Multi-party confidential sorting method for low-altitude cooperative network

Abstract

The invention discloses a multi-party confidential sorting method for a low-altitude cooperative network, and belongs to the technical field of secure multi-party computing. The invention aims to solve the problems of repeated comparison and overflow in the traditional comparative protocol. According to the invention, based on a threshold encryption system, all participants cooperatively generate a threshold ElGamal public key; the participant constructs a prime number coding vector with the length of 1, 2 and 3 according to private input data of the participant, encrypts the prime number coding vector based on a public key and then sends the prime number coding vector to the participant, the participant multiplies each column to obtain a multiplied vector, and the participant selects the first ciphertext component from the ciphertext vectors according to input of the participant; and then obtaining a plaintext based on threshold decryption, carrying out prime factorization on the plaintext, and obtaining a sorting result output by the participant according to the occurrence frequency of the corresponding prime number of the participant in the plaintext prime factorization.

Description

A Multi-Party Confidential Sequencing Method for Low-Altitude Cooperative Networks Technical Field

[0001] This invention belongs to the field of secure multi-party computation technology and relates to a multi-party confidential sorting method. Background Technology

[0002] With the rapid development of the low-altitude economy, drones are being deployed on a large scale in tasks such as emergency rescue, logistics delivery, and airspace management, gradually forming low-altitude collaborative networks (LCNs). In these networks, operators typically need to perform real-time collaborative calculations around core parameters such as mission priority, flight path planning, risk indicators, and resource allocation to ensure overall scheduling efficiency and airspace safety. However, these parameters often contain highly sensitive information, such as corporate operational strategies, mission value assessments, energy costs, and flight paths. If exposed during the collaboration process, it can lead to damage to commercial interests, manipulation of resource scheduling, and even endanger airspace safety.

[0003] In collaborative environments where multiple operators do not fully trust each other, plaintext data exchange cannot meet privacy protection requirements. Furthermore, in the absence of a trusted execution mechanism, malicious nodes may influence the final scheduling order by forging input data, manipulating rerandomization results, or tampering with intermediate parameters, thereby gaining an unfair advantage. This necessitates a privacy-preserving computation method for low-altitude collaborative networks that can "collaborate on computation without disclosing raw data, and resist active attacks" to support the stable and reliable operation of the low-altitude economy. Secure Multi-Party Computation (MPC) can precisely solve these problems. MPC enables all participants to jointly complete function computations without exposing input data, without the need for a trusted third party, and guarantees the privacy and correctness of the computation process through simulation paradigms or ideal-real-world paradigms.

[0004] In actual task scheduling, drone operators often need to jointly determine task priorities, flight path ordering, or resource allocation sequence to ensure overall scheduling efficiency and airspace security. These key decisions can be abstracted into a ranking-based privacy computation problem. For example, different drone operators need to jointly determine task priorities or airspace allocation sequence without disclosing their own task benefits, energy costs, and flight paths. However, traditional ranking algorithms require access to the original input data during execution, making it difficult to meet the dual requirements of privacy and trustworthiness in low-altitude collaborative networks. While existing research on secure multi-party computation has made progress in fields such as finance and healthcare, ranking-based privacy computation for low-altitude collaborative networks is still relatively lacking and lacks the ability to cope with malicious models. Therefore, constructing a confidential ranking protocol suitable for low-altitude collaborative scenarios not only helps to achieve privacy-secure airspace scheduling but also provides a basic guarantee for drone swarm collaboration and cross-domain data fusion. For ease of understanding, Figure 1 shows a typical LCN operating scenario, highlighting the shared airspace that is prone to conflict and the necessity of secure multi-party task priority ranking.

[0005] In recent years, scholars have conducted extensive research on confidential sorting and proposed a series of protocols in different models and application scenarios, providing an important foundation for solving privacy protection issues in the sorting process. In the research direction of multi-party confidential sorting, Li Shundong, Kang Jia, Yang Xiaoyi, and others proposed two multi-party sorting schemes in their paper "Secure Multi-Party Computation for Sorting Multiple Characters." Protocol 1 has a simple structure and low communication cost, but lacks defense against collusion; Protocol 2 introduces secret splitting to enhance security, but correspondingly increases interaction and communication overhead. Subsequently, "A Practical Secure Multi-Party Sorting Scheme Based on Radix Sorting and Homomorphic Encryption" combines homomorphic encryption with radix sorting, achieving high operating efficiency under a semi-honest model; "Efficient Secure Three-Party Sorting with Applications to Data Analysis and Heavy Hitters" optimizes the randomization and ciphertext processing flow, effectively reducing the additional cost caused by repeated re-randomization. Li Shundong, Du Runmeng, Yang Yanjing, and others proposed a multi-party array sorting scheme based on additive homomorphic encryption, which can directly perform numerical operations in the ciphertext domain. Dou Jiawei, Wang Yulin, et al.'s "Secure Sorting Protocol and Its Application" designs sorting protocols for known and unknown range data using parallel bit counting, but the multiple calls to the basic sub-protocol lead to a large overall computational overhead. "A Confidential Multi-Party String Sorting Protocol Based on National Cryptographic Algorithms" combines improved national cryptographic homomorphic encryption and threshold mechanisms to construct a sorting protocol suitable for multiple participants, improving the confidentiality and consistency verification capabilities of inputs from each party during the sorting process.

[0006] Meanwhile, "Secure Word-Level Sorting Based on Fully Homomorphic Encryption" proposes a privacy-preserving sorting mechanism supporting SIMD parallel computation based on fully homomorphic encryption (FHE), enabling batch comparison and reordering in the ciphertext domain. However, its high overhead in noise management and relinearization limits its application in resource-constrained scenarios. The paper "Achieving Efficient Secure Multi-Party Deduplicated Sorting" achieves ciphertext reordering through polynomial multiplication and vectorized ciphertext structures, transforming the sorting process into algebraic operations. However, as the number of participants increases, the ciphertext dimension expands dramatically, significantly increasing communication and storage burdens. "Homomorphic Sorting with Better Scalability" models sorting as a polynomial rank discrimination problem, achieving rank comparison by constructing an encoding matrix, offering certain structural advantages. However, the large matrix size leads to high overall computational and communication costs.

[0007] Overall, existing technologies offer a wealth of solutions to the confidentiality ordering problem under secure multi-party computation, but these solutions generally suffer from the following shortcomings:

[0008] (1) Most protocols are still based on a semi-honest model, which cannot resist malicious input forgery, re-randomization tampering and intermediate result manipulation.

[0009] (2) Many schemes rely on multi-round interactions or large-scale encrypted structures, which are too costly to communicate in the unstable LCN environment.

[0010] (3) The lack of a universal, efficient and verifiable randomization mechanism makes it difficult for the protocol to identify and block malicious behavior internally.

[0011] (4) Some methods rely on high-dimensional matrices, polynomials or FHE structures, which are difficult to deploy in resource-constrained low-altitude collaborative networks. Summary of the Invention

[0012] This invention aims to solve the problems of duplicate comparisons and overflows in traditional comparison protocols.

