The Confidential Judgment Of The Relative Position Relationship Of Some Geometric Objects

Secure multi-party computing was first proposed by professor Yao in 1982 with the frequent occurrence of information interaction in recent years.It is an effective way to ensure the security of interactive information.Simply speaking,it means that multiple participants hold their own private information,exchange information with each other in the case of mutual distrust,and work together to calculate a function to obtain the final calculation result about themselves,and obtain the result in the process of information exchange.There are no personal information about other participants.Secure multi-party computing has become a research hotspot in cryptography,which is widely used in real life and production,and has certain theoretical research significance and practical value.In this thesis some computational geometry problems in secure multi-party computing are studied,that is,the positional relationship between security judgment points and intervals,the positional relationship between security judgment intervals and the positional relationship between security judgment space lines and planes.The specific work is as follows:With regard to the secret calculation of points spacing,this thesis analyzes the development background and current research status of this problem,summarizes the advantages and disadvantages of this research,points out its shortcomings,and make improvements on the basis of previous studies to keep the points spacing secret.The calculation problem is transformed into the confidentiality problem of judging the positive and negative of a polynomial.Based on the Paillier homomorphic encryption system,the corresponding protocol are designed,and the correctness,security and complexity of each protocol are compared and analyzed.Compared with the previous protocol,the protocol in this thesis is lower.The complexity of computation can be better applied to real life.For the secret calculation of interval position relation,the relationship between intervals is an extension of point and interval relation.This thesis analyzes the development background and research status of interval position relation.Based on scholars' research,the interval position relation in ordinary integer field is extended to rational number field.Combined with Paillier homomorphic encryption system,the secret calculation of interval position relation in rational number field is transformed into the problem of comparing the magnitude of linear function value.The relevant protocols are drawn up and analyzed to ensure the correctness and security of the protocol.Compared with the previous protocols,the proposed protocol has lower computational complexity.With regard to the confidential determination of line-plane positional relationship in space,this thesis analyzes the origin,development and research status of line-plane problem,summarizes and sorted out the research process of the positional relationship between lines and planes,summarizes and analyzes its shortcomings from previous research experience,improves and proposes a new coding method,and transforms the determination of line-plane positional relationship into the solution of matrix rank,without using complex modular exponentiation and any homomorphic encryption system,which greatly reduces the computational complexity.At the same time,the correctness and security of the protocol are analyzed.Compared with the existing literature,the protocol designed in this thesis has lower computational complexity.