key generation method and system based on discrete exponential memristor coupling and data encryption method

Abstract

The invention discloses a chaotic key generation method and system based on discrete exponential memristor coupling and a data encryption method. The chaotic key generation method comprises the following steps: firstly, generating a chaotic system composed of a discrete index memristor, a linear controller and one-dimensional chaotic mapping, setting initial input data of the chaotic system and initial state parameters of the discrete index memristor, and then performing multiple iterations through the chaotic system to obtain a chaotic key; and finally, constructing a chaotic key based on the chaotic sequence of each iteration. The chaotic key generation system comprises a chaotic system and a data processing module, and is used for realizing the chaotic key generation method. The data encryption method comprises the following steps: firstly, generating a chaotic key based on a chaotic key generation method, and then encrypting plaintext data to obtain encrypted data. The chaotic dynamics constructed by the method is rich in behavior and excellent in performance index, and the chaotic key constructed based on the chaotic sequence generated by the chaotic system is high in security and can adapt to different hardware signal requirements.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of chaotic key generation methods, in particular to a chaotic key generation method based on discrete exponential memristor coupling, a system and a data encryption method. BACKGROUND

[0002] In the current era of rapid development of digital and networked communication, data security issues are becoming increasingly serious. Chaotic systems, with their extreme sensitivity to initial conditions, unpredictability, and ergodicity, have great potential for application in the field of information security. Since the advent of memristors in 2008, their nanoscale, low power consumption, and strong nonlinearity have made them ideal components for building high-performance chaotic systems.

[0003] Due to their high computational efficiency and simple structure, discrete chaotic maps have become an important direction for research on memristor-based discrete chaotic systems, and many achievements have been made. In existing research, scholars have constructed various chaotic or hyperchaotic systems by coupling discrete memristors with classical maps and extending continuous memristor models to the discrete domain. Chaotic sequences are generated by chaotic systems to construct chaotic keys.

[0004] A discrete cosine memristor model was proposed in a journal paper published in 2020 in IEEE transactions on industrial electronics [Two-Dimensional Memristive Hyperchaotic Maps and Application in Secure Communication]. By coupling it with existing one-dimensional discrete maps (Logistic map, Sine map, Tent map, linear map), a general two-dimensional memristive map model was constructed, resulting in four two-dimensional memristive hyperchaotic maps (2D-MLM, 2D-MSM, 2D-MTM, 2D-SMM). The above paper mainly has the following problems:

[0005] (1) The nonlinearity of the memristor is single, using the cosine memristor model, only introducing nonlinearity through a single cosine function, the ability to regulate system dynamics is limited, and the nonlinearity is insufficient.

[0006] (2) The complexity of the chaotic signal needs to be improved, and the maximum Lyapunov exponent, spectral entropy, and correlation dimension of the hyperchaotic region still have room for optimization.

[0007] (3) The multi-stability characteristics are limited, although the coexisting attractors dependent on initial values are achieved, the homogeneous and heterogeneous multi-stability are not clearly distinguished and implemented simultaneously, and the diversity of multi-stability is insufficient.

[0008] It can be seen that in the current design of the chaotic system based on the memristor, especially in the super chaotic category, the system capable of simultaneously realizing homogeneous and heterogeneous multi-stability is still relatively scarce. The existing system still has room for improvement in terms of nonlinearity, multi-stability diversity, and chaotic signal complexity. Therefore, the chaotic key constructed by the chaotic sequence obtained is difficult to fully meet the high security and high flexibility requirements of the encryption signal in the information security field. SUMMARY The application provides a chaotic key generation method and system based on discrete exponential memristor coupling and a data encryption method to solve the problem that the chaotic key formed by the chaotic sequence output by the discrete cosine memristor model in the prior art cannot meet the encryption signal requirements in terms of nonlinearity, multi-stability diversity, and chaotic signal complexity.

[0009] To achieve the above-mentioned purpose, the technical scheme adopted by the application is as follows:

[0010] A chaotic key generation method based on discrete exponential memristor coupling, the process being as follows:

[0011] Step 1: generating a chaotic system, the chaotic system comprising two links and an adder, wherein the first link comprises a discrete exponential memristor , a linear controller with a linear parameter , and the second link comprises a one-dimensional chaotic mapping ; wherein represents a state parameter of the discrete exponential memristor , represents a control parameter of the one-dimensional chaotic mapping , represents input data of the chaotic system;

[0012] Setting the initial input data of the chaotic system and setting the initial state parameter of the discrete exponential memristor in the chaotic system;

[0013] Step 2: inputting the initial input data to the chaotic system for multiple iteration mappings, the process being as follows:

[0014] When the chaotic system is inputted with the initial input data for the first time, the discrete exponential memristor in the first link of the chaotic system is loaded with the initial state parameter ; the initial input data is inputted to the discrete exponential memristor in the first link which is loaded with the initial state parameter , and the discrete exponential memristor loaded with the initial state parameter Obtain discrete sequence Discrete sequences obtained from discrete exponential memristors The scaled discrete sequence is then obtained by scaling through a linear controller. Furthermore, the initial input data Input to the one-dimensional chaotic mapping in the second link Through one-dimensional chaotic mapping One-dimensional chaotic sequence obtained Finally, the discrete sequence is processed by the adder. and one-dimensional chaotic sequence Adding them together yields the chaotic sequence of the first iteration. After the first iteration, use the initial input data. and the initial state parameters of the discrete exponential memristor The sum is used as the state parameter updated by the discrete exponential memristor after the first iteration. ;

