Fractional chaotic system cloud control stability method and device based on homomorphic encryption

Abstract

This invention discloses a cloud control stabilization method and apparatus for fractional-order chaotic systems based on homomorphic encryption, applied to a client-side application. The method includes: constructing a fractional-order chaotic system model and designing a sampling controller within the model; quantizing the sampled system state using a fixed-point rational number architecture; encrypting the quantized system state using Paillier semi-homomorphic encryption and transmitting it to the cloud; receiving encrypted control input from the cloud based on homomorphic operations performed between the received encrypted state data and the quantized control gain matrix; and decrypting and scaling the encrypted control input to obtain plaintext control input, thereby achieving stable control of the fractional-order chaotic system. This invention, by designing a sampling controller and introducing Paillier semi-homomorphic encryption into the control process, reduces communication frequency while protecting data privacy during cloud control computation, thus mitigating the risk of data leakage.

Description

Technical Field

[0001] This invention relates to the field of next-generation information technology, and in particular to a method, apparatus, terminal equipment, computer-readable storage medium, and computer program product for cloud control stabilization of fractional-order chaotic systems based on homomorphic encryption. Background Technology

[0002] Since meteorologist Edward Lorenz first discovered chaos in 1963, the concept of chaotic systems has been formally proposed and widely studied. Due to their inherent sensitivity to initial conditions and highly nonlinear characteristics, chaotic systems exhibit complex and unpredictable behavior in various practical applications, such as meteorological modeling, financial stock systems, and engineering. Therefore, improving the predictability and controllability of chaotic systems has become an important research goal. Following the pioneering work in 1990 by Edward Ott, Celso Grebogi, and James A. Yorke, which first demonstrated that chaotic systems could be effectively controlled, numerous control strategies have been proposed, such as feedback control, sampled data control, and intermittent control. Among these, sampled data control (SDC) offers advantages such as high reliability, ease of implementation and maintenance, low cost, and applicability to nonlinear system control. Its basic idea is to sample system data at discrete sampling times and transmit the data to the controller within each sampling interval, thereby significantly reducing communication burden and improving bandwidth utilization. It has become an effective method in the stability control analysis of chaotic systems.

[0003] To more accurately characterize the memory and hereditary features prevalent in real-world systems, fractional calculus has been proposed as a generalization of traditional integer calculus. Compared to integer models, fractional models can describe complex dynamic behaviors more flexibly and accurately, thus having broader application prospects in many fields such as secure communication and financial markets. Therefore, research on control methods for fractional chaotic systems is of great significance.

[0004] With the rapid development of cloud computing and communication technologies, cloud-based control architectures have been widely applied in network control systems, such as building automation systems and large-scale cyber-physical systems, due to their excellent scalability, powerful computing capabilities, and flexibility. In this framework, system data is transmitted to a cloud server, where cloud computing and control inputs are fed back to the controlled object via a communication network. While this architecture offers many advantages, it also brings serious security and privacy issues. In particular, system status and control signals transmitted in an open network environment can be intercepted or maliciously exploited, posing a serious threat to the system's security and reliability. Therefore, ensuring data confidentiality while guaranteeing closed-loop system performance has become a critical issue in cloud control systems.

[0005] To address privacy concerns in cloud control systems, cryptographic methods protect data confidentiality by encrypting system signals into ciphertext without altering the signals themselves. Paillier semi-homomorphic encryption, in particular, allows direct computation on the ciphertext without decryption, making it especially suitable for cloud control applications. However, since Paillier encryption operates on the integer domain, quantization errors are inevitably introduced during implementation, a factor that must be carefully considered in controller design and stability analysis.

