| Secure multi-party computational geometry is a hot research topic in the field of cryptography in recent years.As an important problem in secure computational geometry,the point inclusion problem has attracted more and more attention.This article mainly studies the problem of determining the position relationship between points and polygons and the problem of determining the confidentiality of the position relationship between points and circles.The main work is as follows:For the problem of determining the position relationship between points and polygons for privacy protection,this paper proposes a new transformation scheme,which transforms the point inclusion problem into the problem of determining the surrounding direction of triangle boundaries,and achieves direction determination using the cross product of vectors.Based on this transformation method,a protocol for determining the position relationship between points and arbitrary polygons in a semi honest model was designed.Considering the different attacks of adversaries in the actual situation,this paper improves the location determination solution under the semi honest model by using cryptographic primitives such as zero-knowledge proof and segmentation selection scheme,and constructs a determination protocol for the location relationship between points and arbitrary polygons to protect privacy under the malicious model,which can not only prevent the deceptive behavior of malicious adversaries,but also extend the application scope of the protocol from convex polygon to arbitrary polygons.This article demonstrates the security of the protocol using simulated practical examples.Through theoretical analysis and comparison,the protocol proposed in this paper has linear complexity and is efficient.For the problem of determining the confidentiality of the position relationship between points and circles,this paper first designs two Euclidean distance calculation protocols under the semi honest model.Based on the Paillier cryptosystem scheme,the protocol implements negative encryption and decryption by limiting the range of security parameters.Considering that secure multi-party computing protocols based on encryption schemes consume a large amount of communication and computing resources,this scheme entrusts the modular exponentiation operation of random numbers in the encryption stage to cloud servers,and replaces the modular exponentiation operation in the ciphertext calculation stage with equivalent modular multiplication operations,reducing user computing costs.Then,based on this protocol,a privacy preserving protocol for determining the position relationship between points and circles was designed.Transform the point inclusion problem into a secret comparison problem of the size relationship between two numbers.Theoretical analysis has proven that the protocol designed in this article is correct,secure,and efficient. |
