A privacy protection control system for satellite formation based on homomorphic encryption

Abstract

The invention relates to the field of satellite network privacy protection technology, and discloses a satellite formation privacy protection control system based on homomorphic encryption, including a data acquisition module, a homomorphic encryption module, a multi-agent system and a sub-satellite. The system uses a random coupling weight method to ensure that the privacy protection of nodes is achieved without affecting the convergence of the system. After the multi-agent system is modeled as a second-order dynamic system, a communication protocol is designed based on the fully homomorphic cryptographic algorithm SingleMod. The homomorphic encryption has strong privacy protection capabilities. The above algorithm is applied to a specific distributed algorithm to realize distributed control and optimization algorithms under privacy protection. The algorithm is explained for security and compared with numerical simulation to verify the convergence and high performance of the system. The homomorphic encryption module uses Lyapunov function to judge the feasibility of the designed lightweight communication protocol, which ensures that the system converges to the global optimal solution and has an exponential convergence speed, and the resource distributed control has strong stability.

Description

Technical Field

[0001] The present invention relates to the technical field of satellite network privacy protection, and in particular to a satellite formation privacy protection control system based on homomorphic encryption. Background Art

[0002] Satellite formation refers to a certain formation configuration of several satellites, which is a typical multi-agent system. Each satellite in the formation works together and communicates with each other to jointly achieve tasks such as communication, reconnaissance and navigation. The entire constellation constitutes a super-large "virtual satellite" or satellite network system. Through an effective distributed control protocol, it jointly achieves tasks such as near-earth survey, deep space exploration, and interferometry. Each satellite entity has an objective function. Through a suitable local optimization algorithm, it realizes local information interaction with its neighbors, so that each entity can jointly minimize the objective function to achieve the optimal decision of the system, such as minimizing system resource consumption, etc. It has great application prospects in the fields of earth observation, astronomical observation and deep space exploration. However, with the rapid development of satellite formation technology, the communication between nodes in the satellite formation will be eavesdropped by attackers and cause information leakage, which may lead to the destruction of system stability and create huge security risks. How to achieve system goals without leaking key information of each agent has become a new difficulty and challenge. Due to the limitation of satellite's limited computing power resources, a distributed control algorithm with high stability and good performance is crucial for it to achieve the preset goals. At the same time, due to factors such as limited communication bandwidth and scarce power resources, it is necessary to seek the optimal design of the entire system.

[0003] At present, traditional satellite fleet privacy protection control systems generally face two types of security threats: internal and external. Attackers who create internal threats gain control of a certain node by invading the system, obtain the status information of neighboring nodes, and then disrupt the overall operation and decision-making of the system. Attackers who create external threats directly eavesdrop on the communication links between nodes, obtain the status information of each node, or launch a man-in-the-middle attack to tamper with the information sent by a node to its neighbors, thereby affecting the operation and decision-making of the entire system. However, in actual use, the existing privacy protection technology is difficult to meet actual needs. Summary of the invention

[0004] 1. Technical issues to be resolved

[0005] In view of the shortcomings of the existing technology, the present invention provides a satellite formation privacy protection control system based on homomorphic encryption, which has the advantages of strong homomorphic encryption privacy protection capability and strong resource distributed control stability. It solves the problem that the privacy protection technology of traditional satellite formation privacy protection control systems is difficult to meet actual needs and the resource distribution capability is insufficient.

[0006] (II) Technical solution

[0007] To achieve the above-mentioned purpose, the present invention provides the following technical solutions: a satellite formation privacy protection control system based on homomorphic encryption, comprising a data acquisition module, a homomorphic encryption module, a multi-agent system and a sub-satellite;

[0008] The multi-agent system is composed of N sub-satellites, and the topological structure of the multi-agent system is a connected undirected graph containing N nodes, and each node has a local objective function , n represents the dimension of the function input, Represents the real number field Dimensional input vector is converted to 1 dimension;

[0009] The data acquisition module collects a sub-satellite data set through a network connection with the multi-agent system, wherein the sub-satellite data set includes position data and speed control data. The data acquisition module performs second-order dynamic system modeling on the multi-agent system according to the sub-satellite data set. The data acquisition module collects a conditional data set through a network connection with the multi-agent system, wherein the conditional data set includes constraint condition data. The data acquisition module marks some nodes as semi-honest nodes according to the conditional data set and stores them. The data acquisition module transmits the sub-satellite data set and the conditional data set to the homomorphic encryption module through the network.

