Optimal multiple assignments based on integer programming in secret sharing schemes with general access structures

It is known that for any general access structure, a secret sharing scheme (SSS) can be constructed from an (m,m)-threshold scheme by using the so-called cumulative map or from a (t,m)-threshold SSS by a modified cumulative map. However, such constructed SSSs are not efficient generally. In this paper, we propose a new method to construct a SSS from a $(t,m)$-threshold scheme for any given general access structure. In the proposed method, integer programming is used to distribute optimally the shares of (t,m)-threshold scheme to each participant of the general access structure. From the optimality, it can always attain lower coding rate than the cumulative maps except the cases that they give the optimal distribution. The same method is also applied to construct SSSs for incomplete access structures and/or ramp access structures.