A characterisation of ideal weighted secret sharing schemes

Abstract

Abstract Beimel, Tassa and Weinreb [SIAM J. Discrete Math. 22 (2008), 360–397] and Farràs and Padró [Lecture Notes in Comput. Sci. 5978, Springer, 2010, 219–236] partially characterised access structures of ideal weighted secret sharing schemes in terms of the operation of composition. They proved that any weighted ideal access structure is a composition of indecomposable ones. Farràs and Padró gave a list of seven classes of access structures – one unipartite, three bipartite and three tripartite – to which all weighted ideal indecomposable access structures may belong. In this paper we determine exactly which access structures from those classes are indecomposable. We also determine which compositions of indecomposable weighted access structures are again weighted and obtain an if-and-only-if characterisation of ideal weighted secret sharing schemes. We use game-theoretic techniques to achieve this.