[0006] Currently, there is limited research on privacy protection in the stable control of fractional-order chaotic systems. Therefore, researching the stability of fractional-order chaotic systems based on cloud-based encrypted control has significant practical implications. Summary of the Invention

[0007] In view of this, the present invention provides a cloud control stabilization method, system, terminal device, computer-readable storage medium and computer program product for fractional chaotic systems based on homomorphic encryption. By adopting a sampling controller that saves more control costs, and combining fractional calculus and Paillier semi-homomorphic encryption algorithm in fractional chaotic systems, the stability and privacy of fractional chaotic system control are improved.

[0008] The first objective of this invention is to provide a cloud control stabilization method for fractional-order chaotic systems based on homomorphic encryption.

[0009] The second objective of this invention is to provide a cloud control stabilization device for fractional-order chaotic systems based on homomorphic encryption.

[0010] The third objective of this invention is to provide a terminal device.

[0011] A fourth objective of this invention is to provide a computer-readable storage medium.

[0012] The fifth objective of this invention is to provide a computer program product.

[0013] The first objective of this invention can be achieved by adopting the following technical solution:

[0014] A cloud control stabilization method for fractional-order chaotic systems based on homomorphic encryption, applied to a client, the method comprising:

[0015] Construct a fractional-order chaotic system model and design a sampling controller in the fractional-order chaotic system model to determine the sampling and transmission times;

[0016] A fixed-point rational number architecture is used to quantize the sampled system state;

[0017] Paillier semi-homomorphic encryption is used to encrypt the quantized system state and transmit it to the cloud;

[0018] Receive the encrypted control input obtained from the cloud by performing homomorphic operations on the received encrypted state data and the quantized control gain matrix;

[0019] After decrypting and scaling the encrypted control input, the plaintext control input is obtained; based on the control input, stable control of the fractional-order chaotic system is achieved.

[0020] Preferably, the fractional-order chaotic system is:

[0021]

[0022] In the formula, for Caputo's differential of a fractional-order system at time step At the initial moment, Let be the order of the fractional order in the system; for The system state at any given moment. Given a diagonal matrix, The connection weight matrix; The sampling controller to be designed; It is a Lipschitz continuously differentiable nonlinear function. For any two distinct real numbers and There exists a constant satisfy: .

[0023] Preferably, the designed sampling controller is:

[0024]

[0025] In the formula, the sampling time series satisfies For the first time intervals , Indicates the sampling period; This is the feedback gain matrix.

[0026] The second objective of this invention can be achieved by adopting the following technical solution:

[0027] A cloud control stabilization device for a fractional-order chaotic system based on homomorphic encryption, applied to a client, the device comprising:

[0028] The construction and design module is used to build a fractional-order chaotic system model and design the sampling controller in the fractional-order chaotic system model to determine the sampling and transmission times;

[0029] The quantization module is used to quantize the sampled system state using a fixed-point rational number architecture;

[0030] The encryption module is used to encrypt the quantized system state using Paillier semi-homomorphic encryption and transmit it to the cloud;

[0031] The computation module is used to receive the encrypted state data received from the cloud and perform homomorphic operations on the quantization control gain matrix to obtain the encrypted control input for the operation;

[0032] The decryption and control module is used to decrypt and scale the encrypted control input to obtain the plaintext form of the control input; based on the control input, stable control of the fractional-order chaotic system is achieved.

[0033] The third objective of this invention can be achieved by adopting the following technical solution:

[0034] A terminal device includes a processor and a memory for storing a processor-executable program. When the processor executes the program stored in the memory, it implements the above-described method for stabilizing cloud control of a fractional-order chaotic system based on homomorphic encryption.

[0035] The fourth objective of this invention can be achieved by adopting the following technical solution:

[0036] A computer-readable storage medium storing a program that, when executed by a processor, implements the aforementioned method for stabilizing cloud control of fractional-order chaotic systems based on homomorphic encryption.

[0037] The fifth objective of this invention can be achieved by adopting the following technical solution:

[0038] A computer program product includes a computer program that, when executed by a processor, implements the aforementioned method for stabilizing cloud control of fractional-order chaotic systems based on homomorphic encryption.

