On the Composition of Secret Sharing Schemes Related to Codes

TL;DR

This paper introduces a new class of secret sharing schemes based on linear codes, expanding existing structures, and demonstrates their ideal properties and vector space realizations, partially addressing an open conjecture.

Contribution

It constructs a subclass of composite access structures from minimal codeword supports, enlarges the iterated threshold class, and proves vector space realizability for all such structures.

Findings

Constructed a subclass of composite access structures from minimal code supports.

Proved all schemes are ideal and admit vector space constructions.

Provided a partial answer to an existing conjecture.

Abstract

In this paper we construct a subclass of the composite access structure introduced by Mart\'inez et al. based on schemes realizing the structure given by the set of codewords of minimal support of linear codes. This class enlarges the iterated threshold class studied in the same paper. Furthermore all the schemes on this paper are ideal (in fact they allow a vector space construction) and we arrived to give a partial answer to a conjecture stated in the above paper. Finally, as a corollary we proof that all the monotone access structures based on all the minimal supports of a code can be realized by a vector space construction.