In the paper, through a complete description of secret sharing schemes, we give definitions respectively for perfect secret sharing schemes, statistical secret sharing schemes and computational secret sharing schemes in an uniform way. Furthermore, we analyze why and how multiplicative linear secret sharing schemes are used in secure multi-party computation protocols, and explain the security of secure multi-party computation in detail.
In the paper, through a complete description of secret sharing schemes, we give definitions respectively for perfect secret sharing schemes, statistical secret sharing schemes and computational secret sharing schemes in an uniform way. Furthermore, we analyze why and how multiplicative linear secret sharing schemes are used in secure multi-party computation protocols, and explain the security of secure multi-party computation in detail.
Abstract. As an extension of multi-party computation (MPC), we propose the concept of secure parallel multi-party computation which is to securely compute multi-functions against an adversary with multistructures. Precisely, there are m functions f1,..., fm and m adversary structures A1,..., Am, where fi is required to be securely computed against an Ai-adversary. We give a general construction to build a parallel multi-party computation protocol from any linear multi-secret sharing scheme (LMSSS), provided that the access structures of the LMSSS allow MPC at all. When computing complicated functions, our protocol has more advantage in communication complexity than the “direct sum” method which actually executes a MPC protocol for each function. The paper also provides an efficient and generic construction to obtain from any LMSSS a multiplicative LMSSS for the same multi-access structure. 1
The contributions of this paper are three-fold. First, as an abstraction of previously proposed cryptographic protocols we propose two cryptographic primitives: homomorphic shared commitments and linear secret sharing schemes with an additional multiplication property. We describe new constructions for general secure multi-party computation protocols, both in the cryptographic and the information-theoretic (or secure channels) setting, based on any realizations of these primitives. Second, span programs, a model of computation introduced by Karchmer and Wigderson, are used as the basis for constructing new linear secret sharing schemes, from which the two above-mentioned primitives as well as a novel verifiable secret sharing scheme can efficiently be realized. Third, note that linear secret sharing schemes can have arbitrary (as opposed to threshold) access structures. If used in our construction, this yields multi-party protocols secure against general sets of active adversaries, as long as in the cryptographic (information-theoretic) model no two (no three) of these potentially misbehaving player sets cover the full player set. This is a strict generalization of the threshold-type adversaries and results previously considered in the literature. While this result is new for the cryptographic model, the result for the information-theoretic model was previously proved by Hirt and Maurer. However, in addition to providing an independent proof, our protocols are not recursive and have the potential of being more efficient.
While classically cryptography is concerned with the problem of private communication among two entities, say players, in modern cryptography multi-player protocols play an important role.And among these, it is probably fair to say that secret sharing, and its stronger version verifiable secret sharing (VSS), as well as multi-party computation (MPC) belong to the most appealing and/or useful ones.The former two are basic tools to achieve better robustness of cryptographic schemes against malfunction or misuse by "decentralizing" the security from one single to a whole group of individuals (captured by the term threshold cryptography).The latter allows-at least in principle-to execute any collaboration among a group of players in a secure way that guarantees the correctness of the outcome but simultaneously respects the privacy of the participants.In this work, we study three aspects of secret sharing, VSS and MPC, which we denote by fundamentals, generality, and efficiency.By fundamentals we mean the quest for understanding why a protocol works and is secure in abstract (and hopefully simple) mathematical terms.By generality we mean generality with respect to the underlying mathematical structure, in other words, minimizing the mathematical axioms required to do some task.And efficiency of course deals with the improvement of protocols with respect to some meaningful complexity measure.We briefly summarize our main results.(1) We give a complete characterization of blackbox secret sharing in terms of simple algebraic conditions on the integer sharing coefficients, and we propose a black-box secret sharing scheme with minimal expansion factor.Note that, in contrast to the classical field-based secret sharing schemes, a black-box secret sharing scheme allows to share a secret sampled from an arbitrary Abelian group and requires only black-box access to the group operations and to random group elements.Such a scheme may be very useful in the construction of threshold cryptosystems based on groups with secret order (like RSA).(2) We show that without loss of efficiency, MPC can be based on arbitrary finite rings.This is in sharp contrast to the literature where essentially all MPC protocols require a much stronger mathematical structure, namely a field.Apart from its theoretical value, this can lead to efficiency improvements since it allows a greater freedom in the (mathematical) representation of the task that needs to be securely executed.(3) We propose a unified treatment of perfectly secure linear VSS and distributed commitments (a weaker version of the former), and we show that the security of such a scheme can be reduced to a linear algebra condition.The security of all known schemes follows as corollaries whose proofs are pure linear algebra arguments, in contrast v to some hybrid arguments used in the literature.(4) We construct a new unconditionally secure VSS scheme with minimized reconstruction complexity in the setting of a dishonest minority.This improves on the reconstruction complexity of the previously best known scheme without affecting the (asymptotic) complexity of the sharing phase.In the context of MPC with pre-processing, our scheme allows to improve the computation phase of earlier protocols without increasing the complexity of the pre-processing.(5) Finally, we consider commitment multiplication proofs, which allow to prove a multiplicative relation among three commitments, and which play a crucial role in computationally secure MPC.We study a non-interactive solution which works in a distributed-verifier setting and essentially consists of a few executions of Pedersen's VSS.We simplify and improve the protocol, and we point out a (previously overlooked) property which helps to construct non-interactive proofs of partial knowledge in this setting.This allows for instance to prove the knowledge of ℓ out of m given secrets, without revealing which ones.We also construct a noninteractive distributed-verifier proof of circuit satisfiability, which-in principle-allows to prove anything that can be proven without giving away the proof.