[0013] A multi-party confidentiality ordering method for low-altitude cooperative networks, where each participant... Holding private data , , The number of participants; all participants collaboratively generate a threshold ElGamal public key. and each holds a fragment of the private key. The system discloses a set of pairwise coprime prime numbers. Each participant Private selection The unique prime number in the middle is used as its own code, denoted as . The mapping between the prime number and the participating parties remains private and not disclosed; the multi-party confidential sorting process includes:

[0014] Step 1, Participants Based on your own private input data Construction length is A prime number encoding vector, the elements in the vector Participants Encrypting the prime number encoded vector yields the encrypted result of the prime number encoded vector. ;in Indicates threshold ElGamal encryption; As a participant The encryption result for each component in the prime number encoding vector;

[0015] Step 2, All Participants ciphertext vector Send to the participants ;

[0016] Step 3, Participants After receiving all vectors, perform a cumulative multiplication on each column to obtain the result. This leads to the vector obtained after multiplication. , vector Send to all participants;

[0017] Step 4: Decryption of the threshold for collaborative execution by all participants: For each participant Based on its own input From ciphertext vector Selected from Each ciphertext component and to Perform rerandomization to obtain Subsequently, all participating parties... Execution threshold decryption yields plaintext ,in For position The result of the plaintext multiplication, Indicates threshold ElGamal decryption;

[0018] Step 5, Participants right Perform factor decomposition: ,in For in position The set of participants who contribute prime numbers; participants Output its own sorting result , The function is used for ranking, and the sorting result is the number of times the corresponding prime number of the participant appears in the plaintext prime factorization.

[0019] Furthermore, in step 3, a cumulative multiplication is performed on each column to obtain... .

[0020] Furthermore, all participants collaboratively generate a threshold ElGamal public key. The process includes:

[0021] Participants Generate common parameters for the ElGamal cryptosystem ,in For large prime numbers, For model Generators of the multiplicative group; each participant Randomly select private key share , Indicates to The set of integers modulo 1, and published. All participants jointly obtain the public key parameter. Therefore, the threshold ElGamal public key is The joint private key satisfies .

[0022] Furthermore, the participating parties The process of encrypting a prime number encoded vector includes:

[0023] Any participant applies the public key right Encrypt; record a random number. The encryption result is a two-part ciphertext. ;in The first component of the ciphertext is composed of random numbers. generate, This is the second component of the ciphertext.

[0024] Furthermore, For ciphertext The process of decryption to obtain plaintext includes:

[0025] ciphertext Represented as an ElGamal two-component form ;in The first component of the ciphertext. This is the second component of the ciphertext; and for China to the first Elements corresponding to column product:

[0026]

[0027] in As a participant In position The encrypted component;

[0028] Subsequently, each participant Use its private key share Calculate part of the decrypted value. Participants Collect all And calculate Restoring plaintext ,in .

[0029] A multi-party confidentiality ordering method for low-altitude cooperative networks, where each participant... Holding private data , , The number of participants; all participants collaboratively generate a threshold ElGamal public key. and each holds a fragment of the private key. The system discloses a set of pairwise coprime prime numbers. Each participant Private selection The unique prime number in the middle is used as its own code, denoted as . The mapping between the prime number and the participating parties remains private and not disclosed; the multi-party confidential sorting process includes:

[0030] Step (1), Participants According to the input Construction length is prime number encoding vector Participants Select random number Obfuscate the prime number encoding vector Encryption is performed on the obfuscated vector to obtain the encrypted result of the prime number encoded vector. ;in Indicates threshold ElGamal encryption; As a participant The encryption result of each component of the vector in prime number encoding;

[0031] Step (2): All participants transmit their own ciphertext vectors. Send to the participants ;

[0032] Step (3), Participants After receiving all vectors, perform a cumulative multiplication on each column to obtain the result. This leads to the vector obtained after multiplication. Then, a random number is selected for each position. Perform rerandomization, aggregation method Sending vector And for each location Submit zero-knowledge proof If any column fails to prove, it indicates that malicious behavior has occurred, and the agreement will terminate immediately.

[0033] Step (4): All participants on the vector Perform product aggregation ,right Decryption by executing the threshold yields the plaintext. , This indicates ElGamal decryption;

[0034] Perform prime factorization ,in prime number In product The index in prime factorization; participants Local inspection If any participant detects that the agreement is invalid, the agreement will terminate immediately.

[0035] Step (5), Decryption of threshold for collaborative execution by all participants: For each participant Based on its own input From ciphertext vector Selected from Each ciphertext component and to Perform rerandomization to obtain Subsequently, all participating parties... Execution threshold decryption yields plaintext ,in For position The result of the plaintext multiplication, This indicates the threshold for ElGamal decryption. During the decryption process, each participant must use zero-knowledge proof to prove to all participants that their share of the decrypted data is correct. If a participant fails to complete the zero-knowledge proof at this point, it is considered that they are deceiving, and the agreement will terminate immediately.

[0036] Step (6), Participants To its own explicit Perform prime factorization to obtain all prime factors equal to The set of prime factors forms a multiset. Recover the number of true prime numbers Participants Output its own sorting result , The function is used for ranking, and the sorting result is the number of times the corresponding prime number of the participant appears in the plaintext prime factorization.

[0037] Furthermore, in step (3), a cumulative multiplication is performed on each column to obtain... .

[0038] Furthermore, all participants collaboratively generate a threshold ElGamal public key. The process includes:

[0039] Participants First, generate the public parameters for the ElGamal cryptosystem. ,in For large prime numbers, For model Generators of the multiplicative group; each participant Select a random integer , Indicates to The set of integers modulo 1, and published. All participants jointly generate public key parameters. Therefore, the threshold ElGamal public key is The joint private key satisfies .

[0040] Furthermore, the participating parties The process of encrypting a prime number encoded vector includes:

[0041] Any participant applies the public key right Encrypt; record a random number. The encryption result is a two-part ciphertext. ;in The first component of the ciphertext is composed of random numbers. generate, This is the second component of the ciphertext.

[0042] Furthermore, For ciphertext The process of decryption to obtain plaintext includes:

[0043] ciphertext Represented as an ElGamal two-component form ;in The first component of the ciphertext. This is the second component of the ciphertext; and for China to the first Elements corresponding to column product:

[0044]

[0045] in As a participant In position The encrypted component;

[0046] Subsequently, each participant Use its private key share Calculate part of the decrypted value. Participants Collect all And calculate Restoring plaintext ,in .

[0047] Beneficial effects:

[0048] (1) To address the need for order-based privacy computing in low-altitude cooperative networks, this invention proposes a confidential ordering mechanism based on random prime number encoding. This mechanism achieves a unique mapping and rank preservation of input data through prime number encoding, effectively avoiding the problems of repeated comparisons and overflows in traditional comparison protocols; and combined with the ElGamal threshold encryption system, it ensures that any participant with less than the threshold cannot recover the plaintext, thereby resisting input forgery and inference.

[0049] (2) Under the malicious model, a verifiable and operable multi-party confidential sorting protocol was designed. This protocol ensures that malicious participants cannot choose special random masks to destroy the sorting structure by introducing discrete logarithm equality proof and verifiable rerandomization method, thus guaranteeing the correctness of the sorting result and the traceability of the protocol execution process.

[0050] (3) Based on the ideal-real example, this invention proves the semantic security and correctness of the proposed protocol under the malicious model. At the same time, combined with experimental and theoretical analysis, the proposed solution is compared with the existing protocol in terms of computational complexity and communication complexity. The results show that the proposed protocol has significant advantages in efficiency and scalability and can be used as a high-security sorting scheme to resist malicious attacks in low-altitude cooperative networks. Attached Figure Description

[0051] Figure 1 is a schematic diagram of the application scenario of task coordination and privacy protection prioritization in low-altitude collaborative networks.

[0052] Figure 2 is an interactive simulation diagram of a multi-party confidentiality ordering protocol for low-altitude cooperative networks under a semi-honest model.

[0053] Figure 3 is an interactive simulation diagram of a multi-party confidentiality ordering protocol for low-altitude cooperative networks under a malicious model.

[0054] Figure 4 is a comparison chart of the number of participants and operational efficiency.

[0055] Figure 5 is a comparison chart of sorted set size and running time efficiency.

[0056] Figure 6 shows the impact of network latency on total execution time.