[0039] The present invention has the following advantages over the prior art:

[0040] (1) This invention proposes a cloud-based privacy-preserving control framework for achieving stable control of fractional-order chaotic systems. By introducing the Paillier cryptographic algorithm with additive homomorphic properties into the control, the system state can be encrypted during transmission and processed within the encrypted domain, thereby ensuring the privacy of the system state;

[0041] (2) The present invention integrates the sampling data control strategy into the encryption control architecture, which significantly reduces the communication load, network burden and encryption-related computing costs, and is therefore more suitable for actual networked and cloud control environments. Attached Figure Description

[0042] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the structures shown in these drawings without creative effort.

[0043] Figure 1 This is a simplified flowchart of the cloud control stabilization method for fractional-order chaotic systems based on homomorphic encryption according to Embodiment 1 of the present invention;

[0044] Figure 2 This is a detailed flowchart of the cloud control stabilization method for fractional-order chaotic systems based on homomorphic encryption according to Embodiment 1 of the present invention;

[0045] Figure 3 This is a graph showing the changes of various state variables over time in the fractional-order chaotic system of Embodiment 1 of the present invention.

[0046] Figure 4 This is a trajectory diagram of the system under the action of the cloud-based encrypted sampling controller in Embodiment 1 of the present invention;

[0047] Figure 5 In Embodiment 1 of the present invention, the system trajectory is maintained in a set under the action of a cloud-based encrypted sampling controller. Within the range;

[0048] Figure 6 The encrypted system state of Embodiment 1 of the present invention Trajectory diagram in a communication network;

[0049] Figure 7 This is a trajectory diagram of the controller in the communication network according to Embodiment 1 of the present invention;

[0050] Figure 8 This is a graph showing the actual control input curves of the fractional-order chaotic system in Embodiment 1 of the present invention.

[0051] Figure 9 As in Embodiment 1 of the present invention The impact of different quantization accuracies on system performance under certain conditions;

[0052] Figure 10 As in Embodiment 1 of the present invention The impact of different quantization accuracies on system performance under certain conditions;

[0053] Figure 11 This is a structural block diagram of the cloud control stabilization device for a fractional-order chaotic system based on homomorphic encryption, according to Embodiment 2 of the present invention.

[0054] Figure 12 This is a structural block diagram of the terminal device according to Embodiment 3 of the present invention. Detailed Implementation

[0055] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention. It should be understood that the specific embodiments described are merely used to explain this application and are not intended to limit this application.

[0056] Example 1:

[0057] like Figure 1 , 2 As shown, this embodiment provides a cloud control stabilization method for fractional-order chaotic systems based on homomorphic encryption, applied to a client, and includes the following steps:

[0058] S101. Construct a fractional-order chaotic system.

[0059] The fractional-order chaotic system is constructed as follows:

[0060]

[0061] in, yes Caputo's differential of a fractional-order system at time step It is the initial moment. Indicates the order of the fractional order in the system; express The system state at any given moment. These are the initial values ​​of the system; It is a known diagonal matrix, where ; It is the connection weight matrix; This represents the synchronous controller to be designed; It is a Lipschitz continuously differentiable nonlinear function, where For any two distinct real numbers and There exists a constant satisfy:

[0062]

[0063] here and It is a known real number; and , .

[0064] S102. Design a sampling controller based on a fractional-order chaotic system to achieve stability of the fractional-order chaotic system.

[0065] The sampling control law is designed as follows:

[0066]

[0067] Among them, the sampling time series satisfies , For the first time intervals Indicates the sampling period; It is a feedback gain matrix of appropriate dimension.

[0068] S103. After performing fixed-point quantization on the system state, Paillier encryption is applied and the data is transmitted to the cloud. The cloud receives the encrypted control input obtained by performing homomorphic operation on the control gain matrix after fixed-point quantization and the encrypted data. The encrypted control input is decrypted and scaled back to obtain the plaintext control input.

