Hopfield network model of all-memristive synaptic connections, multi-structure attractor and privacy protection method for power internet of things

Abstract

Hopfield network model of full memristive synaptic connections, multi-structure attractors and privacy protection method for power internet of things, through the stability of the equilibrium point analysis and numerical simulation, it is found that the FMSC-HNN considered can produce multi-structure attractors and realize its multi-directional modulation. By switching the strength of the memristive synapse 4, single-structure chaotic attractors with wing, butterfly and vortex types can be generated. On this basis, by adjusting the control parameters of the memristive synapses 1-3, the number of single-structure attractors can be arbitrarily set, thus forming a multi-directional expansion of multi-structure attractors. In addition, under the influence of the initial value of the memristive synapses 1-3, homogeneous extreme multi-stability can be formed, which is manifested as multi-directional regulation of multi-structure attractors. A digital circuit experimental platform based on microcontroller is developed, and complex multi-directional modulation of multi-structure attractors is reproduced, which improves the value of FMSC-HNN in the application of chaos engineering.

Description

Technical Field

[0001] This invention relates to the field of Hopfield network model research, and particularly to a Hopfield network model with full memristor synaptic connections, multi-structure attractors, and a privacy protection method for the power Internet of Things. Background Technology

[0002] Biological neural systems have been proven to be complex nonlinear dynamic systems, and their chaotic dynamics are the essence of the brain's perception, decision-making, and task execution. With the continued deepening of neural network applications, the architecture, dynamic mechanisms, and circuit implementation of biological neural systems have greatly stimulated research enthusiasm among scholars. Hopfield neural networks (HNNs), considering the flexible connectivity and electrical signal transmission mechanisms of neurons, can realistically reflect brain-like structures and dynamics. Therefore, diverse HNNs have been constructed and reproduced brain-like dynamic behaviors, including chaotic dynamics, multistability, synchronization, and collective behavior. On the one hand, this provides theoretical guidance for revealing neural electrical activity, information processing mechanisms, and the mechanisms of neurological diseases; on the other hand, it broadens the application scope of HNNs, including pattern recognition, image compressed sensing, and Sudoku puzzles. Therefore, innovatively proposing more biomimetic HNNs and deeply exploring brain-like chaotic dynamics are key to promoting the development of computational neuroscience and chaos engineering applications.

[0003] The concept of memristors was first pioneered by Tsai Shao-tang in 1971 and confirmed in 2008 by a TiO2 memristor device fabricated by HP Labs. In essence, a memristor is a memory-resistive element that describes the nonlinear constraint relationship between magnetic flux and charge, and can sense voltage changes to generate induced current. Therefore, due to memory effects, ion migration effects, and unique nonlinearities, magnetically controlled memristors have become the most suitable electronic components for describing effects such as neural synapses, electromagnetic radiation, and electromagnetic induction. Based on this fact, by incorporating memristor synapses and memristor electromagnetic radiation, the biomimetic and dynamic complexity of memristor-controlled neural networks (HNNs) has been significantly improved. Using memristors to simulate the synapses and electromagnetic radiation of neurons in HNNs, Yu et al. revealed periodic and chaotic burst oscillation behavior with coexistence and mode transition characteristics. Chen et al. constructed a ReLU-type HNN based on locally active memristor synapses, achieving synchronously adjustable amplitude and offset dynamics, and verified the numerical results using analog and digital circuits. Memristor synaptic crosstalk coupled HNNs and memristor synaptic cascaded clustered HNNs were also established, revealing the coupling interactions and collective dynamics between brain-like functional regions, respectively. Therefore, based on memristor synapses or memristor electromagnetic radiation, brain-like structures and dynamics were powerfully characterized, contributing to revealing the dynamic characteristics of the nervous system and deepening our understanding of neural information patterns.

[0004] To advance the application of memristor HNNs, current research focuses on constructing high-dimensional memristor HNNs to generate complex, multi-directionally expanding, and modulated chaotic attractors with various structural types such as vortices, butterflies, and wings. Segmented memristors are used to control the number of additional equilibrium points in the memristor HNN, acting as connecting bonds to expand the number of chaotic attractors. Li et al. proposed memristor self-synapses based on nested composite functions, inducing unidirectionally multi-vortex chaotic attractors and distance-controllable coexistence attractors in memristor HNNs. To overcome the bottleneck of three-neuron recurrent HNNs struggling to generate chaos, Zhang et al. introduced two memristors to enable memristor HNNs to generate an arbitrary number of multi-vortex hidden chaotic attractors along a single dimension, and designed digital circuits and pseudo-random number generators to demonstrate the complexity of chaos. Ding et al. focused on generating complex hidden multi-winged chaotic attractors in memristor HNNs, which were verified on a digital circuit hardware platform. Lin et al. redefined multi-chaotic attractors as multi-structure attractors and performed multi-directional expansion and modulation of the structural attractor in a high-dimensional memristor HNN composed of four neurons with different topologies. Unlike conventional chaotic attractors, the expanded and modulated attractor can provide any number of high-performance pseudo-random sequences for chaos engineering applications, but all use high-dimensional three / four / multi-neuron HNNs as generators, thus increasing computational complexity. Furthermore, existing research only considers incorporating memristors into the HNN to replace some weights; what kind of dynamic behavior will a fully memristor synaptic HNN exhibit? Therefore, establishing a more biomimetic, lightweight memristor HNN while simultaneously generating multi-directionally expanded and modulated multi-structure chaotic attractors is particularly important and challenging.

[0005] In fact, the multi-directional modulated chaotic attractors generated by the aforementioned high-dimensional memristor HNNs have played a significant role in privacy protection and secure communication in the industrial field. Wang et al. proposed an encryption strategy based on bi-site DNA scrambling and dynamic matrix diffusion, using spatial multi-structure attractors generated by high-dimensional memristor HNNs, and successfully applied it to military image privacy protection. Sun et al. proposed an Arnold mapping diffusion method based on DNA-Cube, combining spatial multi-structure memristor HNNs to address the encryption needs of industrial images. Zhang et al. proposed a medical image secure communication scheme with key update functionality using multi-directional extended multi-vortex chaotic attractors generated by memristor HNNs. Yu et al. proposed an encryption scheme based on high-dimensional memristor HNNs for video and voice information in the Industrial Internet of Things (IIoT) and verified its security performance. Wang et al. constructed a seven-neuron HNN based on three memristor synapses and applied multi-directional hyperchaotic multi-vortex attractors to commercial data encryption communication scenarios. Lai et al. discovered unidirectional extended and modulated multi-vortex attractors in a four-neuron memristor HNN and integrated them into the merging and replacement process of image pixels, protecting the privacy information of the image. The aforementioned literature uses high-dimensional memristor HNNs and complex encryption algorithms to ensure the security of privacy information in multiple scenarios. However, the computational complexity is insufficient to meet the requirements for online encryption at edge devices. Therefore, it is particularly important to continue proposing biomimetic HNN architectures and their encryption algorithms that balance simplicity and reliability for edge computing. Furthermore, to the authors' knowledge, privacy protection and secure transmission across the entire power IoT chain in some critical scenarios reliant on edge computing technology have not yet received sufficient attention.

[0006] In summary, to address the limitations of the aforementioned research, we consider a neural signal generator based on a minimal HNN consisting of two neurons, incorporating four memristor synapses to propose a fully memristor synaptic HNN. The FSMC-HNN more closely resembles the real nervous system, and the memristor synaptic effects can induce more complex chaotic dynamics. Interestingly, the considered FSMC-HNN can generate chaotic attractors with wing, butterfly, and vortex-like patterns. Specifically, under the influence of memristor synaptic parameters and initial values, these attractors can form multi-directionally expanding and modulated multi-structure attractors in unidirectional, bidirectional, and spatial configurations. Subsequently, a microcontroller-based digital circuit was developed for the FSMC-HNN to verify the results of numerical simulations. To the authors' knowledge, this is the first time that three different types of chaotic attractors and complex spatial multi-structure attractors have been simultaneously generated in a memristor two-neuron HNN. Based on this, an extremely simple encryption and transmission scheme for privacy information in the power Internet of Things is proposed, and its effectiveness is verified through security performance discussion and hardware experiments. Summary of the Invention

[0007] To address the problems existing in the prior art, the present invention aims to provide a Hopfield network model with full memristor synaptic connections, a multi-structure attractor, and a privacy protection method for the power Internet of Things.