[0015] The first iteration after the first iteration In the next iteration , with the first Chaotic sequence of the next iteration As a chaotic system, the first The input data at the nth iteration, with the nth After the next iteration, the state parameters of the discrete exponential memristor are updated. As a chaotic system, the first Discrete exponential memristor during the next iteration State parameters; During the nth iteration, the 1st Chaotic sequence of the next iteration Input to the first link to load the first The state parameters after the next iteration Discrete exponential memristor By loading state parameters Discrete exponential memristor Obtain discrete sequence Discrete sequences obtained from discrete exponential memristors The scaled discrete sequence is then obtained by scaling through a linear controller. Furthermore, the first Chaotic sequence of the next iteration Input to the one-dimensional chaotic mapping in the second link Through one-dimensional chaotic mapping One-dimensional chaotic sequence obtained Finally, the discrete sequence is processed by the adder. and one-dimensional chaotic sequence Add them together to get the first one. Chaotic sequence of the next iteration ;No. After the iteration is completed, with the first iteration... The sequence of iterations and the After the next iteration, the state parameters of the discrete exponential memristor are... The sum of, as the first After the next iteration, the state parameters of the discrete exponential memristor are... ;

[0016] Thus, through multiple iterations of the chaotic system, a chaotic sequence for each iteration is obtained;

[0017] Step 3: Construct the chaotic key based on the chaotic sequence obtained from multiple iterations of the chaotic system.

[0018] Furthermore, in step 1, the one-dimensional chaotic mapping It is a linear mapping.

[0019] Furthermore, in step 1, the one-dimensional chaotic mapping This is a Logistic mapping.

[0020] Furthermore, in step 1, the one-dimensional chaotic mapping This is a Tent mapping.

[0021] Furthermore, in step 1, the one-dimensional chaotic mapping This is a Sine mapping.

[0022] Furthermore, in step 1, the initial input data of the chaotic system The initial state parameter of the discrete exponential memristor is set to 0.001. Set it to 0.1.

[0023] Furthermore, in step 3, from the chaotic sequence obtained by multiple iterations of the chaotic system, the chaotic sequence results of the first M unstable iterations are discarded, and the chaotic sequence results of subsequent stable iterations are retained. The retained chaotic sequences of each iteration are converted into positive numbers in the range [0, 255] and then into binary numbers. Then, the binary numbers are concatenated end to end according to the iteration order to obtain a pseudo-random number of the binary sequence, which is used as the chaotic key.

[0024] Furthermore, in step 3, from the chaotic sequence obtained by multiple iterations of the chaotic system, the chaotic sequence results of the first M unstable iterations are discarded, and the chaotic sequence results of subsequent stable iterations are retained. The retained chaotic sequences are then combined into a matrix, which serves as the chaotic key.

[0025] A system for implementing the above-described chaotic key generation method based on discrete exponential memristor coupling includes a chaotic system and a data processing module, wherein:

[0026] The chaotic system includes two links and an adder; the first link includes a discrete exponential memristor. A linear controller, wherein the linear controller is a controller with linear parameters. Proportional controller; the second link includes a one-dimensional chaotic mapping. Discrete exponential memristors in the first link The output of the linear controller is used as the input of the linear controller, and the output of the linear controller is a one-dimensional chaotic mapping in the second link. The output is used as the input of the adder; during the first iteration of the chaotic system, the discrete exponential memristor in the first link... One-dimensional chaotic mapping in the second link Common access to initial input data and discrete exponential memristor Load initial state parameters The chaotic system after its first iteration... At the +1 iteration, the adder's... The chaotic sequence output from each iteration is used as the discrete exponential memristor. One-dimensional chaotic mapping No. Input data for +1 iterations, and discrete exponential memristor Loading the first After the next iteration, the state parameters of the discrete exponential memristor are... ;

[0027] The data processing module collects the chaotic sequence output by the adder in each iteration, and constructs a chaotic key based on the chaotic sequence output by the adder in each iteration.

[0028] A data encryption method, the process of which is as follows:

[0029] Plaintext data is generated, and a chaotic key is generated according to the chaotic key generation method based on discrete exponential memristor coupling described above; then the plaintext data is encrypted using the chaotic key.

[0030] This invention overcomes the limitations of the memristor model by introducing a discrete exponential memristor model. The nonlinearity of the system is enhanced through a composite form of exponential and trigonometric functions, providing a foundation for complex dynamic behaviors. A general two-dimensional discrete memristor coupling framework is constructed within the proposed chaotic system, coupling multiple classical one-dimensional mappings and simultaneously realizing both homogeneous and heterogeneous multistable states. Consequently, the chaotic sequences generated based on this chaotic system exhibit high complexity, good nonlinear strength, and good multistable state diversity. The chaotic keys generated by this system possess advantages such as high security and flexibility, and particularly show potential for application in the field of information security when generating pseudo-random numbers.

[0031] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0032] 1. Richer Dynamic Behavior: The chaotic system generated by this invention can produce hyperchaotic behavior with multiple positive Lyapunov exponents, while simultaneously achieving homogeneous and heterogeneous multistability. Heterogeneous multistability enhances the system's pseudorandomness and initial value sensitivity, while homogeneous multistability supports flexible control of attractor polarity and spatial position. The randomness and complexity of the chaotic signal are significantly better than existing systems.

[0033] 2. Superior performance indicators: Quantitative evaluation shows that the chaotic system of this invention has the highest mapping correlation dimension (2.1024) when using linear mapping, and the best mapping spectral entropy (0.8915) and maximum Lyapunov exponent (0.4963) when using Sine mapping. Compared with the two-dimensional memristive coupled mapping in the existing literature [Two-Dimensional Memristive Hyperchaotic Maps and Application in Secure Communication], the key performance indicators of this invention are significantly improved.

[0034] 3. High Application Value: The chaotic key constructed based on the chaotic sequence generated by the chaotic system of this invention has outstanding advantages such as large key space, strong randomness, and excellent resistance to prediction and statistical attacks. Due to the rich dynamic behavior of the system, high sensitivity to initial conditions, and multi-steady-state characteristics, the generated key sequence has high complexity and long period characteristics, which can greatly improve the security of the encryption system; at the same time, the excellent performance indicators ensure that the key is close to a real random signal in statistical characteristics, making it difficult to crack through phase space reconstruction or statistical analysis.

