Method for securing cloud-embedded cyber-physical systems based on elastic homomorphic encryption

Abstract

The application relates to the field of information physics technology and relates to a cloud-embedded information physical system security protection method based on elastic homomorphism encryption, wherein an encryption key is divided into two security levels based on a partial homomorphism SWHE encryption technology standard design. The second-level key is authorized to the cloud, and the first-level key is only disclosed to the CPS. Therefore, the method designed by the application can not only effectively detect whether an FDI attack is received, but also can elastically recover original system plaintext signals from the polluted system data in response to complex type FDI attacks, eliminates the influence of the FDI attack on the system control signal, effectively solves the defects and deficiencies of the prior art in the privacy protection, FDI attack detection and elastic recovery of the cloud-embedded information physical system, and the security label designed by the application can monitor the key leakage risk in real time, so that the system security performance is effectively improved.

Description

Technical Field

[0001] This invention relates to the field of cyber-physical technology, specifically to a security protection method for cloud-embedded cyber-physical systems based on elastic homomorphic encryption. Background Technology

[0002] With the development of information technology, cyber-physical systems (CPS), which integrate sensing, computing, and communication, meet the requirements of intelligent control systems in the modern era and satisfy people's needs for real-time monitoring, transmission, and processing of system information. Consequently, they have been widely used in fields such as industry, energy, transportation, healthcare, intelligent transportation, and the military. Figure 1 .

[0003] In traditional CPS system architectures, all calculations for control information are performed by the CPS feedback controller, thus concentrating all computational burden on the CPS side. Furthermore, if an attacker intercepts the feedback controller and its calculated feedback control signals, the privacy and security of the control system will face significant risks and challenges.

[0004] The development of cloud computing technology has brought about a revolution in data processing methods and further deepened the integration of the physical layer and the network layer. To alleviate the computational load of Cyber-Physical Systems (CPS) and improve the privacy protection performance of feedback controllers, current solutions involve encrypting feedback controller parameters and control signals, and outsourcing feedback control computation to cloud servers. This framework, known as cloud-embedded cyber-physical systems, leverages the powerful storage and computing resources of cloud servers to effectively alleviate the computational pressure on CPS and prevent the leakage of feedback control information.

[0005] However, with increasingly frequent information exchange within the system, cloud-embedded cyber-physical systems are facing growing security vulnerabilities. Among these, false data injection (FDI) attacks, a typical attack strategy, can mislead and drive the control system away from its normal operating state by tampering with the transmission signals between the cloud server and the cyber-physical system, leading to energy loss and damage to system hardware. To improve the security performance of the control system, current protection strategies combine data encryption mechanisms with FDI attack detectors to identify attackers' tampering with control signals.

[0006] Since the cloud is generally considered insecure, and to prevent feedback control information from being leaked to attackers, data sent to the cloud for processing must be pre-encrypted and then sent back to the Cyber-Physical System (CPS) for decryption after computation. However, the encryption methods currently widely used, based on public-key cryptography such as RSA and Paillier, have a decryption computational complexity that exceeds the carrying capacity of existing cyber-physical systems, making them unsuitable for practical engineering applications.

[0007] More importantly, current security protection methods for cloud-embedded cyber-physical systems have vulnerabilities in detecting FDI attacks. While existing solutions can detect additive FDI attacks, they cannot accurately identify multiplicative and additive-multiplicative FDI attacks. Furthermore, current methods can only detect attacks, not eliminate the interference of FDI attacks on control signals. Therefore, the robustness of cyber-physical systems is severely weakened. Summary of the Invention

[0008] To address the shortcomings of existing technologies, the present invention aims to propose a security protection method for cloud-embedded cyber-physical systems based on elastic homomorphic encryption, comprising:

[0009] Step 1: Construct a Cyber-Physical System (CPS) model, and generate the original signal based on the CPS model. and The raw signals of Cyber-Physical Systems (CPS) and Perform integer mapping to obtain the system plaintext signal after integer mapping. and ;

[0010] Step 2: Set secret parameters ， ， ， ， ,Will ， ， As the first-level key, ， As the second-level key; authorize the first-level key and the second-level key to the CPS, and authorize the second-level key to the cloud server;

[0011] in, Represented as: ,in, They are all positive integers. To represent a dynamic template matrix, Let be a dynamic template vector, where, and All are positive integers. and Used to mask the characteristics of the system's plaintext signals, and and Generated by a strong pseudo-random number generator that iterates over a random seed. It is a positive integer. It is a prime number greater than a preset threshold. It is a static template matrix. It is a static template vector. It is a prime number greater than a preset threshold;

[0012] Step 3: In CPS, based on the partially homomorphic encryption technique SWHE, design a homomorphic encryption algorithm and, based on the secret parameter, process the system's plaintext signal. and Encryption is performed to obtain the system ciphertext signal. and CPS generates security tags The security label The first-level key used in homomorphic encryption and Monitor the leakage situation and include safety labels. Insert system ciphertext signal In the process, the tagged system ciphertext signal is obtained. tagged system encrypted signals and system ciphertext signals Uploaded to the cloud server;

[0013] Step 4: The cloud server receives the encrypted signal from the system. and tagged system encrypted signals According to the tagged system ciphertext signal Obtain security label and system ciphertext signals According to the system's encrypted signal and The cloud server performs feedback control calculations to obtain the encrypted feedback control signal. ; Safety label Add pseudo-random noise The tagged ciphertext feedback control signal is obtained and sent to the CPS.

[0014] Step 5: Determine the key based on the tagged ciphertext feedback control signal received by CPS. and Has it been leaked?

[0015] Step 6: CPS receives the tagged ciphertext feedback control signal. Based on the received tagged ciphertext feedback control signal, the first-level key, and the second-level key, CPS determines whether to upload the tagged system ciphertext signal to the cloud server. and During the process of the cloud server sending tagged encrypted feedback control signals to the CPS, whether it is attacked, and whether the CPS uploads tagged system encrypted signals to the cloud server. and In the event of an attack during the process of the cloud server sending tagged encrypted feedback control signals to the CPS, step 7 is executed, whereby the CPS uploads tagged system encrypted signals to the cloud server. and If no attack occurs during the process of the cloud server sending tagged encrypted feedback control signals to the CPS, proceed to step 8.

[0016] Step 7: Determine if the tagged system encrypted signal is uploaded from CPS to the cloud server. and During the process, and during the process of the cloud server sending tagged encrypted feedback control signals to the CPS, the types of attacks that are carried out;

[0017] Step 8: Decrypt the ciphertext feedback control signal to obtain the system ciphertext signal. and .

[0018] Optionally, step 1 specifically includes:

[0019] Step 1.1: Construct a model of a Cyber-Physical System (CPS). The operation of the Cyber-Physical System model is represented by state equations, specifically by the following formula:

[0020] (1);

[0021] in, Represents the Cyber-Physical System (CPS) k The state variables of the next iteration. Represents the Cyber-Physical System (CPS) k+ The state variables in one iteration This represents the sensor measurement signals of a Cyber-Physical System (CPS). This represents the control input signals of a Cyber-Physical System (CPS). This represents the noise vector applied to a cyber-physical system (CPS). , , , , and These are real-number matrices of different dimensions, used as parameters of a Cyber-Physical System (CPS) to characterize the features of the CPS. Represents the set of real numbers. These are positive integers representing the dimensions of the matrix;

[0022] Control signal in formula (1) The feedback control signal is generated by the feedback controller. The calculation is represented by the following iterative equation:

[0023] (2);

[0024] in, Indicates the first k The state variables of the feedback controller in the next iteration. Indicates the first k The state variables of the feedback controller after +1 iterations , , and This is a real-valued matrix, used as parameters of the feedback controller to characterize its features. It is an integer used to represent the dimension of the state variables of the feedback controller;

[0025] Formula (2) can be rewritten in matrix-vector product form as follows:

[0026] (3);

[0027] in, The parameter matrix of the feedback controller. It is a vector composed of the state variables of the feedback controller and the sensor measurement signals. and As the raw signal of a Cyber-Physical System (CPS); among which, and All are positive integers. For matrix Dimensions For vectors Dimensions ， ;

[0028] Step 1.2: Raw signals from the Cyber-Physical System (CPS) and Integer mapping is specifically represented as:

[0029] (4);

[0030] (5);

[0031] in, Indicates quantization precision. and It is the system plaintext signal after integer mapping. This represents the function for rounding up.