[0057] Figure 7 illustrates the impact of increased participation on delayed execution time. Detailed Implementation

[0058] To address the problems existing in the background technology, this invention proposes a multi-party confidential ordering protocol based on threshold ElGamal encryption and random prime number encoding. This protocol utilizes prime number encoding with unique mapping to achieve rank-preserving privacy representation, employs a verifiable rerandomization mechanism to resist malicious manipulation of the mask by participating parties, and combines discrete logarithm equality proofs to construct a verifiable malicious security framework. Before detailing the specific implementation methods, the knowledge required for this invention will first be explained.

[0059] Security definition:

[0060] The simulation paradigm, proposed by Goldreich, is widely used in proving secure multi-party computation protocols. Compared to other security proof methods, the simulation paradigm is simpler. It can simulate the process by which the parties to the protocol execute the protocol. The proof principle is as follows: each party uses its own inputs and outputs to simulate the execution process of the protocol individually. If it obtains any information that it can obtain from the secure multi-party computation protocol, it means that the party cannot obtain more information from the actual secure multi-party computation protocol than from the ideal secure multi-party computation protocol, thus proving that the protocol is secure.

[0061] Definition 1: Assumption It is a two-sided computation function. and Variables representing input from the two participants. and Let these represent the function values ​​obtained by the two participants. If a probabilistic multinomial-time algorithm exists... and , so that:

[0062]

[0063] in, Indicates the first The party is executing the agreement View at time, .

[0064] This is called an agreement. The function was calculated confidentially. .in, This indicates that the calculations are indistinguishable.

[0065] The generally accepted security proof method under the malicious model is the ideal-practice paradigm, which means that the actual protocol has the same security as the ideal model. The ideal protocol refers to a protocol where participant A possesses the data... Participant B possesses the data The two calculate functions During the function computation process, both parties utilize a trusted third party (TTP) for the computation. At the end of the agreement, both parties obtain... and Furthermore, neither party can obtain any other information. In an ideal protocol, both parties obtain their respective results and no other information. If the result obtained through calculation in the malicious model is the same as the result obtained in the ideal model, then it can be proven that the protocol under the malicious model is secure.

[0066] It is worth noting that, under the malicious model, the execution of the protocol must satisfy the condition that at least one party is honest; otherwise, the security and reliability of the protocol cannot be guaranteed. The definition of security under the malicious model is given in "Protocol for millionaires' problem in malicious models".

[0067] Definition 2: For the ideal protocol In actual agreements, if a mutually agreeable one can be found... , so that:

[0068]

[0069] Therefore, the protocol can safely compute time functions. ,in , These are the probability polynomial functions constructed under the actual protocol and the ideal protocol, respectively. for The time function, It is auxiliary input information. This refers to using strategies under ideal conditions. Calculation output , This refers to the actual agreement and Output pairs generated through interaction.

[0070] Threshold ElGamal encryption algorithm:

[0071] Threshold cryptography is an important tool in MPC to combat collusion attacks. In threshold cryptography, anyone can encrypt a message using a public key, but at least a certain threshold must be met. Decryption requires individual collaboration, less than Individuals cannot obtain any information in plaintext. This invention will apply... structure The threshold cryptography system is as follows:

[0072] The process of jointly generating a public key:

[0073] Participants First, generate the public parameters for the ElGamal cryptosystem. Each participant Select a random integer Calculate and publish The participating parties jointly generate public key parameters.

[0074]

[0075] Therefore, the threshold ElGamal public key is The joint private key satisfies:

[0076]

[0077] in As a participant The share of the private key.

[0078] Encryption: Any participant applies the public key. plaintext Encryption is performed. A random number is recorded. The encryption result is a two-part ciphertext. .in The first component of the ciphertext is composed of random numbers. generate, This is the second component of the ciphertext.

[0079] Threshold decryption: decryption of ciphertext Each participant Use its private key share Calculate the decrypted value in As a participant Based on the first component of the ciphertext Calculated partial decryption value, all decryption values Aggregation

[0080]

[0081] And restore plaintext

[0082]

[0083] in Modulus Multiplicative inverse in the sense of multiplication. (and) Similar to cryptographic systems, the difficulty assumption of the discrete logarithm problem guarantees that the threshold cryptosystem constructed above is semantically secure. This invention mainly applies this threshold cryptosystem to construct protocols.

[0084] Zero-knowledge proof:

[0085] Zero-Knowledge Proof (ZKP) refers to a two-party process: a prover and a verifier. During the interaction, without the prover revealing any information, the verifier, through the interaction, learns that the conclusion is correct. In this process, the proving party is said to have completed the zero-knowledge proof. The verifier only obtains the correctness or incorrectness of the result. For example, a zero-knowledge proof protocol. The secret , and .

[0086] (1) The proving party selects a random number. and calculate The final announcement .

[0087] (2) The verifier needs to verify Whether it is valid or not.

[0088] If the above formula holds true, then the verifier knows the result is correct, and thus knows that the prover knows the secret. .

[0089] Zero-knowledge proofs offer advantages such as privacy protection, confidentiality preservation, reliability, security, and efficiency, making them valuable tools in many fields. They provide innovative solutions to problems related to information verification and privacy protection.

[0090] Prove that the discrete logarithms are equal:

[0091] Jeroen's "An Improved Protocol for Demonstrating Possession of DiscreteLogarithms and Some Generalizations" provides a proof of the equality of discrete logarithms. This equality can be used to prove that elements of two groups have the same discrete logarithmic relationship under different generators, without revealing the secret exponent. In this invention, it is used in a malicious model to verify that the random mask generated during the re-randomization stage is indeed a legitimate non-zero value, and to prevent malicious parties from constructing special random numbers to disrupt the sorting structure.

[0092] make For the order of The cyclic group, whose generator is If a participant holds a random value and calculate Then it needs to prove to the other participating parties:

[0093]

[0094] Correctness:

[0095] The principle behind this proof lies in: the challenge It is publicly available and computable, but if the secret is unknown... Then it is impossible to construct a system that satisfies the condition. response Therefore, verification can confirm that the discrete logarithms of the two are equal.

[0096] The present invention will now be described in detail with reference to specific embodiments.

[0097] The specific implementation method is one:

[0098] This embodiment is a multi-party confidentiality ordering method for low-altitude cooperative networks, which proposes a multi-party confidentiality ordering scheme under a semi-honest model for low-altitude cooperative networks.

[0099] In collaborative applications within the low-altitude economy, different drone operators typically possess confidential key parameters such as task weights, flight priorities, or execution costs. To achieve overall task scheduling or airspace resource allocation, each party often needs to jointly sort its task data without exposing its original inputs to avoid revealing operational logic, flight path planning, or commercially sensitive information. This requirement can be abstracted into the following multi-party confidential sorting model, assuming that the multi-party collaborative computing system has a total of There are 1 participating party, denoted as 1 Each of the participants Holding private input data All parties hope to collaboratively complete the joint dataset without disclosing their own input values. The total order of sorting. After the protocol is executed, each participant... Only its own input can be obtained in the joint sequence The ranking of the data is determined by the data source, and the input values ​​or intermediate calculation results of other participants cannot be obtained.

[0100] To meet the practical needs of low-altitude collaborative networks, such as the non-disclosure of private data, the requirement for correct sorting structure, and the inability of cross-operator trust, this invention proposes a secure multi-party confidential sorting protocol under a semi-honest model, providing a foundation for the subsequent construction of privacy computing mechanisms for low-altitude altitude and collaborative task allocation.

[0101] The core problem of multi-party single-data sorting lies in how to determine the relative sizes of data from multiple participants without revealing the original input. This invention proposes a confidential sorting approach based on random prime number encoding to address this problem. Let each participant... Holding private data Data encoding is achieved by assigning a unique random prime number to each participant. The encoding method is as follows: for each participant... According to privacy data The size of the vector is in front Add a unique prime number to each of the three positions and fill the remaining positions with 1, thereby protecting each piece of private data. Mapped to a length of Prime number encoding vectors.