[0008] To achieve the above objectives, the technical solution of the present invention is as follows: The Hopfield network model for fully memristor synaptic connections can be mathematically established as follows:

[0009] in, and These represent the membrane potentials of the two neurons; Used to adjust the strength of memristor synapses; mem derivative function This refers to the self-synapses and connective synapse weights in the original HNN. After extensive trial and error, the fixed parameters are declared here beforehand. For MS 1 to MS 3, the relevant fixed parameter settings are as follows: , , and For MS 4, settings Secondly, assume the initial value of FMSC-HNN is...

[0010] The construction of multi-structure attractors is as follows: Visualizing the dynamics of FMSC-HNN from a numerical simulation perspective is an effective way to reveal the evolutionary mechanisms of neural dynamics, providing a solid foundation for the development, analysis, and application of brain-like neural systems. This section mainly utilizes 1-D / 2-D bifurcation diagrams, phase trajectory diagrams, and local attractor basins to investigate the multi-directional expansion and regulated multi-structure attractor behavior exhibited by FMSC-HNN. It should be noted that the numerical simulations were performed on Matlab 2021a and Julia platforms, and the ODE45 algorithm was used to obtain the numerical solution of the FMSC-HNN system. In this process, the total simulation time and step size were set to 10000 s and 0.05 s, respectively.

[0011] Step 1: Parameter-dependent 1-D / 2-D bifurcation Performing 1-D / 2-D bifurcation graph simulations with varying parameters can reveal the distribution domain and evolution mechanism of dynamic behavior, facilitating the acquisition of multi-structure chaotic attractors. First, specify... , and And select the relevant parameters for MS 4. and As a control parameter, at this time, it can be... Construct a plane with a specification of The dense parameter matrix Secondly, a Wolf Lyapunov exponent and state sequence-based approach is used. The peak cycle count calculation algorithm is used to traverse each parameter combination, thereby effectively distinguishing a total of 11 dynamic features in FSMC-HNN. Then, after assigning color labels to each dynamic behavior, A 2D bifurcation diagram of the plane can be rendered. The yellow portion marked by CH represents a chaotic parameter domain with a positive LE, while the green portion marked by QP represents a quasi-periodic parameter domain with at least two zero LEs. For LEs less than 0, they are subdivided into parameter domains for the stable point (SP), period 1 (P01), and periods 2 (P02) to 8 (P08), filled with gray, blue, and other colors respectively. This shows that CH occupies a large planar region, implying that FMSC-HNN readily generates multi-structure chaotic attractors. Simultaneously, the intricate distribution of dynamic parameter domains confirms that parameter sensitivity induces complex dynamic behaviors.

[0012] choose Extracted The 1-D bifurcation diagram of the control is obtained, and the corresponding LEs curve can be obtained by calculating LEs. With Increasing from 0, FMSC-HNN in The interval generates stable point type bifurcations, and respectively at... and It splits into P01 and P02 bifurcation types. For trajectory Entering the CH bifurcation interval represented by the red point set from the tangent bifurcation, especially... A chaotic crisis occurred, with two CH bifurcation clusters degenerating into one, signifying a fundamental change in the morphology of the chaotic attractor. Observing the three LEs curves, it can be seen that LEs are less than zero in the periodic bifurcation interval, while LE1 remains greater than zero in the chaotic bifurcation interval, and the evolution trend of LEs conforms to bifurcation characteristics. In summary, the global two-dimensional dynamic distribution is verified from a local perspective, proving the effectiveness of the analysis. Therefore, we can select typical parameters from the 1-D bifurcation diagram and further generate different single-structure chaotic attractors. Observation shows that... , , The planar representation shows chaotic phase trajectories with wing, butterfly, and vortex types, thus FMSC-HNN can generate multi-structure attractors. Of course, this varies with the intensity of MS 4. Switching to 0.6, 0.52, and 0.3 in sequence, the chaotic attractors exhibited single, double, and quadruple structures, respectively.

[0013] Step 2: Multi-directional extended multi-structure chaotic attractor The use of segmented memristors in MS 1 to MS 3 results in multidirectional controllability of the number of EPs. Based on this, the multidirectional expansion problem of single-structure attractors has attracted attention, but there is a lack of relevant research resources in two-neuron HNNs.

[0014] One-way expansion: with Taking the induced single-structure chaotic attractor as an example, when the adjustable parameters , and They are respectively used as control parameters in When changing within an interval, regarding , and The bifurcation point and the first three LE curves. It can be observed that, with... Add, about , and The bifurcation diagrams all show a step-growth trend, and in , , , , and The study presents multiple attractors with structures of 2, 3, 4, 5, 6, and 7. In these cases, the LE curves all change smoothly, and calculations show... , , , , and .because This proves that FMSC-HNN can generate unidirectional ( , and The extended multi-structure chaotic attractor exhibits robust chaotic properties. Next, we select... and The descriptions of FMSC-HNN along the path are as follows: , and The 3 / 4 / 5 / 6-wing, butterfly, and vortex chaotic attractor phase trajectories generated by the direction. Observation reveals that single-structure attractors can extend unidirectionally to ( The multi-structured chaotic attractor proves the validity of the bifurcation diagram and theoretical analysis.

[0015] Planar extension: fixed and ,when At that time, the vortex-type chaotic attractor can be The plane is effectively controlled. Of course, it has been clearly established that the multi-structure attractor is composed of… The controlled multi-EP triggering means that, under the simultaneous action of MS 2 and MS 3, the attractor generated by FMSC-HNN exhibits planar expansion characteristics. Therefore, in The plane was further expanded , , , , and Multi-structured chaotic attractors with vortex type. Since the multi-structured chaotic attractor is connected by a different number of vortex chaotic attractors, and each attractor trajectory has similarity, it is proved that FMSC-HNN has lossless planar expansion properties.

[0016] Spatial Expansion: Because FMSC-HNN contains three memristor synapses with piecewise linear functions, when MS1, MS2, and MS3 act simultaneously, FMSC-HNN... The expansion of space naturally arouses research interest. Fixed ,when At that time, FMSC-HNN in , and It is controllable in both direction, thus forming , and Multi-structure chaotic attractors with vortex type; in fact, , and It can be specified as any integer to form a space expansion. Multi-structured chaotic attractors. To the authors' knowledge, multi-directionally expanding multi-structured chaotic attractors have not yet been reported in two-neuron HNNs, providing controllable and non-decaying chaotic sequences for practical industrial applications.

[0017] Step 3: Multi-directionally modulated coexisting multi-structure attractors The controllability of FMSC-HNN can be increased by using a memristor synaptic model with piecewise functions, i.e., generating multi-structure attractors with multiple extensions. Recent research has reported that realizing novel offset modulation dynamics in memristor HNNs provides insights into increasing network flexibility and multi-stability. Therefore, can multi-structure attractors with unidirectional, planar, and spatial offset modulation be further realized in FMSC-HNNs? This question warrants further investigation. In this section, we maintain... and Unchanged, with , and As a control parameter.

[0018] Unidirectional offset control: changing the internal parameters of MS 1 It is 3.7, depending on the control parameters. A changing 1D coexisting bifurcation diagram. It can be observed that the bifurcation points... Along with an offset of 2 The orientation is copied, causing the position of the attractor in phase space to be modulated while its shape remains unchanged. Therefore, we give... flat Multi-structure attractors with vortex type are spaced at a distance of 2. Regulation. Similarly, by and The controlled 1D coexistence bifurcation diagram shows the phenomenon of bifurcation point offset regulation, and correspondingly, and flat Multi-structure attractors can be separately and The modulation consists of four homogeneous phase trajectories. In addition, the LE curve shows stable fluctuations, indicating that FMSC-HNN can non-destructively generate coexisting multi-structure chaotic attractors with unidirectional offset modulation.

[0019] Planar offset control: The internal parameters of MS 1 and MS 2 are set to... and Based on the position and amplitude information of the sequence generated by FMSC-HNN, it can be used in... The plane is divided into 16 local attraction basins with different attraction domains. Since each basin has the same area and coexists in the same phase space, this means that FMSC-HNN produces homogeneous multistability regulated by plane migration. Therefore, [the following is a separate, unrelated sentence:] ...set A vortex-type 4-structure attractor can be initialized by a memristor synapse. and Nine homogeneous attractors with an offset of 2 were used to verify the effectiveness of the local attraction basin. Furthermore, the internal parameters of MS 2 and MS 3 were set to... and It depicts the and Phase trajectories of modulated planar coexisting homogeneous multi-structure attractors. (Adding...) After the variable, The discovery of coexisting 4-winged chaotic attractors with planar modulation in space indicates that different memristor synapses have varying effects on network dynamics. Therefore, the above analysis demonstrates that FMSC-HNN can generate coexisting multi-structure chaotic attractors with planar modulation.

