On the dealer's randomness required in secret sharing schemes

Abstract

Abstract. The problem we deal with in this paper is the research of upper and lower bounds on the randomness required by the dealer to set up a secret sharing scheme. We give both lower and upper bounds for infinite classes of access structures. Lower bounds are obtained using entropy arguments. Upper bounds derive from a decomposition construction based on combinatorial designs (in particular, t-(v, k, )~) designs). We prove a general result on the randomness needed to construct a scheme for the cycle C,~; when n is odd our bound is tight. We study the access structures on at most four participants and the connected graphs on five vertices, obtaining exact values for the randomness for all them. Also, we analyze the number of random bits required to construct anonymous threshold schemes, giving upper bounds. (Informally, anonymous threshold schemes are schemes in which the secret can be reconstructed without knowledge of which participants hold which shares.)

Optimal Information Rate of Secret Sharing Schemes on Trees

“…We can therefore assume |X| ≥ 2. We use the "independent sequence lemma" from [1], [8] that ensures f (X) ≥ |X| + 1.…”

“…Definition 1. 1 We call a subset X of the vertices of a graph G a core of G if it induces a connected subgraph and one can find a neighbor x ′ / ∈ X of each x ∈ X such that x is the only neighbor of x ′ among the vertices in X and {x ′ | x ∈ X} is an independent set.…”

Geometrical contributions to secret sharing theory

“…Considerable work has gone into trying to find bounds for the proposed efficiency measures for certain families of access structures, for example [19,23]. Since this process mainly using information theoretic techniques we do not discuss it in great depth in this paper (but see Section 4.3).…”

On ASGS framework: general requirements and an example of implementation

“…The other class of extended capabilities focuses on anonymity, randomness and automatization for secret sharing procedures. Anonymous and random secret sharing was studied by Blundo, Giorgio Gaggia, Stinson in [3], [4]. Some of ideas in automatic secret sharing and generation originate from the same field.…”