[0035] In particular, when the pseudo-random number constructed based on the chaotic sequence generated by the chaotic system of this invention is used as a chaotic key, it can pass the NIST SP 800-22 randomness test in all aspects. It has broad application prospects in information security fields such as secure communication, data encryption, and random number generation, and can be adapted to different hardware signal requirements. Attached Figure Description

[0036] Figure 1 This is a schematic diagram of a chaotic system in an embodiment of the present invention.

[0037] Figure 2 The fixed-point stability of the chaotic system in this embodiment of the invention is as follows: and The changes are shown in the following figures: (a) changes in MK mapping, (b) changes in ML mapping, (c) changes in MT mapping, and (d) changes in MS mapping.

[0038] Figure 3 These are the parameter-dependent dynamic graphs of the chaotic system in this embodiment of the invention, wherein: (a) is the parameter-dependent dynamic graph of the MK mapping, (b) is the parameter-dependent dynamic graph of the ML mapping, (c) is the parameter-dependent dynamic graph of the MT mapping, and (d) is the parameter-dependent dynamic graph of the MS mapping.

[0039] Figure 4 This is the chaotic system in the embodiment of the invention that varies with parameter k1. And the bifurcation diagram, where: (a) is the MK mapping as a function of parameter k1. And the bifurcation diagram, (b) shows the variation of the ML mapping with parameter k1. And the bifurcation diagram, (c) shows the MT mapping as a function of parameter k1. And the bifurcation diagram, (d) is the MS mapping as a function of parameter k1. And bifurcation diagram.

[0040] Figure 5 This is the chaotic system in the embodiment of the invention that varies with parameter k2. And the bifurcation diagram, where: (a) is the MK mapping as a function of parameter k2. And the bifurcation diagram, (b) shows the variation of the ML mapping with parameter k2. And the bifurcation diagram, (c) shows the MT mapping as a function of parameter k2. And the bifurcation diagram, (d) is the MS mapping as a function of parameter k2. And bifurcation diagram.

[0041] Figure 6 These are the phase diagrams of four chaotic systems, MK, ML, MT and MS, under different parameters in the embodiments of the present invention, wherein: (a) is the phase diagram of MK under different parameters, (b) is the phase diagram of ML under different parameters, (c) is the phase diagram of MT under different parameters, and (d) is the phase diagram of MS under different parameters.

[0042] Figure 7 It is a multi-cavity chaotic attractor of the MS-mapped chaotic system in the embodiments of the present invention.

[0043] Figure 8 These are the initial value dependent dynamic graphs of four chaotic systems MK, ML, MT and MS in this embodiment of the invention, wherein: (a) is the initial value dependent dynamic graph of MK mapping, (b) is the initial value dependent dynamic graph of ML mapping, (c) is the initial value dependent dynamic graph of MT mapping, and (d) is the initial value dependent dynamic graph of MS mapping.

[0044] Figure 9 In this embodiment of the invention, the four chaotic systems MK, ML, MT, and MS are... changing And bifurcation graph, where: (a) is the MK mapping following changing And the bifurcation graph, (b) is the ML mapping. changing And bifurcation graph, (c) is the MT mapping. changing And bifurcation graph, (d) is the MS mapping. changing And bifurcation diagram. Detailed Implementation

[0045] To enable those skilled in the art to better understand the present invention, the embodiments will be described in detail below with reference to the accompanying drawings and examples. This will allow for a full understanding of how the present invention uses technical means to solve technical problems and achieve corresponding technical effects, and to facilitate its implementation. The embodiments of the present invention and the various features within them can be combined with each other without conflict, and all resulting technical solutions are within the protection scope of the present invention.

[0046] Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort should fall within the scope of protection of the present invention.

[0047] It should be noted that the terms "comprising" and "having," and any variations thereof, in the specification, claims, and accompanying drawings of this invention are intended to cover non-exclusive inclusion.

[0048] This embodiment discloses a chaotic key generation method based on discrete exponential memristor coupling, the process of which is as follows:

[0049] Step 1: Generate a chaotic system. The chaotic system includes two links and an adder, wherein:

[0050] The first link includes a discrete exponential memristor. Linear controller. The linear controller uses linear parameters... Proportional controller. Represents a discrete exponential memristor The state parameters are the charge state parameters.

[0051] The second link includes a one-dimensional chaotic mapping. One-dimensional chaotic mapping It can be a linear mapping, a logistic mapping, a tent mapping, or a sine mapping. Represents a one-dimensional chaotic mapping The control parameters, This represents the input data of a chaotic system.

[0052] Furthermore, the initial input data of the chaotic system is set. Set the initial state parameters of the discrete exponential memristor in the chaotic system. Among them, the initial input data of the chaotic system The initial state parameter of the discrete exponential memristor is preferably set to 0.001. The preferred value is 0.1.

[0053] In the construction of multistable attractors, discrete exponential memristors exhibit significant advantages: on the one hand, their unique exponential memristor characteristics can enhance the nonlinearity of the system, laying the foundation for enriching the dynamic behavior; on the other hand, the trigonometric functions embedded within the memristor value facilitate the realization of coexisting attractors. The mathematical expression of the discrete exponential memristor is shown in formula (1):

[0054] (1)

[0055] In formula (1): and These are the current i(t) parameter value and the charge q(t) parameter value loaded by the discrete exponential memristor at the nth iteration, respectively. It is the voltage v(t) obtained in the nth iteration; It is the updated charge q(t) state value of the discrete exponential memristor after the nth iteration; h is defined as the iteration step size, usually set to h = 1. Indicates loading during the nth iteration Discrete exponential memristors, and have = It describes magnetic flux. Relative to charge The rate of change.

