Homomorphic secret sharing (HSS) is the secret sharing analogue of homomorphic encryption. An HSS scheme supports a local evaluation of functions on shares of one or more secret inputs, such that the resulting shares of the output are short. Some applications require the stronger notion of additive HSS, where the shares of the output add up to the output over some finite Abelian group. While some strong positive results for HSS are known under specific cryptographic assumptions, many natural questions remain open.
We initiate a systematic study of HSS, making the following contributions.
- A definitional framework. We present a general framework for defining HSS schemes that unifies and extends several previous notions from the literature, and cast known results within this framework.
- Limitations. We establish limitations on information-theoretic multi-input HSS with short output shares via a relation with communication complexity. We also show that additive HSS for non-trivial functions, even the AND of two input bits, implies non-interactive key exchange, and is therefore unlikely to be implied by public-key encryption or even oblivious transfer.
- Applications. We present two types of applications of HSS. First, we construct 2-round protocols for secure multiparty computation from a simple constant-size instance of HSS. As a corollary, we obtain 2-round protocols with attractive asymptotic efficiency features under the Decision Diffie Hellman (DDH) assumption. Second, we use HSS to obtain nearly optimal worst-case to average-case reductions in P. This in turn has applications to fine-grained average-case hardness and verifiable computation.
Homomorphic secret sharing (HSS) is the secret sharing analogue of homomorphic encryption. An HSS scheme supports a local evaluation of functions on shares of one or more secret inputs, such that the resulting shares of the output are short. Some applications require the stronger notion of additive HSS, where the shares of the output add up to the output over some finite Abelian group. While some strong positive results for HSS are known under specific cryptographic assumptions, many natural questions remain open.
We initiate a systematic study of HSS, making the following contributions.
A definitional framework. We present a general framework for defining HSS schemes that unifies and extends several previous notions from the literature, and cast known results within this framework.
Limitations. We establish limitations on information-theoretic multi-input HSS with short output shares via a relation with communication complexity. We also show that additive HSS for non-trivial functions, even the AND of two input bits, implies non-interactive key exchange, and is therefore unlikely to be implied by public-key encryption or even oblivious transfer.
Applications. We present two types of applications of HSS. First, we construct 2-round protocols for secure multiparty computation from a simple constant-size instance of HSS. As a corollary, we obtain 2-round protocols with attractive asymptotic efficiency features under the Decision Diffie Hellman (DDH) assumption. Second, we use HSS to obtain nearly optimal worst-case to average-case reductions in P. This in turn has applications to fine-grained average-case hardness and verifiable computation.
Homomorphic secret sharing (HSS) is the secret sharing analogue of homomorphic encryption. An HSS scheme supports a local evaluation of functions on shares of one or more secret inputs, such that the resulting shares of the output are short. Some applications require the stronger notion of additive HSS, where the shares of the output add up to the output over some finite Abelian group. While some strong positive results for HSS are known under specific cryptographic assumptions, many natural questions remain open. We initiate a systematic study of HSS, making the following contributions. - A definitional framework. We present a general framework for defining HSS schemes that unifies and extends several previous notions from the literature, and cast known results within this framework. - Limitations. We establish limitations on information-theoretic multi-input HSS with short output shares via a relation with communication complexity. We also show that additive HSS for non-trivial functions, even the AND of two input bits, implies non-interactive key exchange, and is therefore unlikely to be implied by public-key encryption or even oblivious transfer. - Applications. We present two types of applications of HSS. First, we construct 2-round protocols for secure multiparty computation from a simple constant-size instance of HSS. As a corollary, we obtain 2-round protocols with attractive asymptotic efficiency features under the Decision Diffie Hellman (DDH) assumption. Second, we use HSS to obtain nearly optimal worst-case to average-case reductions in P. This in turn has applications to fine-grained average-case hardness and verifiable computation.
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.
