Bipartite secret sharing and staircases

0582343 - ÚTIA 2025 NL eng J - Journal Article
Csirmaz, Laszlo - Matúš, František - Padró, C.
Bipartite secret sharing and staircases.
Discrete Mathematics. Roč. 347, č. 5 (2024), č. článku 113909. ISSN 0012-365X. E-ISSN 1872-681X
Institutional support: RVO:67985556
Keywords : cryptography * multipartite secret sharing * entropy method * linear secret sharing * submodular optimization
OECD category: Pure mathematics
Impact factor: 0.8, year: 2022
Method of publishing: Limited access
http://library.utia.cas.cz/separaty/2024/MTR/csirmaz-0582343.pdf

Bipartite secret sharing schemes have a bipartite access structure in which the set of participants is divided into two parts and all participants in the same part play an equivalent role. Such a bipartite scheme can be described by a staircase: the collection of its minimal points. The complexity of a scheme is the maximal share size relative to the secret size; and the kappa-complexity of an access structure is the best lower bound provided by the entropy method. An access structure is kappa-ideal if it has kappa-complexity 1. Motivated by the abundance of open problems in this area, the main results can be summarized as follows. First, a new characterization of kappa-ideal multipartite access structures is given which offers a straightforward and simple approach to describe ideal bipartite and tripartite access structures. Second, the kappa-complexity is determined for a range of bipartite access structures, including those determined by two points, staircases with equal widths and heights, and staircases with all heights 1. Third, matching linear schemes are presented for some non-ideal cases, including staircases where all heights are 1 and all widths are equal. Finally, finding the Shannon complexity of a bipartite access structure can be considered as a discrete submodular optimization problem. An interesting and intriguing continuous version is defined which might give further insight to the large-scale behavior of these optimization problems.
Permanent Link: https://hdl.handle.net/11104/0350584