[0020] Spatial offset control: The internal parameters of MS 1, MS 2, and MS 3 are set to... It can be inferred that the initial values ​​of the three segmented memristor synapse models all possess periodic cyclical characteristics, implying that regulatory dynamics can be generated in three dimensions. Therefore, in the set... After selecting the initial value of the memristor, coexisting single-cycle and vortex chaotic attractors with spatial offset control can be generated respectively. In fact, FMSC-HNN can be controlled by spatial offset in... A total of 64 coexisting multi-structure attractors can be generated in the phase space, and the interval All inside All meet Therefore, the piecewise linear functions in the MS 1, MS 2, and MS 3 models possess topological invariance, which is the fundamental reason for offset modulation. Thus, in Under control, the multi-structure attractors of the memristor synaptic initial displacement modulator in FMSC-HNN are distributed in unidirectional, planar, and spatial directions, with corresponding numbers of [number missing]. , and indivual.

[0021] The privacy protection method for the power Internet of Things (IoT) consists of three levels: information collection and encryption, cloud transmission, and information reception and decryption. First, edge devices (UAVs and UVs) conduct real-time inspections of power plants and power lines on the generation side, capturing privacy-preserving image information. This information undergoes online encryption within the edge devices, consuming computing resources. Second, the encrypted privacy information is transmitted over long distances via a cloud server. At this point, attackers can only obtain encrypted images that are difficult to distinguish, meaning the transmission is secure. Finally, when the data center subscribes to the server, the server sends the key and privacy images, among other things. Therefore, the data center can obtain the privacy information through the decryption system, thereby achieving reliable monitoring and evaluation of the secure and stable operation of remote power generation. The encryption algorithm is designed as follows: As the architecture reveals, the core of privacy protection and secure transmission in the power Internet of Things (IoT) lies in the effective encryption of edge devices and the effective decryption process of the operation and maintenance center. However, the real-time and high-speed encryption process of edge devices places high demands on resource utilization. Therefore, proposing secure and lightweight encryption algorithms is a way to overcome hardware limitations. Unlike complex encryption algorithms, this paper designs an encryption algorithm with scrambling, bidirectional diffusion, and substitution structures based on the proposed FMSC-HNN. It utilizes FMSC-HNN to generate multi-directionally expanding / modulating multi-structure chaotic attractors and extracts high-performance pseudo-random sequences to participate in various encryption processes, as detailed below: (1) System configuration: including security key, proposed model, and running algorithm. Select system parameters. and initial conditions The security key is input into the FMSC-HNN, and the simulation time and the fourth-order Runge-Kutta method are set to solve the FMSC-HNN.

[0022] (2) Generate chaotic pseudo-random sequences: when plaintext images The specifications are At that time, FMSC-HNN was iterated This generates six chaotic pseudo-random sequences. , , , , and .

[0023] (3) Key sequence generator: Discard the first 1000 transient data, length is A chaotic pseudo-random sequence is converted into a valid key sequence using a key sequence generator. , , and The details are as follows:

[0024] in, , This represents the modulo operation. express The absolute value, Used to obtain less than or equal to The most recent integer. It can be observed that the security of the key sequence benefits from the chaotic complexity of the sequence and the correlation between the sequences.

[0025] (4) Scrambling process: The key sequence is... Sort and obtain the corresponding index sequence to scramble the plaintext image pixel matrix. The position of the scrambled pixel matrix. as follows:

[0026] (5) Diffusion process: using key sequence and A bidirectional diffusion scheme is proposed to change The pixel value. The forward diffusion process is described as follows:

[0027] in, .

[0028] The reverse diffusion process is described as follows:

[0029] in, and .

[0030] (6) Substitution process: using key sequence With pixel matrix Performing a bitwise XOR operation further enhances security, described as follows:

[0031] At this time, Reorganized into a size of The encrypted ciphertext image can be obtained from the pixel matrix. Furthermore, the decryption process is the reverse of the encryption process. It is worth noting that the proposed encryption algorithm is universal and can guarantee the information security of grayscale, color images, and videos.

[0032] Compared with the prior art, the beneficial effects of the present invention are as follows: (1) A novel HNN with multidirectional modulation dynamics and all-memristor synaptic connections was proposed using the simplest HNN and four magnetically controlled memristors; (2) Complex dynamics were discovered for the first time in the simplest HNN, including chaotic attractors with wing, butterfly, and vortex types, as well as multi-structure attractors with multidirectional expansion / modulation; (3) A microcontroller-based digital circuit platform was developed and circuit experiments were performed; (4) A simple privacy protection architecture and encryption scheme for the power IoT was proposed based on FSMC-HNN, and the expected security indicators were achieved in performance testing and hardware encrypted transmission experiments. Two types of memristor synapses were also proposed and the modeling process of FMSC-HNN was introduced. Multi-structure chaotic dynamic attractors with multidirectional modulation were discovered and revealed through numerical simulation. A comprehensive experiment of digital circuits was performed. Finally, the security of the proposed power IoT privacy protection and transmission scheme was tested through numerical analysis and hardware experiments. Attached Figure Description

[0033] Figure 1 Electrical characteristics of two types of memristors. (a) Frequency-controlled PHLs of model (1); (b) Amplitude-controlled PHLs of model (1); (c) Frequency-controlled model (2); (d) Amplitude-controlled model (2).

[0034] Figure 2 Schematic diagram of the topology of a two-neuron HNN with full memristor synaptic connections; Figure 3 exist A schematic diagram of the equilibrium point generated by the plane; Figure 4 exist Planar generated equilibrium points and multi-structure attractors. (a) (b) (c) ; Figure 5 MS 4-parameter related 1-D / 2-D bifurcation evolution process. (a) (b) 2D bifurcation diagram of the plane; One-dimensional bifurcation diagram of control; (c) Related Figure 6 MS 4 parameter-dependent multi-structure attractors, including wing, butterfly, and spiral types. (a) (b) (c) .

[0035] Figure 7 Follow The changing bifurcation diagram and LEs curve. (a) (b) (c) .

[0036] Figure 8 and Phase diagram of a time-unidirectionally expanding multi-structured chaotic attractor. (a) (b) A directionally expanding multi-winged chaotic attractor; (b) A multi-butterfly chaotic attractor with directional expansion; Directional expansion of multi-vortex chaotic attraction.

[0037] Figure 9 , and Phase trajectory diagram of a planar extended multi-structured chaotic attractor. (a) Structure; (b) Structure; (c) Structure; (d) Structure; (e) Structure; (f) structure.

[0038] Figure 10 At that time, the spatially extended multi-structure attractor. (a) (b) Structural attractor; Structural attractors; (c) Structural attractors Figure 11 One-dimensional coexistence bifurcation diagram and LEs curve induced by memristor synapses. (a) Regulation, (b) Regulation, (c) Regulation, .

[0039] Figure 12 At that time, a coexisting multi-structure chaotic attractor controlled by unidirectional offset. (a) Regulation; (b) Regulation; (c) Regulation.

[0040] Figure 13 Planar offset-controlled coexisting attractor basins with multiple structures and chaotic attractors. (a) (b) Planar localized attraction basin; , Planar displacement control; (c) , Planar displacement control; (d) space Figure 14 Spatially modulated offset-controlled coexisting multi-structure attractors. (a) (a) The periodic attractor of regulation; (b) At that time, the multi-structured chaotic attractor is regulated.

[0041] Figure 15 Digital circuit implementation platform for FMSC-HNN Figure 16 Experimental results of digital circuit implementation platform output Figure 17 Power Internet of Things Privacy Protection and Secure Transmission Architecture Figure 18 A schematic diagram of the proposed encryption method Figure 19 Histogram analysis. (a1-a4) Plaintext image; (b1-b4) Histogram of plaintext image; (c1-c4) Ciphertext image; (d1-d4) Histogram of ciphertext image. Figure 20 Distribution of adjacent pixels. (a1-a4) Adjacent pixels of the plaintext image; (b1-b4) Adjacent pixels of the ciphertext image.

[0042] Figure 21 Decrypted images under different keys. (a1-a4) correspond to the correct key; (b1-b4) show cases where the key is perturbed.

[0043] Figure 22Encryption performance analysis under data loss conditions. (a1-a4) and (c1-c4) are encrypted images cropped to 1 / 16, 1 / 8, 1 / 4, and 1 / 2 respectively; (b1-b4) and (d1-d4) are decrypted images after cropping.

[0044] Figure 23 Encryption performance analysis under data interference conditions. (a1-a4) Decrypted images after applying 1%, 3%, 5%, and 7% salt-and-pepper noise; (b1-b4) Decrypted images after applying 2%, 4%, 6%, and 8% salt-and-pepper noise.

