Secret Sharing in the Rank Metric
Johan Vester Dinesen, Eimear Byrne, Ragnar Freij-Hollanti, Camilla Hollanti
TL;DR
Addresses secret sharing in the rank-metric by extending access structures to vector spaces and modeling them with representable $q$-polymatroids. It develops generalized ports, duality, and minor operations, linking them to rank-metric codes and their induced polymatroids. It shows that MRD rank-metric codes yield perfect threshold secret sharing within this framework and provides an entropy-based interpretation of the polymatroid rank function. The results enable applications to random linear wiretap networks and related coset coding or private information retrieval tasks, with avenues for deeper connections explored in future work.
Abstract
The connection between secret sharing and matroid theory is well established. In this paper, we generalize the concepts of secret sharing and matroid ports to $q$-polymatroids. Specifically, we introduce the notion of an access structure on a vector space, and consider properties related to duality, minors, and the relationship to $q$-polymatroids. Finally, we show how rank-metric codes give rise to secret sharing schemes within this framework.