[0056] Due to the inherent strong nonlinearity of the discrete exponential memristor, it is difficult to perform one-dimensional mappings based on this discrete exponential memristor. The selection of [the appropriate mapping method] is very flexible. Therefore, this embodiment adopts the classic one-dimensional chaotic mapping: linear mapping, logistic mapping, Tent mapping, and Sine mapping, and couples them with discrete exponential memristor and linear controller to obtain a chaotic system that can realize two-dimensional chaotic mapping, such as [example system]. Figure 1 As shown, the hybrid system model is as shown in formula (2):

[0057] (2)

[0058] In formula (2): Represents the chaotic system's first... Input data for the next iteration; Represents the chaotic system's first... Discrete exponential memristor during the next iteration State parameters; Represents the linear parameters of a linear controller; This indicates that the input data is One-dimensional chaotic mapping of time ; The control parameters represent the one-dimensional chaotic mapping; Indicates the first After the next iteration, the discrete exponential memristor Updated status parameters.

[0059] Among them, one-dimensional chaotic mapping Linear mapping, or Logistic mapping, or Tent mapping, or Sine mapping can be used. In this embodiment, a one-dimensional chaotic mapping is used. The chaotic system that realizes a two-dimensional chaotic mapping using linear mapping is named the MK mapping, while the one-dimensional chaotic mapping is... The chaotic system that implements a two-dimensional chaotic mapping using the Logistic mapping is called the ML mapping, while the one-dimensional chaotic mapping is... The chaotic system that implements a two-dimensional chaotic mapping using the Tent mapping is called the MT mapping, while the one-dimensional chaotic mapping is... The chaotic system that realizes the two-dimensional chaotic mapping using the Sine mapping is named the MS mapping. The models of the four chaotic systems, namely the MK mapping, ML mapping, MT mapping, and MS mapping, are shown in equations (3)-(6):

[0060] MK mapping: (3)

[0061] ML mapping: (4)

[0062] MT mapping: (5)

[0063] MS mapping: (6)

[0064] The following analysis focuses on the fixed points, stability, and dynamics of the chaotic system constructed in this embodiment.

[0065] (1) Fixed point and stability analysis

[0066] For a chaotic system as shown in formula (2), its fixed point is denoted as S( According to the definition of a fixed point, this point must satisfy the following equation as shown in formula (7):

[0067] (7)

[0068] in: Indicates input data, This indicates the state of the discrete exponential memristor.

[0069] Solving the system of equations shown in formula (7), we can see that the case of a fixed point can be divided into two types:

[0070] ① When At this point, the chaotic system shown in formula (2) has no fixed point. In this case, the new system exhibits the hidden attractor property.

[0071] ② When At that time, the chaotic system as shown in formula (2) has an infinite number of fixed points, which can be represented as shown in formula (8):

[0072] (8)

[0073] in Let be any constant. Then, the Jacobian matrix at S. As shown in formula (9):

[0074] (9)

[0075] in: .

[0076] The characteristic polynomial P(λ) of the Jacobian matrix in formula (9) is shown in formula (10):

[0077] (10)

[0078] in, Let represent the eigenvalue variables of the Jacobian matrix in formula (9).

[0079] Therefore, two eigenvalues and As shown in formula (11):

[0080] (11)

[0081] From formula (11), we can see that, ,and By function , The stability condition is determined by both μ and eigenvalues, therefore the fixed point of the chaotic system as shown in formula (2) may be unstable or critically stable. Substituting the four two-dimensional memristor chaotic systems into the eigenvalue function and combining the stability criteria of discrete chaotic systems, the stability condition is derived. Figure 2 Give parameters When the parameters are -1.3, 0.5, 0.3, and -0.4 respectively, the four chaotic systems of MK mapping, ML mapping, MT mapping, and MS mapping change with the parameter. and memristor initial value The distribution of fixed-point stability varies, with blue indicating critical stability and red indicating instability at fixed points. Analysis shows that the fixed-point stability of the four two-dimensional mapped chaotic systems exhibits periodicity in the μ direction, a characteristic that predicts the system's multi-coexistence and multi-steady-state properties.

[0082] (2) Dynamic analysis

[0083] This embodiment uses simulation tools such as dynamic diagrams, Lyapunov exponent spectra, bifurcation diagrams, attraction domains, and phase trajectory diagrams to analyze the influence of coupling parameters and initial values ​​on the dynamic behavior of discrete memristor systems, revealing the system's hyperchaotic, multi-coexistence, and multi-steady-state behavior.

[0084] 2.1) Parameter-dependent dynamic behavior

[0085] Set initial conditions ( , Given (0.001, 0.1), a numerical scan of four chaotic systems (MK mapping, ML mapping, MT mapping, and MS mapping) is performed on the two-dimensional parameter plane. The resulting dynamic distribution diagram is shown below. Figure 3 As shown. Figure 3 In the diagram, different colored regions correspond to various steady-state and transient behaviors arising from system evolution, including fixed point (FP), periodic (P2-P8), multi-period (MP), chaotic (CH), and hyperchaotic (HC). Analysis results show that the system exhibits rich dynamic characteristics under different parameter combinations. In particular, significant period-doubling bifurcation behavior can be observed in the parameter space of all four systems, indicating that the system progresses to a chaotic or hyperchaotic state through typical period-doubling bifurcation paths. (Comparison) Figure 3As can be seen from (a)-(d), each system contains an irregular dynamic region, but their distribution patterns and scales differ significantly: the large red area in the MS mapping indicates that it has a broad chaotic and hyperchaotic parameter domain; while the chaotic regions in the other three mappings exhibit a relatively narrow distribution pattern. These phenomena demonstrate that, under the influence of parameters such as coupling strength, the four systems can elicit highly complex and structurally diverse dynamic responses.