[0102] Based on threshold ElGamal public-key encryption, each participant encrypts its prime number encoding vector and submits the resulting ciphertext vector to [the relevant authority / organization]. . Collect all ciphertext vectors, treat the ciphertext vectors sent by each participant as a row to obtain a ciphertext matrix, and perform component-wise multiplication aggregation, that is, multiply the ciphertext in each column of each participant's ciphertext vector to obtain a single aggregated ciphertext at each column position. Through this column-wise homomorphic multiplication operation, the aggregator... The ciphertext on each dimension of the resulting aggregation vector accumulates the contributions of all participants, while the frequency of each prime number in the aggregation result implicitly encodes the size relationship of the corresponding participant's input in the final sorting.

[0103] After aggregation, each participant decrypts its corresponding component from the recovered plaintext vector. By factoring the elements at their size positions, the number of prime numbers is equal to the number of times their encoded data contributes to the cumulative product. Since prime numbers... The number of and The relative size is uniquely matched, and the participants do not need to know the private data of any other participants to deduce their final sort position.

[0104] Table 1 Specific Agreements

[0105] For ease of understanding, the overall workflow of Protocol 1 is shown in Figure 2.

[0106] Correctness analysis:

[0107] First, during the coding phase, the participants Input Mapped to a length of Vector: front Each position is filled with its private prime number. Fill the remaining positions with 1. In this way, the number of times a prime number appears is equal to the size of the input, and the prime numbers used by different participants are different and will not be confused. Then the vector is encrypted by components, which only replaces the plaintext of each position with ciphertext, and does not change the structural information of which prime numbers are in the product at a certain position. Therefore, the order relationship after encoding is well preserved in the ciphertext field.

[0108] Then, in the aggregation phase, the participants Performing homomorphic multiplication on all ciphertext at the same position is equivalent to multiplying the plaintext of all participants in that column; multiplying by the encrypted 1 only performs re-randomization, without changing the plaintext value. Therefore, for any position... In other words, the aggregated plaintext is actually: all that satisfy... The prime numbers of the participants The product of . Next, everyone will discuss vectors. Threshold decryption is performed position by position; each position can only be decrypted into its corresponding product value with the cooperation of all participants. Ultimately, only their own position is considered. plaintext This includes all input values ​​that are not less than The prime numbers of the participants.

[0109] at last, right By performing a prime factorization, we can enumerate all the prime factors that equal the prime factorization. prime numbers, thus obtaining the set Since prime number encoding is injective, different participants have different prime numbers, and each prime number appears exactly once. Therefore, the size of the set is... Exactly equal to any input that is not less than The number of elements is consistent with the rank in the standard sorting definition. Therefore, the protocol outputs... This rank is consistent with the rank obtained by sorting the input vectors from smallest to largest.

[0110] Therefore, the multi-party confidential ordering protocol under the semi-honest model is correct and can correctly calculate the ordering results.

[0111] Under the aforementioned semi-honest model, participants are assumed to execute the protocol correctly, thus enabling joint sorting without revealing private task weights and sorting parameters. However, in real-world low-altitude collaborative networks, participating nodes often come from different operators and lack sufficient trust among themselves. For example, drone operators might conceal true task priorities for competitive advantage or influence the overall sorting results by tampering with re-randomization factors or forging intermediate ciphertexts to seize airspace resources or increase scheduling priority. In such scenarios, relying solely on the semi-honest assumption is insufficient to guarantee the system's security and fairness, making the sorting process highly vulnerable to deliberate attacks.

[0112] Therefore, to adapt to highly sensitive scenarios such as low-altitude mission scheduling, airspace allocation, and cross-enterprise collaborative computing, it is necessary to extend the protocol to malicious models. By incorporating verifiable randomization, zero-knowledge proofs, and other mechanisms, it is ensured that even with deliberate deviations by participating parties, the ordering structure remains correct and privacy is not compromised. Based on this requirement, the design of a multi-party confidential ordering protocol under malicious models will be presented below.

[0113] The specific implementation method is two:

[0114] This implementation method is a multi-party confidentiality ranking method for low-altitude cooperative networks, which proposes a multi-party confidentiality ranking scheme under a malicious model for low-altitude cooperative networks.

[0115] Before discussing potential malicious behaviors, there are three types of malicious behaviors that cannot be prevented even in an ideal multi-party computation protocol: (1) refusal to cooperate; (2) tampering with input information; and (3) termination of the protocol midway. Therefore, in order for the protocol to operate correctly and securely under a malicious model, additional defense mechanisms need to be designed to address potential attacks.

[0116] 1. Possible malicious acts:

[0117] (1) During the coding process, each participant must base their input on their own input. Construct prime number encoding vectors and perform local obfuscation as required. Malicious actors may violate coding rules during this process, such as illegally inserting extra prime numbers where a "1" should be placed. Such structural modifications directly alter the prime contribution of the participant in the global cumulative product, causing an abnormal cumulative product structure and thus directly affecting the sorting results of other participants.

[0118] (2) A malicious participant may set all of its encoding vectors to 1, thus concealing its true sort value. Since the input is in a valid format, the protocol cannot distinguish whether this behavior is malicious concealment or its true input. This situation will be ignored because, in the face of providing false input, neither in the ideal model can resist malicious adversaries, nor in the malicious model can they be prevented. Therefore, this invention does not consider this malicious behavior, and in practice, the applicability of this invention can be ensured by prohibiting such encoding vectors through convention.

[0119] (3) During the rerandomization process, the participating parties Responsible for selecting a non-zero random number for each position. And perform rerandomization on the cumulative ciphertext. Malicious attackers might attempt to select illegal parameters, such as setting... This is done to keep the intermediate ciphertext structure unchanged, so that the ciphertext structure at certain positions remains unchanged or becomes predictable, or even reveals local sorting relationships.

[0120] (4) During the location-based threshold decryption phase, malicious participants may submit incorrect partial decryption shares. The aim is to prevent the target ciphertext from being decrypted correctly, or to cause the participants to obtain incorrect plaintext values, thereby disrupting the final sorting result at the corresponding position.

[0121] To prevent the aforementioned destructive attacks, this protocol employs three types of detection methods:

[0122] 1. To prevent participants from illegally inserting or deleting prime numbers during the encoding phase, the protocol incorporates an exponent consistency verification mechanism after global aggregation. Specifically, the plaintext form of the global product is... Each participant can verify whether the corresponding index is satisfied based on its own input. Any structural alteration, such as adding extra prime numbers or deleting prime numbers that should appear, will cause the corresponding exponents in the result to deviate from this relationship, thus being promptly identified.

[0123] 2. To prevent attackers from choosing illegal random values, the protocol requires generating a zero-knowledge proof for each random factor after rerandomization. This proof demonstrates that the random factor is a valid group element and is non-zero; and that the output ciphertext was indeed generated by the correct random mask. Failure to prove this indicates a problem with the random factor, thus preventing attackers from manipulating random values ​​to maintain the ciphertext structure of certain columns or to reveal local patterns.

[0124] 3. To prevent participants from submitting forged partial decryption values, the protocol requires each decryption share to be accompanied by a consistent zero-knowledge proof demonstrating that it satisfies the correct discrete logarithmic relation, for example: If any party proves that the agreement cannot be established, the agreement shall be terminated.

[0125] Table 2 Specific Agreements

[0126] To facilitate understanding of the overall workflow of Protocol 2, the interaction diagram of the protocol under the malicious model is shown in Figure 3.