[0010] The homomorphic encryption module converts the intermediate variable s i The second-order dynamic system modeling is introduced to converge the state information of the nodes in the multi-agent system. The homomorphic encryption module encrypts and protects the information data of N nodes of the multi-agent system in accordance with the SingleMod scheme in a random coupling weight RCW manner. The homomorphic encryption module performs a truncation operation and sets the precision coefficient Q to design a lightweight communication protocol. The SingleMod scheme is a domain homomorphic algorithm. The homomorphic encryption module applies the lightweight communication protocol based on the SingleMod scheme homomorphic encryption to the DGD distributed optimization algorithm, the EXTRA distributed optimization algorithm and the ZGS distributed optimization algorithm, and uses the Lyapunov function to judge the degree of protection of the designed lightweight communication protocol for information interaction. If the constructed Lyapunov function converges exponentially, the homomorphic encryption module judges that the designed lightweight communication protocol has a high degree of protection for information interaction.

[0011] Preferably, the second-order dynamics system modeling expression is as follows:

[0012]

[0013]

[0014]

[0015] In the formula, represents the derivative of the position information of the i-th node in the multi-agent system, x i Indicates the state value of dimension n corresponding to the position information of the i-th node, represents the derivative of the velocity information of the i-th node in the multi-agent system, v i Indicates the state value of the dimension n corresponding to the velocity information of the i-th node, Indicates that the speed information v of the i-th agent i The value obtained after derivation is used as the input of the multi-agent system. V represents the set of all nodes in the multi-agent system, g1 to g N represents the 1st to Nth agents, f i (x i ) represents the local objective function of the i-th node of the multi-agent system, It means that x is obtained by continuous time dynamics equation i and v i All converge to the optimal solution x j , which are their respective consistent solutions, x j represents the state information of the jth node in the system, and st represents the constraint that represents any i and j.

[0016] Preferably, the conditional data set includes condition 1, condition 2 and condition 3, wherein condition 1 constrains the topological structure diagram of the multi-agent system to be a connected undirected graph without self-loops, and condition 2 constrains the i-th agent g in the multi-agent system to be i The local objective function is a strongly convex function with continuous second-order derivatives, and its local Hessian matrix is Lipschitz continuous, where , Represents the local objective function f i The square of the gradient, the condition three, when each node in the multi-agent system performs calculations and initiates communication under the constraint of condition two, a single node collects and utilizes the obtained information and infers the internal information of other nodes. The nodes in the multi-agent system that simultaneously meet the conditional data set are marked as semi-honest nodes.

[0017] Preferably, the intermediate variable s i The expression introduced for modeling the second-order dynamic system is as follows:

[0018]

[0019] In the formula, , Represents the field of real numbers The set of positive real numbers in a, a represents a random number in the set of positive real numbers, if s i When converging to a consistent solution of the multi-agent system, the state information x i and v i also converge to their respective consistent solutions.

[0020] Preferably, the random coupling weight RCW encrypted expression based on the SingleMod scheme is as follows:

[0021] Set u, v to large secret prime numbers, and , Constitute the private key and Published as a public key, generate a set based on u , , generate a set based on m , ;

[0022] The plaintext k is The kth element in , k represents the plaintext to be encrypted, and then L is randomly selected from the random multiples, and the random coupling weight RCW encryption algorithm multiplies the plaintext k by the random multiple L for encryption protection. The formula of the random coupling weight RCW encryption algorithm is: ;

[0023] After the random coupling weight RCW encryption is completed, the random multiple L is discarded and the decryption party uses the ciphertext Decryption is performed, where the decryption algorithm is ; mod means modular operation.