[0032] Optionally, step 3 specifically includes:

[0033] Step 3.1: In CPS, based on the partially homomorphic encryption technique SWHE, design a homomorphic encryption algorithm and, based on the secret parameter, process the system's plaintext signal. and Encryption is achieved using the following formula:

[0034] (6);

[0035] (7);

[0036] Among them, symbols This represents a multiplication operation between matrices or vectors within a range of elements. Indicates system plaintext signal The system's encrypted signal, vector The system ciphertext signal represents the system plaintext signal;

[0037] Step 3.2: CPS generates security tags The security label Represented as:

[0038] (8);

[0039] in, , and For secret parameters, and All are prime numbers greater than a preset threshold. For matrix The Middle Any element in the row, and It is a positive integer. and Each value is a natural number. Represents the set of natural numbers. This is the serial number used to identify the security label, and d It is an integer;

[0040] Step 3.3: Insert the security tag into the system encrypted signal via synchronization mode I. Specifically, this includes:

[0041] Based on a 1-D chaotic system, a pseudo-random sequence is obtained through multiple iterations according to a preset number of iterations. The 1-D chaotic system is represented as follows:

[0042] (9);

[0043] in, and For the control parameters of a 1-D chaotic system, Indicates the first... via the chaotic system t The pseudo-random numbers generated in the next iteration Indicates the first... via the chaotic system t The pseudo-random number generated in +1 iterations and The values ​​are all less than 1. It is an integer used to identify the number of iterations of the chaotic system;

[0044] The pseudo-random sequence and Multiply and round to obtain the insertion position information. Based on the insertion position information, insert the security tag into the system ciphertext signal. In the process, the tagged system ciphertext signal is obtained. , It is an integer representing the total number of A security tag was inserted into the system's encrypted signal. In the middle, the tagged system encrypted signal and system ciphertext signals Uploaded to the cloud server.

[0045] Optionally, step 4 includes:

[0046] Step 4.1: The cloud server receives the system's encrypted signal. and tagged system encrypted signals In synchronous mode I, from the tagged system ciphertext signal Extraction is performed during this process; specifically, the security tag is inserted into the system's encrypted signal using synchronization mode I. Using a 1-D chaotic system with the same process, a pseudo-random sequence is obtained based on the 1-D chaotic system, and then the insertion position information is obtained. Based on the insertion position information, the security label is extracted at the insertion position. Thus, the system ciphertext signal is obtained. ;

[0047] Step 4.2: Based on the system ciphertext signal and The feedback control calculation is performed by the cloud server, specifically expressed by the following formula:

[0048] (10);

[0049] in, This is a ciphertext feedback control signal;

[0050] Step 4.3: Place the security label Added pseudo-random noise , Specifically, it is expressed by the following formula:

[0051] (11);

[0052] in, , The security tag is obtained after noise processing, wherein the pseudo-random noise is known to the CPS; the security tag is obtained after noise processing. In Synchronization Mode II, it is randomly inserted into the ciphertext feedback control signal. The tagged ciphertext feedback control signal is obtained. The formula for synchronization mode II is the same as that for synchronization mode I, but the specific values ​​of the control parameters of the 1-D chaotic system are different. The tagged ciphertext feedback control signal is then sent to the CPS.

[0053] Optionally, step 5 includes:

[0054] Step 5.1: Receive the tagged encrypted feedback control signal sent by the cloud server at the CPS end, and extract the tagged encrypted feedback control signal using Synchronization Mode II to obtain the security tag. For safety labels The security label authentication calculation is represented by the following formula:

[0055] (12);

[0056] in, The pseudo-random noise added to the cloud server can be removed by CPS. , This is the result obtained after security label authentication calculation, where, The calculation formula for the operator is expressed as:

[0057] (13);

[0058] Among them, parameters Represents a vector consisting of binary numbers. Represents any integer vector. It is a positive integer;

[0059] Step 5.2: Assume a key and The data was leaked, and the attacker uploaded tagged encrypted system signals to the cloud server via CPS. During the process, inject any attack signal Furthermore, an attack signal is injected during the process of the cloud server sending tagged encrypted feedback control signals to the CPS. In this case, the security label received by CPS is represented as:

[0060] (14);

[0061] in, It is a positive integer;

[0062] Solving the equations (14) and (12) simultaneously yields the following results: The specific value, when = At that time, it represents the CPS uploading a tagged system encrypted signal to the cloud server. During the process, and during the process of the cloud server sending tagged encrypted feedback control signals to the CPS, it was not subjected to FDI attacks; when = + At this time, it indicates that CPS is uploading a tagged system encrypted signal to the cloud server. During the process, and during the process of the cloud server sending tagged encrypted feedback control signals to CPS, it was attacked by FDI, and thus executed step 5.3;

[0063] Step 5.3: [The sentence is incomplete and requires more context to be translated accurately.] Substituting into the verification equation, the verification equation is expressed as:

[0064] (15);

[0065] Determine whether the left and right sides of the verification equation are equal. If the left and right sides of the verification equation are equal, it indicates that CPS will determine the key. and The key has already been leaked to the attacker, meaning the assumption in step 5.2 holds true. If the left and right sides of the verification equation are unequal, CPS will determine the key. and It was not leaked to the attacker, meaning the assumption in step 5.2 is not true.

[0066] Optionally, step 6 includes:

[0067] CPS receives the tagged ciphertext feedback control signal, extracts the security tag from the tagged ciphertext feedback control signal, and obtains the tagged ciphertext feedback signal. Therefore, to judge and Whether they are equal, in and If they are equal, it indicates that the CPS is uploading a tagged system ciphertext signal to the cloud server. and During the process, and during the process of the cloud server sending tagged encrypted feedback control signals to the CPS, it was not attacked. and If they are not equal, it indicates that the CPS has uploaded a tagged system ciphertext signal to the cloud server. and During the process, and during the process of the cloud server sending tagged encrypted feedback control signals to the CPS, it was attacked.

[0068] Optionally, step 7 specifically includes:

[0069] Step 7.1: The cloud server calculates the FDI multiplicative attack factor using the second-level key. and Specifically, it is calculated using the following formula:

[0070] (16);

[0071] (17);

[0072] in, and This indicates that the CPS is uploading encrypted signals from the attacked system to the cloud server.

[0073] The calculated multiplicative attack factor and When the value equals 1, it indicates that the tagged system encrypted signal is uploaded from CPS to the cloud server. and During the process, and during the process of the cloud server sending tagged ciphertext feedback control signals to CPS, the attack type is not a multiplicative FDI attack, so proceed to step 7.2.

[0074] The calculated multiplicative attack factor and When the value is not equal to 1, it indicates that the CPS is uploading a tagged system ciphertext signal to the cloud server. and During the process, and during the process of the cloud server sending tagged encrypted feedback control signals to the CPS, the attack type is a multiplicative FDI attack. Therefore, the cloud server filters out multiplicative attacks, specifically through the following formula:

[0075] (18);

[0076] (19);

[0077] The cloud server filters out multiplicative attack factors in the received encrypted feedback control signals. and The multiplicative attack factor in the received ciphertext feedback control signal is filtered out on the cloud server. and The signal after that is represented as:

[0078] (20);

[0079] As can be seen from formula (20), multiplicative attacks are filtered out. and The subsequent signal also contains a multiplicative aggression factor. Calculate the multiplicative attack factor The value of is achieved through the following formula:

[0080] (twenty one);

[0081] Multiplicative attack factor Filtering out multiplicative attacks and Remove the signal from the filtered multiplicative attack to obtain the ciphertext feedback control signal. Substitute the filtered multiplicative attack ciphertext feedback control signal into formula (22). In this process, the original system plaintext signal is obtained. and Formula (22) is expressed as:

[0082] (twenty two);

[0083] Step 7.2: CPS calculates the multiplicative attack factor in an additive-multiplicative FDI attack, specifically using the following formula:

[0084] (twenty three);

