Privacy protection and verifiable federated learning method based on hyper-incremental sequence

Abstract

The invention discloses a privacy protection and verifiable federated learning method based on a hyper-incremental sequence, and aims to improve data security and calculation efficiency in a federated learning process. The method comprises the following steps: (1) system initialization: a trusted authority (TA) generates key parameters required by the system, and distributes related parameters to a client to support subsequent encryption and verification processes; (2) training a model by the clients: training the model by each client based on a local data set, encoding and encrypting gradient data by using a hyper-incremental sequence, and then uploading the gradient data to a cloud for aggregation; (3) central server aggregation: the server aggregates the encrypted gradient data uploaded by the client by using an addition homomorphic encryption technology, returns an aggregation result to the client, and ensures that the gradient information is always kept in an encrypted state at the server side; and (4) local model updating and verification: the client decrypts and decodes the aggregation result, verifies the correctness of the aggregation result, and updates local model parameters based on the aggregation gradient. According to the invention, an efficient encryption and verification mechanism is designed, so that collusion attacks can be resisted, flexible exit of the client is supported, the security, verifiability and calculation efficiency of the system are ensured, and the method is suitable for various privacy protection federated learning application scenes.

Description

Technical Field

[0001] This invention relates to a privacy-preserving and verifiable federated learning method based on super-incrementing sequences. Background Technology

[0002] In a federated learning environment, multiple clients collaboratively train a global model. However, due to data privacy concerns, client data cannot be directly shared, necessitating the protection of data security and the reliability of training results. Therefore, the system needs to employ privacy protection mechanisms to prevent data leakage and malicious attacks. Existing solutions typically use homomorphic encryption, differential privacy, or secure multi-party computation to protect data privacy, or leverage blockchain technology for data traceability. However, the former may suffer from excessive computational overhead, impacting system efficiency, while the latter, although preventing malicious client behavior, offers limited protection against cloud servers, making it difficult to ensure the reliability of aggregated results. Furthermore, some solutions require all clients to be online simultaneously, reducing system flexibility. Therefore, while existing solutions improve the security of federated learning, they still suffer from high communication overhead, high computational costs, or insufficient flexibility, failing to meet practical application needs. To address this issue, a federated learning solution that can both ensure data security and improve computational efficiency while supporting flexible client participation is required. Federal Learning Federated learning is a distributed machine learning technique designed to train models while protecting data privacy. Its underlying technologies primarily involve privacy-preserving computation, distributed optimization, encryption, and secure multi-party computation (MPC). Traditional machine learning typically relies on centralized data storage, while federated learning avoids direct data transfer by allowing participants to train the model on their local devices, sharing only model parameters or gradients. This technology is widely used in finance, healthcare, and smart devices, achieving a balance between data security and personalized needs, while also enhancing security through technologies such as differential privacy and homomorphic encryption. Privacy protection Privacy-preserving technologies aim to prevent the leakage of sensitive information during data processing and analysis. Their background technologies involve cryptography, differential privacy, secure multi-party computation (MPC), homomorphic encryption, and trusted execution environments (TEEs). Differential privacy hides individual information by adding noise to data or query results, thus preventing privacy leaks while ensuring data availability. Homomorphic encryption allows direct computation on encrypted data without decryption, improving data security. MPC enables multiple participants to perform collaborative computations without revealing their individual data, while TEEs protect the confidentiality of the computation process through hardware isolation. These technologies are widely used in scenarios such as finance, healthcare, and federated learning to meet data privacy and compliance requirements. Superincreasing sequence A super-increasing sequence is a special type of integer sequence where each element is greater than the sum of all preceding elements. It is widely used in cryptography and coding theory. Originally used in the Merkle-Hellman knapsack algorithm, it gained widespread adoption due to its uniqueness and ease of decoding. In privacy-preserving federated learning, super-increasing sequences are often used to encode and pack gradient vectors. By combining multiple gradient values ​​into a weighted integer, communication overhead is significantly reduced, improving system efficiency. Furthermore, because its decoding process is simple and unique, combined with encryption mechanisms, it ensures both efficient and secure data transmission, making it particularly suitable for resource-constrained mobile devices or edge environments. Secret sharing Secret sharing is a cryptographic technique that aims to break sensitive information into multiple fragments (called shares) and distribute them to different participants. Only by obtaining a certain number of shares can the original data be recovered. Its underlying technologies mainly involve polynomial interpolation (such as Shamir's Secret Sharing), threshold cryptography, and secure multi-party computation (MPC). Shamir's Secret Sharing, based on the principle of Lagrange interpolation, ensures that even if some shares are lost, the original secret can be recovered with enough shares, while shares below the threshold cannot yield any useful information. This technique is widely used in distributed key management, threshold signatures, decentralized storage, and privacy computing to ensure data security and fault tolerance while reducing the risk of single points of leakage. Summary of the Invention

