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.
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.
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.
Quantum secret sharing (QSS) allows a dealer to distribute a secret quantum state among a set of parties in such a way that certain authorized subsets can reconstruct the secret, while unauthorized subsets obtain no information about it. Previous works on QSS for general access structures focused solely on the existence of perfectly secure schemes, and the share size of the known schemes is necessarily exponential even in cases where the access structure is computed by polynomial size monotone circuits. This stands in stark contrast to the classical setting, where polynomial-time computationally-secure secret sharing schemes have been long known for all access structures computed by polynomial-size monotone circuits under standard hardness assumptions, and one can even obtain shares which are much shorter than the secret (which is impossible with perfect security). While QSS was introduced over twenty years ago, previous works only considered information-theoretic privacy. In this work, we initiate the study of computationally-secure QSS and show that computational assumptions help significantly in building QSS schemes, just as in the classical case. We present a simple compiler and use it to obtain a large variety results: We construct polynomial-time computationally-secure QSS schemes under standard hardness assumptions for a rich class of access structures. This includes many access structures for which previous results in QSS necessarily required exponential share size. In fact, we can go even further: We construct QSS schemes for which the size of the quantum shares is significantly smaller than the size of the secret. As in the classical setting, this is impossible with perfect security. We also apply our compiler to obtain results beyond computational QSS. In the information-theoretic setting, we improve the share size of perfect QSS schemes for a large class of n-party access structures to 1.5^{n+o(n)}, improving upon best known schemes and matching the best known result for general access structures in the classical setting. Finally, among other things, we study the class of access structures which can be efficiently implemented when the quantum secret sharing scheme has access to a given number of copies of the secret, including all such functions in 𝖯 and NP.
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.
Quantum secret sharing (QSS) allows a dealer to distribute a secret quantum state among a set of parties so that certain subsets can reconstruct the secret, while unauthorized subsets obtain no information. While QSS was introduced over twenty years ago, previous works focused only on existence of perfectly secure schemes, and the share size of the known schemes is exponential even for access structures computed by polynomial size monotone circuits. This stands in contrast to the classical case, where efficient computationally-secure schemes have been long known for all access structures in monotoneP, and one can even obtain shares which are much shorter than the secret which is impossible with perfect security. In this work, we initiate the study of computationally-secure QSS and show that computational assumptions help significantly in building QSS schemes. We present a simple compiler and use it to obtain a large variety results: We construct polynomial-time QSS schemes under standard assumptions for a rich class of access structures. This includes many access structures for which previous results in QSS required exponential share size. We also construct QSS schemes for which the size of the shares is significantly smaller than the size of the secret. As in the classical case, this is impossible with perfect security. We also use our compiler to obtain results beyond computational QSS. In the information-theoretic setting, we improve the share size of perfect QSS schemes for a large class of access structures to 1.5n+o(n), improving upon best known schemes and matching the best known result for general access structures in the classical case. Finally, we show construct efficient schemes for all access structures in P and NP when the quantum secret sharing scheme is given multiple of copies of the secret.
Two types of generalizations in secure multi-party computation are described. First, we report on a new general adversary model: while previous protocols were designed to tolerate misbehavior of one of two types (active or passive adversaries) the new protocols tolerate mixed types of adversaries, i.e. an adversary that can corrupt players actively, passively, or can let them crash (i.e. stop execution). We derive the necessary and sufficient conditions under which such a general adversary can be tolerated. Second, we present a generic protocol framework that unifies previous approaches achieving unconditional (or information-theoretic) security and cryptographic security (i.e. security based on a cryptographic intractability assumption). This is achieved by introducing secret sharing schemes with an additional property called the multiplication property and a new type of homomorphic commitment scheme, and by generalizing the concept of commitments.
Introduction In this chapter we will show how the protocols we have seen so far can be generalized to be based on general linear secret-sharing schemes. Doing this generalization has several advantages: first, it allows us to design protocols that protect against general adversary structures, a concept we explain later, and second, it allows us to consider protocols in which the field that is used in defining the secret-sharing scheme can be of size independent of the number of players. This turns out to be an advantage for the applications of multiparty computation (MPC) that we consider later but cannot be done for the protocols that we have seen so far: to use Shamir's scheme for n players based on polynomials as we have seen, the field must be of size at least n +1. For now, we will only consider the theory for linear secret sharing that we need to construct the protocols in this chapter. However, we note already that to get full mileage from the more general applications of secret sharing and MPC, we need additional theory, in particular, about the asymptotic behavior of secret-sharing schemes. This material can be found in Chapter 11. General Adversary Structures The corruption tolerance of the protocols that we have seen so far has been characterized by only a single number t , the maximal number of corruptions that can be tolerated. This is also known as the threshold-t model . One way to motivate this might be to think of corruption of a player as something that requires the adversary to invest some amount of resource, such as time or money. If the adversary has only bounded resources, there should be a limit on the corruptions he or she can do. However, characterizing this limit by only an upper bound implicitly assumes that all players are equally hard to corrupt. In reality, this may be completely false: some players may have much better security than others, so a more realistic model might be that the adversary can corrupt a small number of well-protected players or a large number of poorly protected players. However, the threshold model does not allow us to express this.