[0085] The calculated multiplicative attack factor When the value equals 1, it indicates that the tagged system encrypted signal is uploaded from CPS to the cloud server. and During the process, and during the process of the cloud server sending tagged ciphertext feedback control signals to the CPS, the attack type is not an additive-multiplicative FDI attack, so proceed to step 7.3;

[0086] The calculated multiplicative attack factor When the value is not equal to 1, it indicates that the CPS is uploading a tagged system ciphertext signal to the cloud server. and During the process, and during the process of the cloud server sending tagged ciphertext feedback control signals to the CPS, the attack type is additive-multiplicative FDI attack, which in turn increases the multiplicative attack factor. The purified ciphertext feedback control signal is obtained by removing it from the ciphertext feedback control signal received from the CPS. This will then allow the purified ciphertext to be fed back as a control signal. Substituting into formula (24), the signal injected by the additive FDI attack is removed, and the original system plaintext signal is obtained. and Formula (24) is expressed as:

[0087] (twenty four);

[0088] Step 7.3: Upload tagged system encrypted signals to the cloud server via CPS. and In the process of the cloud server sending tagged ciphertext feedback control signals to the CPS, if the attack type is neither additive-multiplicative FDI nor multiplicative FDI, it indicates that the attack type is additive FDI. Therefore, the ciphertext feedback control signals received by the CPS are substituted into formula (22). In this process, the original system plaintext signal is obtained. and .

[0089] Optionally, step 8 can be implemented using the following formula:

[0090] (25);

[0091] Based on formula (25), the system ciphertext signal is calculated. and .

[0092] The beneficial effects of adopting the above technical solution are as follows:

[0093] This invention is based on the Somewhat Homomorphic Encryption (SWHE) standard and divides the encryption key into two security levels. The lower-level (second-level) key is authorized to the cloud, while the higher-level (first-level) key is only disclosed to the CPS. Compared with existing cloud-embedded cyber-physical system security protection methods, the method designed in this invention, while ensuring the privacy of the cloud-embedded CPS, can not only effectively detect whether it is under FDI attack, but also elastically recover the original plaintext signal from the contaminated system data for complex types of FDI attacks (additive, multiplicative, or additive-multiplicative FDI attacks), eliminating the impact of FDI attacks on system control signals. It effectively solves the shortcomings and deficiencies of existing methods in privacy protection, FDI attack detection, and elastic recovery of cloud-embedded cyber-physical systems. The method proposed in this invention has good detection performance, high-quality reconstructed signal, and provides sufficient protection for CPS privacy. The method designed in this invention is not only highly accurate, but also has fast processing speed and low latency, ensuring the secure, real-time, and stable operation of the cloud-embedded cyber-physical system.

[0094] The method described in this invention has low computational complexity, offering significant advantages over homomorphic encryption algorithms such as RSA and Paillier, and is suitable for practical engineering applications. Furthermore, the security tag designed in this invention can monitor key leakage risks in real time, effectively improving system security. Within the framework of the proposed security protection method, the security performance of cloud-embedded cyber-physical systems will be significantly enhanced. Attached Figure Description

[0095] Figure 1 This is a conceptual diagram of a cloud-embedded cyber-physical system in an embodiment of the present invention;

[0096] Figure 2 This is a schematic diagram of the security protection method for cloud-embedded cyber-physical systems based on elastic homomorphic encryption in an embodiment of the present invention;

[0097] Figure 3 This is a flowchart illustrating the operation of the security tag in an embodiment of the present invention;

[0098] Figure 4 The following are schematic diagrams of the detection results of various types of FDI attacks in the embodiments of the present invention, wherein (a) is a schematic diagram of the detection results of additive FDI attacks, (b) is a schematic diagram of the detection results of multiplicative FDI attacks, (c) is a schematic diagram of the detection results of additive-multiplicative FDI attacks, and (d) is a schematic diagram of the detection results of large-scale additive FDI attacks.

[0099] Figure 5This is a schematic diagram of the removal performance against various types of existing FDI attacks in an embodiment of the present invention, wherein (a) is a schematic diagram of elastic recovery error under additive FDI attack, (b) is a schematic diagram of elastic recovery error under multiplicative FDI attack, and (c) is a schematic diagram of elastic recovery error under additive-multiplicative FDI attack.

[0100] Figure 6 This is a schematic diagram illustrating the assessment of the risk of key leakage by the security tag in an embodiment of the present invention. Detailed Implementation

[0101] The specific embodiments of the present invention will be described in further detail below with reference to the accompanying drawings and examples. The following examples are for illustrative purposes only and are not intended to limit the scope of the invention.

[0102] To address the problems of existing technologies, this invention designs a security protection method for cloud-enabled cyber-physical systems (CECPS) based on resilient homomorphic encryption technology, specifically targeting the security requirements of CECPS. This method aims to solve the problems of computational complexity, FDI attack detection, and resilience in existing CECPS systems. The method designed in this invention features low computational complexity, making it suitable for practical engineering applications. This method can effectively detect various complex types of FDI attacks and recover and reconstruct the original control signals from the attack signals.

[0103] Specifically, this invention provides a security protection method for cloud-embedded cyber-physical systems based on elastic homomorphic encryption. Targeting the characteristics of cloud-embedded cyber-physical systems, it utilizes homomorphic encryption technology to achieve privacy protection of the controller, FDI attack detection, and control signal reconstruction. Figure 2 This may include the following steps:

[0104] Step 1: Construct a Cyber-Physical System (CPS) model, and generate the original signal based on the CPS model. and The raw signals of Cyber-Physical Systems (CPS) and Perform integer mapping to obtain the system plaintext signal after integer mapping. and ;

[0105] Step 1.1: Construct a model of a Cyber-Physical System (CPS). A Cyber-Physical System (CPS) uses deployed sensors to monitor and estimate the system state in real time. The operation of the CPS model is represented by state equations, specifically by the following formula:

[0106] (1);

[0107] in, Represents the Cyber-Physical System (CPS) k The state variables of the next iteration. Represents the Cyber-Physical System (CPS) k+ The state variables in one iteration This represents the sensor measurement signals of a Cyber-Physical System (CPS). This represents the control input signals of a Cyber-Physical System (CPS). This represents the noise vector applied to a cyber-physical system (CPS). , , , , and These are real-number matrices of different dimensions, used as parameters of a Cyber-Physical System (CPS) to characterize the features of the CPS. Represents the set of real numbers. These are positive integers representing the dimensions of the matrix;

[0108] Control signal in formula (1) The feedback control signal is generated by the feedback controller. The calculation is represented by the following iterative equation:

[0109] (2);

[0110] in, Indicates the first k The state variables of the feedback controller in the next iteration. Indicates the first k The state variables of the feedback controller after +1 iterations , , and This is a real-valued matrix, used as parameters of the feedback controller to characterize its features. It is an integer used to represent the dimension of the state variables of the feedback controller;

[0111] Formula (2) can be rewritten in matrix-vector product form as follows:

[0112] (3);

[0113] in, The parameter matrix of the feedback controller. It is a vector composed of the state variables of the feedback controller and the sensor measurement signals. and As the raw signal of a Cyber-Physical System (CPS); among which, and All are positive integers. Let F be the dimension of matrix F. For vectors Dimensions ， ;

[0114] Since the original signals of the aforementioned system need to be transmitted in an open network environment, they need to be encrypted before being uploaded to the cloud server to complete the calculation of feedback control signals in order to prevent the leakage of control system information. However, since non-homomorphic encryption algorithms cannot guarantee the correctness of decryption, it is necessary to design an encryption algorithm that satisfies homomorphic properties. Compared with fully homomorphic encryption (FHE) algorithms, partially homomorphic encryption (SWHE) algorithms can balance encryption security and computational efficiency, and are therefore more suitable for practical engineering applications. To design an SWHE algorithm that meets the above objectives, the original system signals first need to be mapped to integer signals, as detailed in step 1.2.

[0115] Step 1.2: Raw signals from the Cyber-Physical System (CPS) and Integer mapping is performed, directly mapping the original system signal to an integer signal. The mapped value space includes both positive and negative integer spaces. Specifically, let... The quantization accuracy represents the requirement of the control system to meet the quantization error requirements. Based on this, the raw signals of the Cyber-Physical System (CPS) are... and Integer mapping is specifically represented as:

[0116] (4);

[0117] (5);

[0118] in, Indicates quantization precision. and It is the system plaintext signal after integer mapping. This represents the floor function. After the integer mapping is complete... and It will meet the requirements of some homomorphic encryption algorithms.

