Weighted Simple Games as Access Structures of Ideal Secret Sharing Schemes

Beimel, Tassa and Weinreb (2008) and Farras and Padro (2010) partially characterized weighted simple games that are access structures of ideal secret sharing schemes; they did this in terms of the operation of composition of simple games. They classified such indecomposable weighted games, and proved that any other weighted game, which is the access structure of an ideal secret sharing scheme, is a composition of the indecomposable ones. It remained unclear which compositions of indecomposable weighted games of this sort are weighted. In this paper we fill the gap by obtaining an if and only if characterization of weighted simple games that are access structures of ideal secret sharing schemes.