Graph-Theoretic Characterization of Noise Capacity of Conditional Disclosure of Secrets

Abstract

In the Conditional Disclosure of Secrets (CDS) problem, Alice and Bob hold inputs $x\in \mathcal{X}$ and $y\in \mathcal{Y}$ and share a secret. Let $f:\mathcal{X}\times\mathcal{Y}\to\{0,1\}$ be a function such that the secret is revealed to a third party, Carol, if and only if $f(x,y)=1$. To protect the secret when $f(x,y)=0$, Alice and Bob share a common noise variable unknown to Carol. We study the \emph{noise capacity} of CDS, defined as the maximum number of secret bits that can be securely revealed per noise bit. We first derive necessary and sufficient conditions on $f$, represented by a CDS graph, for the extremal case where the noise capacity equals $1$. We then develop converse bounds on the noise rate for all linear schemes: $\frac{(ρ-1)(d-1)}{ρd-1}$ if $ρ$ is finite, and $\frac{d-1}{d}$ if $ρ$ is infinite, where $ρ$ is the covering parameter of the CDS graph and $d$ is the number of unqualified edges in an unqualified path. Under maximal communication efficiency (message size equals secret size), we refine these bounds by analyzing qualified components and their connections. Achievability is shown for CDS instances with cyclic qualified edges and a single unqualified path. This graph-theoretic framework links noise efficiency limits to the unqualified path distance and covering parameter, providing a systematic method to analyze CDS under arbitrary graph topologies.