Characterizing Ideal Weighted Threshold Secret Sharing

Characterization of Threshold Functions: State of the Art, Some New Contributions and Open Problems

“…counting the number of such examples, i.e., asymptotically there are n 3 24 + O(n 2 ) such games. By an exhaustive enumeration we have checked Conjecture 8.4 up to n = 20 voters.…”

“…The study of switching functions goes back at least to Dedekind's 1897 work [9], in which he determined the exact number of simple games with four or fewer players. Since that time these structures have been investigated in a variety of different contexts either theoretically [26,29,28,30,5] in the context of Boolean functions or because of their numerous applications: neural networks [1], simple games [48,34,35,51], threshold logic [13,8,25,33,44], hypergraphs [54], coherent structures [53], learning theory [42], complexity theory [4], and secret sharing [57,59,3]. Several books on neural networks have studied these structures: [49,55,58,52].…”

Roughly weighted hierarchical simple games

“…We note that the only case when we can have four levels is when the top level and the bottom one are both trivial, that is k 1 = 1 and k 4 = k 3 + n 4 . (Beimel et al, 2008) do not get four levels (only three) as they allow trivial levels of one kind but not of another. By duality (Gvozdeva, Hameed, & Slinko, 2011), we obtain a similar characterization in the conjunctive case.…”

A characterisation of ideal weighted secret sharing schemes

“…These include access structures defined by graphs (Brickell & Davenport, 1991), weighted threshold access structures (Beimel, Tassa, & Weinreb, 2008;Farràs & Padró, 2010), hierarchical access structures (Farràs & Padró, 2010), bipartite and tripartite access structures (Padró & Sáez, 1998;Padró & Sáez, 2004;Farràs, Martí-Farré, & Padró, 2012). While in the classes of bipartite and tripartite access structures the ideal ones were given explicitly, for the case of weighted threshold access structures (Beimel et al, 2008) suggested a new kind of description. This method uses the operation of composition of access structures (Martin, 1993).…”

“…Under this approach the first task is obtaining a characterisation of indecomposable structures. Beimel et al (2008) proved that every ideal indecomposable secret sharing scheme is either disjunctive hierarchical or tripartite. Farràs and Padró (2010); later gave a more precise classification which was complete (but some access structures that they viewed as indecomposable later appeared to be decomposable).…”