[0119] Since the cloud is generally considered insecure, system control signals cannot be directly exposed to the cloud server. Therefore, in this invention, all keys are divided into two levels, as detailed in step 2.

[0120] Step 2: Set secret parameters ， ， ， ， ,Will ， ， As the first-level key, ， As the second-level key; authorize the first-level key and the second-level key to the CPS, and authorize the second-level key to the cloud server;

[0121] in, Represented as: ,in, All are positive integers. To represent a dynamic template matrix, Let be a dynamic template vector, where, and All are positive integers. and Used to mask the characteristics of the system's plaintext signals, and and Generated by a strong pseudo-random number generator that iterates over a random seed. It is a positive integer. It is a prime number greater than a preset threshold, which can be 2 to the power of 30. It is a static template matrix. It is a static template vector. It is a prime number greater than a preset threshold;

[0122] Because the first-level key is unknown, even if the cloud server possesses the second-level key, it cannot correctly decrypt the system's plaintext signal. In this situation, the CPS system information remains confidential to the cloud server, thus protecting the privacy of the control system. Based on the aforementioned two-level key allocation authorization, the system's plaintext signal... and Encryption will be performed using a Cyber-Physical System (CPS), which will be accomplished in step 3 below.

[0123] Step 3: In CPS, based on the partially homomorphic encryption technique SWHE, design a homomorphic encryption algorithm and, based on the secret parameter, process the system's plaintext signal. and Encryption is performed to obtain the system ciphertext signal. and CPS generates security tags The security label The first-level key used in homomorphic encryption and Monitor the leakage situation and include safety labels. Insert system ciphertext signal In the process, the tagged system ciphertext signal is obtained. tagged system encrypted signals and system ciphertext signals Uploaded to the cloud server;

[0124] Step 3.1: In CPS, based on the partially homomorphic encryption technique SWHE, design a homomorphic encryption algorithm and, based on the secret parameter, process the system's plaintext signal. and Encryption is achieved using the following formula:

[0125] (6);

[0126] (7);

[0127] Among them, symbols This represents a multiplication operation between matrices or vectors within a range of elements. Indicates system plaintext signal The system's encrypted signal, vector The system ciphertext signal represents the system plaintext signal;

[0128] After the homomorphic encryption operation is completed, because It doesn't need to be transmitted frequently, therefore the risk of attack is low, and it can be sent directly to the cloud. It will be sent to the cloud after a security tag is inserted for further feedback control calculations.

[0129] Step 3.2: Combining Figure 3 CPS generates security labels Security tags are used to secure the first-level key in homomorphic encryption. and Monitor for potential leaks. Once the key... and If the leaked key is given to an attacker, any FDI attacks launched using the leaked key will be detected by CPS. Without loss of generality, let... , It is a positive integer (as previously defined), therefore, the security label Represented as:

[0130] (8);

[0131] in, , and For secret parameters, and All are prime numbers greater than a preset threshold, which can be 2 to the power of 30. For matrix The Middle Any element in the row, and It is a positive integer. and Each value is a natural number. Represents the set of natural numbers. This is the serial number used to identify the security label, and d It is an integer;

[0132] Step 3.3: Insert the security tag into the system encrypted signal via synchronization mode I. In this process, once the system plaintext signal is encrypted into a system ciphertext signal and a security tag is generated by the Cyber-Physical System (CPS), the CPS will transmit the ciphertext signal to the system in Synchronization Mode I. Security tags are randomly inserted. Note that the synchronization mode means that the security tag is inserted with a ciphertext signal. The position within is controllable and can be controlled by a chaotic system.

[0133] Specifically, based on a 1-D chaotic system, a pseudo-random sequence is obtained through multiple iterations according to a preset number of iterations. The 1-D chaotic system is represented as follows:

[0134] (9);

[0135] in, and For the control parameters of a 1-D chaotic system, Indicates the first... via the chaotic system t The pseudo-random numbers generated in the next iteration Indicates the first... via the chaotic system t The pseudo-random number generated in +1 iterations and The values ​​are all less than 1. It is an integer used to identify the number of iterations of the chaotic system;

[0136] In practical implementation, initial values ​​are set for the control parameters of the chaotic system, while satisfying... and Under certain conditions, iteration will cause the control system to enter a chaotic state and generate a pseudo-random sequence.

[0137] The pseudo-random sequence and Multiply and round to obtain the insertion position information. Based on the insertion position information, insert the security tag into the system ciphertext signal. In the process, the tagged system ciphertext signal is obtained. , It is an integer representing the total number of A security tag was inserted into the system's encrypted signal. In the middle, the tagged system encrypted signal and system ciphertext signals Uploaded to the cloud server.

[0138] Step 4: The cloud server receives the encrypted signal from the system. and tagged system encrypted signals According to the tagged system ciphertext signal Obtain security label and system ciphertext signals According to the system's encrypted signal and The cloud server performs feedback control calculations to obtain the encrypted feedback control signal. ; Safety label Add pseudo-random noise The tagged ciphertext feedback control signal is obtained and sent to the CPS.

[0139] Step 4.1: The cloud server receives the system's encrypted signal. and tagged system encrypted signals In synchronous mode I, from the tagged system ciphertext signal Extraction is performed during this process; specifically, the security tag is inserted into the system's encrypted signal using synchronization mode I. Using a 1-D chaotic system with the same process, a pseudo-random sequence is obtained based on the 1-D chaotic system, and then the insertion position information is obtained. Based on the insertion position information, the security label is extracted at the insertion position. Thus, the system ciphertext signal is obtained. ;

[0140] It should be noted that the safety label During the transmission from CPS to the cloud, the data may be tampered with by attackers, therefore the security tags extracted by the cloud are used... This indicates that if the security label has not been subjected to an FDI attack, then .

[0141] Step 4.2: Based on the system ciphertext signal and The feedback control calculation is performed by the cloud server, specifically expressed by the following formula:

[0142] (10);

[0143] in, This is a ciphertext feedback control signal;

[0144] Step 4.3: Place the security label Added pseudo-random noise , This is to mask the computational processing behavior of feedback control, specifically expressed by the following formula:

[0145] (11);

[0146] in, , The security tag is obtained after noise processing, wherein the pseudo-random noise is known to the CPS; the security tag is obtained after noise processing. In Synchronization Mode II, it is randomly inserted into the ciphertext feedback control signal. The tagged encrypted feedback control signal is obtained. The formula for Synchronization Mode II is the same as that for Synchronization Mode I, but the specific values ​​of the control parameters of the 1-D chaotic system are different. Similarly, referring to Synchronization Mode I, a pseudo-random sequence is generated through the chaotic system to determine the insertion position information. Based on the insertion position information, the security tag is... Insert into ciphertext feedback control signal In the process, the tagged ciphertext feedback control signal is obtained and sent to the CPS.

[0147] Pseudo-random noise is known for cyber-physical systems (CPS) and can therefore be eliminated at the CPS end.

[0148] Step 5: Determine the key based on the tagged ciphertext feedback control signal received by CPS. and Has it been leaked?