[0003] To address the aforementioned technical problems, this invention provides a privacy-preserving and verifiable federated learning method based on super-incrementing sequences. The technical solution of this invention to solve the above-mentioned technical problems is: a privacy-preserving and verifiable federated learning method based on super-increasing sequences, comprising the following steps: (1) System Initialization: The Trusted Authority (TA) initializes the entire system and distributes some parameters to the clients. The clients use these parameters to encrypt data; (2) Client-side model training: The client trains the model based on the local dataset, then encodes and encrypts the calculated gradients, and uploads them to the cloud to complete the aggregation update; (3) Central server aggregation: The central server aggregates the received parameters and returns the results to each client; (4) Local model update and verification: The client receives the aggregation results from the cloud, decrypts and decodes the aggregation results, verifies their reliability, and then restores the specific aggregation results and updates the local model. In the aforementioned privacy-preserving and verifiable federated learning method based on super-increasing sequences, step (1) refers to the initialization phase, which involves initializing the entire system and allocating some parameters to the client. During the initialization phase, the TA needs to generate and distribute the following series of parameters. 1-1) First, the TA inputs the security parameters and generates the parameters {N,g,p,q} for the Paillier encryption system. Here, the public key PK = {N,g}, the private key λ = lcm(p-1,q-1), and the private key SK = {λ,p,q} are set. Then, the TA distributes {PK,SK} to all clients, where PK is used to encrypt gradient data, and SK is used to decrypt gradient data. 1-2) Then, TA generates a superincreasing sequence. in Subsequently, TA calculates the encoding parameters. (where i = 1, ..., n+1), and select a random number. Calculate f1 = r *q modN 2 And calculate f2 = g based on g. q modN 2 Then, the TA generates and distributes the parameters. For all clients, f1 is used to encrypt gradient data, f2 is used to verify the aggregation result, and... Responsible for encoding and decoding gradient data. 1-3) In addition, TA generates two constant sequences and These will be used as interpolation parameters to process gradients on the client side. 1-4) Next, TA selects a random number r based on the private key p, satisfying r < p. Then, TA constructs two Shamir secret sharing polynomials G1(x) and G2(x) with different coefficients, and takes the modulus p of these polynomials. Here, pr and r serve as the master keys to be shared in the Shamir secret sharing algorithm. By running this algorithm, TA generates 2m pairs of subkeys, where the subkey pair corresponding to the master key r is represented as: The subkey pair corresponding to the master key pr is represented as follows: Among them, {x i {i = 1, 2, ..., m} are m distinct random values, and {G1(x i {i = 1, 2, ..., m} and {G2(x)} i), i = 1, 2, ..., m} respectively represent substituting these random values ​​into G1(x i ) and G2(x i The calculation results obtained. Finally, TA will... Distribute to client i, and set {x1,x2,...,x m Distribute to all clients. Among them, Used to encrypt gradient data and resist collusion attacks, while {x1,x2,...,x m This is used to recover the master key. At this point, the TA initialization process is complete, providing the system with secure and verifiable federated learning framework support. The above-mentioned privacy-preserving and verifiable federated learning method based on super-increasing sequences, in step (2), at this stage each client uses its local dataset D = {(x j ,y j Each client trains its own model on a dataset {j = 1, 2, ..., T}. Based on this dataset, each client computes the gradient using stochastic gradient descent (SGD). The client randomly selects a subset D. * And calculate the loss function: Client i performs SGD to compute its private gradient w i : w i =▽L f (D *i ,θ) Vector w i It is an n-dimensional vector, denoted as w i =(w i1 ,w i2 ,...,w in Subsequently, client i sends a federated learning request and its corresponding ID to the cloud. When the cloud waits for more than t to receive a request from a client, it collects the IDs of all online clients and returns the ID set to each client. The client then selects the appropriate x based on the ID. i (from the set {x1, x2, ..., x...) m Select from the options in the list, then calculate: Client i further calculates the subkey S i The calculation formula is as follows: S i =β(x i )G1(x i )+β(x i )G2(x i )modp To ensure the security of client data and reduce communication overhead, the client encrypts and packages the gradient data using parameters distributed by the TA. First, the client uses a constant sequence... For private gradient data w i =(w i1 ,w i2 ,...,w in ) and shared subkey S i Interpolation is performed. Then, the client outputs a constant sequence. As input, and through function F i (x) Calculate the corresponding function value F. i (h i Next, client i uses the encryption parameters {g1, g2, ..., g...