A Rank-3 Secret Sharing Scheme over Vector Spaces

Abstract

Secret sharing schemes are a tool used to ensure secure distribution of a secret among a group of participants such that only authorized groups can reconstruct the secret. We explore the relationship between secret sharing schemes and matroids, with emphasis placed on matroid-related schemes. Specifically, we focus on the access structures arising from matroids with rank three. We present the projective plane secret sharing scheme, a reformulation of the scheme of Lopes de Souza, now using projective planes instead of LFSR sequences. We show that this scheme is equivalent to the vector space secret sharing scheme of Brickell. Additionally, we show that the induced subhypergraph isomorphism problem is equivalent to the subgraph isomorphism problem, and use this equivalence in a new method to find realizations of access structures by our scheme. Finally, we give some conditions to find the minimal 𝑞 required for our scheme to realize an access structure in 𝔽_𝑞.