## Cryptology ePrint Archive: Report 2010/348

Lattice-theoretic Characterization of Secret Sharing Representable Connected Matroids

A. N. Alekseychuk

Abstract: Necessary and sufficient conditions for a connected matroid to be secret sharing (ss-)representable are obtained. We show that the flat lattices of ss-representable matroids are closely related with well-studied algebraic objects called linear lattices. This fact implies that new powerful methods (from lattice theory and mathematical logic) for investigation of ss-representable matroids can be applied. We also obtain some necessary conditions for a connected matroid to be ss-representable. Namely, we construct an infinite set of sentences (like to Haiman’s “higher Arguesian identities”) which are hold in all ss-representable matroids.

Category / Keywords: cryptographic protocols / secret sharing