[0149] Step 5.1: Receive the tagged encrypted feedback control signal sent by the cloud server at the CPS end, and extract the tagged encrypted feedback control signal using Synchronization Mode II to obtain the security tag. For safety labels The security label authentication calculation is represented by the following formula:

[0150] (12);

[0151] in, Pseudo-random noise added to cloud servers can be removed by Cyber-Physical Sequence (CPS). , This is the result obtained after security label authentication calculation, where, The calculation formula for the operator is expressed as:

[0152] (13);

[0153] Among them, parameters Represents a vector consisting of binary numbers. Represents any integer vector. It is a positive integer;

[0154] Step 5.2: Assume a key and The data was leaked, and the attacker uploaded tagged encrypted system signals to the cloud server via CPS. During the process, inject any attack signal Furthermore, an attack signal is injected during the process of the cloud server sending tagged encrypted feedback control signals to the CPS. In this scenario, the aforementioned FDI attack injection method can evade attack detectors without a security tag. However, once a security tag is inserted and it is subjected to an FDI attack, the security tag received by the Cyber-Physical System (CPS) will be tampered with, and the security tag received by the CPS will then be as follows:

[0155] (14);

[0156] in, It is a positive integer;

[0157] Solving the equations (14) and (12) simultaneously yields the following results: The specific value, when = At that time, it represents the CPS uploading a tagged system encrypted signal to the cloud server. During the process, and during the process of the cloud server sending tagged encrypted feedback control signals to the CPS, it was not subjected to FDI attacks; when = + At this time, it indicates that CPS is uploading a tagged system encrypted signal to the cloud server. During the process, and during the process of the cloud server sending tagged encrypted feedback control signals to CPS, it was attacked by FDI, and thus executed step 5.3;

[0158] Step 5.3: [The sentence is incomplete and requires more context to be translated accurately.] Substituting into the verification equation, the verification equation is expressed as:

[0159] (15);

[0160] Determine whether the left and right sides of the verification equation are equal. If the left and right sides of the verification equation are equal, it indicates that CPS will determine the key. and The key has already been leaked to the attacker, meaning the assumption in step 5.2 holds true. If the left and right sides of the verification equation are unequal, CPS will determine the key. and It was not leaked to the attacker, meaning the assumption in step 5.2 is not true.

[0161] Step 6: CPS receives the tagged ciphertext feedback control signal. Based on the received tagged ciphertext feedback control signal, the first-level key, and the second-level key, CPS determines whether to upload the tagged system ciphertext signal to the cloud server. and During the process of the cloud server sending tagged encrypted feedback control signals to the CPS, whether it is attacked, and whether the CPS uploads tagged system encrypted signals to the cloud server. and In the event of an attack during the process of the cloud server sending tagged encrypted feedback control signals to the CPS, step 7 is executed, whereby the CPS uploads tagged system encrypted signals to the cloud server. and If no attack occurs during the process of the cloud server sending tagged encrypted feedback control signals to the CPS, proceed to step 8.

[0162] Specifically, CPS receives the tagged ciphertext feedback control signal, extracts the security tag from the tagged ciphertext feedback control signal, and obtains the tagged ciphertext feedback signal. Therefore, to judge and Whether they are equal, in and If they are equal, it indicates that the CPS is uploading a tagged system ciphertext signal to the cloud server. and During the process, and during the process of the cloud server sending tagged encrypted feedback control signals to the CPS, it was not attacked. and If they are not equal, it indicates that the CPS has uploaded a tagged system ciphertext signal to the cloud server. and During the process, and during the process of the cloud server sending tagged encrypted feedback control signals to the CPS, it was attacked.

[0163] It should be noted that, regarding step 6, the determination of whether the CPS is uploading tagged system encrypted signals to the cloud server... and During the process of sending tagged ciphertext feedback control signals from the cloud server to the CPS, whether it is attacked, this invention conducted theoretical analysis on three different attacks in the theoretical design stage: additive FDI attack, multiplicative FDI attack, and additive-multiplicative FDI attack. However, since it is impossible to determine the type of attack when receiving tagged ciphertext feedback control signals in the actual implementation process, these three attacks are only used for theoretical analysis. The specific implementation process still executes step 6. The theoretical analysis designed Table 1, the list of FDI attack detection and elastic recovery conditions for cloud embedded homomorphic encryption scheme.

[0164] Table 1. List of FDI attack detection and elastic recovery conditions for cloud-embedded homomorphic encryption scheme;

[0165]

[0166] Based on Table 1, the theoretical analysis of this invention includes:

[0167] 1. Detection of additive FDI attacks, assuming the CPS uploads tagged encrypted system signals to the cloud server. During the process, and during the process of the cloud server sending tagged ciphertext feedback control signals to the CPS, it was subjected to additive FDI attacks, and the CPS uploaded tagged system ciphertext signals to the cloud server. The attack signal injected during the process is The attack injection signal received by the cloud server during the process of sending tagged encrypted feedback control signals to the CPS is... The tampered ciphertext feedback control signal will then be represented as:

[0168] (26);

[0169] When the additive FDI attack detection condition I given in Table 1 is met, CPS will obtain the following calculation results:

[0170] (27);

[0171] in, Therefore, when At that time, due to Known (as a key), additive FDI attacks will be detected by the attack detector.

[0172] 2. Detection of multiplicative FDI attacks. Assume that the CPS uploads tagged encrypted system signals to the cloud server. During the process, and during the process of the cloud server sending tagged ciphertext feedback control signals to the CPS, it was subjected to multiplicative FDI attacks, and the CPS uploaded tagged system ciphertext signals to the cloud server. The attack signal injected during the process is The attack injection signal received by the cloud server during the process of sending tagged encrypted feedback control signals to the CPS is... ,in, express A set of positive integers of dimension 1 express Given a set of positive integers of dimension 1, the tampered ciphertext feedback control can be represented as:

[0173] (28);

[0174] When the multiplicative FDI attack detection condition II in Table 1 is met, CPS verifies the following formula:

[0175] (29);

[0176] in, = Therefore, when At that time, the multiplicative FDI attack was identified by the attack detector.

[0177] 3. Detection of additive-multiplicative FDI attacks. Assume a CPS uploads a tagged encrypted system signal to a cloud server. During the process, it was subjected to an additive FDI attack, and the attack injection signal was... Suppose that the cloud server is subjected to both additive and multiplicative FDI attacks while sending a tagged ciphertext feedback control signal to the CPS. The injected signal from the additive attack is: The multiplicative attack injected signal is The tampered ciphertext feedback control signal is:

[0178] (30);

[0179] When the additive-multiplicative FDI attack detection conditions in Table 1 are met, CPS verifies the following formula:

[0180] (31);

[0181] when At that time, additive-multiplicative FDI attacks were identified by the attack detector.

[0182] Step 7: Determine if the tagged system encrypted signal is uploaded from CPS to the cloud server. and During the process, and during the process of the cloud server sending tagged encrypted feedback control signals to the CPS, the types of attacks that are carried out;

[0183] Step 7.1: The cloud server calculates the FDI multiplicative attack factor using the second-level key. and Specifically, it is calculated using the following formula:

[0184] (16);

[0185] (17);

[0186] in, and This indicates that the CPS is uploading encrypted signals from the attacked system to the cloud server.

[0187] The calculated multiplicative attack factor and When the value equals 1, it indicates that the tagged system encrypted signal is uploaded from CPS to the cloud server. and During the process, and during the process of the cloud server sending tagged ciphertext feedback control signals to CPS, the attack type is not a multiplicative FDI attack, so proceed to step 7.2.

[0188] The calculated multiplicative attack factor and When the value is not equal to 1, it indicates that the CPS is uploading a tagged system ciphertext signal to the cloud server. and During the process, and during the process of the cloud server sending tagged encrypted feedback control signals to the CPS, the attack type is a multiplicative FDI attack. Therefore, the cloud server filters out multiplicative attacks, specifically through the following formula:

[0189] (18);

[0190] (19);

[0191] The cloud server filters out multiplicative attack factors in the received encrypted feedback control signals. and The multiplicative attack factor in the received ciphertext feedback control signal is filtered out on the cloud server. and The signal after that is represented as:

[0192] (20);

[0193] As can be seen from formula (20), multiplicative attacks are filtered out. and The subsequent signal also contains a multiplicative aggression factor. Calculate the multiplicative attack factor The value of is achieved through the following formula:

[0194] (twenty one);

[0195] Multiplicative attack factor Filtering out multiplicative attacks and Remove the signal from the filtered multiplicative attack to obtain the ciphertext feedback control signal. Substitute the filtered multiplicative attack ciphertext feedback control signal into formula (22). In this process, the original system plaintext signal is obtained. and Formula (22) is expressed as:

[0196] (twenty two);

[0197] Step 7.2: CPS calculates the multiplicative attack factor in an additive-multiplicative FDI attack, specifically using the following formula:

[0198] (twenty three);

[0199] The calculated multiplicative attack factor When the value equals 1, it indicates that the tagged system encrypted signal is uploaded from CPS to the cloud server. and During the process, and during the process of the cloud server sending tagged ciphertext feedback control signals to the CPS, the attack type is not an additive-multiplicative FDI attack, so proceed to step 7.3;

[0200] The calculated multiplicative attack factor When the value is not equal to 1, it indicates that the CPS is uploading a tagged system ciphertext signal to the cloud server. and During the process, and during the process of the cloud server sending tagged ciphertext feedback control signals to the CPS, the attack type is additive-multiplicative FDI attack, which in turn increases the multiplicative attack factor. The purified ciphertext feedback control signal is obtained by removing it from the ciphertext feedback control signal received from the CPS. This will then allow the purified ciphertext to be fed back as a control signal. Substituting into formula (24), the signal injected by the additive FDI attack is removed, and the original system plaintext signal is obtained. and Formula (24) is expressed as:

[0201] (twenty four);

[0202] It should be noted that in the theoretical design section, before step 7.2, this invention needs to determine whether the additive-multiplicative elasticity condition in Table 1 is valid. If it is valid, step 7.2 is executed. However, in the actual implementation process, it is impossible to determine whether the additive-multiplicative elasticity condition is valid, so no determination is made in the actual implementation.