[0024] Preferably, the lightweight communication protocol is expressed as follows:

[0025]

[0026] In the formula, Represents the field of real numbers Any random number in represents the truncation operation of the random number q with the precision coefficient Q, Indicates that the value of q is truncated.

[0027] Preferably, the DGD distributed optimization algorithm is expressed as follows:

[0028]

[0029] In the formula, , is the step length of the rth iteration of the DGD distributed optimization algorithm, Indicates the intermediate variable s in the r+1th iterationi The value of w ij Represents the field of real numbers Symmetric Matrix The element in row i and column j in represents the intermediate variable state value of the jth node in the multi-agent system at the rth iteration, represents the intermediate variable state value of the i-th node in the multi-agent system at the r-th iteration, represents the local objective function f of the i-th node in the multi-agent system i gradient.

[0030] Preferably, the EXTRA distributed optimization algorithm is expressed as follows:

[0031]

[0032]

[0033]

[0034] In the formula, Represents the intermediate variable s during the r+2 iteration i The value of represents the correction term of the EXTRA distributed optimization algorithm, W represents the symmetric matrix in the real number domain, I represents the input vector, Represents the state value of the intermediate variable of the jth node in the system at time t; Indicates the step size of each iteration in the EXTRA algorithm.

[0035] Preferably, the ZGS distributed optimization algorithm is expressed as follows:

[0036]

[0037] In the formula, Represents a constant in the ZGS distributed optimization algorithm, Represents the local objective function of the i-th node in the multi-agent system The inverse matrix of the Hessian matrix is ​​the private information of a single node, a ij Represents a symmetric matrix Elements in a matrix.

[0038] Preferably, the Lyapunov function is as follows:

[0039]

[0040] In the formula, V(s) is a semi-positive definite matrix, represents the unique minimum point in the matrix, s irepresents the i-th data point in the matrix, and T represents the transpose of the matrix.

[0041] Compared with the prior art, the present invention provides a satellite formation privacy protection control system based on homomorphic encryption, which has the following beneficial effects:

[0042] 1. The present invention forms a multi-agent system through N sub-satellites. The topological structure of the multi-agent system is a connected undirected graph containing N nodes, and the local objective function expression of each node is The data acquisition module connects the multi-agent system through the network to collect the position data and speed control data of the sub-satellite, and then models the multi-agent system with a second-order dynamic system. Each node jointly solves the optimization problem in a distributed manner. The data acquisition module connects the multi-agent system through the network to collect the conditional data set. The nodes in the multi-agent system that simultaneously meet the conditional data set are marked as semi-honest nodes and stored. It is not necessary to require each node in the multi-agent system to be connected to at least one legal node, which avoids the privacy inference of semi-honest nodes on other legal nodes. The homomorphic encryption module converts the intermediate variable s i The second-order dynamic system modeling is introduced to converge the state information of the nodes in the multi-agent system. The homomorphic encryption module encrypts and protects the information data of N nodes in the multi-agent system in accordance with the SingleMod scheme using the random coupling weight RCW method, ensuring that the two nodes cannot know each other's state information value while converging to an average consensus. The homomorphic encryption has strong privacy protection capabilities.

[0043] 2. The present invention performs truncation operation through a homomorphic encryption module and selects a precision coefficient Q to design a lightweight communication protocol to overcome the high computational overhead of the traditional Paillier encryption system algorithm. The SingleMod scheme is a domain homomorphic algorithm. The homomorphic encryption module applies a lightweight communication protocol based on the SingleMod scheme homomorphic encryption to the DGD distributed optimization algorithm, the EXTRA distributed optimization algorithm, and the ZGS distributed optimization algorithm to realize distributed control and optimization algorithms under privacy protection. The algorithm is described for security and compared with numerical simulations to verify the convergence and high performance of the system. The homomorphic encryption module uses the Lyapunov function to determine the feasibility of the designed lightweight communication protocol. If the constructed Lyapunov function converges exponentially, the homomorphic encryption module determines that the designed lightweight communication protocol is feasible. While the algorithm execution process always meets the optimization conditions, it also ensures that the system converges to the global optimal solution and has an exponential convergence speed, and the resource distributed control has strong stability. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 It is a schematic diagram of the system flow of the present invention. DETAILED DESCRIPTION