[0086] The Lyapunov exponent (LE) is a key indicator for quantifying the dynamic stability of a system. Its sign and magnitude reflect the system's sensitivity to initial conditions and the chaotic nature of its motion. Bifurcation is a critical phenomenon in which a dynamic system undergoes a qualitative behavioral shift as parameters change. Under initial conditions (… , When ) = (0.001, 0.1), Figure 4 and Figure 5 The Lyapunov exponent spectra and corresponding bifurcation diagrams of four systems are presented, with one parameter fixed and another parameter varying. By comparing and analyzing the Lyapunov exponent spectra and bifurcation diagrams, the evolution of the system's dynamic characteristics with respect to parameters can be clearly revealed, and rich dynamic modes can be identified.

[0087] Under fixed parameters Under the condition that each mapping follows The dynamic characteristics of change, such as Figure 4 As shown: Figure 4 (a) Fixed = -1.3, MK mapping follows As the number of cells increases, its behavior transitions from a periodic state and a quasi-periodic state to chaos, and finally enters a hyperchaotic state. Figure 4 (b) Fixed = 0.5, the ML mapping enters chaos through period-doubling bifurcation, then briefly emerges into hyperchaos, and finally... Within the interval [3.48, 3.6], the system returns to a periodic state due to the outbreak of a crisis. Figure 4 (c) Fixed = 0.3, the dynamic behavior of the MT mapping varies with The changes exhibit alternations between chaos and periodic windows, eventually evolving into hyperchaos; Figure 4 (d) Fixed = -0.4, MS is mapped at It consistently exhibits hyperchaotic behavior with a large Lyapunov exponent within the parameter range [0.4, 0.6], indicating its robustness within this parameter range. (The last sentence appears to be incomplete and possibly refers to a different parameter.) Evolutionary dynamics such as Figure 5 As shown. Wherein, MK mapping ( =0.6), ML mapping ( = 1.35) and MT mapping ( = 1.8) All enter the chaotic state through a typical period-doubling bifurcation path; the MK and MS mappings have a wide hyperchaotic region.

[0088] To intuitively compare the dynamic characteristics of the four systems, under the initial conditions ( , When ) = (0.001, 0.1), Figure 6 (a)-(d) show the attractor phase diagrams under typical parameters, where red, magenta, blue, and black represent hyperchaotic (HC), chaotic (CH), periodic, and quasi-periodic attractors, respectively. Specific parameter configurations and attractor types are as follows: MK is mapped in ( , When the parameters are (0.6, -1.3), (0.507, -1.3), and (0.5, 0), respectively, the map exhibits hyperchaos, chaos, and periodic attractors. The ML mapping corresponds to hyperchaos, chaos, and periodic attractors under the parameter pairs of (-3.465, 0.5), (1.35, -0.785), and (1.35, -0.966), respectively. The MT mapping exhibits hyperchaos, chaos, and periodic attractors at (2.2, -0.15), (2, -0.17), and (3.3, 0.3), respectively. The MS mapping generates hyperchaos, chaos, and periodic attractors under the conditions of (1.5, -0.3), (1.73, -0.25), and (1.4, -0.1), respectively.

[0089] These Figure 6 The phase diagrams shown intuitively verify the strong dependence of the dynamic behavior of each system on the parameters. The periodic, chaotic, hyperchaotic, and quasi-periodic attractors appearing in the MS mapping are all consistent with the evolution mechanism revealed by the bifurcation diagram. Particularly noteworthy is that the hyperchaotic attractors in both the MK and MS mappings exhibit complex left-right fractal structures, reflecting a richer geometric feature of the system in the hyperchaotic state. Specifically, under the initial conditions ( , Given (0.001, 0.1), Figure 7 The MS mapping shown is in the parameter ( = -1, = -0.5) presents a complex multi-cavity attractor, which generates signals with higher randomness and unpredictability, and has clear application prospects in fields such as encrypted communication and random number generation.

[0090] 2.2) Initial value dependent dynamic behavior

[0091] To demonstrate the dynamic mechanism of the memristor initial value enhancement behavior, the chaotic system model shown in equation (2) can be substituted into... Rewritten as an iterative equation as shown in equation (12):

[0092] (12)

[0093] in, Let represent the sequence of the i-th iteration.

[0094] When keep The initial charge state of the discrete exponential memristor remains unchanged. shift ( When is a positive integer, the chaotic system under the new initial conditions is expressed as shown in formula (13):

[0095] (13)

[0096] in: This is the sequence of the i-th iteration under the new initial conditions; for ; This is the sequence of the nth iteration under the new initial conditions; This is the (n+1)th iteration sequence under the new initial conditions; For the new initial conditions, the first The state parameters of the discrete exponential memristor are updated after each iteration; For the new initial conditions, the first The state parameters of the discrete exponential memristor are updated after +1 iterations.

[0097] Since the period of the cosine function is an integer multiple of 2π, we can obtain formula (14) as follows:

[0098] (14)

[0099] It can be concluded that when the initial charge... shift , and The iteration results are the same, and thus formula (15) is derived as follows:

[0100] (15)

[0101] Right now exist Based on the initial charge, The shift will result in a corresponding shift. Therefore, the above results prove that the chaotic system shown in Equation (2) has certainty regarding the initial state of the memristor. It exhibits periodicity, meaning that the dynamic behavior of the chaotic system constructed in this embodiment changes with the initial state of the memristor. It exhibits periodic changes due to its periodic evolution.

[0102] Figure 8 (a)-(d) illustrate the use of dynamic algorithms, with parameters selected respectively ( , The graphs show the dynamics of four chaotic systems (MK mapping, ML mapping, MT mapping, and MS mapping) with initial conditions (x0, q0) for values ​​of (1.3, -1.3), (3.3, 0.5), (2.36, -0.07), and (1.5, -0.3). Different colored areas represent different dynamic behaviors. It can be seen that the four systems exhibit different dynamic behaviors under the given conditions (x0, q0). The changes all exhibit periodic characteristics, and in Within a single period, the system's attractors converge to various dynamical domains, such as hyperchaos (HC), chaos (CH), and multi-periodicity (MP). This indicates that the system... It exhibits multiple coexistence characteristics in different directions and displays multi-steady-state behavior within a single cycle, consistent with theoretical analysis. In particular, Figure 8 In (d), the MS mapping has a large area of ​​dark red on the two-dimensional initial condition plane, indicating that the generated hyperchaotic sequence has strong robustness to the initial conditions.