Nowadays, computation becomes more advanced, and may require more than one parties to participate.Some examples include e-voting system and privacy-preserving data analytics.One of the main concerns here is privacy.Inputs of these systems must be kept secret.To address this concern, we can construct the systems by using secure multi-party protocols, where several parties with their secret inputs perform joint computations in oblivious ways.Many aspects of the protocols can be further improved towards the usage in reallife applications, considering operational requirements and trade-offs.These operational requirements include efficiency, functionality, security, scalability, and verifiability.This thesis consists of three parts; each part focuses on specific improvements.In the first part, we focus on improving security and efficiency.We consider a homomorphic secret sharing scheme, where a secret input from each party is divided into several shares, each share is sent to different computing server, and the servers can perform some computation locally to get shares of the result.It is obvious that homomorphic secret sharing schemes can be used as secure multi-party protocols with small communication.We found that the existing homomorphic secret sharing scheme with public key of Lai et al. (Asiacrypt 2018) is 1-secure, which means if more than one computing servers try to collude, the security of the system will be compromised.They also proposed a t-secure scheme, but it requires solving NP-hard problem.To improve their security level, we use the idea from the secret sharing scheme of Ito et al.Our scheme also supports any number of computing servers without solving NP-hard problem.Using hyper-graph representation of non-access structures, we prove the maximum polynomial degree that our scheme can support.In addition, we prove correctness and security of the scheme, and propose some possible further improvements.In the second part, we focus on improving scalability and efficiency.Most of the secret sharing schemes, including homomorphic secret sharing, require the number of parties to be fixed in advance.This causes an inconvenience if we want to add more parties later in order to change the security setting or improve availability of the system.One existing variant of secret sharing, called evolving secret sharing, allows any number of parties to join the system at any time.However, it is inefficient to perform computation on this kind of secret sharing.To address this problem, we propose the definition of evolving homomorphic secret sharing, which combines evolving and homomorphic properties together.The scheme iii then supports unbounded number of parties, and allows local computation on shares.In addition to the definition, we propose two concrete schemes.The first scheme realizes evolving disjunctive hierarchical access structure supporting multiplication.The second scheme realizes evolving partially disjunctive hierarchical access structure supporting lowdegree polynomial computation.In the third part, we focus on improving functionality, verifiability, and efficiency.We consider the problem of oblivious deterministic finite automata (DFA) evaluation protocol.In this problem, one party holds a secret string, and the other party holds a secret pattern.The goal of this protocol is to output whether the string matches with the pattern of not without revealing both secret inputs.Most of the existing protocols use several communication rounds and require public key primitives, which can be inefficient.As our contribution, we propose an oblivious conditional disclosure of secrets (CDS) scheme for DFA.It is the first CDS scheme that has a functionality of DFA evaluation.We then use the proposed oblivious CDS scheme as a building block to construct an oblivious DFA evaluation protocol, using a multi-client verifiable computation framework.As a result, our protocol is constant-round, verifiable, and does not require any public key primitives.We also propose a standard CDS scheme for DFA as an independent interest.This thesis is expected to be a step of secure multi-party protocols and secret sharing schemes enhancements towards real-life applications.High level ideas for more improvements are also suggested as future prospects.
In this paper, the problem of secure two-party multiplication of shared secrets is considered, and it is one of problems of secure multi-party computation, where the policy for the shared secrets is based on polynomial interpolation. Firstly, a secure two-party protocol for polynomial multiplication is presented. Secondly, based on the first protocol, a protocol for secure two-party multiplication of shared secrets for semi-honest model is proposed, which is employed to multiply two shared secrets without revealing the private information of two parties. Furthermore, based on the definition of secure two-party computation and Composition Theorem, the proofs of security and analysis of the correctness and the complexity of our two protocols are given.
Secure multi-party computation (MPC) is a cryptographic primitive that lets a num- ber N of mutually distrustful parties compute a function f (x1, . . . , xN ) on their private inputs xi such that at the end of the MPC protocol, each honest party obtains the func- tion’s correct output and no adversary controlling a certain subset of parties learns any- thing about the honest parties’ inputs beyond what can be inferred from the function’s output value. In 2002, [Smi01] and [CGS02] introduced the notion of multi-party quan- tum computation (MPQC) in which arbitrary quantum circuits can be evaluated in a distributed manner, secure against a quantum adversary. Few protocols for MPQC are known. We present a fundamental study of the building blocks of MPC and MPQC in the information-theoretic setting with a focus on protocols built on top of (quantum) se- cret sharing schemes. In particular, we point out structural similarities and differences and compile an extensive list of theoretical feasibility results for the maximum number of corrupted parties within classical and quantum secret sharing and MPC protocols.