Multi-party manifold learning method based on secret sharing

Abstract

The invention belongs to research in the field of privacy protection multi-party data collaboration, and particularly relates to a multi-party manifold learning method based on secret sharing. The method comprises the following steps that: a plurality of data owners firstly distribute copy secret shared shares of respective data sets to three computing servers; each computing server interactively computes a k-nearest neighbor distance matrix of all sample pairs on the secret share of the data set based on a novel secure top-k algorithm; then, each computing server interactively computes the distance of a full-source shortest path on the secret share of the k-nearest neighbor distance matrix by adopting a low-round Floyd-Warshall algorithm; and finally, each computing server performs secure multi-dimensional scaling on the shortest path distance matrix shared by the secret to obtain low-dimensional embedding of the data set, and any two servers send the secret share of the low-dimensional embedding to a specified user to locally recover the secret share into a plaintext. According to the invention, the number of distance calculation is reduced, and an efficient and safe top-k and all-source shortest path algorithm is provided.

Description

Technical Field

[0001] The present invention belongs to the research field of privacy-preserving multi-party data collaboration, and specifically relates to a multi-party manifold learning method based on secret sharing. Background Art

[0002] In the era of big data, the exponential growth of data presents significant challenges and opportunities for information extraction and decision-making. Data mining involves developing algorithms and techniques to discover meaningful patterns, relationships, and insights from raw data sets. Data held by a single institution or enterprise often suffers from limited data volume and single-dimensional information, making it difficult to meet the large sample size and diverse information requirements of complex applications. Leveraging data from multiple institutions or enterprises often improves data mining effectiveness. However, as more and more countries emphasize data privacy protection, regulations such as the General Data Protection Regulation have been introduced, resulting in substantial fines for data violations by major companies such as Facebook and Uber. To mitigate risks, many organizations have become more cautious about sharing data externally, exacerbating the phenomenon of "data silos." This prevents the effective integration and utilization of previously highly valuable data from multiple sources, limiting the further realization of data value. Developing secure and efficient data collaboration mechanisms while ensuring data privacy and security is a top priority.

[0003] Secret sharing technology is one of the main approaches to addressing the problem of secure multi-party data collaboration. It provides cryptographically provable security and has attracted widespread attention from academia and industry due to its practicality. Secret sharing schemes typically share input data in the form of secret shares with multiple parties, who then interactively compute the required function. They can efficiently implement a wide range of functions, including but not limited to mixed-mode operations in the arithmetic and Boolean domains such as addition, subtraction, multiplication, division, equality, and comparison; complex scientific operations such as exponentials, square roots, and trigonometric functions; and set operations such as shuffling, permutation, and sorting. Due to its rich function support and high performance, a large number of studies have proposed multi-party manifold learning methods for secret sharing, including principal component analysis, classification, clustering, regression, neural networks, and SQL-like analysis. However, support for manifold learning tasks is currently lacking. A representative method is the isomap. Isomap excels at capturing the inherent nonlinear structure of data, a capability crucial in fields where data exhibits inherent nonlinear relationships. Therefore, manifold learning is widely used in nonlinear dimensionality reduction, denoising, and data visualization in fields such as image and speech processing, bioinformatics, and financial analysis.

[0004] The main problem with existing research is the lack of secure and efficient k-nearest neighbor algorithms and all-source shortest path algorithms to support manifold learning. Current secure k-nearest neighbor computation schemes suffer from various issues. Some schemes sacrifice security for performance, weakening the security model; some schemes calculate approximate results; and some schemes have high preprocessing costs. Current secure shortest path computation schemes are based on the Dijkstra algorithm. They modify the original algorithm to ensure perplexity, but increase the algorithmic complexity, resulting in high communication volume and round complexity in the secret sharing implementation, and poor scalability. Therefore, it is extremely urgent to design secure and efficient k-nearest neighbor algorithms and all-source shortest path algorithms based on secret sharing technology, and then construct a secret sharing multi-party manifold learning method. Summary of the Invention

[0005] The present invention aims, on the one hand, to propose an efficient and secure top-k algorithm and all-source shortest path algorithm based on secret sharing technology, providing an efficient solution for multi-party manifold learning computations in which multiple data owners jointly share secrets using their respective datasets. On the other hand, it aims to protect the privacy of the data owner's private dataset. During the computation process, the computing server cannot infer any information about the data owner's dataset based on its view of communication messages, memory access, and keys.