[0127] Correctness analysis:

[0128] (1) In steps (1)–(2), each participant Based on your own input Construct prime number encoding vector The ciphertext vector is obtained by encrypting element by element using the threshold ElGamal public key. Because the encoding rules stipulate: [The text abruptly ends here, likely due to an incomplete sentence or a formatting error.] The position is a prime number. The remaining positions are 1, and each participant corresponds to only one prime number. Therefore, input A one-to-one correspondence is established between the encoding vector structure and the data structure. If a participant inserts or deletes a prime number without authorization during the encoding or encryption phase, it will inevitably change its exponential contribution in the global product, and will be detected in the exponential consistency verification in subsequent steps. Therefore, steps 1–2 ensure that the structure used in the protocol is consistent with the actual inputs of all parties, without exposing any participant's private data.

[0129] (2) In step (3), the participating parties First, the ciphertext vectors sent by each participant... The sequence of homomorphic cumulative product is obtained Then, a non-zero random number is selected for each position. Perform rerandomization , to obtain vector Rerandomization only changes the randomness of the ciphertext and does not change the multiplication structure of the underlying plaintext. Therefore, the aggregated plaintext remains consistent with the cumulative product of the initial encoded vectors. To prevent malicious parties from selecting... Alternatively, to construct an illegal offset that negates the randomness of the previous round, the aggregator must submit a zero-knowledge proof for the rerandomization process at each position, proving the use of... It is a valid non-zero group element and is related to The generation relation is correct. If any proof fails, the protocol terminates immediately. This ensures the correctness and randomness of step (3) while not revealing the input of any participant.

[0130] (3) In step (4), all participants set the vector Perform product aggregation to obtain and cooperate on Decryption by executing the threshold yields the plaintext. Then on Perform factor decomposition: Because each participant assigns its own prime number during the encoding phase. The frequency of occurrence is confused as Therefore, the following conditions must be met when legally executing the agreement: Participants Check locally whether the above relationship holds; if any exists... Make If this occurs, it indicates that structural tampering or illegal insertion / deletion of prime numbers has taken place during encoding, transmission, or rerandomization, and the protocol will immediately terminate. Since this only verifies prime numbers that are themselves... The relevant exponents do not involve prime factors of other participants, thus avoiding the leakage of other input information. This step ensures the correctness of the global product structure and the consistency of the encoding.

[0131] (4) In step (5), all participants cooperate on the vector Perform threshold decryption based on position to obtain the plaintext vector. During the decryption process, each participant must disclose the portion of the decrypted data they provide. Submit a zero-knowledge proof to prove that it satisfies This ensures that the decryption share is consistent with the corresponding private key share, preventing malicious parties from interfering with the decryption result or causing incorrect output by forging shares.

[0132] In step (6), the participating parties Plaintext at its corresponding position Perform prime factorization to obtain all prime factors equal to set of prime factors And restore the true number of prime numbers. Due to the cumulative multiplication and rerandomization process, The exponent is always equal to Subtract local obfuscation The result obtained later It is precisely the actual input value of this participant that determines its relative size in the overall ranking. This ensures that each participant can correctly reconstruct its own sorting result without knowing the specific inputs of other participants. .

[0133] Security verification:

[0134] Theorem: The multi-party confidential ordering method for low-altitude cooperative networks under the malicious model is secure.

[0135] Proof Summary: Similar to the security proof of Protocol 1, since all participants are equal, we only need to consider the case of the largest colluding attacker set. If the protocol is secure under the largest colluding attacker set, it is also secure for any smaller colluding set. Let the honest participants be... The largest group of colluding attackers is We prove that for the set Any deviation from the control of the participants has a simulated strategy under an ideal model. This makes it computationally indistinguishable for an attacker to view the real protocol and the ideal protocol.

[0136] Analysis: The core process of Protocol 2 includes: prime number encoding, vector encryption, column homomorphic multiplication, rerandomization, consistency checking, position-threshold decryption, and final sorted output. Malicious actors' deviations mainly occur at three key points:

[0137] 1. In step (1) prime number encoding stage: the malicious party attempts to forge or delete the number of prime numbers by constructing an incorrect prime number vector structure in order to create a deviation in the exponent in the final product.

[0138] 2. In steps (3) and (4): the malicious party may choose an illegal re-random number or forge a re-randomized zero-knowledge proof; it may also insert or delete prime factors that are inconsistent with its encoded number in the joint product ciphertext; or forge its own exponential relationship during the consistency check in an attempt to pass the verification.

[0139] 3. In step (5) decryption by position threshold: the malicious party may submit an incorrect partial decryption share or forge a zero-knowledge proof of its partial decryption share, thereby interfering with the final prime number counting result.

[0140] If the agreement is not terminated, it means that all participants have submitted structurally valid prime number encoding vectors, and both the rerandomization verification and threshold decryption verification have passed successfully. In this case, the honest party... The rerandomization and joint decryption operations were successfully executed, and the decryption result of the cumulative ciphertext was correctly obtained. The final sorting result output by the actual protocol is as follows: The view of the attacker set includes: .in: The ciphertext is the product of the columns submitted by the attacker and the honest party in a homomorphic manner. For the The rerandomized ciphertext, Indicates the first in the attacker set A portion of the data is partially decrypted. All zero-knowledge proof information submitted by the attacker.

[0141] Now let's turn to the ideal model. In the ideal model, there exists a trusted third party, TTP, that receives all inputs. Calculate the sorting results and return the results to the honest party. Simulator Only with attackers The interaction, and its output, is denoted as: in These represent the forged ciphertext, rerandomized ciphertext, decryption share, and zero-knowledge proof generated by the simulator, respectively. In the ideal model, there exists a trusted third party (TTP) that receives all inputs. Calculate the sorting results and return the corresponding results to the honest party. Simulator Only with attackers The interaction, its view is: ,in All are generated by simulators to simulate encryption, rerandomization, decryption shares, and zero-knowledge proof structures in real protocols.

[0142] We will demonstrate that the ideal view generated by the simulator is indistinguishable from the real protocol view in terms of distribution, thus showing that the protocol is secure under a malicious model. Perform the following steps:

[0143] (1) Simulator Randomly generate pseudoprime number encoding values And calculate the corresponding encryption matrix: Because ElGamal encryption satisfies semantic security, the real ciphertext and the pseudo-ciphertext are indistinguishable in terms of distribution, so an attacker's set cannot determine whether the simulator is using the real encoding.

[0144] (2) The simulator randomly selects Construct pseudo-re-randomized ciphertext: And generate zero-knowledge proofs that can be simulated for it. Because zero-knowledge proofs are simulable, attackers cannot distinguish between real and simulated proofs.

[0145] (3) If the exponent or product structure submitted by the attacker is incorrect, TTP will detect the inconsistency during the joint exponent check and terminate the protocol; if the attacker's behavior is legitimate, TTP will output the sorting result. Simulator Can be obtained from TTP Automatically construct consistent exponents: This ensures that the structure of the real protocol remains consistent with the ideal model. If an attacker attempts to disrupt the protocol using abnormal exponents or product structures, the simulator can also cause the protocol to terminate directly, maintaining consistency with the real protocol.

[0146] (4) During the threshold decryption phase, if the attacker submits an incorrect partial decryption share... TTP will detect the inconsistency and terminate the protocol; if a legitimate share is submitted, the plaintext at that location can be recovered. The simulator generates a forged partially decrypted share. Furthermore, it generates simulateable zero-knowledge proofs for the honest party to ensure the validity of the structure. Because ZK proofs are simulateable, attackers cannot distinguish between the real and simulated parts of the decryption process.

[0147] Because the simulator can perfectly reproduce all views of the attacker's set in an ideal model, and these views are computationally indistinguishable from those in the real protocol, that is: .

[0148] Therefore, Theorem 2 proves that Protocol 2 is secure under the malicious model, Q.E.D.