[0203] Step 7.3: Upload tagged system encrypted signals to the cloud server via CPS. and In the process of the cloud server sending tagged ciphertext feedback control signals to the CPS, if the attack type is neither additive-multiplicative FDI nor multiplicative FDI, it indicates that the attack type is additive FDI. Therefore, the ciphertext feedback control signals received by the CPS are substituted into formula (22). In this process, the original system plaintext signal is obtained. and .

[0204] It should be noted that in the theoretical design process, step 7.3 requires determining whether the additive elasticity condition in Table 1 is valid. However, since it is impossible to determine whether it is valid in the actual implementation process, this invention does not make such a determination in the actual implementation process.

[0205] Step 8: Decrypt the ciphertext feedback control signal to obtain the system ciphertext signal. and Specifically, this is achieved through the following formula:

[0206] (25);

[0207] Based on formula (25), the system ciphertext signal is calculated. and .

[0208] It is worth noting that although the types of FDI attacks differ, the attack detection mechanism remains consistent. Therefore, system administrators do not need to implement different FDI attack detection strategies for different attack types, which simplifies attack detection implementation. Once an FDI attack is identified by the detector, the cyber-physical system can resiliently recover the system's encrypted signals to filter out FDI attack signals injected by attackers, restore the original system control signals, and ensure the system's secure and stable operation.

[0209] Based on the above steps, the cloud-embedded cyber-physical system security protection method based on elastic homomorphic encryption proposed in this invention, combined with the theoretical analysis, can be implemented by the following algorithm 1.

[0210] Algorithm 1: A security protection method for cloud-embedded cyber-physical systems based on elastic homomorphic encryption;

[0211] Input: Feedback controller matrix , , , Feedback controller state variables Sensor measurement .

[0212] Output: The controller's state variables at the next moment. Feedback control signal .

[0213] CPS side:

[0214] 1: The controller and sensors will respectively feed back the controller matrix. , , , State variables and sensor measurement Integrate into a system matrix and system vector .

[0215] 2: Based on equations (4) and (5), the plaintext signal and Mapped to integer matrices respectively and integer vectors .

[0216] 3: Encrypting integer signals using equations (6) and (7) and And obtain the ciphertext signal and .

[0217] 4: Production A security label Insert it in synchronization mode I The system signals containing security tags are then uploaded to the cloud server.

[0218] Cloud server side:

[0219] 5: Extract using Synchronization Mode I A security label And according to equation (11), pseudo-random noise is inserted to update the security label as follows. .

[0220] 6: Perform multiplicative FDI attack detection. Based on the multiplicative FDI attack detection condition V in Table 1, filter multiplicative FDI attack factors using the authorized secondary key. and .

[0221] 7: Calculate the matrix-vector product This yields a feedback control signal.

[0222] 8: Will A security label Insert feedback control signal in Synchronization Mode II The results are then sent back to CPS.

[0223] CPS side:

[0224] 9: Extract in Synchronization Mode II A security label Decrypt the security tag and assess the risk of key leakage.

[0225] 10: if the key is secure;

[0226] 11: Perform attack detection, capture additive, multiplicative, and additive-multiplicative FDI attacks, and record FDI attack detection information;

[0227] 12: If no FDI attack triggers the detector alarm;

[0228] 13: Decrypt according to equation (25). Perform inverse integer mapping and The feedback controller state is extracted according to equation (3). and control input .

[0229] 14: else if the FDI attack identified by the detector is within the system's resilience range;

[0230] 15: Remove various types of FDI attacks and correctly decrypt the obtained data. .

[0231] 16: end if;

[0232] 17: end if the security tag evaluation key has been compromised;

[0233] 18: Terminate decryption operation, reset key security parameters, and request data retransmission.

[0234] 19: end if.

[0235] This invention addresses the security requirements of cloud-embedded cyber-physical systems (CPS) by designing a security protection method based on resilient homomorphic encryption. It enables the detection of erroneous data injection (FDI) attacks and resilient signal recovery for CPS. This method is designed based on the partially homomorphic encryption (SWHE) standard, dividing the encryption key into two security levels. The lower-level (second-level) key is authorized to the cloud, while the higher-level (first-level) key is only disclosed to the CPS. Compared to existing security protection methods for cloud-embedded CPS, the method designed in this invention, while ensuring the privacy of the cloud-embedded CPS, can not only effectively detect various complex types of FDI attacks (additive, multiplicative, or additive-multiplicative FDI attacks), but also eliminate the impact of FDI attacks on system control signals.

[0236] The method described in this invention has low computational complexity, offering significant advantages over homomorphic encryption algorithms such as RSA and Paillier, and is suitable for practical engineering applications. Furthermore, the security tag designed in this invention can monitor key leakage risks in real time, effectively improving system security. Within the framework of the proposed security protection method, the security performance of cloud-embedded cyber-physical systems will be significantly enhanced.

[0237] This invention utilizes homomorphic encryption technology to construct an attack detector through dynamic secret parameters while ensuring system privacy. This enables the detection and identification of various complex types of FDI attacks and allows for the resilient recovery of the original plaintext signal from contaminated system data. This effectively addresses the shortcomings and deficiencies of existing methods in privacy protection, FDI attack detection, and resilient recovery for cloud-embedded cyber-physical systems. The proposed method exhibits excellent detection performance, high-quality reconstructed signal, and comprehensive protection of CPS privacy. Furthermore, the method designed in this invention not only boasts high accuracy but also fast computation speed and low latency, ensuring the secure, real-time, and stable operation of cloud-embedded cyber-physical systems.

[0238] To verify the safety performance of this invention, MATLAB was used as the simulation test platform, and a classic four-cylinder water tank control system was used as the simulation object for testing. The state of the control system was the water level in the four-cylinder water tank, and the control input was the pressure in the water pump. In the experiment, the sampling interval of the sensor measurement signals was set to 1 second. Specifically, the parameters of the four-cylinder water tank system were quantified as follows:

[0239] ;

[0240] ;

[0241] The parameters of the feedback controller are quantized as follows:

[0242] ;

[0243] ;

[0244] In the simulation, the key , The parameters p are set within a reasonable range to ensure that the aforementioned key has sufficient security distance, thereby correctly decrypting the control system signals. Specifically, the secret parameters... , , By using a strong pseudo-random number generator with a random seed, the dynamic modulus matrix is ​​obtained. and dynamic modulus vector It can be generated. Experimental results are rounded to two decimal places, therefore... .

[0245] Figure 4 The invention demonstrates the detection effectiveness of the security protection method designed in this invention against various existing FDI attacks. Among them, (a) is a schematic diagram of the detection results of additive FDI attacks, (b) is a schematic diagram of the detection results of multiplicative FDI attacks, (c) is a schematic diagram of the detection results of additive-multiplicative FDI attacks, and (d) is a schematic diagram of the detection results of large-scale additive FDI attacks.