[0006] In order to design an efficient and secure top-k algorithm, we construct it based on the plaintext fast selection algorithm based on the idea of ​​"shuffle and open". in The bit value is used to ensure that each element in the input vector is unique. Subsequently, the input vector is randomly shuffled using a secure shuffling protocol, ensuring that the comparison results in subsequent partitioning operations are independent of the original input vector and can be securely disclosed. In this way, data exchange operations can be conveniently completed locally on the server. To design an efficient and secure all-source shortest path algorithm, the present invention expands the two-layer loop in the original Floyd-Warshall algorithm (i.e., the Floyd algorithm) and adopts batch communication technology in combination with data dependencies, thereby significantly reducing the number of communication rounds. At the same time, combined with the characteristics of Isomap, the number of secret sharing operations is effectively reduced.

[0007] In order to achieve the above object, the present invention adopts the following technical solutions:

[0008] A multi-party manifold learning method based on secret sharing includes the following steps:

[0009] Step 1: The data owner encrypts and shards their respective datasets using a secret sharing protocol, and uploads the secret shares of the sharded datasets to three computing servers respectively.

[0010] Step 2: Each computing server calculates the secret share of the k-nearest neighbor distance matrix of all sample pairs based on the secret share of the dataset, using the Euclidean distance as the distance metric.

[0011] Step 3: Each computing server uses the low-round Floyd-Warshall algorithm on the secret shared k-nearest neighbor distance matrix to calculate the distance of the all-source shortest path, and stores the calculation results in the secret shared shortest path distance matrix;

[0012] Step 4: Each computing server performs secure multi-dimensional scaling on the secret-shared shortest path distance matrix to obtain a low-dimensional embedding of the data set. Any two computing servers send the low-dimensional embedded secret shares to the designated user and restore them to plaintext locally.

[0013] This technical solution is further optimized in that the number of data owners in step 1 is not limited, and the data owner data set is secretly shared with the computing server using a three-party semi-honest replication secret sharing scheme.

[0014] In a further optimization of the present technical solution, the secure k-nearest neighbor calculation in step 2 utilizes matrix symmetry to halve the number of secret sharing operations in the Euclidean distance calculation.

[0015] This technical solution is further optimized. The secure k-nearest neighbor calculation in step 2 uses the secure top-k algorithm to find the smallest distance in each row of the distance matrix. distance, that is, each sample corresponds to nearest neighbors, and set the distances corresponding to the remaining non-neighbors to .

[0016] This technical solution is further optimized. In step 3, the low-round Floyd-Warshall algorithm does not leak the number of edges. By expanding the two-layer loop in the algorithm, the operations are divided into three batches according to the data dependency relationship, and the communication in each batch is parallelized, which significantly reduces the number of communication rounds. is the input distance matrix, express The distance value of the position, let They are the pointers of the two inner loops, is a specific value of the outermost loop pointer, is the size of the distance matrix, and three batches are defined as follows: 1) Batch 1 includes all The secret sharing operation at this time is , 2) Batch 2 includes (1) all The secret sharing operation at this time, that is, for all ; (2) all The secret sharing operation at this time, that is, for all ,3) Batch 3 includes all The secret sharing operation at this time, that is, for all .

[0017] This technical solution is further optimized. Step 2 is based on the idea of ​​"opening after shuffling" and an efficient and secure top-k algorithm is designed on the basis of the plaintext fast selection algorithm.

[0018] This technical solution is further optimized. The low-round Floyd-Warshall algorithm in step 3 reduces the number of secret sharing operations based on the characteristics of Isomap: 1) the operation of updating the diagonal elements of the matrix can be omitted, including the operations on the diagonal in batches 1 and 3; 2) the operations in batch 2 can be omitted; 3) the operations in the lower triangle of the matrix in batch 3 can be omitted.

[0019] This technical solution is further optimized. The secure top-k algorithm in step 2 is appended to the end of each element of the input vector. in The bit value makes the elements of the input vector unique, and the input vector is randomly shuffled using a secure shuffling protocol. The comparison results in subsequent partitioning operations are independent of the input vector and can be safely opened, so that data exchange can be conveniently completed locally on the server.

