Weighted threshold secret sharing schemes with applications to blockchain technology

Abstract

Secret Sharing Schemes are cryptographic tools for securely distributing a secret among participants, ensuring that only authorized subsets can reconstruct it while unauthorized coalitions cannot, a property known as information-theoretic security. A key challenge in designing such schemes is reducing share size, which impacts efficiency and scalability in distributed systems. This thesis studies this problem in structured access structures. After reviewing threshold schemes and their ideality, it focuses on weighted threshold access structures, analyzing classical constructions and methods, including approximation techniques, to reduce share size. Their relevance is illustrated in Proof-of-Stake blockchain protocols, where influence is proportional to staked resources. Ideal hierarchical access structures are then characterized using matroid theory, Boolean polymatroids, and lattice path matroids, and related to applications in multi-level blockchain networks such as Polkadot.