[0149] The proposed protocol will be comprehensively analyzed and evaluated from both theoretical and experimental perspectives. First, through theoretical derivation of computational and communication complexity, an upper bound on the protocol's operating cost under different parameter scales will be given. Then, the complexity of this protocol will be compared with existing representative sorting protocols, theoretically demonstrating the improvement in computational overhead and communication rounds, providing a foundation for subsequent experimental verification. Finally, experimental testing will further evaluate the protocol's performance in a real-world environment, thus comprehensively showcasing its efficiency.

[0150] Computational complexity analysis:

[0151] In the paper "Secure Multi-Party Computation of Multiple Character Sorting" by Li Shundong, Kang Jia, Yang Xiaoyi, et al. (hereinafter referred to as reference [7]), each participant needs to encrypt the encoded sequence elements. Each participating party needs to execute Modulo exponentiation; finally, decryption is required. Therefore, reference [7] requires a total of several modulo exponentiation operations. Submodulus exponent calculation.

[0152] In the paper "Secure Multi-Party Multi-Data Sorting" by Li Shundong, Du Runmeng, Yang Yanjing, et al. (hereinafter referred to as reference

[10] ), the participating parties... In its Execution when 3D vector is encrypted Secondary exponentiation; during the ciphertext propagation process, the remaining participants... Perform a re-encryption or re-randomization operation once for each component, totaling approximately Secondary exponentiation; furthermore, randomization and partial decryption operations are required in the final comparison phase. Submodulo exponentiation. Considering all the above overheads, the overall computational complexity of this protocol is O(n log n). .

[0153] In the "Secure Ordering Protocol and Its Application" by Dou Jiawei, Wang Yulin, et al. (hereinafter referred to as reference

[11] ), each participating party First, regarding one dimensional vector Encryption is performed; subsequently, For a single value Encryption is performed, and the ciphertext is obtained by ciphertext multiplication. Finally, all participants jointly decrypt the encrypted result. Therefore, a total of [number] steps are required. Submodulus exponent calculation.

[0154] In Protocol 1 of this invention, all participants jointly generate a threshold ElGamal public key and complete the distribution of key shares. This process involves... The modulo exponentiation is then performed. Subsequently, each participant constructs a sequence of length... The encoding vector is generated, and ElGamal encryption is performed once for each element. Each encryption involves two modulo exponentiation operations, so the computational complexity of the encoding and encryption stages is O(n log n). During protocol execution, all participants send their ciphertext vectors to the aggregator. The aggregator performs homomorphic multiplication on each column, followed by a rerandomization operation by each participant. This rerandomization requires... The exponentiation operation is repeated, while homomorphic multiplication generates additional exponentiation. Modulo exponentiation. Finally, in the threshold decryption stage, each position requires... Each participant calculates their partial decryption share once, thus generating The second-order exponentiation operation. In summary, under the semi-honest model, the overall computational complexity of this protocol is O(n log n). .

[0155] In Protocol 2 of this invention, all participants first collaboratively generate a threshold ElGamal public key and complete the key share allocation. This process involves... Modulo exponentiation. During the vector encryption phase, each participant needs to perform a vector encryption operation on a length of... The prime-encoded vector is encrypted component-by-component using ElGamal encryption. Each ciphertext requires two modulo exponentiation operations, therefore this stage involves a total of [number missing] operations. Submodular exponent calculation; subsequently, in the aggregation and rerandomization phase, the participants... The ciphertext vectors uploaded by all participants are aggregated column-wise using homomorphic multiplication. Then, a re-randomization is performed on each position of the resulting aggregated vector. Each position's re-randomization requires two modulo exponentiation operations, thus adding extra complexity to this stage. Submodular exponentiation; during the consistency check phase, all participants verify the single aggregate ciphertext. To collaboratively execute a threshold decryption, each participant needs to perform one modulo exponentiation operation, totaling [amount missing]. Modulo exponentiation; finally, in the stage of recovering the true number of primes by position, it is necessary to perform vector operations. of Each position requires threshold decryption to be performed separately. The decryption share is divided into several parts, therefore this stage requires a total of [number] decryption shares. The modulo exponentiation operation is performed. Adding the above parts together, the total number of modulo exponentiation operations for this protocol under the malicious model is: In addition, additional steps are required to verify the correctness of the rerandomization. A zero-knowledge proof, and a method for verifying the validity of a partial decryption share. Each zero-knowledge proof requires a total of [number] proofs. Therefore, Protocol 2 requires a total of zero-knowledge proofs. .

[0156] Communication complexity analysis:

[0157] In the protocol in reference [7], each participant divides its own ciphertext into... Distribute and generate approximately Round-robin communication; then summarizing and sending to the coordinator for decryption. Round, then announce the results Wheels, totaling approximately wheel.

[0158] In the protocol of reference

[10] , during the communication process, each participant needs to send the ciphertext vector, the intermediate re-randomization result, and the final partial decryption result in sequence. Therefore, each participant needs to publish 3 sets of information, and the number of communications of the protocol is 10. wheel.

[0159] In the protocol of reference

[11] , each participant needs to send three types of information to the other participants in sequence: its own encryption vector, the aggregated result ciphertext, and the corresponding partial decryption result. Therefore, each participant conducts three rounds of communication, and the total number of communications in the entire system is [number missing]. wheel.

[0160] In Protocol 1 of this invention, firstly, each participating party needs to send its locally encrypted encoded vector to the aggregator, constituting the first round of communication; secondly, after homomorphic aggregation or result summarization, the computer needs to send the aggregated ciphertext to each participating party, forming the second round of communication; finally, in the decryption phase, each participating party sends the decryption result or verification information to complete joint decryption, forming the third round of communication. Therefore, the total number of communication rounds required for protocol execution is approximately 3.

[0161] In Protocol 2 of this invention, each participant needs to send an encrypted vector sequentially, requiring one round of communication; forwarding the aggregated ciphertext and rerandomization result requires one round of communication; and the threshold decryption share and corresponding zero-knowledge proof require one round of communication. Therefore, the total number of communication rounds is approximately 3. All participants send their encrypted prime number encoded vectors to... One round of communication is required; Broadcasting the re-randomized ciphertext vector and its corresponding zero-knowledge proof requires one round of communication; each participant performs a threshold decryption on the aggregated ciphertext for encoding verification and broadcasts their respective partial decryption share, requiring one round of communication; after the encoding passes verification, each participant performs threshold decryption on the sorted output according to position and broadcasts their partial decryption share again, requiring one round of communication. Therefore, in a malicious model, this protocol requires a total of four rounds of communication.

[0162] For ease of analysis, this invention evaluates the computational complexity of the protocol from the perspective of the number of modulo exponentiation operations, wherein... This represents a single modulo exponentiation operation. Communication complexity is measured in the number of communication rounds. Throughout the analysis, Indicates the size of the entire set. This indicates the number of participants. Furthermore, This represents the computational cost required for a single zero-knowledge proof.

[0163] Table 3 Comparison of Protocol Efficiency Analysis

[0164]

[0165] As shown in Table 1, Protocol 1 and Protocol 2 proposed in this invention have better overall performance than the schemes given in references [7],

[10] and

[11] . Protocol 1 significantly reduces the number of modular exponentiation operations in the encryption and decryption stages while maintaining computational correctness and the security of the semi-honest model. The number of communication rounds is also consistent with references

[10] and

[11] , requiring only 3 rounds of interaction, which is more efficient than the complex communication method of reference [7] which requires multiple rounds of aggregation. Protocol 2 introduces threshold decryption and zero-knowledge proof mechanisms on the basis of Protocol 1, realizing security protection against malicious models. Compared with the schemes in references

[10] and

[11] that only support the semi-honest model, Protocol 2 of this invention can not only resist attacks from malicious participants, but also maintain high execution efficiency under the same number of participants and data scale. In addition, combined with the application background of this invention for low-altitude cooperative networks and UAV task scheduling, Protocol 1 and Protocol 2 can realize secure sorting and matching judgment between UAV nodes while ensuring task connectivity and privacy, ensuring the fairness and reliability of the task allocation process. In summary, the two protocols proposed in this invention achieve a good balance between efficiency and security, and are particularly suitable for high-privacy scenarios such as low-altitude collaboration, the Internet of Things, and distributed decision-making.