Abstract. Fully homomorphic encryption (FHE) enables secure computation over the encrypted data of a single party. We explore how to extend this to multiple parties, using threshold fully homomorphic encryption (TFHE). In such scheme, the parties jointly generate a common FHE public key along with a secret key that is shared among them; they can later cooperatively decrypt ciphertexts without learning anything but the plaintext. We show how to instantiate this approach efficiently, by extending the recent FHE schemes of Brakerski, Gentry and Vaikuntanathan (CRYPTO ’11, FOCS ’11, ITCS ’12) based on the (ring) learning with errors assumption. Our main tool is to exploit the property that such schemes are additively homomorphic over their keys. Using TFHE, we construct simple multiparty computation protocols secure against fully malicious attackers, tolerating any number of corruptions, and providing security in the universal composability framework. Our protocols have the following properties: Low interaction: 3 rounds of interaction given a common random string, or 2 rounds with a public-key infrastructure. Low communication: independent of the function being computed (proportional to just input and output sizes). Cloud-assisted computation: the bulk of the computation can be efficiently outsourced to an external entity (e.g. a cloud service) so that the computation of all other parties is independent of the complexity of the evaluated function. 1
Cryptography grew to be much more than "the study of secret writing". Modern cryptography is concerned with establishing properties such as privacy, integrity and authenticity in protocols for secure communication and computation. This comes at a price: Cryptographic tools usually introduce an overhead, both in terms of communication complexity (that is, number and size of messages transmitted) and computational efficiency (that is, time and memory required). As in many settings communication between the parties involved is the bottleneck, this thesis is concerned with improving communication complexity in cryptographic protocols. One direction towards this goal is scalable cryptography: In many cryptographic schemes currently deployed, the security degrades linearly with the number of instances (e.g. encrypted messages) in the system. As this number can be huge in contexts like cloud computing, the parameters of the scheme have to be chosen considerably larger - and in particular depending on the expected number of instances in the system - to maintain security guarantees. We advance the state-of-the-art regarding scalable cryptography by constructing schemes where the security guarantees are independent of the number of instances. This allows to choose smaller parameters, even when the expected number of instances is immense. - We construct the first scalable encryption scheme with security against active adversaries which has both compact public keys and ciphertexts. In particular, we significantly reduce the size of the public key to only about 3% of the key-size of the previously most efficient scalable encryption scheme. (Gay,Hofheinz, and Kohl, CRYPTO, 2017) - We present a scalable structure-preserving signature scheme which improves both in terms of public-key and signature size compared to the previously best construction to about 40% and 56% of the sizes, respectively. (Gay, Hofheinz, Kohl, and Pan, EUROCRYPT, 2018) Another important area of cryptography is secure multi-party computation, where the goal is to jointly evaluate some function while keeping each party’s input private. In traditional approaches towards secure multi-party computation either the communication complexity scales linearly in the size of the function, or the computational efficiency is poor. To overcome this issue, Boyle, Gilboa, and Ishai (CRYPTO, 2016) introduced the notion of homomorphic secret sharing. Here, inputs are shared between parties such that each party does not learn anything about the input, and such that the parties can locally evaluate functions on the shares. Homomorphic secret sharing implies secure computation where the communication complexity only depends on the size of the inputs, which is typically much smaller than the size of the function. A different approach towards efficient secure computation is to split the protocol into an input-independent preprocessing phase, where long correlated strings are generated, and a very efficient online phase. One example for a useful correlation are authenticated Beaver triples, which allow to perform efficient multiplications in the online phase such that privacy of the inputs is preserved and parties deviating the protocol can be detected. The currently most efficient protocols implementing the preprocessing phase require communication linear in the number of triples to be generated. This results typically in high communication costs, as the online phase requires at least one authenticated Beaver triple per multiplication. We advance the state-of-the art regarding efficient protocols for secure computation with low