Common information, matroid representation, and secret sharing for matroid ports

Abstract

Linear information and rank inequalities as, for instance, Ingleton inequality, are useful tools in information theory and matroid theory. Even though many such inequalities have been found, it seems that most of them remain undiscovered. Improved results have been obtained in recent works by using the properties from which they are derived instead of the inequalities themselves. We apply here this strategy to the classification of matroids according to their representations and to the search for bounds on secret sharing for matroid ports.

On ideal and weakly-ideal access structures

“…We recall that a function ε : N → R is said to be negligible if ε(k) = k −ω (1) . Statistical security .…”

“…We will review known results in Section 1. 1. GC schemes are generalizations of several classes of SSSs, including multi-linear, abelian and homomorphic 1 schemes and probably a rich subclass of non-linear ones.…”

“…Ideal SSSs are not only equivalent to entropic matroids, but also to p-representable matroids [30] and almost affine codes [38]. Despite several important works (e.g., see [38,30,7,3,31,29,1]), the characterization and classification of ideal SSSs is still an open problem. Simonis and Ashikhmin [38] proved that the class of ideal multi-linear SSSs is strictly larger than that of the linear ones (because, e.g., the non-Pappus matroid is not representable) and raised the question if every ideal access structure admits an ideal multi-linear SSS.…”