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

TL;DR

This paper introduces a $q$-polymatroid framework to analyze information leakage in secure linear network coding, linking it to rank-metric codes and extending classical concepts to the $q$-analogue setting.

Contribution

It establishes a novel $q$-polymatroid approach for characterizing information leakage and extends key coding theory results to the rank-metric and $q$-analogue context.

Findings

Leakage characterized by conditional rank function of a $q$-polymatroid.

Introduction of $q$-ports and $q$-access structures with structural properties.

Extension of Massey's minimal codewords correspondence to rank-metric codes.

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.