Threshold Changeable Secret Sharing Schemes Without The Trusted Center

The extensive applications of information technology not only greatly accelerate the development of the economy and society,but also promote the emergence of cyberspace.However,the cyberspace security is becoming severe.The country’s politics,economy,culture,society,national defense security and citizens’ legitimate rights in cyberspace are facing severe risks and challenges.Cryptography provides key technologies for the security of the cyberspace,and secret sharing,as one of the most important cryptographic primitives,is widely used in threshold cryptography,secure multi-party computation,cloud computing,and blockchain.The(t,r,n)-ramp scheme is a secret sharing scheme with two thresholds,where the privacy threshold and the reconstruction threshold are the indicators of the security levels of the secret.During the life of a secret,it often faces unpredictable changes,and requires to adjust thresholds to ensure the security levels of the secret.In this thesis,we focus on the scenario with no trusted center and no interactions among participants during the change of thresholds.Based on the knowledge of finite fields,algebraic coding and polynomials,we construct three Threshold Changeable Secret Sharing(TCSS)schemes with good performances.These three TCSS schemes have two different threshold change regimes that are designed to meet the needs of different scenarios.We further investigate the full threshold change range of optimal TCSS schemes.More precisely,we make the following four contributions.(1)Reed-Solomon Code-based Communication Efficient Secret Sharing SchemesThe communication efficient secret sharing scheme is a TCSS scheme in which the initial scheme and the resulting scheme have the same privacy threshold.It aims to reconstruct the secret with the smallest decoding bandwidth.Previous communication efficient secret sharing schemes divide each secret into a vector with multiple components,which lead to the decoding delay and big share size.In this work,we use the good properties of Reed-Solomon codes to express each secret as an element of the extended field of the base field,and construct a communication efficient secret sharing scheme without decoding delay for the first time.Its share size is smaller than all previous communication efficient secret sharing schemes.Furthermore,for a given base field,and assume that each participant provides the same amount of information to reconstruct the secret,we determine the tight lower bound on the share size of communication efficient secret sharing schemes.It is found that our new scheme’s share size reaches the lower bound.(2)Bivariate Polynomial-based Secret Sharing with Secure Secret ReconstructionThe Secret Sharing with Secure Secret Reconstruction(SSR)is a TCSS scheme in which both the initial scheme and the resulting scheme are perfect.It can be used to resist the attack from a kind of outside adversaries defined by Tompa and Woll.More precisely,the outside adversary is not a participant of the scheme,but the adversary can pretend to be a participant of the scheme,and take part in the reconstruction phase of the secret.As a result,the adversary can derive the information released by other participants and try to reveal the secret.It is found that all previous bivariate polynomial-based SSR schemes are insecure.Therefore,we present a theoretical model for constructing secure SSR schemes.Based on this model,we construct a symmetric bivariate polynomial-based SSR scheme and an asymmetric bivariate polynomial-based SSR scheme,respectively.The schemes here are easier to be constructed compared to the SSR scheme based on the Chinese Reminder Theorem(CRT),and their share sizes are almost the same.(3)CRT for Polynomial Ring-based Secret Sharing with Secure Secret ReconstructionAsmuth-Bloom scheme is a secret sharing scheme based on CRT for integer ring.Compared to Shamir’s scheme,Asmuth-Bloom scheme has a smaller information rate,but it has a smaller computational complexity,which make it to be widely used to construct other cryptographic algorithms.How to improve the information rates of CRT-based secret sharing schemes is always a hot topic.In this work,a CRT for polynomial ring-based ramp scheme is constructed for the first time.Its information rate is greater than the scheme of Blakley and Meadows based on CRT for integer ring,and achieves the maximum information rate of ramp schemes.By using multiple ramp schemes of this type,we construct an SSR scheme based on CRT for polynomial ring,where its information rate is greater than that of the bivariate polynomial-based SSR schemes and the CRT for integer ring-based SSR scheme.(4)Optimal Threshold Changeable Secret Sharing Schemes with New Threshold Change RegimesOptimal TCSS schemes are a kind of TCSS schemes in which both the initial scheme and the resulting scheme have the maximum information rate.They have good performances,and have attracted much attention.For example,all the known communication efficient secret sharing schemes with the maximum information rate are optimal.Previous works propose two methods for constructing optimal TCSS schemes.One method is to take multiple shares of multiple secret sharing schemes as a share of the optimal TCSS scheme,which realizes the increase of the reconstruction threshold,while the privacy threshold remains the same.The other method is to take multiple shares of a given secret sharing scheme as a share of the optimal TCSS scheme,where the privacy threshold and reconstruction threshold can change proportionally.In this work,we propose a new method and obtain an optimal TCSS scheme with a new threshold change regime.We further determine the full threshold change range of optimal TCSS schemes,which are covered by our new threshold change regime and other known threshold change regimes.This also shows why none of the known SSR schemes are optimal.