[0045] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0046] See also Figure 1 , a satellite formation privacy protection control system based on homomorphic encryption, including a data acquisition module, a homomorphic encryption module, a multi-agent system and a sub-satellite;

[0047] The multi-agent system consists of N sub-satellites. The topological structure of the multi-agent system is a connected undirected graph with N nodes, and each node has a local objective function f i , each node local objective function , n represents the dimension of the function input, It means converting the n-dimensional input vector in the real number field into 1-dimensional one. The size of N depends on the specific application scenario. The output can represent the optimization processing result of node i.

[0048] The data acquisition module connects the multi-agent system through the network to collect sub-satellite data sets. The sub-satellite data sets include position data and speed control data. The data acquisition module performs second-order dynamic system modeling on the multi-agent system based on the sub-satellite data sets. The second-order dynamic system modeling expression is as follows:

[0049]

[0050]

[0051]

[0052] In the formula, represents the derivative of the position information of the i-th node in the multi-agent system, x i Indicates the state value of dimension n corresponding to the position information of the i-th node, represents the derivative of the velocity information of the i-th node in the multi-agent system, v i Indicates the state value of the dimension n corresponding to the velocity information of the i-th node, Indicates that the speed information v of the i-th agent i The value obtained after derivation is used as the input of the multi-agent system. V represents the set of all nodes in the multi-agent system, g1 to g N represents the 1st to Nth agents, f i (x i) represents the local objective function of the i-th node of the multi-agent system, It means that x is obtained by continuous time dynamics equation i and v i All converge to the optimal solution x j , which are their respective consistent solutions, x j represents the state information of the jth node in the system, and st represents the constraint that Denotes an optimization problem that any i and j are solved by all nodes in a distributed manner;

[0053] The data acquisition module collects conditional data sets through the network connection of the multi-agent system. The conditional data sets include constraint data. The conditional data sets include condition one, condition two and condition three. The topological structure diagram of the multi-agent system constrained by condition one is a connected undirected graph without self-loops. Since in the distributed control and optimization of the multi-agent system, two-way communication is often required between nodes to achieve the purpose of information interaction, a connected undirected graph is used for modeling. In addition, it is not necessary for nodes to communicate with themselves, so it is assumed that there is no self-loop in the graph.

[0054] Condition 2 constrains any node in the multi-agent system , local objective function is a strongly convex function that is twice continuously differentiable, and the strongly convex coefficient , its local Hessian matrix (the Hessian matrix is ​​a second-order partial derivative matrix used to describe the local curvature of a function) is Lipschitz continuous (Lipschitz continuity is a concept that describes how smoothly a function changes) and is consistent with the local objective function have the same domain , the strong convexity of the local objective function ensures that it has a unique local minimum, thus ensuring that the formula has a unique global minimum , the necessary and sufficient conditions for optimization are as follows:

[0055]

[0056] Condition three: when each node in the multi-agent system performs calculations and initiates communication under condition two constraints, a single node collects and utilizes the obtained information and infers the internal information of other nodes, and the nodes in the multi-agent system that simultaneously meet the conditional data set are marked as semi-honest nodes. The data acquisition module marks some nodes according to the conditional data set, and the nodes in the multi-agent system that simultaneously meet the conditional data set are marked as semi-honest nodes and stored. It is not necessary to require each node in the multi-agent system to be connected to at least one legitimate node, thereby avoiding the privacy inference of semi-honest nodes on other legitimate nodes.

