Paper 2019/575

On Abelian Secret Sharing: duality and separation

Amir Jafari and Shahram Khazaei

Abstract

Unlike linear secret sharing, very little is known about abelian secret sharing. In this paper, we present two results on abelian secret sharing. First, we show that the information ratio of access structures (or more generally access functions) remain invariant for the class of abelian schemes with respect to duality. Then, we prove that abelian secret sharing schemes are superior to the linear ones. New techniques and insight are used to achieve both results. Our result on abelian duality is proved using the notion of Pontryagin duality. The intuition behind the usefulness of this tool is to work with an equivalent definition of linear secret sharing, which is less prevalent in the literature, to make it possible to extend the result on linear duality to abelian duality. We develop a new method for proving lower bound on the linear information ratio of access structures that can work not only for general linear secret sharing but also for linear schemes on finite fields with a specific characteristic. Unlike the common lower bound techniques, which are usually either based on rank/information inequalities or based on counting/combinatorial-algebraic arguments, our method is linear algebraic in essence. We apply our method to the Fano and non-Fano access structures for the characteristics on which they are not ideal. We then show in a straightforward way that for their union---a well-known 12-participant access structure---the abelian schemes are superior to the linear ones.