} distributed by TA. n+1 ,f1,N} and subkey S i To encrypt the value {F i (h1),F i (h2),...,F i (h n+1 )}: Ultimately, client i obtains the encrypted value C. i Afterwards, the encrypted data is uploaded to the cloud server, waiting for the cloud to complete the aggregation and update of federated learning. In the above privacy-preserving and verifiable federated learning method based on super-increasing sequences, in step (3), after the aggregation server receives the ciphertext uploaded by all clients, the server uses the additive homomorphic property to perform aggregation calculation on the ciphertext, as shown below: Here, C represents the aggregated ciphertext. The server returns the aggregated ciphertext to all clients. It's important to note that the gradient information remains encrypted; the server cannot access the plaintext gradient data. Since the central cloud server is typically untrusted, it may forge aggregation results to conserve its computing resources. Therefore, clients should have the ability to verify the correctness of the aggregation results returned by the cloud server. Furthermore, because clients may be offline within the system, the encryption and verification mechanisms must support flexible client participation to ensure system integrity and security. In the above privacy-preserving and verifiable federated learning method based on super-increasing sequences, in step (4), after receiving the aggregation result, each client first uses the Paillier decryption algorithm to decrypt the aggregated ciphertext C, and the calculation formula is as follows: Subsequently, each client utilizes a superincreasing sequence. Decoding the aggregation results yields the final calculation result {F(h1),F(h2),...,F(h... n+1 The decoding process is as follows: Set the initial value X. j =M, then calculate X sequentially from j=n+1 to 2. j-1 =X j mod a j and obtain Finally, calculation The complete decoding result {F(h1),F(h2),...,F(h...} is obtained. n+1 After obtaining these calculation results, the client performs Lagrange interpolation calculations based on the dataset to construct the interpolation function: Subsequently, the client sets x = k n+1 As input, calculate F(k) n+1 And verify using the secret parameter f2. If it satisfies The client then confirms the aggregation result is correct. If the verification passes, the client calculates the gradient aggregation result w = {F(k1), F(k2), ..., F(k...}. n The result of F(x) calculation corresponds to the gradient aggregation result of each model. In this process, the client first decrypts the ciphertext C to obtain the aggregation result M, and then uses a super-increasing sequence... Decoding yields {F(h1),F(h2),...,F(h... n+1 Then, construct the function using Lagrange interpolation: And calculate F(k) n+1 The algorithm performs validation. If validation succeeds, it calculates and returns w; if validation fails, it returns an error message. Finally, each client updates the model parameter M locally using the aggregation result, calculated as follows: Where η is the learning rate and m is the number of clients participating in the training. The beneficial effects of this invention are as follows: 1. This invention innovatively utilizes super-increasing sequences to efficiently encode gradient data, which not only improves data manageability but also significantly reduces system communication overhead, thereby optimizing the overall performance of federated learning and making it suitable for large-scale distributed computing scenarios. 2. This invention designs a highly efficient and secure privacy-preserving federated learning encryption mechanism. This mechanism, combined with advanced cryptographic techniques, can effectively resist collusion attacks and prevent malicious participants from inferring private data. Furthermore, this mechanism supports dynamic client exit, making the system more flexible and robust in practical applications and suitable for complex network environments. 3. This invention proposes a highly efficient privacy-preserving federated learning verification mechanism. Even when multiple clients simultaneously disconnect, this mechanism can still execute stably, ensuring the integrity and reliability of the computation results. Through this mechanism, abnormal or malicious behavior can be effectively detected, data tampering can be prevented, and the system's credibility can be enhanced. 4. This invention has conducted in-depth analysis of the performance of the proposed scheme through extensive simulation experiments. The experimental results show that the scheme ensures high accuracy and efficiency while keeping communication and computational overhead within acceptable ranges. Furthermore, the scheme demonstrates good adaptability and stability in different scenarios, providing reliable technical support for the application of federated learning in the field of privacy computing. Attached Figure Description

