Information Rates of Minimal Non-Matroid-Related Access Structures

TL;DR

This paper determines the exact information rates for all minor-minimal, non-matroid-related access structures in secret sharing schemes, expanding understanding of their efficiency limits.

Contribution

It provides the first complete characterization of information rates for all remaining non-matroid-related access structures, building on prior partial results.

Findings

Exact information rates for all minor-minimal, non-matroid-related access structures are established.

All such structures have information rates at or below two-thirds.

The results confirm the infinite diversity of these structures with specific efficiency bounds.

Abstract

In a secret sharing scheme, shares of a secret are distributed to participants in such a way that only certain predetermined sets of participants are qualified to reconstruct the secret. An access structure on a set of participants specifies which sets are to be qualified. The information rate of an access structure is a bound on how efficient a secret sharing scheme for that access structure can be. Marti-Farre and Padro showed that all access structures with information rate greater than two-thirds are matroid-related, and Stinson showed that four of the minor-minimal, non-matroid-related access structures have information rate exactly two-thirds. By a result of Seymour, there are infinitely many remaining minor-minimal, non-matroid-related access structures. In this paper we find the exact information rates for all such structures.