Secret Sharing and Non-Shannon Information Inequalities

The known secret-sharing schemes for most access structures are not efficient; even for a one-bit secret the length of the shares in the schemes is 2(O(n)), where n is the number of participants in the access structure. It is a long standing open problem to improve these schemes or prove that they cannot be improved. The best known lower bound is by Csirmaz, who proved that there exist access structures with n participants such that the size of the share of at least one party is n/log n times the secret size. Csirmaz's proof uses Shannon information inequalities, which were the only information inequalities known when Csirmaz published his result. On the negative side, Csirmaz proved that by only using Shannon information inequalities one cannot prove a lower bound of omega(n) on the share size. In the last decade, a sequence of non-Shannon information inequalities were discovered. In fact, it was proved that there are infinity many independent information inequalities even in four variables. This raises the hope that these inequalities can help in improving the lower bounds beyond n. However, we show that any information inequality with four or five variables cannot prove a lower bound of omega(n) on the share size. In addition, we show that the same negative result holds for all information inequalities with more than five variables that are known to date.