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

TL;DR

This work analyzes the noise efficiency of conditional disclosure of secrets (CDS) using a graph-theoretic framework. It derives a necessary and sufficient condition for achieving unit noise capacity and establishes general linear-noise-rate bounds that depend on the CDS graph's covering parameter ρ and the residing unqualified path distance d, with refinements under maximal communication efficiency (N = L). The authors also prove achievability for specific graph constructions featuring cyclic qualified edges and a single unqualified path, linking structural graph properties to noise-optimal strategies. Collectively, the results provide a systematic method to analyze and design CDS protocols across arbitrary topologies, with implications for noise-efficient secure computation in resource-constrained settings.

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.

Figures (7)

• Figure 1: Illustration of the conditional disclosure of secrets (CDS) problem. Alice and Bob have private inputs $x$ and $y$, respectively, i.e., Alice does not know about $y$ and Bob does not know about $x$, and share a secret $S$ along with a common noise variable $Z$ that protects the secret. A publicly known function $f(x, y)$ specifies the disclosure condition: when $f(x, y)=1$, they encode $S$ into their transmitted messages so that Carol can recover it; when $f(x, y)=0$, the messages depend only on $Z$, ensuring that Carol learns nothing about $S$.
• Figure 2: The secret is disclosed if and only if $x=y=1$ (i.e., from $A_1, B_1$).
• Figure 3: A CDS instance with the coding scheme that achieves noise rate $1$.
• Figure 4: A CDS instance that has an internal qualified edge $\{B_2, A_2\}$ in a residing unqualified path $(B_2, A_1, B_3, A_2)$. The residing unqualified path distance is $d=3$, i.e., $\{A_2,B_3\},\{B_3,A_1\},\{A_1,B_2\}$. The secret consists of $L = 2$ symbols, $S_1$ and $S_2$, while the noise consists of $L_Z = 3$ independent uniform symbols, $Z_0$, $Z_1$, and $Z_2$. The achieved noise rate is $R = L / L_Z = 2 / 3$. The scheme achieves this rate by allowing non-uniform message sizes across nodes and therefore does not satisfy the constraint that all nodes have message size $N$.
• Figure 5: A CDS instance contains an internal qualified edge $\{B_2, A_2\}$ located in an unqualified path $(B_2, A_1, B_3, A_2)$ within a qualified component $G_f$. The unqualified path has a distance $d = 3$, and the size of the connected edge cover is $\rho = 5$. The achievable noise rate is calculated as $(\rho - 1)(d - 1) / (\rho d - 1) = 4/7$. In this instance, the secret consists of $L = 4$ symbols, denoted as $S_1, S_2, S_3, S_4$, while the noise includes $L_Z = 7$ independent uniform symbols, represented as $Z_0, Z_1, \dots, Z_6$. Each message contains $N = 5$ symbols. Thus, the achieved rate is $R = L / L_Z = 4/7$.

Theorems & Definitions (24)

• Definition 1: Qualified/Unqualified Path and Component
• Definition 2: Internal Qualified Edge and Residing Unqualified Path
• Definition 3: Residing Unqualified Path Distance
• Definition 4: Connected Edge Cover
• Definition 5: Components of Residing Unqualified Path
• Lemma 1
• Theorem 1
• Example 1: Achievability of ${R_Z}=1$
• Example 2: Counterexample with Violation
• Theorem 2