Characterization and Optimization of Ideal Secret Sharing Schemes via Algebraic Matroid Theory

Abstract

This paper establishes a rigorous algebraic characterization of Perfect Secret Sharing Schemes by generalizing the classical linear framework to the domain of Algebraic Matroids defined over field extensions. We demonstrate that the class of access structures admitting an Ideal Scheme with information rate ρ = 1 is canonically isomorphic to the class of matroids representable over a field extension L/K via transcendental and algebraic dependencies. To overcome the geometric limitations imposed by the Ingleton Inequality on linear representations, we introduce a constructive methodology utilizing the module of Kähler differentials. as a linearization functor that transforms algebraic independence into vector space independence. We prove that the security of a scheme is equivalent to the non-vanishing of the Jacobian determinant associated with the unauthorized sets in the realization variety. Furthermore, we explicitly construct ideal schemes for non-linear matroids, specifically the Vámos matroid, by exploiting the properties of purely inseparable extensions and the Frobenius endomorphism in fields of positive characteristic p. These constructions confirm that the capacity of algebraic secret sharing strictly exceeds that of linear schemes, providing a complete structural resolution to the problem of ideal realizability through the geometry of field extensions and algebraic curves.