A Characterization of Ideal Weighted Secret Sharing Schemes

Beimel, Tassa and Weinreb (2008) and Farras and Padro (2010) partially characterized access structures of ideal weighted threshold secret sharing schemes in terms of the operation of composition. They classified indecomposable ideal weighted threshold access structures, and proved that any other ideal weighted threshold access structure is a composition of indecomposable ones. It remained unclear which compositions of indecomposable weighted threshold access structures are weighted. In this paper we fill the gap. Using game-theoretic techniques we determine which compositions of indecomposable ideal access structures are weighted, and obtain an if and only if characterization of ideal weighted threshold secret sharing schemes.