[0045] Figure 24 Experimental Platform and Results for Privacy Protection Hardware in the Power Internet of Things Detailed Implementation

[0046] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments: like Figure 1-24 As shown, the Hopfield network model with all-memristor synaptic connections, multi-structure attractors, and privacy protection methods for the power Internet of Things are detailed below: Example 1: Two-neuron HNN with total memristor synaptic connection This embodiment proposes two types of magnetically controlled memristor synaptic models and tests the electrical characteristics of the hysteresis loop. By characterizing memristor-connected synapses and self-synapses, a novel two-neuron Hopfield neural network with fully memristor synaptic connections is constructed. Next, the effective distribution characteristics and stability of the equilibrium point set are analyzed and simulated to align with the generation of attractors.

[0047] 1. Two types of memristors and their electrical characteristics The memconductance of a magnetically controlled memristor can record the magnetic flux flowing through it, which in turn can generate an induced current under the influence of voltage. Utilizing this memory characteristic, the memristor-simulated memristor synapses and memristor electromagnetic radiation effects have played a significant role in constructing biomimetic nervous systems and generating complex dynamics.

[0048] Memristors modeled by different differential equations will have different effects on the dynamic control of the memristor system. Therefore, based on the definition and theory of memristors, a first-type flux-controlled memristor model is considered to increase the dynamic complexity of a two-neuron HNN network, as follows: in, This is an adjustable parameter for controlling the equilibrium point or the number of attractors. Unlike other segmented memristors, this model only requires a single switching function to control either an odd or even number of attractors.

[0049] Before incorporating memristors into the nervous system by simulating synaptic properties, it is necessary to verify whether the electrical characteristics of the two types of models conform to the three characteristic fingerprints of memristors. First, the control parameters are set as follows: , , and And apply a periodic voltage excitation to the model described by equations (1) and (2). Next, regarding and The hysteresis loop of the model shown in equation (1) shows a contraction trend as the frequency increases, as... Figure 1 As shown in (a). For and The sidelobe area of ​​the hysteresis loop also gradually increases, such as Figure 1 As shown in (b). Similarly, Figure 1 (c) and Figure 1 (d) Display model (2) exhibits frequency dependence under parameter settings ( and ) and amplitude dependence ( and Hysteresis loop. Since the electrical properties prove that the inclined hysteresis loop passes through the origin and is adjusted by frequency and amplitude, it means that the two types of models are memristor models with memory characteristics.

[0050] 2. Construction of the Memristor HNN Model HNN is widely recognized as the most reliable model for simulating brain-like topology and dynamic evolution mechanisms, and its generated dynamic attractors are extensively used in optimization and pattern recognition tasks. It is a complex artificial neural network composed of multiple neurons connected by synapses, and can be described using differential dynamic equations:

[0051] in, and This is an index for Hopfield neurons; The number of neurons; and These are the neuron's internal membrane potential and output potential, respectively. The integral capacitance used to characterize the cell membrane is set to... ; Specify as transmembrane resistance, set to ;in addition, Representing the effect of external electrical stimulation on neurons, set to ; Defined as the first The first neuron to the second The synaptic weight coefficients between neurons are represented by their electrical conductance values.

[0052] On the one hand, traditional HNN models, due to their low degree of nonlinearity, struggle to represent rich neural system dynamics. On the other hand, due to fixed resistive weights, HNN circuits must replace resistive elements when handling complex tasks, implying a gap between the functional structure of HNNs and real neural networks. Two-neuron HNNs, as the simplest type of neural network, struggle to generate multi-directional, multi-structure attractors similar to high-dimensional memristor HNNs. To address this issue, a two-neuron HNN is used as the base network, and its basic mathematical model is described as follows:

[0053] Benefiting from the synaptic-like properties and nonlinearity of memristors, establishing memristor-based HNNs is an effective way to overcome the aforementioned limitations. Therefore, we innovatively propose a two-neuron HNN with fully memristor synaptic connections (FMSC-HNN), the topology of which is shown in the schematic diagram below. Figure 2 As shown. It contains two neurons, two self-feedback synapses, and two connecting synapses, where MS 1, MS 2, and MS 3 are derived from the memristor model (2), while MS 4 is actually represented by the memristor model (1). At this point, combining equations (1), (2), and (5), a mathematical model of a two-neuron HNN with full memristor synaptic connections can be established as follows:

[0054] in, and These represent the membrane potentials of the two neurons; Used to adjust the strength of memristor synapses; mem derivative function This refers to the self-synapses and connective synapse weights in the original HNN. After extensive trial and error, the fixed parameters are declared here beforehand. For MS 1 to MS 3, the relevant fixed parameter settings are as follows: , , and For MS 4, settings Secondly, assume the initial value of FMSC-HNN is...

[0055] 3. Stability analysis of the equilibrium point For non-hidden neural systems, the evolutionary trajectories of kinetic attractors are all excited by system equilibrium points (EPs), and the stability analysis of EPs can be performed from both theoretical and simulation perspectives. When the right-hand side equation of the FMSC-HNN model (6) is 0, i.e. It can be assumed We can directly obtain a special set of equilibrium points, represented as:

[0056] because , and Equivalent to the piecewise nonsmooth function described in equation (3), therefore according to and The waveform curve can be found The plane determines the distribution of EPs, such as Figure 3 As shown in the figure. The figure contains three types of equilibrium points: EP1, EP2, and EP3, and these equilibrium points change with the control parameters. As the function increases, its periodicity causes the number of EP values ​​to increase as well, following a multi-directional growth pattern. For example, Dominate One EP1, two EP2s and two EP3s, and Dominate EP1, 3 EP2s, and 3 EP3s. Therefore, the number of EPs is related to... The relationship can be naturally summarized as follows: , It can be inferred that the EP of FMSC-HNN can follow , and Three dimensions are controlled parameters adjust.

[0057] For the general form of EPs, the relationships between the elements in EPs must satisfy:

[0058] To determine the distribution and stability of the equilibrium point set generated by equation (8), we set... , and At this point, specify Then, the ezplot function can be used to plot the graph. and The intersection points of the curves are thus determined and are called EPs, such as... Figure 4 As shown. Analysis shows that, Figure 3 and Figure 4 The coordinate distribution of EPs in the equations corresponds perfectly, meaning that FMSC-HNN only contains the set of equilibrium points represented by equation (8). Under the premise that FMSC-HNN can Planar multi-structure attractors are generated, and their phase trajectories are represented by pink curves. Interestingly, since EP1, indicated by the purple square, is located within the cavity of each vortex attractor, EP1 is used to generate the trajectory of a single vortex. EP2, indicated by the green circle, and EP3, indicated by the yellow triangle, are located between two chaotic attractors, thus EP2 and EP3 play a crucial connecting role in expanding the multi-structure attractor. Therefore, with... In addition, FMSC-HNN can produce results in three dimensions. Each EP1 is connected via corresponding EP2 units, thus forming a multi-directional extension. -Structural attractors.

[0059] Furthermore, clarifying the properties of EPs is a crucial procedure for exploring the reasons for attractor generation. To this end, the Jacobian matrix of the associated EPs can be calculated as follows:

[0060] Therefore, combine equations (10) and (11) and solve for... The characteristic roots can be obtained, and their unified standard form is expressed as:

[0061] Because of system parameters Therefore, the stability of EPs is determined by and Decision. When and When EPs satisfy The total count is 21. Substituting other parameters into the calculation, we can now see that all particular roots at EPs are real roots and This implies that EPs are unstable and that the multi-structure attractor of FMSC-HNN is excited by unstable self-excited EPs. Therefore, the above analysis verifies the consistency between theoretical analysis and numerical simulation.

[0062] Example 2: Dynamic Analysis Based on Numerical Simulation Visualizing the dynamics of FMSC-HNN from a numerical simulation perspective is an effective way to reveal the evolutionary mechanisms of neural dynamics, providing a solid foundation for the development, analysis, and application of brain-like neural systems. This section mainly utilizes 1-D / 2-D bifurcation diagrams, phase trajectory diagrams, and local attractor basins to investigate the multi-directional expansion and regulated multi-structure attractor behavior exhibited by FMSC-HNN. It should be noted that the numerical simulations were performed on Matlab 2021a and Julia platforms, and the ODE45 algorithm was used to obtain the numerical solution of the FMSC-HNN system. In this process, the total simulation time and step size were set to 10000 s and 0.05 s, respectively.