[0057] The data acquisition module transmits the sub-satellite data set and the conditional data set to the homomorphic encryption module through the network;

[0058] The homomorphic encryption module converts the intermediate variable s i Introduce second-order dynamic system modeling, converge the state information of nodes in the multi-agent system, and the intermediate variable s i The expression introduced for modeling the second-order dynamic system is as follows:

[0059]

[0060] In the formula, , Represents the field of real numbers The set of positive real numbers in a, a represents a random number in the set of positive real numbers, if s i When converging to a consistent solution of the multi-agent system, the state information x i and v i They also converge to their respective consistent solutions;

[0061] The homomorphic encryption module encrypts and protects the information data of N nodes in the multi-agent system in a random coupling weight RCW manner (the weight value is randomly generated) according to the SingleMod scheme (referring to a single module or a single function scheme). The random coupling weight RCW encryption expression based on the SingleMod scheme is as follows:

[0062] Set u,v to large secret prime numbers, and , Constitute the private key and Published as a public key, generate a set based on u , , generate a set based on m , ;

[0063] The plaintext k is The kth element in , k represents the plaintext to be encrypted, and then L is randomly selected from the random multiples, and the random coupling weight RCW encryption algorithm multiplies the plaintext k by the random multiple L for encryption protection. The formula of the random coupling weight RCW encryption algorithm is: ;

[0064] After the random coupling weight RCW encryption is completed, the random multiple L is discarded and the decryption party uses the ciphertext Decryption is performed, where the decryption algorithm is ; mod means modular operation, node g i Just according to the node g j Whether it is its neighbor can know a ijIf some random information can be added to the value of the weight, the two communicating parties will not be able to know the exact value of the weight, and will not be able to calculate the state information value of the other party, thus achieving the purpose of avoiding privacy leakage. i With node g j Each generates a random value z known only to itself i and z j Through communication interaction supported by homomorphic encryption algorithm, both parties can calculate the random weight value , and then the random weight value is used in the iterative calculation, which can ensure that the two nodes cannot know each other's state information value while converging to an average consensus. Homomorphic encryption has strong privacy protection capabilities;

[0065] Homomorphic encryption module settings Perform truncation operation and set precision coefficient Q to design lightweight communication protocol to overcome the high computational overhead of traditional Paillier encryption system algorithm. The lightweight communication protocol expression is as follows:

[0066]

[0067] In the formula, Represents the field of real numbers Any random number in represents the truncation operation of the random number q with the precision coefficient Q, Indicates that the value of q is truncated. In the actual satellite formation system, the status information of each node is stored in decimal form. The truncation operation can be used to control the decimal precision of these data. For scenarios with higher precision requirements, a larger Q value can be selected, but it will also put forward higher requirements on the computing power of the node accordingly.

[0068] The SingleMod scheme is a domain homomorphic algorithm. The homomorphic encryption module applies the lightweight communication protocol based on the homomorphic encryption of the SingleMod scheme to the DGD distributed optimization algorithm (DGD (Distributed Gradient Descent) distributed optimization algorithm is an algorithm used to optimize large-scale data sets. By distributing computing tasks to multiple nodes, the gradient descent process is completed collaboratively. Its goal is to accelerate the training process and improve computing efficiency), the EXTRA distributed optimization algorithm (EXTRA (EXtreme Gradient Descent with Adaptive step-size) is an algorithm for distributed optimization. It is mainly used to solve optimization problems in large-scale machine learning tasks. The EXTRA algorithm aims to improve the efficiency of the distributed gradient descent method) and the ZGS distributed optimization algorithm (ZGS (Zero-Gradient-Sparsification) distributed optimization algorithm is a method to improve the efficiency of distributed optimization by sparse gradients) to realize distributed control and optimization algorithms under privacy protection. The security of the algorithms is explained and compared with numerical simulations to verify the convergence and high performance of the system.

[0069] The DGD distributed optimization algorithm expression is as follows:

[0070]

[0071] In the formula, , is the step length of the rth iteration of the DGD distributed optimization algorithm, Indicates the intermediate variable s in the r+1th iteration i The value of w ij Represents the field of real numbers Symmetric Matrix The element in row i and column j in represents the intermediate variable state value of the jth node in the multi-agent system at the rth iteration, represents the intermediate variable state value of the i-th node in the multi-agent system at the r-th iteration, represents the local objective function f of the i-th node in the multi-agent system i The gradient of

[0072] The EXTRA distributed optimization algorithm expression is as follows:

[0073]

[0074]

[0075]

