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

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 relate minimal codewords in the rank-metric setting to minimal reconstructing spaces and prove a $q$-analogue of the Brickell--Davenport theorem.