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arXiv:2601.07567

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

Eimear Byrne, Johan Vester Dinesen, Ragnar Freij-Hollanti, Camilla Hollanti

Published Jan 12, 2026• 0 citations
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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.

DOI
10.48550/arxiv.2601.07567

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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 qqq-polymatroid associated with the underlying rank-metric code pair. Building on this connection, we introduce the notions of qqq-polymatroid ports and qqq-access structures and describe their structural properties. Moreover, we relate minimal codewords in the rank-metric setting to minimal reconstructing spaces and prove a qqq-analogue of the Brickell--Davenport theorem.
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Secret-sharing schemes are a tool used in many cryptographic protocols. In these schemes, a dealer holding a secret string distributes shares to the parties such that only authorized subsets of participants can reconstruct the secret from their shares. The collection of authorized sets is called an access structure. An access structure is ideal if there is a secret-sharing scheme realizing it such that the shares are taken from the same domain as the secrets. Brickell and Davenport (Journal of Cryptology, 1991) have shown that ideal access structures are closely related to matroids. They give a necessary condition for an access structure to be ideal-the access structure must be induced by a matroid. Seymour (Journal of Combinatorial Theory B, 1992) has proved that the necessary condition is not sufficient: There exists an access structure induced by a matroid that does not have an ideal scheme. The research on access structures induced by matroids is continued in this work. The main result in this paper is strengthening the result of Seymour. It is shown that in any secret-sharing scheme realizing the access structure induced by the Vamos matroid with domain of the secrets of size k, the size of the domain of the shares is at least k + Omega(radic(k)). The second result considers nonideal secret-sharing schemes realizing access structures induced by matroids. It is proved that the fact that an access structure is induced by a matroid implies lower and upper bounds on the size of the domain of shares of subsets of participants even in nonideal schemes (as long as the shares are still relatively short). This generalized results of Brickell and Davenport for ideal schemes. Finally, an example of a nonideal access structure that is nearly ideal is presented.
Optimizing the maximum, or average, length of the shares in relation to the length of the secret for every given access structure is a difficult and long-standing open problem in cryptology. Most of the known lower bounds on these parameters have been obtained by implicitly or explicitly using that every secret sharing scheme defines a polymatroid related to the access structure. The best bounds that can be obtained by this combinatorial method can be determined by using linear programming, and this can be effectively done for access structures on a small number of participants. By applying this linear programming approach, we improve some of the known lower bounds for the access structures on five participants and the graph access structures on six participants for which these parameters were still undetermined. Nevertheless, the lower bounds that are obtained by this combinatorial method are not tight in general. For some access structures, they can be improved by adding to the linear program non-Shannon information inequalities as new constraints. We obtain in this way new separation results for some graph access structures on eight participants and for some ports of non-representable matroids. Finally, we prove that, for two access structures on five participants, the combinatorial lower bound cannot be attained by any linear secret sharing scheme.
Quantum secret sharing (QSS) schemes can be constructed from quantum error-correcting codes, but characterizing their access structures and implementation costs becomes challenging for finite-length, high-rate quantum low-density parity-check (qLDPC) codes. We develop an exact computational framework for QSS induced by CSS qLDPC codes. The framework characterizes authorized, forbidden, and intermediate coalitions using restricted parity-check matrices and binary linear algebra, and introduces the accessible logical dimension κ(A)κ(A)κ(A), defined through the symplectic rank of locally supported logical classes, to refine the characterization of intermediate coalitions. We further incorporate physical network constraints through separate Tanner-incidence, Steiner-routing, and scheduling-depth costs, and treat the partitioning of physical qubits into participants as an explicit design variable. Numerical experiments on four CSS qLDPC codes verify the predicted complement duality and monotonicity properties and show substantial variation in access transitions and network costs. For the codes studied, the bivariate bicycle instance exhibits the sharpest transition, while network embedding costs vary by more than an order of magnitude across physical topologies. We also show that participant partitioning can substantially improve block-level security and reconstruction thresholds. These results provide a practical bridge between the algebraic structure of CSS qLDPC codes, the graded accessibility of quantum secrets, and the physical constraints of networked QSS implementations.
Hierarchical secret sharing is among the most natural generalizations of threshold secret sharing, and it has attracted a lot of attention since the invention of secret sharing until nowadays. Several constructions of ideal hierarchical secret sharing schemes have been proposed, but it was not known what access structures admit such a scheme. We solve this problem by providing a natural definition for the family of the hierarchical access structures and, more importantly, by presenting a complete characterization of the ideal hierarchical access structures, that is, the ones admitting an ideal secret sharing scheme. Our characterization is based on the well-known connection between ideal secret sharing schemes and matroids and, more specifically, on the connection between ideal multipartite secret sharing schemes and integer polymatroids. In particular, we prove that every hierarchical matroid port admits an ideal linear secret sharing scheme over every large enough finite field. Finally, we use our results to present a new proof for the existing characterization of the ideal weighted threshold access structures.