Optimal Multiple Assignments with (m, m)-Scheme for General Access Structures

Given the number n of the participants, one can solve an integer programming on 2 variables to construct an optimal multiple assignment with threshold schemes for general access structure. In this paper, we focus on finding optimal multiple assignments with (m,m)-schemes. We prove that most of the variables in the corresponding integer programming take the value of 0, while the remaining variables take the values of either 0 or 1. We also show that given a complete access structure, an optimal scheme may be obtained directly from the scheme by Ito, Saito, and Nishizeki (Secret sharing scheme realizeing any access structure, in Globecom 1987). Keywords—multiple assignments, threshold scheme, integer programming, access structure.