[0063] 1. Parameter-dependent 1-D / 2-D bifurcation Performing 1-D / 2-D bifurcation graph simulations with varying parameters can reveal the distribution domain and evolution mechanism of dynamic behavior, facilitating the acquisition of multi-structure chaotic attractors. First, specify... , and And select the relevant parameters for MS 4. and As a control parameter, at this time, it can be... Construct a plane with a specification of The dense parameter matrix Secondly, a Wolf Lyapunov exponent and state sequence-based approach is used. The peak cycle count calculation algorithm is used to traverse each parameter combination, thereby effectively distinguishing a total of 11 dynamic features in FMSC-HNN. Then, after assigning a color code to each dynamic behavior, A 2D bifurcation diagram in a plane can be rendered, such as... Figure 5 As shown in (a), the yellow portion marked by CH represents a chaotic parameter domain with a positive LE, while the green portion marked by QP represents a quasi-periodic parameter domain with at least two zero LEs. For LEs less than 0, the domains are further subdivided into the stable point (SP), period 1 (P01), and period 2 (P02) to period 8 (P08) parameter domains, filled with gray, blue, and other colors respectively. This demonstrates that CH occupies a large planar region, indicating that FMSC-HNN readily generates multi-structure chaotic attractors. Furthermore, the intricate distribution of dynamic parameter domains confirms that parameter sensitivity induces complex dynamic behaviors.

[0064] choose From Figure 5 Extracted from (a) The 1-D bifurcation diagram of the control is obtained, and the corresponding LEs curves can be obtained by calculating LEs, as shown below. Figure 5 As shown in (b) and (c). With Increasing from 0, FMSC-HNN in The interval generates stable point type bifurcations, and respectively at... and It splits into P01 and P02 bifurcation types. For trajectory Entering the CH bifurcation interval represented by the red point set from the tangent bifurcation, especially... A chaotic crisis has occurred, with the two CH bifurcation clusters degenerating into one cluster, signifying a fundamental change in the morphology of the chaotic attractor. Observing the three LEs curves, it can be seen that LEs are less than zero in the periodic bifurcation interval, while LE1 remains greater than zero in the chaotic bifurcation interval, and the evolution trend of LEs conforms to bifurcation characteristics. In summary, Figure 5 (a) and (b) verify the global two-dimensional dynamic distribution from a local perspective, proving the effectiveness of the analysis. Thus, we can select typical parameters from the 1-D bifurcation diagram and further generate different single-structure chaotic attractors, whose phase trajectories are as follows... Figure 6 As shown. Observation shows that, in , , The planar representation shows chaotic phase trajectories with wing, butterfly, and vortex types, thus FMSC-HNN can generate multi-structure attractors. Of course, this varies with the intensity of MS 4. Switch to 0.6, 0.52, and 0.3 in sequence. Figure 6 (a), (b), and (c) respectively present single, double, and quadruple structures of chaotic attractors.

[0065] 2 Multi-directional extended multi-structure chaotic attractor The use of segmented memristors in MS 1 to MS 3 results in multidirectional controllability of the number of EPs. Based on this, the multidirectional expansion problem of single-structure attractors has attracted attention, but there is a lack of relevant research resources in two-neuron HNNs.

[0066] One-way expansion: with Induced Figure 6 Taking the single-structure chaotic attractor shown in (c) as an example, when the adjustable parameter... , and They are respectively used as control parameters in When changing within an interval, regarding , and The bifurcation point and the first three LE curves are as follows Figure 7 As shown in (a), (b), and (c), it can be observed that, with... Add, about , and The bifurcation diagrams all show a step-growth trend, and in , , , , and The study presents multiple attractors with structures of 2, 3, 4, 5, 6, and 7. In these cases, the LE curves all change smoothly, and calculations show... , , , , and .because This proves that FMSC-HNN can generate unidirectional ( , and The extended multi-structure chaotic attractor exhibits robust chaotic properties. Next, we select... and , Figure 8 (a), (b), and (c) describe the FMSC-HNN along... , and The 3 / 4 / 5 / 6-wing, butterfly, and vortex chaotic attractor phase trajectories generated by the direction. Observation reveals that... Figure 6 (c) shows a single-structure attractor that can be unidirectionally extended to ( The multi-structured chaotic attractor proves the validity of the bifurcation diagram and theoretical analysis.

[0067] Planar extension: fixed and ,when At that time, the vortex-type chaotic attractor can be The plane is effectively controlled. Of course, by Figure 4 The multi-structure attractor has been clearly defined by The controlled multi-EP triggering means that, under the simultaneous action of MS 2 and MS 3, the attractor generated by FMSC-HNN exhibits planar expansion characteristics. Therefore, Figure 9 (a) to (f) in The plane was further expanded , , , , and Multi-structured chaotic attractors with vortex type. Since the multi-structured chaotic attractor is connected by a different number of vortex chaotic attractors, and each attractor trajectory has similarity, it is proved that FMSC-HNN has lossless planar expansion properties.

[0068] Spatial Expansion: Because FMSC-HNN contains three memristor synapses with piecewise linear functions, when MS1, MS2, and MS3 act simultaneously, FMSC-HNN... The expansion of space naturally arouses research interest. Fixed ,when At that time, FMSC-HNN in , and It is controllable in both direction, thus forming , and Multi-structured chaotic attractors with vortex types, respectively as follows Figure 10 As shown in (a), (b), and (c). In fact, , and It can be specified as any integer to form a space expansion. Multi-structured chaotic attractors. To the authors' knowledge, multi-directionally expanding multi-structured chaotic attractors have not yet been reported in two-neuron HNNs, providing controllable and non-decaying chaotic sequences for practical industrial applications.

[0069] 3. Coexisting multi-structure attractors with multi-directional control The controllability of FMSC-HNN can be increased by using a memristor synaptic model with piecewise functions, i.e., generating multi-structure attractors with multiple extensions. Recent research has reported that realizing novel offset modulation dynamics in memristor HNNs provides insights into increasing network flexibility and multi-stability. Therefore, can multi-structure attractors with unidirectional, planar, and spatial offset modulation be further realized in FMSC-HNNs? This question warrants further investigation. In this section, we maintain... and Unchanged, with , and As a control parameter.

[0070] Unidirectional offset control: changing the internal parameters of MS 1 It is 3.7, depending on the control parameters. The changing 1D coexistence bifurcation diagram is as follows: Figure 11 As shown in (a), the bifurcation point can be observed. Along with an offset of 2 The orientation is copied, causing the position of the attractor in phase space to be modulated while its shape remains unchanged. Therefore, Figure 12 (a) gives flat Multi-structure attractors with vortex type are spaced at a distance of 2. Regulation has validated Figure 11 The validity of (a). Similarly, by and The controlled 1D coexistence bifurcation diagram shows the phenomenon of bifurcation point offset regulation, such as... Figure 11 As shown in (b) and (c). Accordingly, and flat Multi-structure attractors can be separately and The regulation is based on four sets of homogeneous phase trajectories, such as Figure 12 As shown in (b) and (c). In addition, the LE curves show stable fluctuations, indicating that FMSC-HNN can non-destructively generate coexisting multi-structure chaotic attractors with unidirectional offset modulation.

[0071] Planar offset control: The internal parameters of MS 1 and MS 2 are set to... and Based on the position and amplitude information of the sequence generated by FMSC-HNN, it can be used in... The plane is divided into 16 local attraction basins with different attraction domains. ),like Figure 13As shown in (a). Since each basin has the same area and coexists in the same phase space, it means that FMSC-HNN produces homogeneous multistability regulated by plane migration. Therefore, [the following is set...] 4-structure attractors with vortex type can be initialized by memristor synapses. and Nine groups of homogeneous attractors are controlled with an offset of 2, and their phase trajectories are as follows: Figure 13 As shown in (b), the effectiveness of the localized suction basin was verified. Furthermore, the internal parameters of MS 2 and MS 3 were set to... and , Figure 13 (c) also depicts the... and Phase trajectories of modulated planar coexisting homogeneous multi-structure attractors. (Adding...) After the variable, in Figure 13 (d) Characterized The discovery of coexisting 4-winged chaotic attractors with planar modulation in space indicates that different memristor synapses have varying effects on the network's dynamics. Therefore, the above analysis demonstrates that FMSC-HNN can generate coexisting multi-structure chaotic attractors with planar modulation.

[0072] Spatial offset control: The internal parameters of MS 1, MS 2, and MS 3 are set to... It can be inferred that the initial values ​​of the three segmented memristor synapse models all possess periodic cyclical characteristics, implying that regulatory dynamics can be generated in three dimensions. Therefore, in the set... After selecting the initial value of the memristor, coexisting single-cycle and vortex chaotic attractors with spatial offset control can be generated respectively, such as Figure 14 As shown in (a) and (b). In fact, FMSC-HNN can be modulated by spatial offset in... A total of 64 coexisting multi-structure attractors can be generated in phase space. In fact, from... Figure 3 It can be seen that the interval All inside All meet Therefore, the piecewise linear functions in the MS 1, MS 2, and MS 3 models possess topological invariance, which is the fundamental reason for offset modulation. Thus, in Under control, the multi-structure attractors of the memristor synaptic initial displacement modulator in FMSC-HNN are distributed in unidirectional, planar, and spatial directions, with corresponding numbers of [number missing]. , and This novel dynamic behavior has not been reported in two-neuron HNNs.

