A $q$-Polymatroid Framework for Information Leakage in Secure Linear Network Coding

2026-01-12

AI Summary

This study addresses the problem of information leakage in secure linear network coding caused by an eavesdropper observing a subset of network links. By constructing a secure coding scheme based on nested rank-metric codes, the work establishes—for the first time—a connection between rank-metric codes and representable $q$-polymatroids. It introduces the notions of $q$-polymatroid ports and $q$-access structures to characterize information leakage precisely. The classical Massey correspondence and the Brickell–Davenport theorem are generalized to the rank-metric setting, leveraging conditional rank functions and combinatorial security analysis to quantify leakage in network coding rigorously. This approach significantly extends the applicability of secret sharing theory to rank-metric spaces.

Technology Category

• Machine Learning: Quantum Machine Learning
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Knowledge Representation and Reasoning: Qualitative Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study information leakage in secure linear network coding schemes based on nested rank-metric codes. We show that the amount of information leaked to an adversary that observes a subset of network links is characterized by the conditional rank function of a representable $q$-polymatroid associated with the underlying rank-metric code pair. Building on this connection, we introduce the notions of $q$-polymatroid ports and $q$-access structures and describe their structural properties. Moreover, we extend Massey's correspondence between minimal codewords and minimal access sets to the rank-metric setting and prove a $q$-analogue of the Brickell--Davenport theorem.

Problem

• information leakage
• secure linear network coding
• rank-metric codes
• q-polymatroid
• adversary

Innovation

• q-polymatroid
• rank-metric codes
• information leakage
• secure network coding
• q-access structures