Secret Sharing, Rank Inequalities and Information Inequalities

Abstract

Abstract. 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 negative result about the power of information inequalities in the search for lower bounds in secret sharing. Namely, we prove that all information inequalities on a bounded number of variables only can provide lower bounds that are polynomial on the number of participants.

Improving the Linear Programming Technique in the Search for Lower Bounds in Secret Sharing

“…That solution is related to the one used by Csirmaz [15] to prove that κ(Γ) is at most the number of participants. Another negative result about the power of information inequalities to provide asymptotic lower bounds was presented in [49]. Namely, every lower bound that is obtained by using rank inequalities on at most r variables is O(n r−2 ), and hence polynomial on the number n of participants.…”

“…Some limitations of the LP-technique in the search for asymptotic lower bounds have been found. Namely, the best lower bound that can be obtained by using all information inequalities that were known at the beginning of this decade is linear in the number of participants [8,15], while at most polynomial lower bounds can be found by using all known or unknown inequalities on a bounded number of variables [49].…”

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

“…Linear information and rank inequalities are fundamental in the linear programming technique that has been used to find bounds on the information ratio of secret sharing schemes [6,7,36,44,48] and on the achievable rates in network coding [17,56,60]. An improvement to that technique has been recently proposed [22].…”

Entropic Matroids and Their Representation

“…Matroids, secret sharing and linearity are also discussed in several papers as mentioned in part earlier. Reference [ 25 ] gave the first example of an access structure (i.e., the parties that can recover the secret from their share) induced by a matroid, namely the Vamos matroid, that is non-ideal (a measure of optimality of the secret shares lengths); Reference [ 26 ] presented the first non-trivial lower bounds on the size of the domain of the shares for secret-sharing schemes realizing an access structure induced by the Vamos matroid and this is later improved in Reference [ 27 ] using using non-Shannon inequalities for the entropy function. As mentioned earlier, an important line of work is also dedicated to understanding the representation of entropic polymatroids for a fixed ground set cardinality [ 9 ], which is well-understood for cardinality 2 and 3 and more complicated for larger cardinality with the non-Shannon inequalities emerging.…”