[0166] This invention further and intuitively verifies the protocol execution efficiency through experimental simulation. Protocols in references [7] and

[10] -

[11] were selected for comparison, which also solve similar computational problems of sorting under secure multi-party computation.

[0167] Experimental Environment: The experimental environment consisted of a Windows 11 (64-bit) operating system, an Intel Core i9-12900H processor, and 32 GB of memory. The experimental program was implemented using Python in PyCharm. The underlying encryption mechanism adopted the ElGamal threshold homomorphic encryption system, with a security parameter set to 1024 bits.

[0168] Experimental Methods: To evaluate the performance of the proposed multi-party confidentiality ordering protocol under two different participant models, execution efficiency tests were conducted on Protocol 1 and Protocol 2, respectively. Since the runtime of both protocols is primarily affected by the number of participants... Size of sorted sets To mitigate the impact of random fluctuations, the experiment employed a controlled variable approach. Under the condition of fixing some parameters, the change in protocol execution time was observed by altering a single variable. Each experiment was repeated 100 times, and the average execution time was taken as the final result to reduce random volatility and ensure the reliability and comparability of the test results. The experimental design is as follows:

[0169] (1) To evaluate the performance of the proposed ordering protocol under different numbers of participants, this experiment fixed all other parameters and only changed the number of participants. The size of the sorted set remains constant. The number of participants is set sequentially as follows: For each The values ​​of were used to perform 100 experiments on Protocol 1 and Protocol 2 respectively, and the average execution time was recorded. The collected data was used to analyze how the protocol runtime changes as the number of participants increases. The experimental results are shown in Figure 4.

[0170] (2) To examine the impact of increasing data size on sorting performance, this experiment fixed the number of participants to be [number missing]. and the size of the sorted set Set to the following order For each The values ​​of were used to perform 100 experiments on Protocol 1 and Protocol 2 respectively, and the average execution time was used as the result to evaluate the performance under different data scales. The experimental results are shown in Figure 5.

[0171] (3) In LCN scenarios, communication between UAVs often faces problems such as unstable wireless links and latency fluctuations. To simulate these real-world conditions, round-trip latency is injected in each encrypted exchange phase. The latency parameter is set to... For each latency configuration, 100 experiments were performed on Protocol 1 and Protocol 2 respectively, and the average total execution time was recorded to evaluate the robustness of the protocols under different network conditions. The experimental results are shown in Figure 6.

[0172] (4) In large-scale LCN collaborative tasks, more UAV nodes mean higher inter-node message processing overhead, which may further amplify communication latency even in the same network environment. To characterize this impact, the experiment introduced processing latency related to the number of participants. Each participant adds a [something] with An additional processing delay, proportional to the actual processing time, was introduced to simulate fluctuations in a real network. Subsequently, 100 experiments were performed on both Protocol 1 and Protocol 2, and the average execution time was recorded as the final result. The experimental results are shown in Figure 7.

[0173] (5) To evaluate the robustness of Protocol 2 under the malicious model, this experiment simulates three typical malicious behaviors: encoding vector tampering; illegal re-randomization; and forgery of part of the decryption share.

[0174] Each attack was tested 10,000 times. If Protocol 2 could detect the attack through its built-in verification mechanism, the defense was considered successful. The experiments recorded the detection success rate of Protocol 2 and the extremely low probability of attack success to verify the effectiveness of its defense mechanism. The specific experimental results are shown in Table 4.

[0175] Table 4. Success rate of malicious behavior detection under different attack types

[0176]

[0177] Based on the experimental results in Figure 4–7, the performance differences of different protocols under the three dimensions of the number of participants, the size of the sorted set, and the network latency can be systematically observed. Overall, Reference [7] is the most lightweight in terms of computation, and therefore maintains the fastest execution time when the number of participants and the data size are small; however, due to the significant increase in its communication complexity with the number of participants, its total execution latency increases the steepest in Experiment 3 (network latency) and Experiment 4 (latency as the number of participants increases), showing a high sensitivity to network conditions. Reference

[11] has the most computational steps, and its execution time is much longer than other protocols in all experiments, and it shows a clear exponential growth trend with the increase of the number of participants and the data size, making it the slowest overall solution. The performance of Reference

[10] is between the two: it is faster than Reference

[11] , but due to the inclusion of more complex matrix operations, its execution efficiency is still significantly weaker than Protocol 1 and Protocol 2 of this invention.

[0178] In comparison with existing solutions, Protocol 1 maintained a stable execution speed under all experimental conditions, demonstrating good scalability. When the number of participants or the data scale is large, its growth curve is significantly lower than that of References

[10] and

[11] . Even in network latency environments with different RTTs and scenarios where latency increases with the number of participants, the total execution time of Protocol 1 still maintains a low growth rate, indicating that its communication overhead is stable and its sensitivity to network latency is low. In contrast, although Protocol 2 is slightly slower in computation than Protocol 1 due to the introduction of security enhancement mechanisms such as zero-knowledge proofs, it is still significantly better than References

[10] and

[11] in all experiments. Especially in environments with large-scale participants or high latency, the growth rate of Protocol 2 remains flat, demonstrating that it can still maintain reasonable efficiency even after malicious security enhancements.

[0179] Overall, Reference [7] still has the fastest execution performance in environments with no network latency or low latency, but its growth rate increases significantly with the deterioration of network conditions, indicating that its communication layer is more vulnerable; References

[10] and

[11] are high-latency protocols in all dimensions. In contrast, Protocol 1 and Protocol 2 show linear and stable growth under changes in the number of participants, data size, and network latency, proving that their design has excellent scalability and stability in multi-party ordering scenarios.

[0180] Low-Altitude Collaborative Networks (LCNs) are playing an increasingly important role in applications such as drone logistics, urban sensing, and emergency detection. However, task scheduling, ranking comparison, and resource allocation operations in these collaborative environments often involve the sharing of sensitive data. Without effective protection mechanisms, they face serious risks of privacy breaches and malicious behavior. To address these challenges, this invention proposes a privacy-preserving ranking scheme for LCNs. Compared to existing research that largely focuses on semi-honest models, this study further introduces security issues under malicious models into the multi-party ranking protocol, enabling the identification and resistance to malicious behavior while ensuring privacy, thereby improving the overall system's credibility and security.

[0181] In the semi-honest model, the protocol utilizes algebraic randomization and threshold homomorphic encryption to achieve lightweight global sorting computation, enabling secure sorting tasks without exposing participant input. In the malicious model, this invention further designs a verifiable mechanism to ensure the protocol can detect and prevent deliberate tampering, thereby guaranteeing the correctness, verifiability, and robustness of the sorting results. Theoretical analysis and experimental evaluation both demonstrate that, under different parameter scales, the protocol maintains good performance in terms of computational and communication costs and can still stably output correct sorting results even when facing malicious participants.

[0182] From a practical application perspective, the malicious security sorting protocol proposed in this invention possesses strong environmental adaptability and engineering feasibility. In typical scenarios such as low-altitude logistics scheduling, drone swarm collaboration, and collaborative industrial equipment inspection, nodes often exhibit characteristics such as limited resources and unstable communication. For example, in drone task allocation scenarios, this protocol can perform privacy-based sorting of task priorities or risk indicators to prevent scheduling errors caused by abnormal nodes, and ensures the authenticity and consistency of input through a secure and verifiable mechanism, thereby improving the security and business continuity of the collaborative network.