[0020] This technical solution is further optimized, the secure top-k algorithm in step 2 is equal to the reference value or The probability of the algorithm stopping after each partition doubles, so the average number of partitions is reduced.

[0021] Differentiating from existing technologies, this paper proposes a multi-party manifold learning method based on secret sharing. This method does not restrict the number of data owners; instead, data owners outsource their datasets to a computing server in the form of secret sharing. The computing server interactively performs computations on the secret shares, ensuring high security and preventing the server from inferring any information related to the data owner's private data. This method leverages matrix symmetry to halve the number of secret sharing operations in Euclidean and distance-squared calculations. Based on the "shuffle and open" principle, an efficient and secure top-k algorithm is designed to support k-nearest neighbor computations, building on the plaintext fast selection algorithm. A low-round Floyd-Warshall algorithm based on secret sharing is designed for all-source shortest path computation. This algorithm maintains the number of edges, significantly reduces the number of communication rounds by expanding two loops within the algorithm and employing batch communication techniques based on data dependencies. Furthermore, the number of secret sharing operations is reduced by leveraging the properties of Isomap. In secure multidimensional scaling, a parallel Jacobi method is used to decompose matrix features, and a secure top-k algorithm is used to extract the largest eigenvalues ​​and their corresponding eigenvectors. BRIEF DESCRIPTION OF THE DRAWINGS

[0022] Figure 1 This is the overall flow chart of this program;

[0023] Figure 2 This is the system architecture diagram of this solution;

[0024] Figure 3 It is a diagram of the steps of the secure top-k algorithm;

[0025] Figure 4 This is a diagram of the steps of the low-round Floyd-Warshall algorithm;

[0026] Figure 5 This is a diagram of the steps of the secure multidimensional scaling algorithm. DETAILED DESCRIPTION

[0027] The technical content, structural features, implementation objectives and effects of this technical solution are further explained below with reference to specific embodiments and accompanying drawings.

[0028] It should be emphasized that in order to avoid unnecessary details interfering with the understanding of this application, the drawings only show structures and / or processing steps closely related to the scheme of this application, and omit other details not related to this application.

[0029] This paper proposes a multi-manifold learning method based on secret sharing, which uses three computing servers and multiple users. Figure 1 As shown. Specifically, we focus on a commonly used manifold learning algorithm, isomap. First, the data owner negotiates predefined parameters, the number of neighbors and dimensionality reduction Each computing server calculates the Euclidean distance between all pairs of samples based on the secret share multiplication and square root operation on the secret share of the complete dataset. The secure top-k algorithm based on secret share shuffling, secret share comparison and secret share opening operation finds the top k distance of all samples. The distances to the nearest neighbors are calculated, and the distances to the remaining non-neighbor samples are set to their maximum values ​​to obtain a neighbor distance matrix. The all-source shortest path distance matrix is ​​then calculated using the low-round Floyd-Warshall algorithm based on the secret shares of the neighbor distance matrix. Finally, the all-source shortest path distance matrix is ​​securely multi-scaled using the secure parallel Jacobi algorithm and the secure top-k algorithm to obtain the secret shares of the low-dimensional embedding of the dataset. Any two computing servers send the secret shares of the low-dimensional embedding to a designated user, who then opens the result locally as plaintext.

[0030] The present invention architecture refers to Figure 2 As shown in the figure, the data owner and computing service are deconstructed. There is no limit on the number of data owners, but there are three computing servers. The computing servers do not collude with each other, but may attempt to infer the data owner's dataset based on their own perspective. The data owner sends the secret share of their respective dataset to the three computing servers. The computing servers interactively calculate the low-dimensional embedding of the dataset based on the secret share and finally send the secret share to the designated user to convert it into a plaintext low-dimensional embedding. This method can provide accurate computation results to the designated user. At the same time, the computing server cannot infer any information about the data owner's dataset based on communication messages, memory accesses, and keys from its perspective, thus ensuring high privacy.

[0031] The details are described in the following examples.

[0032] Step 1: Multiple data owners select three non-colluding untrusted computing servers and use the replication secret sharing scheme to secretly share their data sets with the servers: any variable Will be split into three random parts , and is defined as a secret sharing form ,server Holding a secret share ,server Holding a secret share ,server Holding a secret share , the secret share of any server cannot be restored to its original value.