[0004] Figure 1 This is the overall flowchart of the present invention. Figure 2 This is a system framework diagram of the present invention. Detailed Implementation

[0005] The present invention will be further described below with reference to the accompanying drawings and embodiments. like Figure 2 As shown, Figure 2 The system framework diagram of this invention shows that the Trusted Authorization Authority (TA) is responsible for initializing the network model, generating the necessary keys and system parameters, and distributing them to all clients. Clients, as participating entities, each hold an independent dataset and flexibly participate in the federated learning process based on their own state. Even when some clients are offline, the correctness of the aggregation results can still be verified, ensuring the robustness and flexibility of the system. In the system model, a central aggregation server is introduced, responsible for receiving and aggregating the ciphertext uploaded by each client in each round of federated learning, and then distributing the aggregation results to each client. Considering the possibility of malicious behavior by the server, this invention specifically designs mechanisms to prevent privacy leaks and forged results, ensuring that clients can still securely complete model updates and verification operations even when facing untrusted servers. like Figure 1 As shown, a method for generating multi-user access control keys based on elliptic curves in a mobile edge computing environment includes the following steps: (1) System Initialization: The Trusted Authority (TA) initializes the entire system and distributes some parameters to the clients. The clients use these parameters to encrypt data; (2) Client-side model training: The client trains the model based on the local dataset, then encodes and encrypts the calculated gradients, and uploads them to the cloud to complete the aggregation update; (3) Central server aggregation: The central server aggregates the received parameters and returns the results to each client; (4) Local model update and verification: The client receives the aggregation results from the cloud, decrypts and decodes the aggregation results, verifies their reliability, and then restores the specific aggregation results and updates the local model. In the aforementioned privacy-preserving and verifiable federated learning method based on super-increasing sequences, step (1) refers to the initialization phase, which involves initializing the entire system and allocating some parameters to the client. During the initialization phase, the TA needs to generate and distribute the following series of parameters. 1-1) First, the TA inputs the security parameters and generates the parameters {N,g,p,q} for the Paillier encryption system. Here, the public key PK = {N,g}, the private key λ = lcm(p-1,q-1), and the private key SK = {λ,p,q} are set. Then, the TA distributes {PK,SK} to all clients, where PK is used to encrypt gradient data, and SK is used to decrypt gradient data. 1-2) Then, TA generates a superincreasing sequence. in Subsequently, TA calculates the encoding parameters. (where i = 1, ..., n+1), and select a random number. Calculate f1 = r *q modN 2 And calculate f2 = g based on g. q modN 2 Then, the TA generates and distributes the parameters. For all clients, f1 is used to encrypt gradient data, f2 is used to verify the aggregation result, and... Responsible for encoding and decoding gradient data. 1-3) In addition, TA generates two constant sequences and These will be used as interpolation parameters to process gradients on the client side. 1-4) Next, TA selects a random number r based on the private key p, satisfying r < p. Then, TA constructs two Shamir secret sharing polynomials G1(x) and G2(x) with different coefficients, and takes the modulus p of these polynomials. Here, pr and r serve as the master keys to be shared in the Shamir secret sharing algorithm. By running this algorithm, TA generates 2m pairs of subkeys, where the subkey pair corresponding to the master key r is represented as: The subkey pair corresponding to the master key pr is represented as follows: Among them, {x i {i = 1, 2, ..., m} are m distinct random values, and {G1(x i {i = 1, 2, ..., m} and {G2(x)} i ), i = 1, 2, ..., m} respectively represent substituting these random values ​​into G1(x i ) and G2(x i The calculation results obtained. Finally, TA will... Distribute to client i, and set {x1,x2,...,x m Distribute to all clients. Among them, Used to encrypt gradient data and resist collusion attacks, while {x1,x2,...,x m This is used to recover the master key. At this point, the TA initialization process is complete, providing the system with secure and verifiable federated learning framework support. The above-mentioned