[0069] Cloud-based control gain matrix based on fixed-point quantization With encrypted data Encryption control input obtained by homomorphic operation The encrypted control input is transmitted to the client via the network; the client decrypts and scales the encrypted control input to obtain the plaintext control input. .

[0070] Further, step S103 includes:

[0071] (1) The sampled system state is quantized using a fixed-point rational number architecture.

[0072] To achieve encryption, real-valued signals are represented using binary signed fixed-point numbers. Let there be a set of rational numbers. Indicates having The decimal part A set of fixed-point numbers, covering an interval The resolution is Real numbers Quantified in the following ways:

[0073]

[0074] To support the application of the Paillier cryptosystem, this embodiment defines bijections and inverse mappings as follows:

[0075]

[0076]

[0077] in, Modulus The remaining class ring.

[0078] During the controlled computation process, the sum and product of fixed-point numbers may exceed the original word length. This can be addressed by selecting numbers that satisfy... number of digits This ensures its closure property under addition and scaling multiplication operations, thus yielding:

[0079]

[0080]

[0081] Among them, rational numbers , Indicates having The decimal part The set of rational numbers with fixed points; and .

[0082] Therefore, fixed-point arithmetic can be equivalently used as ( Modulus Integer operations are performed in the remaining class ring, making it suitable for cryptographic computation.

[0083] (2) Use Paillier semi-homomorphic encryption to encrypt the quantized system state.

[0084] In the Paillier encryption control scheme, the Paillier encryption algorithm is based on integer rings. Run, where the public key is , They are two large prime numbers of similar size that satisfy... The private key is defined as follows: For plaintext The encryption process is defined as follows:

[0085]

[0086] in, It was selected randomly.

[0087] (3) Decryption.

[0088] The corresponding decryption process:

[0089]

[0090] in, , This makes For all Both are valid.

[0091] Paillier cryptosystems possess additive homomorphism. In particular, for all... Make ,but:

[0092]

[0093] It also supports semi-homomorphic multiplication operations, suitable for every... Make ,but:

[0094] .

[0095] S104. Based on the plaintext control input, achieve stable control of the fractional-order chaotic system.

[0096] After the above encryption and decryption process, the control gain and system state inevitably lead to quantization errors in the actual control input, resulting in the following: and Therefore, the above sampling control law can be expressed as:

[0097]

[0098] in, They are bounded, and represent the gain matrix respectively. and system status The quantization error. Therefore, the system in step S101 can be rewritten as:

[0099] .

[0100] Next, this embodiment transforms the system into a non-fragile control of a fractional-order chaotic system with perturbations, namely:

[0101]

[0102] in, Treated as a bounded perturbation, and , These are the parameters to be designed. Control gain quantization error. satisfy , The known real constant matrix has an appropriate dimension, while the unknown matrix... satisfy .

[0103] Then, by using the input time delay method, the discrete-time control law is expressed as a time-delayed control signal between adjacent sampling times, so that the sampled data system can be modeled as a continuous-time system with input delay, as follows:

[0104]

[0105] in, .

[0106] To further advance the derivation, the following definitions and lemmas will help in deriving the main results.

[0107] Definition 1: Within the Banach space Continuously differentiable function of order of The Caputo fractional derivative is defined as:

[0108]

[0109] Where, integer satisfy , , It is the initial time;

[0110] Definition 2: For a Lebesgue integrable function , The fractional integral of order is defined as:

[0111]

[0112] The stability of fractional-order chaotic systems is studied using the Lyapunov functional method. Theorem 1: For any given scalar... and Under sampling control, fractional-order chaotic systems can asymptotically achieve mean-square bounded stability in the mean square. If and only if there exists a positive definite matrix and its arbitrary diagonal matrix , and positive scalar The following linear matrix inequalities must be satisfied:

[0113]

[0114] in, , , , , .also, The always in the collection In, it is given by the following formula:

[0115] .

