Secret sharing is a cryptographic discipline in which the goal is to distribute information about a secret over a set of participants in such a way that only specific authorized combinations of participants together can reconstruct the secret. Thus, secret sharing schemes are systems of variables in which it is very clearly specified which subsets have information about the secret. As such, they provide perfect model systems for information decompositions. However, following this intuition too far leads to an information decomposition with negative partial information terms, which are difficult to interpret. One possible explanation is that the partial information lattice proposed by Williams and Beer is incomplete and has to be extended to incorporate terms corresponding to higher-order redundancy. These results put bounds on information decompositions that follow the partial information framework, and they hint at where the partial information lattice needs to be improved.
Secret sharing is a cryptographic discipline in which the goal is to distribute information about a secret over a set of participants in such a way that only specific authorized combinations of participants together can reconstruct the secret. Thus, secret sharing schemes are systems of variables in which it is very clearly specified which subsets have information about the secret. As such, they provide perfect model systems for information decompositions. However, following this intuition too far leads to an information decomposition with negative partial information terms, which are difficult to interpret. One possible explanation is that the partial information lattice proposed by Williams and Beer is incomplete and has to be extended to incorporate terms corresponding to higher-order redundancy. These results put bounds on information decompositions that follow the partial information framework, and they hint at where the partial information lattice needs to be improved.
A secret sharing scheme typically requires secure communications in each of two distribution phases: (1) a dealer distributes shares to participants (share distribution phase); and later (2) the participants in some authorised subset send their share information to a combiner (secret reconstruction phase). While problems on storage required for participants, for example, the size of shares, have been well studied, problems regarding the communication complexity of the two distribution phases seem to have been mostly neglected in the literature so far. In this correspondence, we deal with several communication related problems in the secret reconstruction phase. Firstly, we show that there is a tradeoff between the communication costs and the number of participants involved in the secret reconstruction. We introduce the communication rate as the ratio of the secret size and the total number of communication bits transmitted from the participants to the combiner in the secret reconstruction phase. We derive a lower bound on the communication rate and give constructions that meet the bound. Secondly, we show that the point-to-point secure communication channels for participants to send share information to the combiner can be replaced with partial broadcast channels. We formulate partial broadcast channels as set systems and show that they are equivalent to the well-known combinatorial objects of cover-free family. Surprisingly, we find that the number of partial broadcast channels can be significantly reduced from the number of point-to-point secure channels. Precisely, in its optimal form, the number of channels can be reduced from n to O(log n), where is the number of participants in a secret sharing scheme. We also study the communication rates of partial broadcast channels for the secret reconstruction.
The following standard relaxations of perfect security for secret sharing schemes (SSSs) exist in the literature:quasi-perfect,almost-perfect, andstatistical. Understanding the power of these relaxations on the efficiency of SSSs, measured via a parameter calledinformation ratio, is a long-standing open problem. In this article, we introduce and study an extremely relaxed security notion, calledpartial security, for which it is only required that any qualified set gains strictly more information about the secret than any unqualified one. To get a meaningful efficiency measure, we normalize the (standard) information ratio of such schemes by an appropriate parameter and refer to the new measure aspartial information ratio. We present three main results in this paper.First, we prove that partial and perfect information ratios coincide for the class of linear SSSs.Second, we prove that for the general (i.e., non-linear) class of SSSs, partial and statistical information ratios are equal.Third, we show that partial and almost-perfect information ratios do not coincide for the class of mixed-linear schemes (i.e., schemes constructed by combining linear schemes with different underlying finite fields). We also use the notion of partial secret sharing to strengthen and unify the previousdecompositiontheorems for constructing SSSs.
Abstract. We construct a perfect extrinsic secret sharing scheme for any case in which a set of participants can gain access to the secret if and only if the set contains a pair of members from some given list of pairs. A secret sharing scheme is a way by which a dealer rnay distribute secret information to individuals (call participants). There is associated '-<U.lh'.lcH"""F- of subsets of a set of can if they constitute a member of the access structure. usual to demand that the access structure be monotone: If A contains a subset which to the structure, then A itself must also belong to the structure. A secret scheme is called perfect if a set of pa.rticipants can no information by their shares unless the set is member of the access structure. An structure is called extrinsic if each participant '8 share contains the same amount of information. Simmons [4] has pointed out the desirability of a scheme which is and the access structure of which extrinsic. Observe
/spl acute/A secret sharing scheme permits a secret to be shared among participants in such a way that only qualified subsets of participants can recover the secret, but any nonqualified subset has absolutely no information on the secret. We derive new limitations on the information rate of secret sharing schemes, that measures how much information is being distributed as shares as compared to the size of the secret key, and the average information rate, that is the ratio between the secret size and the arithmetic mean of the size of the shares. By applying the substitution technique, we are able to construct many new examples of access structures where the information rate is bounded away from 1. The substitution technique is a method used to obtain a new access structure by replacing a participant in a previous structure with a new access structure.>
Secret sharing is a cryptographic scheme to encode a secret to multiple shares being distributed to participants, so that only qualified sets of participants can restore the original secret from their shares. When we encode a secret by a secret sharing scheme and distribute shares, sometimes not all participants are accessible, and it is desirable to distribute shares to those participants before a secret information is determined. Secret sharing schemes for classical secrets have been known to be able to distribute some shares before a given secret. Lie et al. found a ((2, 3))-threshold secret sharing for quantum secrets can distribute some shares before a given secret. However, it is unknown whether distributing some shares before a given secret is possible with other access structures of secret sharing for quantum secrets. We propose a quantum secret sharing scheme for quantum secrets that can distribute some shares before a given secret with other access structures.
Secret sharing is a mechanism by which a trusted dealer holding a secret “splits ” the secret into many “shares ” and distributes the shares to a collection of parties. Associated with the sharing is a monotone access structure, that specifies which parties are “qualified ” and which are not: any qualified subset of parties can (efficiently) reconstruct the secret, but no unqualified subset can learn anything about the secret. In the most general form of secret sharing, the access structure can be any monotone NP language. In this work, we consider two very natural extensions of secret sharing. In the first, which we call distributed secret sharing, there is no trusted dealer at all, and instead the role of the dealer is distributed amongst the parties themselves. Distributed secret sharing can be thought of as combining the features of multiparty non-interactive key exchange and standard secret sharing, and may be useful in settings where the secret is so sensitive that no one individual dealer can be trusted with the secret. Our second notion is called functional secret sharing, which incorporates some of the features of functional encryption into secret sharing by providing more fine-grained access to the secret. Qualified subsets of parties do not learn the secret, but instead