privacy-preserving and verifiable federated learning method based on super-increasing sequences, in step (2), at this stage each client uses its local dataset D = {(x j ,y j Each client trains its own model on a dataset {j = 1, 2, ..., T}. Based on this dataset, each client computes the gradient using stochastic gradient descent (SGD). The client randomly selects a subset D. * And calculate the loss function: Client i performs SGD to compute its private gradient w i : w i =▽L f (D *i ,θ) Vector w i It is an n-dimensional vector, denoted as w i =(w i1 ,w i2 ,...,w in Subsequently, client i sends a federated learning request and its corresponding ID to the cloud. When the cloud waits for more than t to receive a request from a client, it collects the IDs of all online clients and returns the ID set to each client. The client then selects the appropriate x based on the ID. i (from the set {x1, x2, ..., x...) m Select from the options in the list, then calculate: Client i further calculates the subkey S i The calculation formula is as follows: S i =β(x i )G1(x i )+β(x i )G2(x i )modp To ensure the security of client data and reduce communication overhead, the client encrypts and packages the gradient data using parameters distributed by the TA. First, the client uses a constant sequence... For private gradient data w i =(w i1 ,w i2 ,...,w in ) and shared subkey S i Interpolation is performed. Then, the client outputs a constant sequence. As input, and through function F i (x) Calculate the corresponding function value F. i (h i Next, client i uses the encryption parameters {g1, g2, ..., g...} distributed by TA. n+1 ,f1,N} and subkey S i To encrypt the value {F i (h1),F i (h2),...,F i (h n+1 )}: Ultimately, client i obtains the encrypted value C. i Afterwards, the encrypted data is uploaded to the cloud server, waiting for the cloud to complete the aggregation and update of federated learning. In the above privacy-preserving and verifiable federated learning method based on super-increasing sequences, in step (3), after the aggregation server receives the ciphertext uploaded by all clients, the server uses the additive homomorphic property to perform aggregation calculation on the ciphertext, as shown below: Here, C represents the aggregated ciphertext. The server returns the aggregated ciphertext to all clients. It's important to note that the gradient information remains encrypted; the server cannot access the plaintext gradient data. Since the central cloud server is typically untrusted, it may forge aggregation results to conserve its computing resources. Therefore, clients should have the ability to verify the correctness of the aggregation results returned by the cloud server. Furthermore, because clients may be offline within the system, the encryption and verification mechanisms must support flexible client participation to ensure system integrity and security. In the above privacy-preserving and verifiable federated learning method based on super-increasing sequences, in step (4), after receiving the aggregation result, each client first uses the Paillier decryption algorithm to decrypt the aggregated ciphertext C, and the calculation formula is as follows: Subsequently, each client utilizes a superincreasing sequence. Decoding the aggregation results yields the final calculation result {F(h1),F(h2),...,F(h... n+1 The decoding process is as follows: Set the initial value X. j =M, then calculate X sequentially from j=n+1 to 2. j-1 =X j mod a j and obtain Finally, calculation The complete decoding result {F(h1),F(h2),...,F(h...} is obtained. n+1 After obtaining these calculation results, the client performs Lagrange interpolation calculations based on the dataset to construct the interpolation function: Subsequently, the client sets x = k n+1 As input, calculate F(k) n+1 And verify using the secret parameter f2. If it satisfies The client then confirms the aggregation result is correct. If the verification passes, the client calculates the gradient aggregation result w = {F(k1), F(k2), ..., F(k...}. n The result of F(x) calculation corresponds to the gradient aggregation result of each model. In this process, the client first decrypts the ciphertext C to obtain the aggregation result M, and then uses a super-increasing sequence... Decoding yields {F(h1),F(h2),...,F(h... n+1 Then, construct the function using Lagrange interpolation: And calculate F(k) n+1 The algorithm performs validation. If validation succeeds, it calculates and returns w; if validation fails, it returns an error message. Finally, each client updates the model parameter M locally using the aggregation result, calculated as follows: Where η is the learning rate and m is the number of clients participating in the training. At this point, the entire federated learning process is complete.