[0033] Step 2: The three computing servers interactively perform secure k-nearest neighbor computation on the secret shared input dataset. is the number of samples in the dataset, is the sample dimension, The distance matrix is ​​initialized with zeros for the number of neighbors defined by the user. , and interactively calculate the Euclidean distance between all sample pairs based on secret shared multiplication and square root operations. Utilizing the symmetry of the distance matrix, the distance calculation operation is reduced by half. Samples to The distance from the first sample is equal to Samples to The distance of samples is calculated only The distance in the upper triangle of , and adding the matrix to its transpose, the position The value assigned to the position .

[0034] Use the safe top-k algorithm to find the smallest value in each row of the matrix distance, that is, each sample corresponds to nearest neighbors, and set the distances corresponding to the remaining non-neighbors to Specifically, for each row of the matrix, the permutation generated by the secure top-k algorithm will be the smallest Distance to front Position, then The distance of the position is set to The distances are rearranged back to their original positions by applying the corresponding permutation in reverse to each row. Finally, the distance matrix is ​​symmetrized by setting the value at a given position and its transposed position to the minimum of the two, based on the minimum operation on the secret share.

[0035] Based on the idea of ​​"opening after shuffling", a secure top-k algorithm is designed based on the original quick selection algorithm. The algorithm requires that the elements of the input vector are unique, otherwise it can be solved by appending The additional value contains bits. The input vector is randomly shuffled at the beginning of the algorithm, so that the comparison results in subsequent partitions are independent of the input vector and can be safely opened. Furthermore, data exchange can be conveniently completed locally. For detailed steps of the secure top-k algorithm, see Figure 3 .vector Record the permutation between the input vector and the output vector. The computing server first uses the secure shuffle protocol Interactively implement random shuffling of a secret-shared vector. After shuffling, the vector is repeatedly partitioned into shrinking ranges. The partitioning operation is based on a comparison operation on the secret-shared value and an open operation on the secret-shared value. The partitioning operation randomly selects a pivot value, shifts elements smaller than the pivot value to the left, and shifts elements larger than the pivot value to the right. The pivot value position is then calculated, with element swapping performed locally on the compute server based on the open comparison results.

[0036] In the standard quick selection algorithm, when the position of the base value is equal to The algorithm stops when , because its goal is to find the Small or The largest element, in this case, is to find the Although this stopping condition also applies to finding the previous elements, i.e. the smallest or largest elements, but the present invention proposes a better stopping condition: when the position of the reference value is equal to or The partition stops when . This stop condition is valid because when is the reference value, for any and ,have Therefore, for any , both This means It is a valid top-k result. Since the randomly selected benchmark position allows the algorithm to stop in two cases, that is, the benchmark position is equal to or The algorithm stops when , and the probability of stopping after each partition doubles, so the average number of partitions is expected to decrease.

[0037] The average time complexity of the original quick selection algorithm is Therefore, the number of secret sharing comparisons of the secure top-k algorithm of the present invention is , the corresponding communication complexity is bits. Similar to quick sort, the average number of partitions is Since the communication in the secret sharing comparison operation of each partition is performed in parallel, the round complexity of the secure top-k algorithm is The cost of a secure shuffle is Bit and Wheel, this overhead is concealed.

[0038] Step 3: Each computing server uses the round-efficient secure Floyd-Warshall algorithm on the secret shared k-nearest neighbor distance matrix to calculate the distance of the all-source shortest path, and the calculation results are stored in the secret shared shortest path distance matrix. The detailed steps of the round-efficient secure Floyd-Warshall algorithm proposed in this invention can be found in Figure 4 , the derivation process is as follows. Input matrix of points ,The original Floyd-Warshall algorithm uses the following three nested loops.

[0039] for :

[0040] for :

[0041] for :

[0042] .

[0043] This structure leads to Since the number of communication rounds of the secret sharing protocol is proportional to the circuit depth, this overhead is impractical. The present invention first expands the two loops inside the Floyd-Warshall algorithm, analyzes its data dependencies, and batch processes the communication of independent secret sharing operations, significantly reducing the number of communication rounds. For each specific value of , the operations are divided into the following three batches:

[0044] Batch 1: Includes all The secret sharing operation at this time is .

[0045] Batch 2: includes (1) all The secret sharing operation at this time, that is, for all ; (2) all The secret sharing operation at this time, that is, for all .

