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Journal of Cryptology

An Optimal Distributed Discrete Log Protocol with Applications to Homomorphic Secret Sharing

Itai Dinur, Nathan Keller, Ohad Klein

Published Sep 26, 2019• 5 citations
Publisher
DOI
10.1007/s00145-019-09330-2

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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.
Secret sharing (SS) has been extensively studied as for both secure data storage and a fundamental building block for multiparty computation (MPC). Recently, Kikuchi et al. proposed a passively and unconditionally secure conversion protocol that converts from a share of a ramp scheme to another of homomorphic SS scheme. The share-size of the ramp scheme is small, and the homomorphic SS scheme is a class of SS schemes that includes Shamir's and replicated SS schemes, which are convenient for MPC. Therefore, their protocol is a conversion from an SS scheme whose share-size is small to MPC-friendly SS schemes, and can be applied to reduce the amount of data storage while maintaining extendibility to MPC. We propose five unconditionally and actively secure protocols in the honest majority. In this paper, we consider a privacy and correctness as security requirement and does not consider a robustness: A cheat caused by an active adversary must be detected. These protocols consist of two conversion protocols, two reveal protocols and a protocol generating specific randomness. Main protocols among them are two conversion protocols for bilateral conversion between a ramp scheme and linear SS scheme, and the others are building blocks of the main protocols. Linear SS scheme is a subset of homomorphic SS scheme but includes both Shamir's and replicated SS schemes. Therefore, these main protocols are conversions between an SS scheme whose share-size is small to MPC-friendly SS schemes. These main protocols are unconditionally and actively secure so if MPC protocols used after the conversion are actively secure, the whole system involving SS scheme, conversion, and MPC protocols can be unconditionally and actively secure by using our main protocols. One of our two main protocols is the first to convert from MPC-friendly SS schemes to the ramp scheme. This enhances applications, such as secure backup, of the conversion protocol. Other than the two main protocols, we propose a protocol for generating specific randomnesses and two reveal protocols as building blocks. The latter two reveal protocols are actively and unconditionally secure in the honest majority and requires O(n||F||)-bit communication per revealing, and we believe that it is independently interest.