Linear Secret Sharing and the Automatic Search of Linear Rank Inequalities

We study the problem of computing linear rank inequalities, and the related problem of computing lower bounds on the linear share complexity of access structures. We prove that if one knows a generating set for the cone of linear rank inequalities on n + 1 variables, then he can use this generating set and linear programming (or semi-infinite programming) to compute the exact linear optimal information ratio of any access structure on n parties. This theorem shows that it is useful and important to compute generating sets for the cones of linear rank inequalities on a given number of variables. Then, we study the only method we know to cope with the later task, the so called common information method. We investigate the completeness of this method and we prove some preliminary results suggesting that the method is not complete (i.e. there are linear rank inequalities which cannot be obtained via the studied method). Mathematics Subject Classification: 68P30