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IEEE Information Theory Workshop (ITW)

Hybrid Multiplicative Secret Sharing

Maki Yoshida

Published Oct 17, 2021• 1 citation
Publisher

Abstract

A secret-sharing scheme is d-multiplicative if it allows the players to multiply d (rather than two) shared secrets (without recovering them) by locally converting their shares into an additive sharing of the product. In this work, the d-multiplicative secret-sharing (MSS) is extended to a hybrid MSS (HMSS), which is mainly designed for sharing d secrets against different access structures. A necessary and sufficient condition for n-player d-HMSS schemes to exist is presented. The condition is necessary for arbitrary (possibly inefficient or even nonlinear) secret-sharing schemes.

DOI
10.1109/itw48936.2021.9611361

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•Aug 1, 2013•0 citations
A secret-sharing scheme is d-multiplicative if it allows the players to multiply d (rather than two) shared secrets (without recovering them) by locally converting their shares into an additive sharing of the product. In this work, the d-multiplicative secret-sharing (MSS) is extended to a hybrid MSS (HMSS), which is mainly designed for sharing d secrets against different access structures. A necessary and sufficient condition for n-player d-HMSS schemes to exist is presented. The condition is necessary for arbitrary (possibly inefficient or even nonlinear) secret-sharing schemes.
A secret sharing scheme is said to be d-multiplicative if the scheme allows the players to multiply shared d secrets by locally converting their shares into an additive sharing of the product. This paper considers the most general setting where d secrets are independently shared by different adversary structures (hybrid) and a partial information can be leaked (non-perfect) and presents a necessary and sufficient condition for d-multiplication to be possible. In particular, even for the extreme case that the i-th secret is known by all but the i-th player who has partial information on the secret (e.g., all but one bit), d-multiplication is impossible.
A secret sharing scheme is said to be d-multiplicative if the scheme allows the players to multiply shared d secrets by locally converting their shares into an additive sharing of the product. In the previous work, the following negative result for perfect secret sharing has been shown: The d-multiplicative secret sharing for d players is impossible. This paper extends the impossibility result to non-perfect secret sharing. Our main result is a proof that d-multiplicative secret sharing for d players is impossible even if every player has partial information on the secret (e.g., all but one bit). This result means that there is no need to relax the privacy requirement with leakage of partial information only for the purpose of d-multiplication.
Secret sharing schemes are said to be d-multiplicative if the i-th shares of any d secrets s^(j), j∈[d] can be converted into an additive share of the product ∏_{j∈[d]}s^(j). d-Multiplicative secret sharing is a central building block of multiparty computation protocols with minimum number of rounds which are unconditionally secure against possibly non-threshold adversaries. It is known that d-multiplicative secret sharing is possible if and only if no d forbidden subsets covers the set of all the n players or, equivalently, it is private with respect to an adversary structure of type Q_d. However, the only known method to achieve d-multiplicativity for any adversary structure of type Q_d is based on CNF secret sharing schemes, which are not efficient in general in that the information ratios are exponential in n. In this paper, we explicitly construct a d-multiplicative secret sharing scheme for any 𝓁-partite adversary structure of type Q_d whose information ratio is O(n^{𝓁+1}). Our schemes are applicable to the class of all the 𝓁-partite adversary structures, which is much wider than that of the threshold ones. Furthermore, our schemes achieve information ratios which are polynomial in n if 𝓁 is constant and hence are more efficient than CNF schemes. In addition, based on the standard embedding of 𝓁-partite adversary structures into ℝ^𝓁, we introduce a class of 𝓁-partite adversary structures of type Q_d with good geometric properties and show that there exist more efficient d-multiplicative secret sharing schemes for adversary structures in that family than the above general construction. The family of adversary structures is a natural generalization of that of the threshold ones and includes some adversary structures which arise in real-world scenarios.
Multiplicative linear secret sharing is a fundamental notion in the area of secure multiparty computation and, since recently, in the area of two-party cryptography as well. In a nutshell, this notion guarantees that the product of two secrets is obtained as a linear function of the vector consisting of the coordinatewise product of two respective share-vectors. This paper focuses on the following foundational question, which is novel to the best of our knowledge. Suppose we abandon the latter linearity condition and instead require that this product is obtained by some, not-necessarily-linear “product reconstruction function.” Is the resulting notion equivalent to multiplicative linear secret sharing? We show the (perhaps somewhat counterintuitive) result that this relaxed notion is strictly more general. Concretely, fix a finite field Fq{\mathbb F}_qFq​ as the base field over which linear secret sharing is considered. Then we show there exists an (exotic) linear secret sharing scheme with an unbounded number of players nnn such that it has ttt-privacy with t=Ω(n)t = \Omega(n)t=Ω(n) and such that it does admit a product reconstruction function, yet this function is necessarily nonlinear. In addition, we determine the minimum number of players for which those exotic schemes exist. Our proof is based on combinatorial arguments involving quadratic forms. It extends to similar separation results for important variations, such as strongly multiplicative secret sharing.
The characterization of access structure is one of the main open problems in secret sharing schemes. We are particularly interested in determining which Q 3 access structures can be realized and the requirement should be achieved even in the presence of an adversary who is able to corrupt some players in an ideal secret sharing scheme. The problems are studied in our paper. Above all, we present a concrete scheme that can meet all above requirements. In our scheme, it is not only a dynamic multi-secret and multi-use sharing protocol, but participants from access structure can recover the secret using their shares, while the same number of participants from adversary structure can't recover the secret using their shares, which is different from general access structure schemes. Moreover, in the whole dynamic process, the coefficients of polynomial and private secret shares always stay the same. Therefore, it enjoys high practical value.
Multiplicative linear secret sharing is a fundamental notion in the area of secure multiparty computation (MPC) and, since recently, in the area of two-party cryptography as well. In a nutshell, this notion guarantees that “the product of two secrets is obtained as a linear function of the vector consisting of the coordinate-wise product of two respective share-vectors”. This paper focuses on the following foundational question, which is novel to the best of our knowledge. Suppose we abandon the latter linearity condition and instead require that this product is obtained by some, not-necessarily-linear “product reconstruction function”. Is the resulting notion equivalent to multiplicative linear secret sharing? We show the (perhaps somewhat counter-intuitive) result that this relaxed notion is strictly more general. Concretely, fix a finite field Fq as the base field Fq over which linear secret sharing is considered. Then we show there exists an (exotic) linear secret sharing scheme with an unbounded number of players n such that it has t-privacy with t ≈ √ n and such that it does admit a product reconstruction function, yet this function is necessarily nonlinear. Our proof is based on combinatorial arguments involving bilinear forms. It extends to similar separation results for important variations, such as strongly multiplicative secret sharing.