Optimal information rate of secret sharing schemes on trees

TL;DR

This paper precisely determines the optimal information rate for secret sharing schemes on trees, linking it to the largest core size, and provides bounds that match for trees due to a novel graph-theoretic correspondence.

Contribution

It establishes an exact formula for the information rate of secret sharing on trees based on the largest core, advancing understanding of secret sharing efficiency on graph structures.

Findings

Exact formula for trees: 1/(2-1/c) where c is largest core size

Bounds on information rate coincide for trees due to a new graph correspondence

Provides a method to compute optimal secret sharing rates on tree graphs

Abstract

The information rate for an access structure is the reciprocal of the load of the optimal secret sharing scheme for this structure. We determine this value for all trees: it is 1/(2-1/c), where c is the size of the largest core of the tree. A subset of the vertices of a tree is a core if it induces a connected subgraph and for each vertex in the subset one finds a neighbor outside the subset. Our result follows from a lower and an upper bound on the information rate that applies for any graph and happen to coincide for trees because of a correspondence between the size of the largest core and a quantity related to a fractional cover of the tree with stars.