Eprints of Rényi Institute

On an infinite family of graphs with information ratio 2 − 1/k

Csirmaz, László and Ligeti, Péter (2009) On an infinite family of graphs with information ratio 2 − 1/k. Computing, 85 (1-2). pp. 127-136. ISSN 0010-485X

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Abstract

In this paper we consider the secret sharing problem on special access structures with minimal qualified subsets of size two, i.e. secret sharing on graphs. This means that the participants are the vertices of the graph and the qualified subsets are the subsets of V(G) spanning at least one edge. The information ratio of a graph G is denoted by R(G) and is defined as the ratio of the greatest size of the shares a vertex has to remember and of the size of the secret. Since the determination of the exact information ratio is a non-trivial problem even for small graphs (i.e. for V(G) = 6), every construction can be of particular interest. Let k be the maximal degree in G. In this paper we prove that R(G) = 2 − 1/k for every graph G with the following properties: (A) every vertex has at most one neighbour of degree one; (B) vertices of degree at least 3 are not connected by an edge; (C) the girth of the graph is at least 6. We prove this by using polyhedral combinatorics arguments and the entropy method.

Item Type:Article
Keywords and phrases:Secret sharing, graph, entropy, linear programming
Subjects:90C05 Linear programming
94A60 Cryptography
68R10 Graph theory
94A17 Measures of information, entropy
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Divisions:Research Groups > Theoretical cryptography
Last Modified:16 Apr 2010 09:52

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