Optimal Multiple Assignments Based on Integer Programming in Secret Sharing Schemes with General Access Structures

摘要

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, a new method is proposed 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 derive the optimal (t, m)-threshold scheme and the optimal distribution of the shares to minimize the average or maximum size of the distributed shares to participants. From the optimality, it can always attain lower coding rate than the cumulative maps because the cumulative maps cannot attain the optimal distribution in many cases. The same method is also applied to construct SSSs for incomplete access structures and/or ramp access structures.