communication complexity as follows. - We construct the first homomorphic secret sharing scheme for computing arbitrary functions in NC 1 (that is, functions that are computably by circuits with logarithmic depth) which supports message spaces of arbitrary size, has only negligible correctness error, and does not require expensive multiplication on ciphertexts. (Boyle, Kohl, and Scholl, EUROCRYPT, 2019) - We introduce the notion of a pseudorandom correlation generator for general correlations. Pseudorandom correlation generators allow to locally extend short correlated seeds into long pseudorandom correlated strings. We show that pseudorandom correlation generators can replace the preprocessing phase in many protocols, leading to a preprocessing phase with sublinear communication complexity. We show connections to homomorphic secret sharing schemes and give the first instantiation of pseudorandom correlation generators for authenticated Beaver triples at reasonable computational efficiency. (Boyle, Couteau, Gilboa, Ishai, Kohl, and Scholl, CRYPTO, 2019)
Cramer, Damgard, and Nielsen [CDN01] show how to construct an efficient secure multiparty computation scheme using a threshold homomorphic encryption scheme that has four properties i) a honest-verifier zero-knowledge proof of knowledge of encrypted values, ii) proving multiplications correct iii) threshold decryption and iv) trusted shared key setup. Naor and Nissim [NN01a] show how to construct secure multi-party protocols for a function f whose communication is proportional to the communication required to evaluate f without security, albeit at the cost of computation that might be exponential in the description of f . Gentry [Gen09a] shows how to combine both ideas with fully homomorphic encryption in order to construct secure multi-party protocol that allows evaluation of a function f using communication that is independent of the circuit description of f and computation that is polynomial in | f |. This paper addresses the major drawback’s of Gentry’s approach: we eliminate the use of non-black box methods that are inherent in Naor and Nissim’s compiler. To do this we show how to modify the fully homomorphic encryption construction of van Dijk et al. [vDGHV10] to be threshold fully homomorphic encryption schemes. We directly construct (information theoretically) secure protocols for sharing the secret key for our threshold scheme (thereby removing the setup assumptions) and for jointly decrypting a bit. All of these constructions are constant round and we thoroughly analyze their complexity; they address requirements (iii) and (iv). The fact that the encryption scheme is fully homomorphic addresses requirement (ii). To address the need for an honest-verifier zero-knowledge proof of knowledge of encrypted values, we instead argue that a weaker solution suffices. We provide a 2-round blackbox protocol that allows us to prove knowledge of encrypted bits. Our protocol is not zeroknowledge, but it provably does not release any information about the bit being discussed, and this is sufficient to prove the correctness of a simulation in a method similar to Cramer et al. Altogether, we construct the first black-box secure multi-party computation protocol that allows evaluation of a function f using communication that is independent of the circuit description of f .
We construct homomorphic secret sharing (HSS) and pseudorandom correlation functions (PCFs) in a general group-theoretic framework, with security based on the subgroup indistinguishability (SgI) assumption introduced by Brakerski and Goldwasser (Crypto 2010). Under certain instantiations of this framework, SgI corresponds to decisional composite residuosity (DCR), but other instantiations are possible as well. Hence, our work expands the set of assumptions that imply HSS and PCFs. Our constructions crucially rely on the existence of an efficient distributed discrete logarithm (DDLog) algorithm for the utilized group. We construct a new DDLog algorithm that works in general groups as long as the order t of the base is smooth. In particular, our DDLog supports the prime-power case, which was previously an open problem. Moreover, we apply a divide-and conquer optimization that significantly improves the performance from O(log2 t/loglogt) group operations (Abram et al., Crypto 2022) to O(logtloglogt). Toward constructing PCFs, we introduce a new notion of public-coin subgroup indistinguishability (PC-SgI). We show that, in certain instantiations, SgI implies PC-SgI, so no new assumption is required. We then obtain PCFs for VOLE, OT, OLE, and degree-two correlations, all generically under PC-SgI. Our PCFs for OLE and degree-two correlations additionally relyon the sparse LPN assumption. Along the way, we discover and patch two security issues in prior work: first, an attack on decisional Diffie–Hellman (DDH) in the Joye–Libert instantiation by Abram et al. (Crypto 2022), and second, a flaw in the security proofs of optimized variants of the IKNP OT extension protocol and of many silent OT extension protocols.