Our results on abelian and homomorphic SSSs have been motivated by the following concerns and questions. All known linear rank inequities have been derived using the so-called common information property of random variables [Dougherty, Freiling and Zeger, 2009], and it is an open problem that if common information is complete for deriving all such inequalities (Q1). The common information property has also been used in linear programming to find lower bounds for the information ratio of access structures [Farràs, Kaced, Molleví and Padró, 2018] and it is an open problem that if the method is complete for finding the optimal information ratio for the class of multi-linear schemes (Q2). Also, it was realized by the latter authors that the obtained lower bound does not have a good behavior with respect to duality and it is an open problem that if this behavior is inherent to their method (Q3).
Our first result provides a negative answer to Q2. Even though, we are not able to completely answer Q1 and Q3, we have some observations about them.
Category / Keywords: foundations / Access structure, Duality, Abelian secret sharing, Homomorphic secret sharing, Ideal secret sharing Date: received 27 May 2019, last revised 26 Feb 2020 Contact author: shahram khazaei at gmail com Available format(s): PDF | BibTeX Citation Version: 20200226:105723 (All versions of this report) Short URL: ia.cr/2019/575