[0103] To highlight the four systems in To address the multi-coexistence phenomenon in different directions, the numerical simulation algorithms for the bifurcation diagram and Lyapunov exponent spectrum described above are adjusted to an initial value-dependent mode. Parameters ( , Let x0 be set to (0.6, -1.3), (1.35, -0.785), (1.26, -0.4), and (1.5, -0.3) respectively, and fix x0 = 0.001 to obtain the following values: Variations of LEs and bifurcation diagrams, such as Figure 9 As shown in the diagram, it is clear from the bifurcation diagram that all four systems exhibit a tendency to follow the trend. The phenomenon of periodic evolution, and the fact that this result is highly consistent with the LEs result, further verifies the accuracy of the obtained results.

[0104] The following describes the performance evaluation of the chaotic system constructed in this embodiment.

[0105] Maximum Ley-Leys exponent, sample entropy (SampEn), and spectral entropy (SE) are important tools for evaluating the complexity of chaotic systems. They reflect the dynamic characteristics of the system from different perspectives. This embodiment uses various quantitative analysis methods to evaluate the chaotic or hyperchaotic sequences generated by four chaotic systems: MK mapping, ML mapping, MT mapping, and MS mapping.

[0106] As shown in Table 1, quantitative performance analysis of four chaotic systems (MK, ML, MT, and MS maps) under typical parameters reveals the following: ML map has a high correlation dimension and slightly low sample entropy, making it suitable for scenarios requiring high complexity but moderate determinism; MT map has the lowest sample entropy, making it suitable for simplified applications with low complexity; MK map has the highest correlation dimension, demonstrating significant advantages in processing high-dimensional complex data; and MS map exhibits the best sample entropy performance, showing strong adaptability to uncertain data, and its high correlation dimension indicates its ability to effectively capture the multi-dimensional complex structure of data. Table 1 is as follows:

[0107] Table 1. Complexity Analysis Results

[0108]

[0109] As shown in Table 2, compared with the system proposed in the 2020 IEEE Transactions on Industrial Electronics paper [Two-Dimensional Memristive Hyperchaotic Maps and Application in Secure Communication], the chaotic system proposed in this embodiment shows advantages in key indicators such as the maximum Lyapunov exponent, SE, and correlation dimension. Its larger Lyapunov exponent and higher sample entropy (SE) indicate stronger uncertainty and randomness, making it suitable for scenarios such as encryption and random number generation. The higher correlation dimension (CorDim) reveals greater complexity in the system dynamics, indicating its ability to effectively characterize complex structures in multivariate data. Table 2 is as follows:

[0110] Table 2 Comparison with existing technical literature

[0111]

[0112] In summary, this embodiment uses a discrete exponential memristor as the core, coupling linear, logistic, tent, and sine maps to obtain four one-dimensional maps (MK, ML, MT, and MS) forming four chaotic systems. Numerical simulations reveal that the four hybrid systems enter chaos / hyperchaos through multiple paths under parameter dependence. Simultaneously, the dynamic behavior of the hybrid systems under initial value dependence exhibits periodic recurrence of coexisting attractors, and multiple different types of attractors exist within a single period, achieving homogeneous and heterogeneous multi-stable state synergy. Quantitative evaluation and comparison with the system proposed in the journal article [Two-Dimensional Memristive Hyperchaotic Maps and Application in Secure Communication] show that the MK-mapped hybrid system constructed in this embodiment has the highest correlation dimension (2.1024), and the MS-mapped hybrid system has the best spectral entropy (0.8915) and maximum Lyapunov exponent (0.4963), demonstrating superior performance.

[0113] Step 2: Input initial data The input is fed into the chaotic system for multiple iterative mappings, as follows:

[0114] During the first iteration, the chaotic system inputs the initial input data. Discrete exponential memristors in the first link of a chaotic system Load the initial state parameters Initial input data Input the initial state parameters into the first link Discrete exponential memristor By loading initial state parameters Discrete exponential memristor Obtain discrete sequence Discrete sequences obtained from discrete exponential memristors The scaled discrete sequence is then obtained by scaling through a linear controller. Furthermore, the initial input data Input to the one-dimensional chaotic mapping in the second link Through one-dimensional chaotic mapping One-dimensional chaotic sequence obtained Finally, the discrete sequence is processed by the adder. and one-dimensional chaotic sequence Adding them together yields the chaotic sequence of the first iteration. After the first iteration, use the initial input data. and the initial state parameters of the discrete exponential memristor The sum of these values ​​serves as the state parameter of the discrete exponential memristor after the first iteration. .

[0115] The first iteration after the first iteration In the next iteration , with the first Chaotic sequence of the next iteration As a chaotic system, the first The input data at the nth iteration, with the nth After the next iteration, the state parameters of the discrete exponential memristor are... As a chaotic system, the first Discrete exponential memristor during the next iteration State parameters; During the nth iteration, the 1st Chaotic sequence of the next iteration Input to the first link to load the first The state parameters after the next iteration Discrete exponential memristor By loading state parameters Discrete exponential memristor Obtain discrete sequence Discrete sequences obtained from discrete exponential memristors The scaled discrete sequence is then obtained by scaling through a linear controller. Furthermore, the first Chaotic sequence of the next iteration Input to the one-dimensional chaotic mapping in the second link Through one-dimensional chaotic mapping One-dimensional chaotic sequence obtained Finally, the discrete sequence is processed by the adder. and one-dimensional chaotic sequence Add them together to get the first one. Chaotic sequence of the next iteration ;No. After the iteration is completed, with the first iteration... The next iteration sequence and the After the next iteration, the state parameters of the discrete exponential memristor are... The sum of, as the first After the next iteration, the state parameters of the discrete exponential memristor are... .