[0046] Batch 3: Includes all The secret sharing operation at this time, that is, for all .

[0047] We can observe the data dependencies: (1) operations in batch 1 do not depend on the computation results of other batches; (2) operations in batch 2 only depend on batch 1: (3) The operations in batch 3 only depend on batch 2: and Therefore, as long as batches 1, 2, and 3 are executed sequentially, the operations in each batch can be executed in parallel. This reduces the circuit depth of the Floyd-Warshall algorithm to .

[0048] The Floyd-Warshall algorithm is further optimized based on the characteristics of the manifold learning algorithm Isomap to reduce the number of secret sharing operations.

[0049] Optimization 1: In Isomap, due to the non-negativity of the distance metric, all distances in the neighbor matrix are non-negative. In particular, the distance from any sample to itself is zero. Therefore, the distance of the shortest path from any sample to itself is also zero. Therefore, (1) operations to update the diagonal elements of the matrix can be saved, including operations on the diagonal in batches 1 and 3, because their values ​​are known to be zero; (2) batch 2 can be saved because , the calculation will not change the result.

[0050] Optimization 2: The output of the nearest neighbor algorithm is an undirected graph, which means that the nearest neighbor matrix is ​​symmetric. Apparently, the Floyd-Warshall algorithm maintains the symmetry of the matrix, allowing us to eliminate the operations in the lower triangle in batch 3. We only perform operations in the upper triangle, and copy the results from the upper triangle to the corresponding transposed positions in the lower triangle.

[0051] Compared with the original Floyd-Warshall algorithm, the optimized algorithm reduces the number of secret sharing operations from Reduce to .

[0052] Step 4: Each server calculates the shortest path distance matrix of the secret sharing Perform secure multidimensional scaling to obtain a low-dimensional embedding of the dataset. For detailed steps of the secure multidimensional scaling algorithm, see Figure 5 Multidimensional scaling first calculates the shortest path distance matrix The square matrix of distances is then converted into an inner product matrix by applying double centering. ,

[0053] The square distance matrix is ​​defined as , in express The square of the distance between the locations, is the center matrix, defined as ,in Is the identity matrix. Multidimensional scaling matrix Perform eigendecomposition and calculate dimensional embedding, where is the dimension after dimensionality reduction: in is The diagonal matrix consisting of the largest eigenvalues, is a matrix consisting of corresponding eigenvectors, It is calculated dimensional embedding.

[0054] The secure version of multidimensional scaling performs secret-shared distance square operations only in the upper triangle of the matrix and uses a secure parallel Jacobi protocol. To calculate the eigenvalues ​​and eigenvectors of the matrix. Experiments have shown that this method is more efficient than the QR-based method in a secret sharing environment. In this case, the secure top-k algorithm will minimize Arrange the values ​​to the front of the vector locations, the goal of secure multidimensional scaling is to find The largest eigenvalue can be converted to a secure top-k algorithm to find The smallest eigenvalue, then the vector finally The eigenvalues ​​at the positions correspond to The largest eigenvalue.

[0055] Finally, any two computing servers will embed the low-dimensional The secret share is sent to the specified user and restored locally as a plaintext result.

[0056] The present invention was experimentally validated using the Wisconsin Breast Cancer Dataset and a synthetic dataset. The scheme was run on three LEGION REN9000K-34IAZ compute servers, each equipped with a 12th Gen Intel(R) Core(TM) i9-12900KF 3.20 GHz CPU and 32GB of RAM. The servers performed computations on a single thread, connected to each other via a network with 5Gbps bandwidth and a 0.2ms RTT. Since the semi-honest three-party replication secret sharing scheme does not utilize a preprocessing paradigm, the end-to-end runtime of the scheme was measured. In all experiments, the ring size of the secret shares was set to 128 bits.

[0057] First, we tested the time performance of the secure top-k and all-source shortest path algorithms. In both tests, all input data was randomly synthesized. Data distribution does not affect computational or communication overhead because all algorithms are blind. Regardless of whether the input is real or random, the computation server performs computations on uniformly random secret shares.