[0076] In the formula, Represents the intermediate variable s during the r+2 iteration i The value of represents the correction term of the EXTRA distributed optimization algorithm, W represents the symmetric matrix in the real number domain, I represents the input vector, Represents the state value of the intermediate variable of the jth node in the system at time t; Indicates the step size of each iteration in the EXTRA algorithm;

[0077] The ZGS distributed optimization algorithm is expressed as follows:

[0078]

[0079] In the formula, Represents a constant in the ZGS distributed optimization algorithm, Represents the local objective function of the i-th node in the multi-agent system The inverse matrix of the Hessian matrix is ​​the private information of a single node, a ij Represents a symmetric matrix The elements in the matrix are not known to other nodes, which avoids multi-node collusion attacks, and homomorphic encryption has strong privacy protection capabilities;

[0080] The homomorphic encryption module uses the Lyapunov function to determine the feasibility of the designed lightweight communication protocol. The Lyapunov function is as follows:

[0081]

[0082] In the formula, V(s) is a semi-positive definite matrix, represents the unique minimum point in the matrix, s i represents the i-th data point in the matrix, and T represents the transpose of the matrix;

[0083] If the constructed Lyapunov function converges exponentially, the homomorphic encryption module determines that the designed lightweight communication protocol is feasible. While the algorithm execution process always meets the optimization conditions, it also ensures that the system converges to the global optimal solution with an exponential convergence speed, and the resource distributed control has strong stability.

[0084] Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions and variations may be made to the embodiments without departing from the principles and spirit of the present invention, and that the scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. A satellite formation privacy protection control system based on homomorphic encryption, characterized by: It includes data acquisition module, homomorphic encryption module, multi-agent system and sub-satellite; The multi-agent system is composed of N sub-satellites, and the topological structure of the multi-agent system is a connected undirected graph containing N nodes, and each node has a local objective function , n represents the dimension of the function input, Indicates converting the n-dimensional input vector in the real number field into 1 dimension; The data acquisition module collects a sub-satellite data set through a network connection with the multi-agent system, wherein the sub-satellite data set includes position data and speed control data. The data acquisition module performs second-order dynamic system modeling on the multi-agent system according to the sub-satellite data set. The data acquisition module collects a conditional data set through a network connection with the multi-agent system, wherein the conditional data set includes constraint condition data. The data acquisition module marks some nodes as semi-honest nodes according to the conditional data set and stores them. The data acquisition module transmits the sub-satellite data set and the conditional data set to the homomorphic encryption module through the network. The homomorphic encryption module converts the intermediate variable s i The second-order dynamic system modeling is introduced to converge the state information of the nodes in the multi-agent system. The homomorphic encryption module encrypts and protects the information data of N nodes in the multi-agent system in accordance with the SingleMod scheme in a random coupling weight RCW manner. The homomorphic encryption module performs a truncation operation and sets the precision coefficient Q to design a lightweight communication protocol. The SingleMod scheme is a domain homomorphic algorithm. The homomorphic encryption module applies the lightweight communication protocol based on the SingleMod scheme homomorphic encryption to the DGD distributed optimization algorithm, the EXTRA distributed optimization algorithm and the ZGS distributed optimization algorithm, and uses the Lyapunov function to judge the degree of protection of the designed lightweight communication protocol for information interaction. If the constructed Lyapunov function converges exponentially, the homomorphic encryption module judges that the designed lightweight communication protocol has a high degree of protection for information interaction; wherein, the lightweight communication protocol expression is as follows: In the formula, Represents the field of real numbers Any random number in represents the truncation operation of the random number q with the precision coefficient Q, Indicates that the value of q is truncated; The modeling expression of the second-order dynamic system is as follows: ； ； ； In the formula, represents the derivative of the position information of the i-th node in the multi-agent system, x i Indicates the state value of dimension n corresponding to the position information of the i-th node, represents the derivative of the velocity information of the i-th node in the multi-agent system, v i Indicates the state value of the dimension n corresponding to the velocity information of the i-th node, ,u i Indicates that the speed information v of the i-th agent i The value obtained after derivation is used as the input of the multi-agent system. V represents the set of all nodes in the multi-agent system, g1 to g N represents the 1st to Nth agents, f i (x i ) represents the local objective function of the i-th node of the multi-agent system, It means that x is obtained by continuous time dynamics equation i and v i All converge to the optimal solution x j , which are their respective consistent solutions, x j represents the state information of the jth node in the system, and st represents the constraint that represents any i and j; The intermediate variable s i The expression introduced for modeling the second-order dynamic system is as follows: ； In the formula, , Represents the field of real numbers The set of positive real numbers in a, a represents a random number in the set of positive real numbers, if s i When converging to a consistent solution of the multi-agent system, the state information x i and v i also converge to their respective consistent solutions.