[0116] Thus, through multiple iterations of the chaotic system, a chaotic sequence for each iteration is obtained.

[0117] Step 3: Construct the chaotic key based on the chaotic sequence obtained from multiple iterations of the chaotic system.

[0118] In this specific embodiment, from the chaotic sequence obtained through multiple iterations of the chaotic system, the chaotic sequence results of the first M=2000 unstable iterations are discarded, and the chaotic sequence results of subsequent stable iterations are retained. Then, the retained chaotic sequences of each iteration are input into MATLAB software, and the value of each retained chaotic sequence is denoted as X (where X is a real number vector). Using MATLAB software, X is substituted into the following formula:

[0119] ,

[0120] This converts the chaotic sequence of each iteration into a positive number Y in the interval [0, 255].

[0121] Next, the positive numbers obtained from the chaotic sequence in each iteration are converted into binary numbers using the built-in conversion function dec2bin.m in MATLAB. The binary numbers corresponding to all the chaotic sequences in each iteration are then concatenated end to end in the order of the iterations to obtain a pseudo-random number of the binary sequence, which is used as the chaotic key.

[0122] In this embodiment, the NIST SP 800-22 statistical testing standard published by the National Institute of Standards and Technology (NIST) is used to test the pseudo-random numbers constructed based on the chaotic sequences generated by the four chaotic systems MK, ML, MT, and MS in this embodiment.

[0123] NIST SP 800-22 is one of the most commonly used testing standards in the world. NIST SP 800-22 tests the balance, rate of change, periodicity, and linear complexity of a sequence to provide a reliable and comprehensive test of binary sequences. When all test results are greater than 0.01, it indicates that the sequence being tested is random.

[0124] The NIST SP 800-22 statistical test results of pseudorandom numbers constructed based on the chaotic sequences generated by the four chaotic systems MK, ML, MT, and MS in this embodiment are shown in Table 3. All test results are greater than 0.01, indicating that the chaotic sequences generated by the four chaotic systems MK, ML, MT, and MS in this embodiment have good randomness. Therefore, pseudorandom numbers generated based on chaotic sequences can be used as chaotic keys. Table 3 is as follows:

[0125] Table 3 Statistical Test Results of NIST SP 800-22

[0126]

[0127] Furthermore, in this embodiment, from the chaotic sequence obtained through multiple iterations of the chaotic system, the chaotic sequence results of the first M unstable iterations can be discarded, while the chaotic sequence results of subsequent stable iterations can be retained. The retained chaotic sequences can then be arranged into a matrix as a chaotic key. This chaotic key can be used to encrypt image data, video data, and audio data.

[0128] This embodiment also discloses a system for implementing the above-mentioned chaotic key generation method based on discrete exponential memristor coupling, including a chaotic system and a data processing module, wherein:

[0129] The chaotic system consists of two links and an adder; the first link includes a discrete exponential memristor. A linear controller, wherein the linear controller is a controller with linear parameters. The proportional controller. The second link includes a one-dimensional chaotic map. .

[0130] Discrete exponential memristor in the first link The output of the linear controller is used as the input of the linear controller, and the output of the linear controller is a one-dimensional chaotic mapping in the second link. The output is used as the input of the adder.

[0131] During the first iteration of the chaotic system, the discrete exponential memristor in the first link One-dimensional chaotic mapping in the second link Common access to initial input data And discrete exponential memristor Load initial state parameters .

[0132] The chaotic system after its first iteration At the +1 iteration, the adder's... The chaotic sequence output from each iteration is used as the discrete exponential memristor. One-dimensional chaotic mapping No. Input data for +1 iterations, and discrete exponential memristor Loading the first After the next iteration, the state parameters of the discrete exponential memristor are... .

[0133] The data processing module collects the chaotic sequences output by the adder in each iteration, and constructs pseudo-random numbers or chaotic matrices based on the chaotic sequences output by the adder in each iteration as chaotic keys.

[0134] This embodiment also discloses a data encryption method, the process of which is as follows:

[0135] Generate plaintext data, and generate pseudo-random numbers or chaotic matrices as chaotic keys using the chaotic key generation method based on discrete exponential memristor coupling described above.

[0136] When the generated chaotic key is a pseudo-random number, it is used to encrypt vectorized plaintext data. The encrypted data is obtained by XORing the pseudo-random number with the plaintext data.

[0137] When the generated chaotic key is a chaotic matrix, it is used to encrypt plaintext image data, video data, or audio data. The encrypted data is obtained by scrambling and spreading the positions of the plaintext data using the chaotic matrix.

[0138] The preferred embodiments of the present invention have been described in detail above with reference to the accompanying drawings. These embodiments are merely descriptions of preferred embodiments and are not intended to limit the scope or concept of the invention. The specific technical features described in the above embodiments can be combined in any suitable manner without contradiction. Such combinations, as long as they do not violate the spirit of the present invention, should also be considered as part of this disclosure. To avoid unnecessary repetition, the present invention will not further describe the various possible combinations.

[0139] This invention is not limited to the specific details of the above embodiments. Within the scope of the technical concept of this invention and without departing from the design idea of ​​this invention, all modifications and improvements made by those skilled in the art to the technical solutions of this invention should fall within the protection scope of this invention. The technical content for which protection is sought in this invention has been fully described in the claims.

