Secret Sharing, Rank Inequalities, and Information Inequalities

Beimel and Orlov proved that all information inequalities on four or five variables, together with all information inequalities on more than five variables that are known to date, provide lower bounds on the size of the shares in secret sharing schemes that are at most linear on the number of participants. We present here another two negative results about the power of information inequalities in the search for lower bounds in secret sharing. First, we prove that all information inequalities on a bounded number of variables can only provide lower bounds that are polynomial on the number of participants. Second, we prove that the rank inequalities that are derived from the existence of two common informations can provide only lower bounds that are at most cubic in the number of participants.