[0183] Theoretical and simulation results demonstrate that this protocol, while maintaining higher privacy, offers significant advantages over existing solutions in terms of computational and communication complexity, particularly in critical multi-party collaboration stages such as low-altitude airspace scheduling and task prioritization. This invention enables efficient and reliable task prioritization computation without disclosing private information, providing a practical solution for building a secure and controllable low-altitude economic cooperation system.

[0184] Overall, the privacy-preserving sorting scheme proposed in this invention demonstrates excellent performance in terms of security, efficiency, and scalability. It can meet the dual requirements of privacy protection and trusted computing in low-altitude collaborative networks, IoT edge computing, and cross-domain collaborative systems, providing a feasible and robust technical path for building secure and efficient collaborative intelligent agent systems.

[0185] This invention may have other embodiments. Without departing from the spirit and essence of this invention, those skilled in the art can make various corresponding changes and modifications according to this invention, but these corresponding changes and modifications should all fall within the protection scope of the appended claims.

Claims

1. A multi-party confidentiality sequencing method for low-altitude cooperative networks, characterized in that: Each participant Holding private data ， ， The number of participants; all participants collaboratively generate a threshold ElGamal public key. and each holds a fragment of the private key. The system discloses a set of pairwise coprime prime numbers. Each participant Private selection The unique prime number in the middle is used as its own code, denoted as . ； The mapping between the prime number and the participating parties remains private and not disclosed; the multi-party confidential sorting process includes: Step 1, Participating parties Based on your own private input data Construction length is A prime number encoding vector, the elements in the vector Participants Encrypting the prime number encoded vector yields the encrypted result of the prime number encoded vector. ;in Indicates threshold ElGamal encryption; As a participant The encryption result for each component in the prime number encoding vector; Step 2, all participants ciphertext vector Send to the participants ； Step 3, Participants After receiving all vectors, perform a cumulative multiplication on each column to obtain the result. This leads to the vector obtained after multiplication. ， vector Send to all participants; Step 4, all participants collaboratively execute threshold decryption: For each participant Based on its own input From ciphertext vector Selected from Each ciphertext component and to Perform rerandomization to obtain Subsequently, all participating parties... Execution threshold decryption yields plaintext ,in For position The result of the plaintext multiplication, Indicates threshold ElGamal decryption; Step 5, Participants right Perform factor decomposition: ,in For in position The set of participants who contribute prime numbers; participants Output its own sorting result ， The function is used for ranking, and the sorting result is the number of times the corresponding prime number of the participant appears in the plaintext prime factorization.

2. The multi-party confidentiality sequencing method for low-altitude cooperative networks according to claim 1, characterized in that: Step 3 involves performing a cumulative multiplication on each column to obtain the result. 。 3. A multi-party confidentiality sequencing method for low-altitude cooperative networks according to claim 1 or 2, characterized in that: All participants collaborate to generate a threshold ElGamal public key. The process includes: participating parties Generate common parameters for the ElGamal cryptosystem ,in For large prime numbers, For model Generators of the multiplicative group; each participant Randomly select private key share ， Indicates to The set of integers modulo 1, and publish it. All participants jointly obtain the public key parameter. Therefore, the threshold ElGamal public key is The joint private key satisfies 。 4. The multi-party confidentiality sequencing method for low-altitude cooperative networks according to claim 3, characterized in that: Participants The process of encrypting a prime number encoded vector includes: any participating party applying a public key. right Encrypt; record a random number. The encryption result is a two-part ciphertext. ;in The first component of the ciphertext is composed of random numbers. generate, This is the second component of the ciphertext.

5. A multi-party confidentiality sequencing method for low-altitude cooperative networks according to claim 4, characterized in that: For ciphertext The process of decrypting to obtain plaintext includes: taking the ciphertext... Represented as an ElGamal two-component form ;in The first component of the ciphertext. This is the second component of the ciphertext; and for China to the first Elements corresponding to column product: ;in As a participant In position The encrypted component; subsequently, each participant Use its private key share Calculate part of the decrypted value. Participants Collect all And calculate Restoring plaintext ,in 。 6. A multi-party confidentiality sequencing method for low-altitude cooperative networks, characterized in that: Each participant Holding private data ， ， The number of participants; all participants collaboratively generate a threshold ElGamal public key. and each holds a fragment of the private key. The system discloses a set of pairwise coprime prime numbers. Each participant Private selection The unique prime number in the middle is used as its own code, denoted as . ； The mapping between the prime number and the participating parties remains private and not disclosed; the multi-party confidential sorting process includes: step (1), participating parties According to the input Construction length is prime number encoding vector Participants Select random number Obfuscate the prime number encoding vector Encryption is performed on the obfuscated vector to obtain the encrypted result of the prime number encoded vector. ;in Indicates threshold ElGamal encryption; As a participant Encryption results for each component of the vector in prime number encoding; Step (2): All participants transmit their own ciphertext vectors. Send to the participants ； Step (3), Participants After receiving all vectors, perform a cumulative multiplication on each column to obtain the result. This leads to the vector obtained after multiplication. Then, a random number is selected for each position. Perform rerandomization, aggregation method Sending vector And for each location Submit zero-knowledge proof If any column fails to prove, it indicates malicious behavior has occurred, and the agreement terminates immediately; Step (4), all participants test the vector Perform product aggregation ,right Decryption by executing the threshold yields the plaintext. ， Indicates ElGamal decryption; performs prime factorization. ,in prime number In product The index in prime factorization; participants Local inspection If any participant detects a failure, the agreement terminates immediately; Step (5): All participants collaboratively execute threshold decryption: For each participant Based on its own input From ciphertext vector Selected from Each ciphertext component and to Perform rerandomization to obtain Subsequently, all participating parties... Execution threshold decryption yields plaintext ,in For position The result of the plaintext multiplication, This indicates the threshold for ElGamal decryption; during the decryption process, each participant must use zero-knowledge proof to prove to all participants that their share of the decrypted data is correct. If a participant fails to complete the zero-knowledge proof at this point, it indicates that they are deceiving, and the agreement will terminate immediately; Step (6), Participants To its own explicit Perform prime factorization to obtain all prime factors equal to The set of prime factors forms a multiset. Recover the number of true prime numbers Participants Output its own sorting result ， The function is used for ranking, and the sorting result is the number of times the corresponding prime number of the participant appears in the plaintext prime factorization.

7. A multi-party confidentiality sequencing method for low-altitude cooperative networks according to claim 6, characterized in that: In step (3), the cumulative multiplication is performed on each column to obtain... 。 8. A multi-party confidentiality sequencing method for low-altitude cooperative networks according to claim 6 or 7, characterized in that: All participants collaborate to generate a threshold ElGamal public key. The process includes: participating parties First, generate the public parameters for the ElGamal cryptosystem. ,in For large prime numbers, For model Generators of the multiplicative group; each participant Select a random integer ， Indicates to The set of integers modulo 1, and publish it. All participants jointly generate public key parameters. Therefore, the threshold ElGamal public key is The joint private key satisfies 。 9. A multi-party confidentiality sequencing method for low-altitude cooperative networks according to claim 8, characterized in that: Participants The process of encrypting a prime number encoded vector includes: any participating party applying a public key. right Encrypt; record a random number. The encryption result is a two-part ciphertext. ;in The first component of the ciphertext is composed of random numbers. generate, This is the second component of the ciphertext.

10. A multi-party confidentiality sequencing method for low-altitude cooperative networks according to claim 9, characterized in that: For ciphertext The process of decrypting to obtain plaintext includes: taking the ciphertext... Represented as an ElGamal two-component form ;in The first component of the ciphertext. This is the second component of the ciphertext; and for China to the first Elements corresponding to column product: ;in As a participant In position The encrypted component; subsequently, each participant Use its private key share Calculate part of the decrypted value. Participants Collect all And calculate Restoring plaintext ,in 。

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