Claims

1. A privacy-preserving and verifiable federated learning method based on super-increasing sequences, characterized in that, Includes the following steps: (1) System Initialization: The Trusted Authority (TA) initializes the entire system and distributes some parameters to the clients. The clients use these parameters to encrypt data; (2) Client-side model training: The client trains the model based on the local dataset, then encodes and encrypts the calculated gradients, and uploads them to the cloud to complete the aggregation update; (3) Central server aggregation: The central server aggregates the received parameters and returns the results to each client; (4) Local model update and verification: The client receives the aggregation results from the cloud, decrypts and decodes the aggregation results, verifies their reliability, and then restores the specific aggregation results and updates the local model.

2. The privacy-preserving and verifiable federated learning method based on super-increasing sequences according to claim 1, characterized in that, In step (1), the initialization phase refers to the initialization work performed on the entire system and the allocation of some parameters to the client. During the initialization phase, the TA needs to generate and distribute the following series of parameters. 1-1) First, the TA inputs the security parameters and generates the parameters {N,g,p,q} for the Paillier encryption system. Here, the public key PK = {N,g}, the private key λ = lcm(p-1,q-1), and the private key SK = {λ,p,q} are set. Then, the TA distributes {PK,SK} to all clients, where PK is used to encrypt gradient data, and SK is used to decrypt gradient data. 1-2) Then, TA generates a superincreasing sequence. in Subsequently, TA calculates the encoding parameters. (where i = 1, ..., n+1), and select a random number. Calculate f1 = r *q mod N 2 And calculate f2 = g based on g. q mod N 2 Then, the TA generates and distributes the parameters. For all clients, f1 is used to encrypt gradient data, f2 is used to verify the aggregation result, and... Responsible for encoding and decoding gradient data. 1-3) In addition, TA generates two constant sequences and These will be used as interpolation parameters to process gradients on the client side. 1-4) Next, TA selects a random number r based on the private key p, satisfying r < p. Then, TA constructs two Shamir secret sharing polynomials G1(x) and G2(x) with different coefficients, and takes the modulus p of these polynomials. Here, pr and r serve as the master keys to be shared in the Shamir secret sharing algorithm. By running this algorithm, TA generates 2m pairs of subkeys, where the subkey pair corresponding to the master key r is represented as: The subkey pair corresponding to the master key pr is represented as follows: Among them, {x i {i = 1, 2, ..., m} are m distinct random values, and {G1(x i {i = 1, 2, ..., m} and {G2(x)} i ), i = 1, 2, ..., m} respectively represent substituting these random values ​​into G1(x i ) and G2(x i The calculation results obtained. Finally, TA will... Distribute to client i, and set {x1,x2,...,x m Distribute to all clients. Among them, Used to encrypt gradient data and resist collusion attacks, while {x1,x2,...,x m This is used to recover the master key. At this point, the TA initialization process is complete, providing the system with secure and verifiable federated learning framework support.

