Symmetries in Linear Programming for Information Inequalities

TL;DR

This paper explores the use of symmetries in linear programming to derive improved bounds on share sizes in secret sharing schemes, leveraging non-Shannon-type information inequalities and symmetry considerations to simplify complex optimization problems.

Contribution

It introduces a novel approach combining symmetry analysis with linear programming to strengthen bounds in secret sharing, extending previous methods with new information inequalities.

Findings

Improved lower bounds on share sizes for secret sharing schemes.

Effective use of symmetry to reduce linear programming complexity.

Extension of non-Shannon-type information inequalities in this context.

Abstract

We study the properties of secret sharing schemes, where a random secret value is transformed into shares distributed among several participants in such a way that only the qualified groups of participants can recover the secret value. We improve the lower bounds on the sizes of shares for several specific problems of secret sharing. To this end, we use the method of non-Shannon-type information inequalities going back to Z. Zhang and R.W. Yeung. We employ and extend the linear programming technique that allows to apply new information inequalities indirectly, without even writing them down explicitly. To reduce the complexity of the problems of linear programming involved in the bounds we extensively use symmetry considerations.