[0073] Example 3: Digital Circuit Implementation Based on Microcontroller In practical applications, analog or digital circuits are needed to generate robust chaotic pseudo-random sequences. Since HNN networks use the hyperbolic tangent function as the activation function, the analog circuits that need to be developed suffer from drawbacks such as large circuit area, numerous components, and low accuracy. Therefore, this research develops a digital circuit implementation platform for FMSC-HNN based on a high-precision STM32 microcontroller unit, promoting the engineering application value of all network models. Compared to analog circuits, the developed digital circuits can arbitrarily configure network parameters and initial values, offering significant advantages in terms of accuracy, portability, and computational efficiency.

[0074] The developed digital circuit implementation platform, such as Figure 15 As shown, its operation steps are described below: (1) Programming. In the pre-installed Keil... Write a program in the Vision5 software host computer to solve and port the FMSC-HNN based on the fourth-order Runge-Kutta method. The digital circuit model of FMSC-HNN is as follows:

[0075] in, These correspond to the six state variables in the FMSC-HNN model (4); Represents six dimensions; For the first in model (4) One equation; Defined as about The recursive parameter matrix; The discretization sampling interval is set to 0.01 s.

[0076] (2) Program execution. The program runs on the high-performance MCU STM32F407VET6 development board with 32-bit ARM, Cortex-M4 processor, 168 MHz main frequency, 512 KB embedded Flash and 92 KB embedded SRAM, and outputs the digital values ​​corresponding to the status in the range of 0~65535.

[0077] (3) Signal processing. The digital signal output from the development board is converted into an analog output voltage ranging from 0 to 5 V through the output ports OUTA and OUTB of the two DAC8563 modules. Then, the voltage conversion circuit further standardizes the output voltage range to -10 to 10 V.

[0078] (4) Results display. The output voltage was captured by an oscilloscope of model YOKOGAWA DL850E.

[0079] To verify the reliability of the microcontroller digital circuit implementation platform, fixed parameters were set as follows: , , , , , and The initial value is .when , At that time, the analog voltage sequence output by the platform is combined into the X / Y mode of the oscilloscope. , , Phase trajectories on a plane, such as Figure 16 As shown in (a1), (b1), and (c1). Compare... Figure 6 From (c1), (c2), and (c3), it can be seen that the phase trajectories maintain consistency in terms of contour, position, size, and type, verifying the chaotic attractor with wing, butterfly, and vortex types. At this point, respectively set... , and , Figure 16 (a2), (b2), and (c2) demonstrate the extension of different types of chaotic attractors into 4-structure chaotic attractors, and... Figure 8 (a), (b), and (c) are in perfect agreement. Notably, by changing the initial values, multi-structure attractors with multi-directional offset modulation can be captured. Therefore, the MCU-based digital circuit implementation platform can accurately output multi-structure chaotic attractor sequences, verifying the reliability of the circuit implementation paradigm and the effectiveness of the numerical simulation.

[0080] Example 4: Application in the Power Internet of Things Currently, countries are integrating wireless communication technology into smart grids, driving the rapid development of the power Internet of Things (IoT). Especially when transmitting important images of power lines, workstations, and equipment obtained through drone and unmanned vehicle inspections, data is easily leaked and intercepted. This not only increases the risk of cyberattacks on the power IoT but may also endanger national security. Therefore, protecting the privacy and ensuring secure transmission of sensitive power image data in the power IoT is crucial for the safe operation of the smart grid. Unlike other IoT encryption requirements, unmanned devices have limited storage and computing resources, making it difficult for traditional encryption algorithms such as DNA and AES to balance lightweight design, security, and reliability. To address this, this paper proposes a novel end-to-end privacy protection and secure transmission scheme for power IoT image data, based on FMSC-HNN, which possesses a multi-directionally expandable / controllable multi-structure chaotic attractor. 1 The constructed power Internet of Things security architecture, such as Figure 17As shown, it comprises three layers: information collection and encryption, cloud transmission, and information reception and decryption. First, edge devices (UAVs and UVs) conduct real-time inspections of power plants and power lines on the power generation side, capturing private image information. This information undergoes online encryption within the edge devices, consuming computing resources. Second, the encrypted private information is transmitted over long distances via a cloud server. At this point, attackers can only obtain encrypted images that are difficult to distinguish, meaning the transmission is secure. Finally, after the data center subscribes to the server, the server sends the key and the private image, among other things. Therefore, the data center can obtain the private information through the decryption system, thereby achieving reliable monitoring and evaluation of the secure and stable operation of remote power generation. Thus, the core of the power IoT security architecture lies in the reliability of the encryption algorithm, which will be discussed in detail in the next section.

[0082] 2. Encryption Algorithm Design As the architecture reveals, the core of privacy protection and secure transmission in the power Internet of Things (IoT) lies in the effective encryption of edge devices and the effective decryption process of the operation and maintenance center. However, the real-time and high-speed encryption process of edge devices places high demands on resource utilization, making the development of secure and lightweight encryption algorithms a way to overcome hardware limitations. Unlike complex encryption algorithms, this paper designs an encryption algorithm based on the proposed FMSC-HNN with a scrambling, bidirectional diffusion, and substitution structure, such as... Figure 18 As shown, FMSC-HNN is used to generate multi-directional extended / controlled multi-structure chaotic attractors, and high-performance pseudo-random sequences are extracted to participate in various encryption processes. A detailed description follows: (1) System configuration: including security key, proposed model, and running algorithm. Select system parameters. and initial conditions The security key is input into the FMSC-HNN, and the simulation time and the fourth-order Runge-Kutta method are set to solve the FMSC-HNN.

[0083] (2) Generate chaotic pseudo-random sequences: when plaintext images The specifications are At that time, FMSC-HNN was iterated This generates six chaotic pseudo-random sequences. , , , , and .

[0084] (3) Key sequence generator: Discard the first 1000 transient data, length is A chaotic pseudo-random sequence is converted into a valid key sequence using a key sequence generator. , , and The details are as follows:

[0085] in, , This represents the modulo operation. express The absolute value, Used to obtain less than or equal to The most recent integer. It can be observed that the security of the key sequence benefits from the chaotic complexity of the sequence and the correlation between the sequences.

[0086] (4) Scrambling process: The key sequence is... Sort and obtain the corresponding index sequence to scramble the plaintext image pixel matrix. The position of the scrambled pixel matrix. as follows:

[0087] (5) Diffusion process: using key sequence and A bidirectional diffusion scheme is proposed to change The pixel value. The forward diffusion process is described as follows:

[0088] (6) Substitution process: using key sequence With pixel matrix Performing a bitwise XOR operation further enhances security, described as follows:

[0089] At this time, Reorganized into a size of The encrypted ciphertext image can be obtained from the pixel matrix. Furthermore, the decryption process is the reverse of the encryption process. It is worth noting that the proposed encryption algorithm is universal and can guarantee the information security of grayscale, color images, and videos.

[0090] 3. Discussion on Safety Performance To verify the security performance of the encryption algorithm in ensuring the security of the power Internet of Things, relevant images taken by unmanned equipment were specifically selected as test benchmarks, such as images of power equipment, power lines, substations, and photovoltaic stations. These images were uniformly processed to a size of [size missing]. Standard grayscale images, such as Figure 19As shown in (a), the security performance test of the encryption algorithm was performed on the MATLAB R2021a platform. In fact, multi-directional expansion and regulated multi-structure attractors can both be used to generate key sequences, but to avoid confusion, we will use [the specific method here]. Figure 10 (c) shows - Structural attractors; relevant parameters can be found in Section 3.4.3. A comprehensive discussion of safety performance follows.

[0091] (1) Key space: A large key space is an effective measure to resist brute-force attacks, and the key space benchmark for reliable encryption algorithms should be no less than [amount missing]. Since the proposed encryption algorithm requires a double-precision floating-point key consisting of 17 parameters and 6 initial values, it is limited in terms of finite precision. Under constraints, the key space is approximately Therefore, it is much larger than the baseline value of the key space, proving that the encryption algorithm has strong resistance to brute-force attacks.

