More information theoretical inequalities to be used in secret sharing?

Sebastià Martı́n, Carles Padró, Yang An

Amos Beimel, Ilan Orlov

Renato M. Capocelli, Alfredo De Santis, Luisa Gargano +1 more

Emirhan Gürpınar

Secret sharing is a central topic in information-theoretical cryptography. In a secret sharing scheme a random secret is distributed among participants so that only the qualified subsets of parties can recover the secret. In this paper we improve previously known [13] lower bounds on the size of the largest share for several specific access structures. Our approach is based on non-Shannon-type information inequalities. We employ the linear programming (LP) technique that permits to apply new information inequalities indirectly. To reduce computational complexity of the involved linear programs we extensively use symmetry considerations.

Sebastià Martı́n, Carles Padró, Yang An

Beimel and Orlov proved that all information inequalities on four or five variables, together with all information inequalities on more than five variables that are known to date, provide lower bounds on the size of the shares in secret sharing schemes that are at most linear on the number of participants. We present here another two negative results about the power of information inequalities in the search for lower bounds in secret sharing. First, we prove that all information inequalities on a bounded number of variables can only provide lower bounds that are polynomial on the number of participants. Second, we prove that the rank inequalities that are derived from the existence of two common informations can provide only lower bounds that are at most cubic in the number of participants.

Heling Xiao, Wang-Mei Guo, Xiao Wang

Quantum secret sharing (QSS) schemes are analyzed from an information theoretical perspective centered on the Araki—Lieb inequality. Based on this inequality, mathematical characterizations of QSS schemes and quantum error-correcting codes (QECCs) are given. Furthermore, we present a proof of the relation between QSS schemes and QECCs. This information theoretic description of QSS schemes is used to derive the quantum Singleton bound.