3. The privacy-preserving and verifiable federated learning method based on super-increasing sequences according to claim 1, characterized in that... In step (2), at this stage each client determines the local dataset D = {(x j ,y j Each client trains its own model on a dataset {j = 1, 2, ..., T}. Based on this dataset, each client computes the gradient using stochastic gradient descent (SGD). The client randomly selects a subset D. * And calculate the loss function: Client i performs SGD to compute its private gradient w i : Vector w i It is an n-dimensional vector, denoted as w i =(w i1 ,w i2 ,...,w in Subsequently, client i sends a federated learning request and its corresponding ID to the cloud. When the cloud waits for more than t to receive a request from a client, it collects the IDs of all online clients and returns the ID set to each client. The client then selects the appropriate x based on the ID. i (from the set {x1, x2, ..., x...) m Select from the options in the list, then calculate: Client i further calculates the subkey S i The calculation formula is as follows: S i =β(x i )G1(x i )+β(x i )G2(x i )modp To ensure the security of client data and reduce communication overhead, the client encrypts and packages the gradient data using parameters distributed by the TA. First, the client uses a constant sequence... For private gradient data w i =(w i1 ,w i2 ,...,w in ) and shared subkey S i Interpolation is performed. Then, the client outputs a constant sequence. As input, and through function F i (x) Calculate the corresponding function value F. i (h i Next, client i uses the encryption parameters {g1, g2, ..., g...} distributed by TA. n+1 ,f1,N} and subkey S i To encrypt the value {F i (h1),F i (h2),...,F i (h n+1 )}: Ultimately, client i obtains the encrypted value C. i Afterwards, the encrypted data is uploaded to the cloud server, waiting for the cloud to complete the aggregation and update of federated learning.

4. The privacy-preserving and verifiable federated learning method based on super-increasing sequences according to claim 1, characterized in that, In step (3), after the aggregation server receives the ciphertext uploaded by all clients, the server uses the additive homomorphic property to perform aggregation calculation on the ciphertext, as shown below: Here, C represents the aggregated ciphertext. The server returns the aggregated ciphertext to all clients. It's important to note that the gradient information remains encrypted; the server cannot access the plaintext gradient data. Since the central cloud server is typically untrusted, it may forge aggregation results to conserve its computing resources. Therefore, clients should have the ability to verify the correctness of the aggregation results returned by the cloud server. Furthermore, because clients may be offline within the system, the encryption and verification mechanisms must support flexible client participation to ensure system integrity and security.

5. The privacy-preserving and verifiable federated learning method based on super-increasing sequences according to claim 1 is characterized in that... In step (4), after receiving the aggregation result, each client first uses the Paillier decryption algorithm to decrypt the aggregated ciphertext C. The calculation formula is as follows: Subsequently, each client utilizes a superincreasing sequence. Decoding the aggregation results yields the final calculation result {F(h1),F(h2),...,F(h... n+1 The decoding process is as follows: Set the initial value X. j =M, then calculate X sequentially from j=n+1 to 2. j-1 =X j mod a j and obtain Finally, calculation The complete decoding result {F(h1),F(h2),...,F(h...} is obtained. n+1 After obtaining these calculation results, the client performs Lagrange interpolation calculations based on the dataset to construct the interpolation function: Subsequently, the client sets x = k n+1 As input, calculate F(k) n+1 And verify using the secret parameter f2. If it satisfies The client then confirms the aggregation result is correct. If the verification passes, the client calculates the gradient aggregation result w = {F(k1), F(k2), ..., F(k...}. n The result of F(x) calculation corresponds to the gradient aggregation result of each model. In this process, the client first decrypts the ciphertext C to obtain the aggregation result M, and then uses a super-increasing sequence... Decoding yields {F(h1),F(h2),...,F(h... n+1 Then, construct the function using Lagrange interpolation: And calculate F(k) n+1 Perform verification; if verification is successful, calculate and return w. If verification fails, an error message is returned. Finally, each client updates the model parameter M locally using the aggregation results, calculated as follows: Where η is the learning rate and m is the number of clients participating in the training.

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