Extending Brickell–Davenport theorem to non-perfect secret sharing schemes

Abstract

One important result in secret sharing is the Brickell-Davenport Theorem: every ideal perfect secret sharing scheme defines a matroid that is uniquely determined by the access structure. Even though a few attempts have been made, there is no satisfactory definition of ideal secret sharing scheme for the general case, in which non-perfect schemes are considered as well. Without providing another unsatisfactory definition of ideal non-perfect secret sharing scheme, we present a generalization of the Brickell-Davenport Theorem to the general case. After analyzing that result under a new point of view and identifying its combinatorial nature, we present a characterization of the (not necessarily perfect) secret sharing schemes that are associated to matroids. Some optimality properties of such schemes are discussed.

On the Information Ratio of Non-perfect Secret Sharing Schemes

“…Moreover, we present in Section 8 a new definition for ideal non-perfect secret sharing scheme. Even though our new definition is equivalent to the one proposed in previous works [39,49], the use of access functions and the results in [29] make it clear that this is the right way to extend the corresponding concept for perfect schemes. In particular, the new framework provides a better insight on the connection between ideal non-perfect schemes and matroids.…”

“…The extension to the non-perfect case of the connection between ideal perfect secret sharing schemes and matroids discovered by Brickell and Davenport [12] has attracted some attention [39,49]. A thorough analysis of this extension is provided in a recent work [29]. A secret sharing scheme is called uniform if the values of its access function depend only on the number of players.…”

Optimal Non-perfect Uniform Secret Sharing Schemes

“…This connection between secret sharing schemes and matroids was first extended to non-perfect schemes by Kurosawa et al [24], who characterized the non-perfect secret sharing schemes that define a matroid. Recently, a characterization with weaker conditions has been presented [15]. Similarly to the results in this paper, its proof is based on the connection between secret sharing and polymatroids.…”

“…for every access function Φ [15,29,30]. In particular, this implies the well known fact that the information ratio of every perfect secret sharing scheme is at least 1.…”

“…On the basis of the connections between information theory, matroid theory, and secret sharing found by Fujishige [16,17], Brickell and Davenport [9], and Csirmaz [13], matroids and polymatroids have appeared to be a powerful tool, as it can be seen from several recent works [2-4, 25, 26]. Similar questions have been considered for non-perfect secret sharing schemes too [15,18,24,29,30,33], but the research is much less developed in this direction. In particular, only basic bounds on the information ratio of non-perfect secret sharing schemes are known [29,30].…”