[0246] In the simulation, FDI attacks were scheduled to launch during periods of 31-50 seconds, 101-130 seconds, and 151-160 seconds, respectively. (FDI attack components...) , , and These were all randomly generated by the simulation testing platform. To quantify the detection results for the three types of FDI attacks, the following detection metrics are given:

[0247] (32);

[0248] (33);

[0249] (34);

[0250] in, , and These represent the detection metrics for additive, multiplicative, and additive-multiplicative FDI attacks, respectively. Figure 2 As can be seen, although attackers attempt to construct various complex FDI attack injection forms to evade the detector, the FDI attacks are effectively identified and captured under the attack detection method designed in this invention.

[0251] Figure 5 The performance of removing various types of existing FDI attacks is demonstrated. (a) shows a schematic diagram of the elastic recovery error under an additive FDI attack, (b) shows the elastic recovery error under a multiplicative FDI attack, and (c) shows the elastic recovery error under an additive-multiplicative FDI attack. CPS and the cloud, by running homomorphic decryption filtering attack algorithms respectively, can extract and recover the original system information from the control signal contaminated by the FDI attack. In the simulation, the cloud server filters the multiplicative FDI attack signal after obtaining a secondary authorization key. Under this condition, the quality of the reconstructed control system signal is guaranteed.

[0252] Figure 6 This demonstrates the results of security labels monitoring the risk of key leakage. The simulation assumes an attacker has stolen the key. and When an FDI attacker injects random attacks during the 0-10 second and 31-60 second periods, and a covert additive attack during the 11-30 second period, the system administrator can analyze and assess the risk of key leakage based on the verification results of the security labels, and accurately infer that the attack during the 11-30 second period was related to the key leakage. and It was launched after the information was leaked. Figure 6 In the equation, represents the number of security tags under attack, represents the number of security tags whose decrypted content does not match the preset pseudo-plaintext, and represents the number of security tags whose decrypted content satisfies the key leakage judgment condition given in equation (15). Since the simulation simulates the situation where an attacker accurately obtains the key and launches an FDI attack, and given that the perturbation of security tags by random attacks can be filtered by the decryption algorithm, therefore... Figure 6 The points marked in the text coincide.

[0253] The above description is merely a preferred embodiment of this disclosure and an explanation of the technical principles employed. Those skilled in the art should understand that the scope of the invention involved in the embodiments of this disclosure is not limited to technical solutions formed by specific combinations of the above-described technical features, but should also cover other technical solutions formed by arbitrary combinations of the above-described technical features or their equivalents without departing from the above-described inventive concept. For example, technical solutions formed by substituting the above-described features with (but not limited to) technical features with similar functions disclosed in the embodiments of this disclosure.

Claims

1. A security protection method for cloud-embedded cyber-physical systems based on elastic homomorphic encryption, characterized in that, include: Step 1: Construct a Cyber-Physical System (CPS) model, and generate the original signal based on the CPS model. and The raw signals of Cyber-Physical Systems (CPS) and Perform integer mapping to obtain the system plaintext signal after integer mapping. and ; Step 1.1: Construct a model of a Cyber-Physical System (CPS). The operation of the Cyber-Physical System model is represented by state equations, specifically by the following formula: (1)； in, Represents the Cyber-Physical System (CPS) k The state variables of the next iteration. Represents the Cyber-Physical System (CPS) k+ The state variables in one iteration This represents the sensor measurement signals of a Cyber-Physical System (CPS). This represents the control input signals of a Cyber-Physical System (CPS). This represents the noise vector applied to a cyber-physical system (CPS). , , , , and These are real-number matrices of different dimensions, used as parameters of a Cyber-Physical System (CPS) to characterize the features of the CPS. Represents the set of real numbers. These are positive integers representing the dimensions of the matrix; Control input signal in formula (1) The control input signal is generated by the feedback controller. The calculation is represented by the following iterative equation: (2)； in, Indicates the first k The state variables of the feedback controller in the next iteration. Indicates the first k The state variables of the feedback controller after +1 iterations , , and This is a real-valued matrix, used as parameters of the feedback controller to characterize its features. It is an integer used to represent the dimension of the state variables of the feedback controller; Formula (2) can be rewritten in matrix-vector product form as follows: (3)； in, The parameter matrix of the feedback controller, It is a vector composed of the state variables of the feedback controller and the sensor measurement signals. and As the raw signal of a Cyber-Physical System (CPS); among which, and All are positive integers. For matrix Dimensions For vectors Dimensions ， ; Step 1.2: Raw signals from the Cyber-Physical System (CPS) and Integer mapping is specifically represented as: (4)； (5)； in, Indicates quantization precision. and It is the system plaintext signal after integer mapping. This represents the floor function; Step 2: Set secret parameters ， ， ， ， ,Will ， ， As the first-level key, ， As the second-level key; authorize the first-level key and the second-level key to the CPS, and authorize the second-level key to the cloud server; in, Represented as: ,in, All are positive integers. To represent a dynamic template matrix, Let be a dynamic template vector, where, and All are positive integers. and Used to mask the characteristics of the system's plaintext signals, and and Generated by a strong pseudo-random number generator that iterates over a random seed. It is a positive integer. It is a prime number greater than a preset threshold. It is a static template matrix. It is a static template vector. It is a prime number greater than a preset threshold; Step 3: In CPS, based on the partially homomorphic encryption technique SWHE, design a homomorphic encryption algorithm and, based on the secret parameter, process the system's plaintext signal. and Encryption is performed to obtain the system ciphertext signal. and CPS generates security tags The security label The first-level key used in homomorphic encryption and Monitor the leakage situation and include safety labels. Insert system ciphertext signal In the process, the tagged system ciphertext signal is obtained. tagged system encrypted signals and system ciphertext signals Uploaded to the cloud server; Step 4: The cloud server receives the encrypted signal from the system. and tagged system encrypted signals According to the tagged system ciphertext signal Obtain security label and system ciphertext signals According to the system's encrypted signal and The cloud server performs feedback control calculations to obtain the encrypted feedback control signal. ; Safety label Add pseudo-random noise The tagged ciphertext feedback control signal is obtained and sent to the CPS. Step 5: Determine the key based on the tagged ciphertext feedback control signal received by CPS. and Has it been leaked? Step 6: CPS receives the tagged ciphertext feedback control signal. Based on the received tagged ciphertext feedback control signal, the first-level key, and the second-level key, CPS determines whether to upload the tagged system ciphertext signal to the cloud server. and During the process of the cloud server sending tagged encrypted feedback control signals to the CPS, whether it is attacked, and whether the CPS uploads tagged system encrypted signals to the cloud server. and In the event of an attack during the process of the cloud server sending tagged encrypted feedback control signals to the CPS, step 7 is executed, whereby the CPS uploads tagged system encrypted signals to the cloud server. and If no attack occurs during the process of the cloud server sending tagged encrypted feedback control signals to the CPS, proceed to step 8. Step 7: Determine if the tagged system encrypted signal is uploaded from CPS to the cloud server. and During the process, and during the process of the cloud server sending tagged encrypted feedback control signals to the CPS, the types of attacks that are carried out; Step 8: Decrypt the ciphertext feedback control signal to obtain the system ciphertext signal. and .

2. The cloud-embedded cyber-physical system security protection method based on elastic homomorphic encryption according to claim 1, characterized in that, Step 3 specifically includes: Step 3.1: In CPS, based on the partially homomorphic encryption technique SWHE, design a homomorphic encryption algorithm and, based on the secret parameter, perform encryption on the system's plaintext signal. and Encryption is achieved using the following formula: (6)； (7) ； Among them, symbols This represents a multiplication operation between matrices or vectors within a range of elements. Indicates system plaintext signal The system's encrypted signal, vector Indicates system plaintext signal The system's encrypted signal; Step 3.2: CPS generates security tags The security label Represented as: (8)； in, , and For secret parameters, and All are prime numbers greater than a preset threshold. For matrix The Middle Any element in the row, and It is a positive integer. and Each value is a natural number. Represents the set of natural numbers. This is the serial number used to identify the security label, and d It is an integer; Step 3.3: Insert the security tag into the system encrypted signal via synchronization mode I. Specifically, this includes: Based on a 1-D chaotic system, a pseudo-random sequence is obtained through multiple iterations according to a preset number of iterations. The 1-D chaotic system is represented as follows: (9)； in, and For the control parameters of a 1-D chaotic system, Indicates the first... via the chaotic system t The pseudo-random numbers generated in the next iteration Indicates the first... via the chaotic system t The pseudo-random number generated in +1 iterations and The values ​​are all less than 1. It is an integer used to identify the number of iterations of the chaotic system; The pseudo-random sequence and Multiply and round to obtain the insertion position information. Based on the insertion position information, insert the security tag into the system ciphertext signal. In the process, the tagged system ciphertext signal is obtained. , It is an integer representing the total number of A security tag was inserted into the system's encrypted signal. In the middle, the tagged system encrypted signal and system ciphertext signals Uploaded to the cloud server.

