On the Linear Capacity of Conditional Disclosure of Secrets

TL;DR

This paper extends the understanding of the maximum efficiency in conditional disclosure of secrets by characterizing the linear capacity for a broader class of instances using graph-based alignment methods.

Contribution

It advances the alignment approach to determine the linear capacity of CDS instances beyond the highest capacity cases, introducing a new formula involving a graph covering parameter.

Findings

Linear capacity for certain CDS instances is ( ho) where  ho is a graph covering parameter.

Characterization of linear capacity extends previous results for maximum capacity cases.

Graph representation and alignment techniques are key to analyzing CDS capacity.

Abstract

Conditional disclosure of secrets (CDS) is the problem of disclosing as efficiently as possible, one secret from Alice and Bob to Carol if and only if the inputs at Alice and Bob satisfy some function $f$. The information theoretic capacity of CDS is the maximum number of bits of the secret that can be securely disclosed per bit of total communication. All CDS instances, where the capacity is the highest and is equal to $1/2$, are recently characterized through a noise and signal alignment approach and are described using a graph representation of the function $f$. In this work, we go beyond the best case scenarios and further develop the alignment approach to characterize the linear capacity of a class of CDS instances to be $(\rho-1)/(2\rho)$, where $\rho$ is a covering parameter of the graph representation of $f$.