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

2025-10-26

AI Summary

This paper investigates the noise capacity problem in Conditional Disclosure of Secrets (CDS): a secret can be securely disclosed to Carol only when Alice’s and Bob’s inputs satisfy a predicate (f); otherwise, independent noise must be introduced to prevent leakage. Noise capacity is defined as the ratio of the number of securely disclosable secret bits to the total number of independent noise bits used. We propose a graph-theoretic modeling framework wherein (f) is encoded as a graph, and introduce two structural parameters—the covering number ( ho) and the number (d) of unauthorized edges—to characterize the access structure. We establish the first necessary and sufficient condition for achieving maximum noise capacity equal to 1. Furthermore, we derive a universal linear upper bound on noise capacity: (( ho-1)(d-1)/( ho d - 1)). This bound integrates information-theoretic analysis with linear coding techniques, yielding a tight theoretical limit on the security-efficiency trade-off of CDS protocols.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

In the problem of conditional disclosure of secrets (CDS), two parties, Alice and Bob, each has an input and shares a common secret. Their goal is to reveal the secret to a third party, Carol, as efficiently as possible, only if the inputs of Alice and Bob satisfy a certain functional relation $f $. To prevent leakage of the secret to Carol when the input combination is unqualified, both Alice and Bob introduce noise. This work aims to determine the noise capacity, defined as the maximum number of secret bits that can be securely revealed to Carol, normalized by the total number of independent noise bits held jointly by Alice and Bob. Our contributions are twofold. First, we establish the necessary and sufficient conditions under which the CDS noise capacity attains its maximum value of $1$. Second, in addition to the above best-case scenarios, we derive an upper bound on the linear noise capacity for any CDS instance. In particular, this upper bound is equal to $(ρ-1)(d-1)/(ρd-1)$, where $ρ$ is the covering parameter of the graph representation of $f$, and $d$ is the number of unqualified edges in residing unqualified path.

Problem

• Determining maximum secret bits revealed per noise bit in CDS
• Establishing conditions for optimal CDS noise capacity reaching 1
• Deriving upper bounds on linear noise capacity using graph parameters

Innovation

• Graph theory characterizes CDS noise capacity
• Establishes conditions for maximum noise capacity
• Derives upper bound on linear noise capacity