3. The cloud-embedded cyber-physical system security protection method based on elastic homomorphic encryption according to claim 2, characterized in that, Step 4 includes: Step 4.1: The cloud server receives the system's encrypted signal. and tagged system encrypted signals In synchronous mode I, from the tagged system ciphertext signal Extraction is performed during this process; specifically, the security tag is inserted into the system's encrypted signal using synchronization mode I. Using a 1-D chaotic system with the same process, a pseudo-random sequence is obtained based on the 1-D chaotic system, and then the insertion position information is obtained. Based on the insertion position information, the security label is extracted at the insertion position. Thus, the system ciphertext signal is obtained. ; Step 4.2: Based on the system ciphertext signal and The feedback control calculation is performed by the cloud server, specifically expressed by the following formula: (10)； in, This is a ciphertext feedback control signal; Step 4.3: Place the security label Added pseudo-random noise , Specifically, it is expressed by the following formula: (11)； in, , The security tag is obtained after noise processing, wherein the pseudo-random noise is known to the CPS; the security tag is obtained after noise processing. In Synchronization Mode II, it is randomly inserted into the ciphertext feedback control signal. The tagged ciphertext feedback control signal is obtained. The formula for synchronization mode II is the same as that for synchronization mode I, but the specific values ​​of the control parameters of the 1-D chaotic system are different. The tagged ciphertext feedback control signal is then sent to the CPS.

4. The cloud-embedded cyber-physical system security protection method based on elastic homomorphic encryption according to claim 3, characterized in that, Step 5 includes: Step 5.1: Receive the tagged encrypted feedback control signal sent by the cloud server at the CPS end, and extract the tagged encrypted feedback control signal using Synchronization Mode II to obtain the security tag. For safety labels The security label authentication calculation is represented by the following formula: (12)； in, The pseudo-random noise added to the cloud server can be removed by CPS. , This is the result obtained after security label authentication calculation, where, The calculation formula for the operator is expressed as: (13)； Among them, parameters Represents a vector consisting of binary numbers. Represents any integer vector. It is a positive integer; Step 5.2: Assume a key and The data was leaked, and the attacker uploaded tagged encrypted system signals to the cloud server via CPS. During the process, inject any attack signal Furthermore, an attack signal is injected during the process of the cloud server sending tagged encrypted feedback control signals to the CPS. In this case, the security label received by CPS is represented as: (14)； in, It is a positive integer; Solving the equations (14) and (12) simultaneously yields the following results: The specific value, when = At that time, it represents the CPS uploading a tagged system encrypted signal to the cloud server. During the process, and during the process of the cloud server sending tagged encrypted feedback control signals to the CPS, it was not subjected to FDI attacks; when = + At this time, it indicates that CPS is uploading a tagged system encrypted signal to the cloud server. During the process, and during the process of the cloud server sending tagged encrypted feedback control signals to CPS, it was attacked by FDI, and thus executed step 5.3; Step 5.3: [The sentence is incomplete and requires more context to be translated accurately.] Substituting into the verification equation, the verification equation is expressed as: (15)； Determine whether the left and right sides of the verification equation are equal. If the left and right sides of the verification equation are equal, it indicates that CPS will determine the key. and The key has already been leaked to the attacker, meaning the assumption in step 5.2 holds true. If the left and right sides of the verification equation are unequal, CPS will determine the key. and It was not leaked to the attacker, meaning the assumption in step 5.2 is not true.

5. The cloud-embedded cyber-physical system security protection method based on elastic homomorphic encryption according to claim 4, characterized in that, Step 6 includes: CPS receives the tagged ciphertext feedback control signal, extracts the security tag from the tagged ciphertext feedback control signal, and obtains the tagged ciphertext feedback signal. Therefore, to judge and Whether they are equal, in and If they are equal, it indicates that the CPS is uploading a tagged system ciphertext signal to the cloud server. and During the process, and during the process of the cloud server sending tagged encrypted feedback control signals to the CPS, it was not attacked. and If they are not equal, it indicates that the CPS has uploaded a tagged system ciphertext signal to the cloud server. and During the process, and during the process of the cloud server sending tagged encrypted feedback control signals to the CPS, it was attacked.

6. The cloud-embedded cyber-physical system security protection method based on elastic homomorphic encryption according to claim 5, characterized in that, Step 7 specifically includes: Step 7.1: The cloud server calculates the FDI multiplicative attack factor using the second-level key. and Specifically, it is calculated using the following formula: (16)； (17)； in, and This indicates that the CPS is uploading encrypted signals from the attacked system to the cloud server. The calculated multiplicative attack factor and When the value equals 1, it indicates that the tagged system encrypted signal is uploaded from CPS to the cloud server. and During the process, and during the process of the cloud server sending tagged ciphertext feedback control signals to CPS, the attack type is not a multiplicative FDI attack, so proceed to step 7.

2. The calculated multiplicative attack factor and When the value is not equal to 1, it indicates that the CPS is uploading a tagged system ciphertext signal to the cloud server. and During the process, and during the process of the cloud server sending tagged encrypted feedback control signals to the CPS, the attack type is a multiplicative FDI attack. Therefore, the cloud server filters out multiplicative attacks, specifically through the following formula: (18)； (19)； The cloud server filters out multiplicative attack factors in the received encrypted feedback control signals. and The multiplicative attack factor in the received ciphertext feedback control signal is filtered out on the cloud server. and The signal after that is represented as: (20)； As can be seen from formula (20), multiplicative attacks are filtered out. and The subsequent signal also contains a multiplicative aggression factor. Calculate the multiplicative attack factor The value of is achieved through the following formula: (21)； Multiplicative attack factor Filtering out multiplicative attacks and Remove the signal from the filtered multiplicative attack to obtain the ciphertext feedback control signal. Substitute the filtered multiplicative attack ciphertext feedback control signal into formula (22). In this process, the original system plaintext signal is obtained. and Formula (22) is expressed as: (22)； Step 7.2: CPS calculates the multiplicative attack factor in an additive-multiplicative FDI attack, specifically using the following formula: (23)； The calculated multiplicative attack factor When the value equals 1, it indicates that the tagged system encrypted signal is uploaded from CPS to the cloud server. and During the process, and during the process of the cloud server sending tagged ciphertext feedback control signals to the CPS, the attack type is not an additive-multiplicative FDI attack, so proceed to step 7.3; The calculated multiplicative attack factor When the value is not equal to 1, it indicates that the CPS is uploading a tagged system encrypted signal to the cloud server. and During the process, and during the process of the cloud server sending tagged ciphertext feedback control signals to the CPS, the attack type is additive-multiplicative FDI attack, which in turn increases the multiplicative attack factor. Remove from the ciphertext feedback control signal received from CPS to obtain the purified ciphertext feedback control signal. This will then allow the purified ciphertext to be fed back as a control signal. Substituting into formula (24), the signal injected by the additive FDI attack is removed, and the original system plaintext signal is obtained. and Formula (24) is expressed as: (24)； Step 7.3: Upload tagged system encrypted signals to the cloud server via CPS. and In the process of the cloud server sending tagged ciphertext feedback control signals to the CPS, if the attack type is neither additive-multiplicative FDI nor multiplicative FDI, it indicates that the attack type is additive FDI. Therefore, the ciphertext feedback control signals received by the CPS are substituted into formula (22). In this process, the original system plaintext signal is obtained. and .

7. The cloud-embedded cyber-physical system security protection method based on elastic homomorphic encryption according to claim 6, characterized in that, Step 8 is specifically implemented using the following formula: (25)； Based on formula (25), the system ciphertext signal is calculated. and .

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