[0092] (2) Histogram analysis: In terms of resistance statistical analysis, uniformly distributed pixels are promising. Figure 19 The histograms corresponding to the plaintext images shown in (a1-a4) display pixels with a relatively concentrated distribution, such as... Figure 19 (b1-b4) are given. However, after applying the encryption algorithm, Figure 19 The encrypted image represented by (c1-c4) has a uniform pixel distribution, making it difficult to read pixel distribution information, such as... Figure 19 As shown in (d1-d4). Based on the differences in the histograms, it is demonstrated that the algorithm can effectively avoid the risk of statistical attacks.

[0093] (3) Correlation analysis: measures the degree of correlation between pixels. In terms of quantitative analysis, the intensity values ​​of adjacent pixels on the horizontal, vertical, and diagonal lines can be calculated. correlation coefficient ,as follows:

[0094] in, and represent Adjacent pixels The average pixel intensity value. When When the value is close to 1, the correlation between pixels is high, and vice versa. Therefore, when randomly selecting from three directions... The average correlation coefficients for adjacent pixels are shown in Table 1. It can be observed that for plaintext images, For encrypted images, This directly proves that the encryption algorithm effectively reduces the correlation between pixels.

[0095]

[0096] In terms of qualitative analysis, plaintext and ciphertext images The distribution results of adjacent pixels in three directions were respectively... Figure 20 (a1-a4) and Figure 20 (b1-b4) Representation. The comparison clearly shows that after applying the encryption algorithm, the strong correlation distribution between pixels is switched to a random distribution, which means that the privacy information of the plaintext image is effectively protected.

[0097] (4) Information entropy analysis: The amount of information presented by an image is characterized by information entropy, and its calculation criteria are as follows:

[0098] In the formula, pixel The probability of occurrence and the bit depth are respectively and For an ideal encrypted image, and ,so Comparing the information entropy calculation results of the original and encrypted power images in Table 1, it can be found that the information entropy of the encrypted image is close to 8, indicating that it presents a huge amount of information. Therefore, the proposed encryption algorithm effectively masks the privacy information of the image.

[0099] (5) Differential Attack Analysis: Differential attacks place higher demands on the sensitivity of encryption algorithms, which can be measured using the Number of Pixel Change Rate (NPCR) and Unified Average Change Intensity (UACI). The calculation criteria for NPCR and UACI are as follows:

[0100] in, and These are the two ciphertext images corresponding to any pixel in the plaintext image before and after adding 1. This refers to the specifications of grayscale images. When four power images are set to have single-pixel differences, the NPCR and UACI values ​​calculated according to equation (20) are very close to the ideal values ​​of 99.6094% and 33.4635%, respectively, as shown in Table 1. Therefore, the proposed algorithm is highly sensitive to small pixel changes and can effectively resist differential attacks.

[0101] (6) Key Sensitivity Analysis: To further demonstrate the sensitivity of the proposed encryption algorithm to the key, encryption / decryption experiments were conducted by applying a small perturbation to the key. When the correct key was used, all ciphertext images were recovered, and the results are shown in... Figure 21 In (a1-a4). However, setting the key to finite precision... , , and At that time, the encrypted image could not be decrypted correctly, such as Figure 21 As shown in (b1-b4). The comparison shows that the proposed encryption algorithm maintains high sensitivity to extremely small key perturbations, that is, it has key uniqueness.

[0102] (7) Robustness Analysis: During transmission, altering the structure of the ciphertext image is a common method to compromise privacy information, which places higher demands on the anti-interference capabilities of encryption algorithms. Therefore, after cropping the ciphertext image by 1 / 16, 1 / 8, 1 / 4, and 1 / 2, the decrypted image recovered the privacy information of the power equipment and power lines, such as... Figure 22 As shown. Additionally, after applying salt-and-pepper noise, by Figure 23 It is evident that significant information about substations and photovoltaic stations can still be decrypted. Therefore, the proposed encryption algorithm exhibits strong robustness against data loss and interference, ensuring the accurate acquisition of private information.

[0103] 4. Hardware Experiment for Privacy Protection in the Power Internet of Things To further promote the application of privacy protection solutions in practice, a full-chain privacy protection experimental platform for the power Internet of Things was built, such as... Figure 23 As shown. At the hardware level, a Gen Intel® Core™ i7 CPU 2.1GHz PC was used to create the development environment, a router established a wireless network, and three Raspberry Pis were used to simulate the publisher (IP 192.168.37.174), intermediate server (IP 192.168.37.127), and subscriber (IP 192.168.37.188). A monitor was used to display the hardware encryption results. At the software level, Python 3.7 was used to program the EMQX 4.3.10 MQTT protocol for sending and receiving privacy information, and the encryption and decryption algorithms were downloaded to RPI. At the operational level, within the same local area network, the subscriber subscribed to the publisher for an image and its key. Next, after receiving the subscription request, the publisher encrypted the image using the provided encryption algorithm and key and sent it to the subscriber. Finally, the subscriber received the key and the privacy image, decrypted it, and recovered the original information. The encryption result is shown below. Figure 24As shown, (a), (b), and (c) represent the image to be sent, the encrypted image, and the decrypted image, respectively. Therefore, the developed hardware experimental platform for privacy protection in the power IoT not only demonstrates the effectiveness of the encryption algorithm but also provides a solution for information security in the power industry.

Claims

1. A Hopfield network model with fully memristor synaptic connections, characterized in that, The mathematical model can be established as follows: ； in, and These represent the membrane potentials of the two neurons; Used to adjust the strength of memristor synapses; mem derivative function Refers to the self-synapses and connection synapse weights in the original HNN; after extensive trial and error experiments, the fixed parameters are declared here in advance; for MS 1~MS 3, the relevant fixed parameter settings are as follows: , , and For MS 4, set Secondly, assume the initial value of FMSC-HNN is... .