2. According to claim 1, a satellite formation privacy protection control system based on homomorphic encryption is characterized in that: The conditional data set includes condition 1, condition 2 and condition 3. The condition 1 constrains the topological structure diagram of the multi-agent system to be a connected undirected graph without self-loops. The condition 2 constrains the i-th agent g in the multi-agent system to be i The local objective function is a strongly convex function with continuous second-order derivatives, and its local Hessian matrix is Lipschitz continuous, where , represents the local objective function f i The square of the gradient, the condition three, when each node in the multi-agent system performs calculations and initiates communication under the constraint of condition two, a single node collects and utilizes the obtained information and infers the internal information of other nodes. The nodes in the multi-agent system that simultaneously meet the conditional data set are marked as semi-honest nodes.

3. According to claim 2, a satellite formation privacy protection control system based on homomorphic encryption is characterized in that: The random coupling weight RCW encryption steps based on the SingleMod scheme are as follows: set up is a large secret prime number, and , Constitute the private key and Published as a public key, generate a set based on u , , generate a set based on m , ; The plaintext k is The kth element in , k represents the plaintext to be encrypted, and then L is randomly selected from the random multiples, and the random coupling weight RCW encryption algorithm multiplies the plaintext k by the random multiple L for encryption protection. The formula of the random coupling weight RCW encryption algorithm is: ; After the random coupling weight RCW encryption is completed, the random multiple L is discarded and the decryption party uses the ciphertext Decryption is performed, where the decryption algorithm is ; mod means modular operation.

4. According to claim 3, a satellite formation privacy protection control system based on homomorphic encryption is characterized in that: The DGD distributed optimization algorithm expression is as follows: ； In the formula, , is the step length of the rth iteration of the DGD distributed optimization algorithm, Indicates the intermediate variable s in the r+1th iteration i The value of w ij Represents the field of real numbers Symmetric Matrix The element in row i and column j in represents the intermediate variable state value of the jth node in the multi-agent system at the rth iteration, represents the intermediate variable state value of the i-th node in the multi-agent system at the r-th iteration, represents the local objective function f of the i-th node in the multi-agent system i gradient.

5. According to claim 4, a satellite formation privacy protection control system based on homomorphic encryption is characterized in that: The EXTRA distributed optimization algorithm is expressed as follows: ； ； ； In the formula, Represents the intermediate variable s during the r+2 iteration i The value of represents the correction term of the EXTRA distributed optimization algorithm, W represents a symmetric matrix in the real number domain, I represents the input vector, s j (t) represents the state value of the intermediate variable of the jth node in the system at time t; α represents the step size of each iteration in the EXTRA algorithm.

6. According to claim 5, a satellite formation privacy protection control system based on homomorphic encryption is characterized in that: The ZGS distributed optimization algorithm is expressed as follows: ； In the formula, β represents the constant in the ZGS distributed optimization algorithm. represents the local objective function f of the i-th node in the multi-agent system i (s i ) is the inverse matrix of the Hessian matrix, which is the private information of a single node, a ij Represents a symmetric matrix Elements in a matrix.

7. A satellite formation privacy protection control system based on homomorphic encryption according to claim 6, characterized in that: The Lyapunov function is as follows: ； In the formula, V(s) is a semi-positive definite matrix, represents the unique minimum point in the matrix, s i represents the i-th data point in the matrix, and T represents the transpose of the matrix.

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