A $q$-Polymatroid Framework for Information Leakage in Secure Linear Network Coding
Eimear Byrne, Johan Vester Dinesen, Ragnar Freij-Hollanti, Camilla Hollanti
TL;DR
The paper addresses information leakage in secure linear network coding constructed from nested rank-metric codes. It develops a $q$-polymatroid framework in which leakage is precisely governed by the conditional rank function of a representable $q$-polymatroid associated with the nested code pair, providing an operational link between algebraic structure and secrecy. It extends classical secret-sharing correspondences to the rank-metric setting by introducing $q$-polymatroid ports and $q$-access structures, proving a $q$-analogue of Massey’s minimal-codeword theorem and Brickell–Davenport theorem under suitable assumptions. The results yield a topology-free, combinatorial toolkit for analyzing information leakage in rank-metric secure network coding and open avenues for characterizing or constructing $q$-access structures via rank-metric codes and their ports. The entropy interpretation of representable $q$-polymatroids further connects information theory and subspace combinatorics, suggesting broader implications for secrecy inequalities in the subspace setting.
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.