Secret Sharing in the Rank Metric

2025-04-25

AI Summary

Classical matroid-based secret sharing lacks analogues in the q-analogue setting—specifically, no notions of secret sharing schemes or ports exist for q-matroids. Method: This work extends matroid-driven secret sharing to the q-matroid framework by introducing *q-access structures* over vector spaces, systematically characterizing their duality, minors, and rank-function properties, and establishing their intrinsic connection to rank-metric codes. A constructive method based on rank-metric codes is proposed to generate linear secret sharing schemes. Results: The resulting schemes achieve optimal information rate and strong security, unifying both perfect and ramp (non-perfect) access models. The core contribution is a tripartite correspondence among q-matroids, rank-metric codes, and q-access structures, wherein duality and minors acquire precise access-control semantics. This framework provides a novel structural foundation for key management in post-quantum cryptography.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Web search models and ranking

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.

Problem

• Generalize secret sharing to q-polymatroids
• Introduce access structures on vector spaces
• Link rank-metric codes to secret sharing

Innovation

• Generalize secret sharing to q-polymatroids
• Introduce access structure on vector spaces
• Use rank-metric codes for secret sharing