Additional democratic secret sharing schemes and maximal non-u-qualifying sets

Develop and classify additional democratic secret sharing schemes (as defined via monomial-Cartesian code coset constructions in Definition 14) beyond the currently proven class covered by Proposition 17, and for each such scheme determine the structure and cardinalities of maximal non-u-qualifying participant sets for all integers u in the range 1 to l.

Background

The paper introduces democratic secret sharing, focusing on a second layer of security: systematic structures of maximal non‑u‑qualifying sets that prevent recovery of u q‑bits of information even when significantly more than t_u participants collude. Using monomial‑Cartesian codes and a reformulation of information transfer (Theorem 3), the authors construct families exhibiting these properties and provide sufficient conditions (Proposition 17) under which schemes are democratic.

The explicit open question seeks to extend these results by identifying further democratic schemes outside the parameter regime guaranteed by Proposition 17 and to rigorously characterize their maximal non‑u‑qualifying sets, thereby broadening the applicability and design space of democratic ramp secret sharing.