2. The construction of the multi-structure attractor based on claim 1, characterized in that, Specifically as follows: Visualizing the dynamic behavior of FMSC-HNN from the perspective of numerical simulation is an effective way to reveal the evolutionary mechanism of neural dynamics, providing a solid foundation for the development, analysis and application of brain-like neural systems. This study investigated the multi-directional expansion and regulated multi-structure attractor behavior of FMSC-HNN using 1-D / 2-D bifurcation diagrams, phase trajectory diagrams, and local attractor basins. Numerical simulations were performed on Matlab 2021a and Julia platforms, and the ODE45 algorithm was used to obtain the numerical solution of the FMSC-HNN system. The total simulation time and step size were set to 10000 s and 0.05 s, respectively. Step 1: Parameter-dependent 1-D / 2-D bifurcation: Performing 1-D / 2-D bifurcation diagram simulations with varying parameters can reveal the distribution domain and evolution mechanism of dynamic behavior, providing convenience for obtaining multi-structure chaotic attractors; firstly, specifying , and And select the relevant parameters for MS 4. and As a control parameter, at this time, it can be Construct a plane with a specification of The dense parameter matrix Secondly, a method based on the Wolf Lyapunov exponent and state sequences is used. The peak cycle number calculation algorithm is used to traverse each parameter combination, thereby effectively distinguishing a total of 11 dynamic features in FSMC-HNN; then, after assigning a color code to each dynamic behavior, The 2D bifurcation diagram of the plane can be rendered; the yellow part marked by CH represents the chaotic parameter domain with a positive LE, while the green part marked by QP has a quasi-periodic parameter domain with at least two zero LEs; for LEs less than 0, they are subdivided into the parameter domains of the stable point (SP), period 1 (P01), and period 2 (P02) to period 8 (P08) according to the period number, and filled with gray, blue and other colors respectively; it can be seen that CH occupies a large planar region, which means that FMSC-HNN is very likely to generate multi-structure chaotic attractors; at the same time, the intricate distribution of dynamic parameter domains also confirms that parameter sensitivity will induce complex dynamic behavior. choose Extracted The 1-D bifurcation diagram of the control is obtained, and the corresponding LEs curve can be obtained by calculating LEs; with Increasing from 0, FMSC-HNN in The interval generates stable point type bifurcations, and respectively at... and The splits into P01 and P02 bifurcation types; for trajectory Entering the CH bifurcation interval represented by the red point set from the tangent bifurcation, especially... A chaotic crisis occurred, with two CH bifurcation clusters degenerating into one, signifying a fundamental change in the morphology of the chaotic attractor. Observing the three LEs curves, it can be seen that LEs are less than zero in the periodic bifurcation interval, while LE1 remains greater than zero in the chaotic bifurcation interval, and the evolution trend of LEs conforms to bifurcation characteristics. In summary, the global two-dimensional dynamic distribution was verified from a local perspective, proving the effectiveness of the analysis. Therefore, we can select typical parameters from the 1-D bifurcation diagram and further generate different single-structure chaotic attractors. Observation shows that... , , The planar representation shows chaotic phase trajectories with wing, butterfly, and vortex types, thus FMSC-HNN can generate multi-structure attractors; of course, with the strength of MS 4... Switching to 0.6, 0.52, and 0.3 in sequence, the resulting chaotic attractors exhibited single, double, and quadruple structures, respectively. Step 2: Multi-directionally expanding multi-structured chaotic attractor: MS 1 to MS 3 employ segmented memristors, resulting in multidirectional controllability of the number of EPs. Based on this, the multidirectional expansion problem of single-structure attractors has attracted attention, and there is a lack of relevant research resources in two-neuron HNNs. One-way expansion: with Taking the induced single-structure chaotic attractor as an example, when the adjustable parameters , and They are respectively used as control parameters in When changing within an interval, regarding , and The bifurcation point and the first three LE curves; it can be observed that, as... Add, about , and The bifurcation diagrams all show a step-growth trend, and in , , , , and The study presents multiple attractors with structures of 2, 3, 4, 5, 6, and 7; in this case, the LE curves all change smoothly, and calculations show that... , , , , and ;because This proves that FMSC-HNN can generate unidirectional ( , and The extended multi-structure chaotic attractor exhibits robust chaotic properties; next, select... and The descriptions of FMSC-HNN along the path are as follows: , and The 3 / 4 / 5 / 6-wing, butterfly, and vortex chaotic attractor phase trajectories generated by the direction; observation reveals that single-structure attractors can extend unidirectionally to ( The multi-structured chaotic attractor proves the validity of the bifurcation diagram and theoretical analysis; Planar extension: fixed and ,when At that time, the vortex-type chaotic attractor can be The plane is effectively controlled; of course, the multi-structure attractor has been clearly defined by... The controlled multi-EP triggering means that, under the simultaneous action of MS 2 and MS 3, the attractor generated by FMSC-HNN exhibits planar expansion characteristics; therefore, in The plane was further expanded , , , , and Multi-structured chaotic attractors with vortex type; since the multi-structured chaotic attractor is connected by a different number of vortex chaotic attractors and each attractor trajectory has similarity, it is proved that FMSC-HNN has lossless planar expansion properties; Spatial Expansion: Because FMSC-HNN contains three memristor synapses with piecewise linear functions, when MS 1, MS 2, and MS 3 act simultaneously, FMSC-HNN... The expansion of space naturally arouses research interest; fixed ,when At that time, FMSC-HNN in , and It is controllable in both direction, thus forming , and Multi-structured chaotic attractors with vortex type; in fact, , and It can be specified as any integer to form a space expansion. Multi-structured chaotic attractor; to the authors’ knowledge, a multi-directionally extended multi-structured chaotic attractor has not been reported in a two-neuron HNN, which provides a controllable and non-degrading chaotic sequence for practical industrial applications. Step 3: Coexistence of multi-structure attractors with multi-directional control: The controllability of FMSC-HNN can be increased by using a piecewise function-based memristor synaptic model, i.e., generating multi-structure attractors with multiple extensions. Recent research has reported that realizing novel offset modulation dynamics in memristor HNNs provides insights into increasing network flexibility and multi-stability. Therefore, can multi-structure attractors with unidirectional, planar, and spatial offset modulation be further realized in FMSC-HNNs? This question warrants further investigation. In this section, we maintain... and Unchanged, with , and As a control parameter; Unidirectional offset control: changing the internal parameters of MS 1 It is 3.7, depending on the control parameters. A changing 1D coexistence bifurcation diagram; it can be observed that the bifurcation points... Along with an offset of 2 The orientation is copied, causing the position of the attractor in phase space to be modulated while its shape remains unchanged; for this purpose, we give... flat Multi-structure attractors with vortex type are spaced at a distance of 2. Regulation; similarly, by and The controlled 1D coexistence bifurcation diagram shows the phenomenon of bifurcation point offset regulation, and correspondingly, and flat Multi-structure attractors can be separately and The regulation is based on four sets of homogeneous phase trajectories. In addition, the LE curve shows stable fluctuations, indicating that FMSC-HNN can non-destructively generate coexisting multi-structure chaotic attractors with unidirectional offset regulation. Planar offset control: The internal parameters of MS 1 and MS 2 are set to... and Based on the position and amplitude information of the sequence generated by FMSC-HNN, it can be used in... The plane is divided into 16 local attraction basins with different attraction domains. Since each basin has the same area and coexists in the same phase space, it means that FMSC-HNN produces homogeneous multistability regulated by plane migration; therefore, a setting is made. A vortex-type 4-structure attractor can be initialized by a memristor synapse. and Nine homogeneous attractors with an offset of 2 were used to verify the effectiveness of the local attraction basin. Furthermore, the internal parameters of MS 2 and MS 3 were set to... and It depicts the and Phase trajectories of modulated planar coexisting homogeneous multi-structure attractors; addition After the variable, The discovery of coexisting 4-winged chaotic attractors with planar control in space indicates that different memristor synapses have different effects on the network dynamics; therefore, the above analysis shows that FMSC-HNN can generate coexisting multi-structure chaotic attractors with planar offset control. Spatial offset control: The internal parameters of MS 1, MS 2, and MS 3 are set to... It can be inferred that the initial values ​​of the three segmented memristor synapse models all have periodic cyclical characteristics, implying that regulatory dynamics can be generated in three dimensions; therefore, in the set After selecting the initial value of the memristor, coexisting single-cycle and vortex chaotic attractors with spatial offset control can be generated respectively. In fact, FMSC-HNN can be controlled by spatial offset in... A total of 64 coexisting multi-structure attractors can be generated in the phase space, and the interval All inside All meet Therefore, the piecewise linear functions in the MS 1, MS 2, and MS 3 models possess topological invariance, which is the fundamental reason for offset modulation; thus, in Under control, the multi-structure attractors of the memristor synaptic initial displacement modulator in FMSC-HNN are distributed in unidirectional, planar, and spatial directions, with corresponding numbers of [number missing]. , and indivual.

3. The privacy protection method for the power Internet of Things based on claim 2, characterized in that, It consists of three levels: information collection and encryption, cloud transmission, and information reception and decryption. First, edge devices UAV and UV conduct real-time inspections of power plants and power lines on the power generation side and capture privacy image information. This information is encrypted online in edge devices, consuming computing resources; secondly, the encrypted private information is transmitted over long distances via cloud servers, at which point attackers can only obtain encrypted images that are difficult to distinguish, meaning that the transmission is secure; finally, when the data center requests a subscription from the server, the server sends the key and the private image, among other things. Therefore, data centers can obtain private information through the decryption system, thereby enabling reliable monitoring and evaluation of the safe and stable operation of remote power generation. The encryption algorithm is designed as follows: As the architecture shows, the core of privacy protection and secure transmission in the power Internet of Things (IoT) lies in the effective encryption of edge devices and the effective decryption process of the operation and maintenance center. However, the real-time and high-speed encryption process of edge devices has high resource utilization requirements, and proposing a secure and lightweight encryption algorithm is a way to overcome hardware limitations. Unlike complex encryption algorithms, this paper designs an encryption algorithm with scrambling, bidirectional diffusion, and substitution structures based on the proposed FMSC-HNN. It utilizes FMSC-HNN to generate multi-directional expansion / regulation of multi-structure chaotic attractors and extracts high-performance pseudo-random sequences to participate in various encryption processes, as detailed below: (1) System configuration: including security key, proposed model and running algorithm; select system parameters and initial conditions The security key is input into FMSC-HNN, and the simulation time and fourth-order Runge-Kutta method are set to solve FMSC-HNN. (2) Generate chaotic pseudo-random sequences: when plaintext images The specifications are At that time, FMSC-HNN was iterated This generates six chaotic pseudo-random sequences. , , , , and ; (3) Key sequence generator: Discard the first 1000 transient data, length is A chaotic pseudo-random sequence is converted into a valid key sequence using a key sequence generator. , , and The details are as follows: ； in, , This represents the modulo operation. express The absolute value, Used to obtain less than or equal to The nearest integer; it can be found that the security of the key sequence benefits from the chaotic complexity of the sequence and the correlation between the sequences; (4) Scrambling process: The key sequence is... Sort and obtain the corresponding index sequence to scramble the plaintext image pixel matrix. Position; Scrambled pixel matrix as follows: ； At this time, Reorganized into a size of The encrypted ciphertext image can be obtained from the pixel matrix. Furthermore, the decryption process is the reverse of the encryption process; it is worth noting that the proposed encryption algorithm is universal and can guarantee the information security of grayscale, color images, and videos.

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