[0116] On the other hand, this embodiment proposes the following theorem to guarantee the security of encrypted data. Safe computation. This theorem shows that, with reasonable design of fixed quantization parameters... and Under the premise that quantization is required, a closed-loop dynamic system can be equivalently represented as a non-fragile control model with a perturbation system.

[0117] Theorem 2: Let ,like and Satisfy any All of them are:

[0118]

[0119]

[0120] Then it exists , This makes the state trajectories of systems 1.1 and 1.2 completely consistent.

[0121] Theorem 3: If and , making Assuming the bit length With Paillier modulus satisfy:

[0122]

[0123]

[0124] Then the first encryption control input Each component is given by the following formula:

[0125]

[0126] Among them, for any randomly generated , any , .

[0127] Theorems 2 and 3 in this embodiment establish parameter selection criteria related to the Paillier encryption scheme, avoiding data overflow and ensuring the feasibility of the security control scheme;

[0128] In a practical application case, the following three-dimensional fractional chaotic system is first selected to verify the validity of the proposed theorem:

[0129] , ,

[0130] From this we can obtain , .when In this embodiment, the initial system value is selected as... .like Figure 3 As shown, the fractional-order chaotic system exhibits chaotic behavior and has two attractors.

[0131] This embodiment selects The sampling period is set to Assuming It satisfies the requirements regarding The condition is given. According to Theorem 1, the controller gain can be obtained using the LMI toolbox. ,as well as The upper bound of the perturbation is taken as... .according to , can be obtained This value is much smaller than the sampling period. Therefore, we can obtain... Furthermore, determine the minimum quantization parameter that satisfies Theorem 2. To obtain the corresponding minimum length To implement Paillier homomorphic encryption, according to Theorem 3, we choose... The minimum value is Furthermore, it is known that the modulus size must satisfy... For safety reasons, the simulation ultimately adopted... Bit key length.

[0132] Within the framework of the sampling data control scheme proposed in this embodiment, combined with Paillier homomorphic encryption, stable control of a fractional-order chaotic system is achieved, such as... Figure 4 As shown. Figure 5 This indicates that the system state norm gradually converges and remains within the preset error bound, thus verifying the effectiveness of the proposed stabilization strategy. Figure 6 Display system status When Paillier encryption is used for transmission in a communication network, the encrypted result cannot reveal any valid information about the original state. The corresponding encrypted control inputs calculated by the cloud controller are as follows: Figure 7 As shown, the decryption control signal applied to fractional-order chaotic systems is as follows: Figure 8 As shown in the figure. These results demonstrate that the scheme proposed in this embodiment can complete control computation within the encrypted domain while ensuring data confidentiality. Figure 7 and Figure 8 This further demonstrates that the encrypted system state and encrypted control input also exhibit randomness, thus effectively preventing information from being extracted. Figure 9 and Figure 10Further demonstrates that in the same The impact of different quantization accuracies on system performance under certain conditions was investigated. Although quantization errors may cause system fluctuations, the system state norm remained within the preset limits, thus verifying the robustness of the control scheme proposed in this embodiment under quantization effects.

[0133] Those skilled in the art will understand that all or part of the steps in the methods of the above embodiments can be implemented by a program instructing related hardware, and the corresponding program can be stored in a computer-readable storage medium.

[0134] It should be noted that although the method operations of the above embodiments are described in a specific order in the accompanying drawings, this does not require or imply that these operations must be performed in that specific order, or that all the operations shown must be performed to achieve the desired result. On the contrary, the order of execution of the described steps may be changed. Additionally or alternatively, certain steps may be omitted, multiple steps may be combined into one step, and / or one step may be broken down into multiple steps.