Claims

1. A chaotic key generation method based on discrete exponential memristor coupling, characterized in that, The process is as follows: Step 1: Generate a chaotic system, which includes two links and an adder, wherein the first link includes a discrete exponential memristor. The linear parameter is A linear controller, the second link of which includes a one-dimensional chaotic mapping. ;in Represents a discrete exponential memristor State parameters, Represents a one-dimensional chaotic mapping The control parameters, This represents the input data of a chaotic system; Set the initial input data of the chaotic system And set the initial state parameters of the discrete exponential memristor in the chaotic system. ; Step 2: Input initial data The input is fed into the chaotic system for multiple iterative mappings, as follows: During the first iteration, the chaotic system inputs the initial input data. Discrete exponential memristors in the first link of a chaotic system Load the initial state parameters Initial input data Input the initial state parameters into the first link Discrete exponential memristor By loading initial state parameters Discrete exponential memristor Obtain discrete sequence Discrete sequences obtained from discrete exponential memristors The scaled discrete sequence is then obtained by scaling through a linear controller. Furthermore, the initial input data Input to the one-dimensional chaotic mapping in the second link Through one-dimensional chaotic mapping One-dimensional chaotic sequence obtained Finally, the discrete sequence is processed by the adder. and one-dimensional chaotic sequence Adding them together yields the chaotic sequence of the first iteration. ; After the first iteration, with the initial input data and the initial state parameters of the discrete exponential memristor The sum is used as the state parameter updated by the discrete exponential memristor after the first iteration. ; The first iteration after the first iteration In the next iteration , with the first Chaotic sequence of the next iteration As a chaotic system, the first The input data at the nth iteration, with the nth After the next iteration, the state parameters of the discrete exponential memristor are updated. As a chaotic system, the first Discrete exponential memristor during the next iteration State parameters; During the nth iteration, the 1st Chaotic sequence of the next iteration Input to the first link to load the first The state parameters after the next iteration Discrete exponential memristor By loading state parameters Discrete exponential memristor Obtain discrete sequence Discrete sequences obtained from discrete exponential memristors The scaled discrete sequence is then obtained by scaling through a linear controller. Furthermore, the first Chaotic sequence of the next iteration Input to the one-dimensional chaotic mapping in the second link Through one-dimensional chaotic mapping One-dimensional chaotic sequence obtained Finally, the discrete sequence is processed by the adder. and one-dimensional chaotic sequence Add them together to get the first one. Chaotic sequence of the next iteration ;No. After the iteration is completed, with the first iteration... The sequence of iterations and the After the next iteration, the state parameters of the discrete exponential memristor are... The sum of, as the first After the next iteration, the state parameters of the discrete exponential memristor are... ; Thus, through multiple iterations of the chaotic system, a chaotic sequence for each iteration is obtained; Step 3: Construct the chaotic key based on the chaotic sequence obtained from multiple iterations of the chaotic system.

2. The chaotic key generation method based on discrete exponential memristor coupling according to claim 1, characterized in that, In step 1, the one-dimensional chaotic mapping It is a linear mapping.

3. The chaotic key generation method based on discrete exponential memristor coupling according to claim 1, characterized in that, In step 1, the one-dimensional chaotic mapping This is a Logistic mapping.

4. The chaotic key generation method based on discrete exponential memristor coupling according to claim 1, characterized in that, In step 1, the one-dimensional chaotic mapping This is a Tent mapping.

5. The chaotic key generation method based on discrete exponential memristor coupling according to claim 1, characterized in that, In step 1, the one-dimensional chaotic mapping This is a Sine mapping.

6. The chaotic key generation method based on discrete exponential memristor coupling according to claim 1, characterized in that, In step 1, the initial input data of the chaotic system The initial state parameter of the discrete exponential memristor is set to 0.

001. Set it to 0.

1.

7. The chaotic key generation method based on discrete exponential memristor coupling according to claim 1, characterized in that, In step 3, from the chaotic sequence obtained by multiple iterations of the chaotic system, the chaotic sequence results of the first M unstable iterations are discarded, and the chaotic sequence results of subsequent stable iterations are retained. The retained chaotic sequences of each iteration are converted into positive numbers in the range [0, 255] and then into binary numbers. Then, the binary numbers are concatenated end to end according to the iteration order to obtain a pseudo-random number of the binary sequence, which is used as the chaotic key.

8. The chaotic key generation method based on discrete exponential memristor coupling according to claim 1, characterized in that, In step 3, from the chaotic sequence obtained by multiple iterations of the chaotic system, the chaotic sequence results of the first M unstable iterations are discarded, and the chaotic sequence results of subsequent stable iterations are retained. The retained chaotic sequences are then combined into a matrix, which serves as the chaotic key.

9. A system for implementing the chaotic key generation method based on discrete exponential memristor coupling as described in any one of claims 1-8, characterized in that, Includes a chaotic system and a data processing module, among which: The chaotic system includes two links and an adder; the first link includes a discrete exponential memristor. A linear controller, wherein the linear controller is a controller with linear parameters. Proportional controller; the second link includes a one-dimensional chaotic mapping. Discrete exponential memristors in the first link The output of the linear controller is used as the input of the linear controller, and the output of the linear controller is a one-dimensional chaotic mapping in the second link. The output is used as the input of the adder; during the first iteration of the chaotic system, the discrete exponential memristor in the first link... One-dimensional chaotic mapping in the second link Common access to initial input data And discrete exponential memristor Load initial state parameters The chaotic system after its first iteration... At the +1 iteration, the adder's... The chaotic sequence output from each iteration is used as the discrete exponential memristor. One-dimensional chaotic mapping No. Input data for +1 iterations, and discrete exponential memristor Loading the first After the next iteration, the state parameters of the discrete exponential memristor are... ; The data processing module collects the chaotic sequence output by the adder in each iteration, and constructs a chaotic key based on the chaotic sequence output by the adder in each iteration.

10. A data encryption method, characterized in that, The process is as follows: Plaintext data is generated, and a chaotic key is generated according to the chaotic key generation method based on discrete exponential memristor coupling as described in any one of claims 1-8; then the plaintext data is encrypted using the chaotic key.

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