Democratic Ramp Secret Sharing

In this work we revisit the fundamental findings by Chen et al. in [5] on general information transfer in linear ramp secret sharing schemes to conclude that their method not only gives a way to establish worst case leakage [5, 25] and best case recovery [5, 19], but can also lead to additional insight on non-qualifying sets for any prescribed amount of information. We then apply this insight to schemes defined from monomial-Cartesian codes and by doing so we demonstrate that the good schemes from Sec.\ IV in [14] have a second layer of security. Elaborating further, when given a designed recovery number, in a new construction the focus is entirely on ensuring that the access structure possess desirable second layer security, rather on what is the worst case information leakage in terms of number of participants. The particular structure of largest possible sets being not able to determine given amount of information suggests that we call such schemes democratic