[0135] Example 2:

[0136] like Figure 11 As shown, this embodiment provides a cloud control stabilization device for a fractional-order chaotic system based on homomorphic encryption. This device is applied to a client and includes a construction and design module 1001, a quantization module 1002, an encryption module 1003, a computation module 1104, and a decryption module 1105, wherein:

[0137] The construction and design module 1101 is used to construct a fractional-order chaotic system model and design a sampling controller in the fractional-order chaotic system model to determine the sampling and transmission times;

[0138] Quantization module 1102 is used to quantize the sampled system state using a fixed-point rational number architecture;

[0139] Encryption module 1103 is used to encrypt the quantized system state using Paillier semi-homomorphic encryption and transmit it to the cloud;

[0140] The computing module 1104 is used to receive the encrypted state data received from the cloud and perform homomorphic operations on the quantization control gain matrix to obtain the encrypted control input for the operation;

[0141] The decryption and control module 1105 is used to decrypt and scale the encrypted control input to obtain the plaintext form of the control input; based on the control input, stable control of the fractional-order chaotic system is achieved.

[0142] The specific implementation of each module in this embodiment can be found in Embodiment 1 above, and will not be repeated here. It should be noted that the device provided in this embodiment is only illustrated by the division of the above functional modules. In practical applications, the above functions can be assigned to different functional modules as needed, that is, the internal structure can be divided into different functional modules to complete all or part of the functions described above.

[0143] Example 3:

[0144] This embodiment provides a terminal device, which can be a computer, such as... Figure 12 As shown, the processor 1202, memory, input device 1203, display 1204, and network interface 1205 are connected via system bus 1201. The processor provides computing and control capabilities. The memory includes a non-volatile storage medium 1206 and an internal memory 1207. The non-volatile storage medium 1206 stores the operating system, computer programs, and database. The internal memory 1207 provides an environment for the operation of the operating system and computer programs in the non-volatile storage medium. When the processor 1202 executes the computer programs stored in the memory, it implements the cloud control stabilization method for fractional-order chaotic systems based on homomorphic encryption in Embodiment 1 described above.

[0145] Example 4:

[0146] This embodiment provides a computer-readable storage medium storing a computer program. When the computer program is executed by a processor, it implements the cloud control stabilization method for fractional-order chaotic systems based on homomorphic encryption as described in Embodiment 1 above.

[0147] It should be noted that the computer-readable storage medium in this embodiment can be a computer-readable signal medium or a computer-readable storage medium, or any combination thereof. The computer-readable storage medium can be, for example, but not limited to, an electrical, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, or device, or any combination thereof. More specific examples of a computer-readable storage medium may include, but are not limited to: an electrical connection having one or more wires, a portable computer disk, a hard disk, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), optical fiber, portable compact disk read-only memory (CD-ROM), optical storage device, magnetic storage device, or any suitable combination thereof.

[0148] Example 5:

[0149] This embodiment provides a computer program product, including a computer program that, when executed by a processor, implements the cloud control stabilization method for fractional-order chaotic systems based on homomorphic encryption described in Embodiment 1 above.

[0150] The above description is merely a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any equivalent substitutions or modifications made by those skilled in the art within the scope disclosed in the present invention, based on the technical solution and inventive concept of the present invention, shall fall within the scope of protection of the present invention.

Claims

1. A cloud control stabilization method for fractional-order chaotic systems based on homomorphic encryption, applied to a client-side application, characterized in that... The method includes: Construct a fractional-order chaotic system model and design a sampling controller in the fractional-order chaotic system model to determine the sampling and transmission times; A fixed-point rational number architecture is used to quantize the sampled system state; Paillier semi-homomorphic encryption is used to encrypt the quantized system state and transmit it to the cloud; Receive the encrypted control input obtained from the cloud by performing homomorphic operations on the received encrypted state data and the quantized control gain matrix; After decrypting and scaling the encrypted control input, the plaintext control input is obtained; based on the control input, stable control of the fractional-order chaotic system is achieved.