[0058] The secure top-k algorithm of our invention was compared with the tripartite radix sort algorithm, which is currently considered the most practical solution. Table 1 shows the runtime (in seconds) of the secure top-k algorithm and tripartite radix sort using a synthetic dataset with varying input sizes. For input data sizes ranging from 200,000 to 1,000,000, the secure top-k algorithm of our invention was 9 to 13.6 times faster than tripartite radix sort. The secure top-k algorithm completed the top-k computation for 200,000 secret-shared input data in less than 0.8 seconds, demonstrating that top-k computation is nearly instantaneous for smaller amounts of data.

[0059] Table 1. Running time (seconds) of the secure top-k algorithm and tripartite radix sort using synthetic datasets

[0060] The low-round Floyd algorithm of the present invention was compared with the state-of-the-art secure Dijkstra algorithm for all-source shortest path computation. To ensure a fair comparison, the secure Dijkstra algorithm was instantiated using the same three-party replicated secret sharing scheme, but the secure shuffle protocol used was replaced with the state-of-the-art three-party protocol. Table 2 shows the runtime (in seconds) of the secure Floyd and secure Dijkstra algorithms using synthetic datasets with varying data sizes. The low-round Floyd algorithm of the present invention significantly improves the efficiency of all-source shortest path computation, being 684.9 to 1818.5 times faster than the secure Dijkstra algorithm. This performance improvement is primarily due to the reduction in communication overhead, particularly the reduction in the number of communication rounds.

[0061] Table 2. Running time (seconds) of the safe Floyd algorithm and the safe Dijkstra algorithm using synthetic datasets

[0062] Finally, the time performance of the secret-sharing multi-party manifold learning Isomap algorithm was tested using the Wisconsin Breast Cancer dataset, assuming a dataset is jointly owned by multiple data owners. The algorithm reduced the dimensionality of the original 30-dimensional dataset to 5 dimensions, with the number of nearest neighbors set to 20. The baseline scheme used by the present invention for comparison used the secure Dijkstra algorithm to calculate the all-source shortest path. Table 3 shows the runtime (in seconds) of the present and baseline schemes for manifold learning on the Wisconsin Breast Cancer dataset at various data sizes. The present invention ran approximately 11.1 to 28.2 times faster than the baseline scheme. The performance improvement became more significant with increasing sample size.

[0063] Table 3. Running time (seconds) of this scheme and the baseline scheme on the Wisconsin breast cancer dataset

[0064] The advantages of the present invention are as follows: 1. Data owners outsource data sets to computing servers in the form of secret sharing, with no limit on the number of data owners; 2. It has high security, and the computing server cannot infer any information related to the data owner's private data; 3. It uses matrix symmetry to halve the number of secret sharing operations in Euclidean distance calculation and distance squared calculation; 4. It accurately calculates secure top-k and k-nearest neighbors, and the secure top-k algorithm is at least 9 times faster than the tripartite radix sort; 5. The low-round Floyd-Warshall algorithm for secret sharing uses batch communication technology based on data dependencies to achieve low communication rounds, and uses the characteristics of Isomap to reduce the number of secret sharing operations. The algorithm does not leak the number of edges.

[0065] It should be noted that the relational terms such as "first" and "second" mentioned in this article are only used to distinguish different entities or operations, and do not necessarily require or imply the existence of an actual relationship or order between these entities or operations. In addition, the terms "include", "comprise" and any other variations thereof are intended to express non-exclusive inclusion, that is, the process, method, article or terminal device may include other elements not explicitly listed while including the listed elements, or include elements inherent to the process, method, article or terminal device itself. In the absence of further qualification, the elements defined by "include..." or "comprise..." do not exclude the presence of other elements in the process containing these elements.

[0066] In addition, the terms "exceed", "greater than", "less than" and the like in this article should be understood as excluding the number itself; while the terms "above", "below", "within" and the like should be understood as including the number itself.

[0067] Although the above embodiments have been described in detail, those skilled in the art may make various changes and improvements to these embodiments after understanding the basic creative concept of the present invention. Therefore, the above contents are only some specific embodiments of the present invention and should not be used to limit the scope of patent protection of the present invention. Any modification of equivalent structures or equivalent processes based on the contents of this specification and the drawings, or any direct or indirect application of the principles of the present invention in other related technical fields, shall be covered by the scope of patent protection of the present invention.

