Research On Lattice-based Proxy Re-encryption Schemes

With the improvement of computing power,especially the development of quantum computing,traditional cryptographic schemes suffer more and more challenges.In response to these potential security threats,people are eager to find some more secure(such as anti-quantum attacks)cryptographic primitives.Lattice cryptography has gained more and more attention for its beneficial features,such as efficient,provably security and supporting homomorphic encryption.And how to obtain lattice cryptography schemes that satisfied equivalent security requirements as the traditional password scheme is an important research direction,such as lattice-based public key encryption schemes(PKE)and lattice-based proxy re-encryption(PRE)schemes,etc.Based on the research of lattice cryptography and hard problems,on the condition of constructing lattice-based public key encryption scheme,on the purpose of constructing latticebased proxy re-encryption scheme,a series of provable security lattice cryptographic have been obtained.The main results are as follows:1.To construct lattice-based cryptography schemes that are security enough,a series of hard problems have been analysed and studied.As we know,the hard problems are divided into the worst-case and average-case,where worst-case problems mainly contain the shortest vector problems(SVP),the closest vector problems(CVP)and their various,such as the shortest independent vector problems(SIVP),and average-case problems mainly contain the small integer solutions(SIS)and learning with errors(LWE)problems.Through the complexity of the worst-case hard problems,a complexity table of the SVP,CVP,and SIVP based on the size of the parameters have been constructed,under different parameters and norms.Through the reduction between hard problems,a reduction tables of SIS and LWE have been constructed,according to the size of the parameters.Finally,a reduction relationship diagram between these hard problems has been constructed according to these reductions.2.By the homomorphic encryption operations of lattice-based cryptography,a single-layer homomorphic cloud computing scheme s LHCC that supports addition and multiplication homomorphic has been constructed successfully based on the learning with errors(LWE).The scheme can perform remote operations and controls without revealing operational requirements(ciphertexts).Compared with other lattice-based homomorphic encryption schemes,the public key size of the scheme is reduced from O(n2log q)to O(n log q),and this reduces the ciphertext sizes and cloud storage requirements.3.Compared with traditional cryptographic schemes,lattice-based cryptographic schemes are inefficient while encrypt multi-bit message.This paper proposes a series of multibit PKE schemes to improve the efficiency through orthogonal lattice.Without using the traditional multi-bit improvement method,the schemes can encrypt multi-bit(vector)once time.Compared with other PKE schemes,they are more efficient while encrypt multi-bit messages.Besides,the efficiency of them can be further increased through some improvements when the messages are single-bit.4.Proxy re-encryption(PRE)enable users to authorize legitimate users to decrypt legitimate ciphertexts that they can decrypt without revealing their private keys.However,there are few lattice-based proxy re-encryption schemes so far.We have analysed the feasibility of constructing a proxy re-encryption scheme through SIS and LWE.The analysis results show that it is not feasible to construct a proxy re-encryption scheme through a single problem directly.5.Collusion-resistant proxy re-encryption scheme has important research value cause the proxy is usually semi-trust in practice.By studying the existing collusion-resistant PRE scheme from lattice,we finds that the two existing collusion-resistant PRE schemes are structurally incorrect.We seeks out and points out the common causes of mistakes in the program firstly.And we also construct a novel collusion-resistant unidirectional PRE scheme from lattice.The new scheme fixs the mistakes in previous schemes and maintains useful properties in both schemes,such as collusion-resistant,chosen plaintext attack(CPA)secure and unidirectional.In terms of correctness of the scheme,this new scheme is the first collusion-resistant unidirectional proxy re-encryption scheme from lattice.