2. The cloud control stabilization method for fractional-order chaotic systems according to claim 1, characterized in that, The fractional-order chaotic system is: ； In the formula, for Caputo differential of a fractional-order system at time step At the initial moment, Let be the order of the fractional order in the system; for The system state at any given moment. Given a diagonal matrix, The connection weight matrix; The sampling controller to be designed; It is a Lipschitz continuously differentiable nonlinear function. For any two distinct real numbers and There exists a constant satisfy: .

3. The cloud control stabilization method for fractional-order chaotic systems according to claim 2, characterized in that, The designed sampling controller is as follows: ； In the formula, the sampling time series satisfies For the first time intervals , Indicates the sampling period; This is the feedback gain matrix.

4. The cloud control stabilization method for fractional-order chaotic systems according to claim 3, characterized in that, After encryption and decryption, a quantization error occurs in the actual control input, which leads to the following: and , , These represent the gain matrices respectively. and system status The quantization errors are all bounded; The fractional-order chaotic system model can then be rewritten as: ； The rewritten fractional-order chaotic system model is transformed into nonfragile control of a perturbation-based fractional-order chaotic system: ； In the formula, Treated as a bounded perturbation, and , These are the parameters to be designed; control gain quantization error. satisfy , Given a known real constant matrix, and an unknown matrix. satisfy ; The method of achieving stable control of a fractional-order chaotic system based on control input includes: For any given scalar and Under sampling control, fractional-order chaotic systems can asymptotically achieve mean-square bounded stability in the mean square. If and only if there exists a positive definite matrix and its arbitrary diagonal matrix , and positive scalar The following linear matrix inequalities must be satisfied: ； in, , , , , , ;also, The always in the collection middle, .

5. The cloud control stabilization method for fractional-order chaotic systems according to claim 4, characterized in that, The discrete-time control law is expressed as a time-delay control signal between adjacent sampling times, allowing the sampled data system to be modeled as a continuous-time system with input delay: ； In the formula, ; The method of achieving stable control of a fractional-order chaotic system based on control input further includes: To ensure secure computation based on encrypted data, fixed quantization parameters are designed. and Under the premise that the following conditions are met, the state trajectories of systems 1.1 and 1.2 will be completely identical: set up ,like and Satisfy any All of them are: ； ； Then it exists , .

6. The cloud control stabilization method for a fractional-order chaotic system according to any one of claims 4 to 5, characterized in that, , For having The decimal part The set of rational numbers with fixed points; The method of achieving stable control of a fractional-order chaotic system based on control input further includes: Assuming bit length With Paillier modulus satisfy: ； ； Then the first encryption control input Each component is given by the following formula: ； Among them, for any randomly generated , any , .

7. A cloud control stabilization device for a fractional-order chaotic system based on homomorphic encryption, applied to a client, characterized in that, The device includes: The construction and design module is used to build a fractional-order chaotic system model and design the sampling controller in the fractional-order chaotic system model to determine the sampling and transmission times; The quantization module is used to quantize the sampled system state using a fixed-point rational number architecture; The encryption module is used to encrypt the quantized system state using Paillier semi-homomorphic encryption and transmit it to the cloud; The computation module is used to receive the encrypted state data received from the cloud and perform homomorphic operations on the quantization control gain matrix to obtain the encrypted control input for the operation; The decryption and control module is used to decrypt and scale the encrypted control input to obtain the plaintext form of the control input; based on the control input, stable control of the fractional-order chaotic system is achieved.

8. A terminal device, comprising a processor and a memory for storing a processor-executable program, characterized in that, When the processor executes the program stored in the memory, it implements the cloud control stabilization method for the fractional-order chaotic system as described in any one of claims 1 to 6.

9. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by the processor, it implements the cloud control stabilization method for fractional-order chaotic systems as described in any one of claims 1 to 6.

10. A computer program product, characterized in that, It includes a computer program that, when executed by a processor, implements the cloud control stabilization method for a fractional-order chaotic system as described in any one of claims 1 to 6.

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