Claims

1. A multi-party manifold learning method based on secret sharing, characterized in that: The steps include: Step 1: The data owner encrypts and shards their respective datasets using a secret sharing protocol, and uploads the secret shares of the sharded datasets to three computing servers respectively. Step 2: Each computing server calculates the secret share of the k-nearest neighbor distance matrix of all sample pairs based on the secret share of the dataset, using the Euclidean distance as the distance metric. Step 3: Each computing server uses the low-round Floyd algorithm on the secret shared k-nearest neighbor distance matrix to calculate the distance of the all-source shortest path, and the calculation results are stored in the secret shared shortest path distance matrix; Step 4: Each computing server performs secure multi-dimensional scaling on the secret-shared shortest path distance matrix to obtain a low-dimensional embedding of the data set. Any two computing servers send the low-dimensional embedded secret shares to the designated user and restore them to plaintext locally.

2. The multi-party manifold learning method based on secret sharing according to claim 1, characterized in that: In step 1, the number of data owners is unlimited, and the data owner datasets are secretly shared with each computing server using a three-party semi-honest replication secret sharing scheme.

3. The multi-party manifold learning method based on secret sharing according to claim 2, characterized in that: The three-party semi-honest replication secret sharing scheme is as follows: Will be split into three random parts , and is defined as a secret sharing form ,server Holding a secret share ,server Holding a secret share ,server Holding a secret share .

4. The multi-party manifold learning method based on secret sharing according to claim 1, characterized in that: The k-nearest-neighbor distance matrix calculation in step 2 utilizes matrix symmetry to reduce the number of secret sharing operations in the Euclidean distance calculation by half.

5. The multi-party manifold learning method based on secret sharing according to claim 1, characterized in that: The k-nearest-neighbor distance matrix in step 2 is calculated using the secure top-k algorithm to find the smallest distance matrix in each row. distance, that is, each sample corresponds to nearest neighbors, and set the distances corresponding to the remaining non-neighbors to .

6. The multi-party manifold learning method based on secret sharing according to claim 5, characterized in that: The secure top-k algorithm in step 2 is equal to the reference value or The probability of the algorithm stopping doubles after each partition.

7. The multi-party manifold learning method based on secret sharing according to claim 5, characterized in that: The secure top-k algorithm in step 2 appends the end of each element of the input vector in The bit value is used to make the elements of the input vector unique, and the input vector is randomly shuffled using a secure shuffling protocol. The comparison results in subsequent partitioning operations are independent of the input vector, so that data exchange can be conveniently completed locally on the server.

8. The multi-party manifold learning method based on secret sharing according to claim 1, characterized in that: The low-round Floyd algorithm in step 3 does not leak the number of edges. By expanding the two-layer loop in the algorithm, the operations are divided into three batches according to the data dependency relationship, and the communication in each batch is parallelized, which significantly reduces the number of communication rounds. is the input distance matrix, express The distance value of the position, let They are the pointers of the two inner loops, is a specific value of the outermost loop pointer, is the size of the distance matrix, and the three batches are defined as follows: 1) Batch 1: includes all The secret sharing operation at this time is , 2) Batch 2: including (1) all The secret sharing operation at this time, that is, for all ; (2) all The secret sharing operation at this time, that is, for all ,3) Batch 3: including all The secret sharing operation at this time, that is, for all .

9. The multi-party manifold learning method based on secret sharing according to claim 8, characterized in that: The low-round Floyd algorithm in step 3 reduces the number of secret sharing operations based on the characteristics of isometric mapping: 1) operations to update the diagonal elements of the matrix can be omitted, including operations on the diagonals in batches 1 and 3; 2) operations in batch 2 can be omitted; 3) operations in the lower triangle of the matrix in batch 3 can be omitted.

10. The multi-party manifold learning method based on secret sharing according to claim 1, characterized in that: The secure multi-dimensional scaling steps in step 4 are as follows: Multidimensional scaling first calculates the shortest path distance matrix Then apply double centering to convert the square matrix into an inner product matrix , The square distance matrix is ​​defined as ,in express The square of the distance between the locations, is the center matrix, defined as ,in is the identity matrix, the multidimensional scaling matrix Perform eigendecomposition and calculate dimensional embedding, where is the dimension after dimensionality reduction: in is The diagonal matrix consisting of the largest eigenvalues, is a matrix consisting of corresponding eigenvectors, It is calculated dimensional embedding.

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