Two-partition-symmetrical entropy function regions

Consider the entropy function region for discrete random variables X_i, i ϵ N and partition N into N₁ and N₂ with 0 ≤ |N₁| ≤ |N₂|. An entropy function h is called (N₁, N₂)-symmetrical if for all A, B ⊂ N, h(A) = h(B) whenever |A ∩ N₁| = |B ∩N₁|, i = 1,2. We prove that for |N₁| = 0 or 1, the closure of the (N₁, N₂)-symmetrical entropy function region is completely characterized by Shannon-type information inequalities. Applications of this work include threshold secret sharing and distributed data storage, where symmetry exists in the structure of the problem.

I. Introduction

Let . For a (discrete) random vector , for , let and , where by convention. For a fixed can be regarded as a set function from to with , and hence is called the entropy function of .