Research On Homomorphic Encryption From The Learning With Errors Problem On Lattice

In the context of the big data era,the collection,storage,and analysis of information plays an important role in the development of various fields.While leveraging the enormous value of data,it also faces many security risks.Therefore,how to effectively protect data during the processing has become an important issue.In the field of cryptography,one of the methods to solve this problem is homomorphic encryption technology,which can process ciphertext data without exposing plaintext.Therefore,this technology can effectively protect information and has broad application prospects in data sharing and privacy protection.Since Gentry constructed the first-generation fully homomorphic encryption scheme on ideal lattices in 2009,the current design of fully homomorphic encryption algorithms is mainly based on the Learning With Errors(LWE)problem on lattices,which has gone through continuous development.There are many variants of cryptographic constructions based on this hardness problem.Therefore,research on the LWE problem has high theoretical significance for the construction and development of homomorphic encryption algorithms.In addition,the current basic fully homomorphic encryption schemes are mostly constructed under a single key or a single bit,which cannot meet the needs of multi-user participation and multi-bit encryption.Therefore,designing and improving homomorphic encryption algorithms that support multi-key or multi-bit processing is a key issue in this field.In response to the above content,this work is mainly carried out from the following three aspects: 1.Research on the LWE problem,proposing the concept of generalized LWE problem and designing a password-based identity authentication protocol based on the multi-instance LWE problem without the participation of third-party key service organizations.The design of this algorithm mainly applies the construction idea of the multi-instance LWE problem,providing a basic idea for the subsequent construction of multi-bit homomorphic encryption algorithms;2.Considering the structural characteristics of the NTRU-type cryptographic system based on the polynomial ring,this paper designs a homomorphic encryption scheme of NTRU type under the ring LWE problem.The existing homomorphic encryption schemes under the NTRU system require more complex modular and key exchange technologies,while the algorithm proposed in this paper mainly uses the bit expansion-recomposition technology to get rid of the dependence on the modular and key exchange process.At the same time,the construction of this algorithm supports multi-key processing,so it can meet the needs of multiple users participating in secret computation together;3.Considering the problem that the original GSW scheme does not support multi-bit encryption,based on the work of Li and Tang,this paper designs a homomorphic encryption scheme of GSW type under the LWE problem that supports multi-bit processing.This scheme simplifies the key structure and optimizes the key sampling process using the key expansion idea,aiming at the problem of large key size in the existing designs and improves efficiency.