TY - JOUR

T1 - Secret-Sharing Schemes for Very Dense Graphs

AU - Beimel, Amos

AU - Farràs, Oriol

AU - Mintz, Yuval

N1 - Funding Information:
This work was supported by ISF Grant 938/09. A preliminary version of this paper appeared in Advances in Cryptology CRYPTO 2012, volume 7417 of Lecture Notes in Computer Science, pages 144–161. Springer-Verlag, 2012.
Funding Information:
Partly supported by the Spanish Government through projects TIN201127076-C03-01, 2010 CSD2007-00004, and by the Catalan Government through Grant 2009 SGR 1135. Part of his work was done while at Ben Gurion University.
Publisher Copyright:
© 2014, International Association for Cryptologic Research.

PY - 2016/4/1

Y1 - 2016/4/1

N2 - A secret-sharing scheme realizes a graph if every two vertices connected by an edge can reconstruct the secret while every independent set in the graph does not get any information on the secret. Similar to secret-sharing schemes for general access structures, there are gaps between the known lower bounds and upper bounds on the share size for graphs. Motivated by the question of what makes a graph “hard” for secret-sharing schemes (that is, they require large shares), we study very dense graphs, that is, graphs whose complement contains few edges. We show that if a graph with (Formula presented.) vertices contains (Formula presented.) edges for some constant (Formula presented.) , then there is a scheme realizing the graph with total share size of (Formula presented.). This should be compared to (Formula presented.) , the best upper bound known for the total share size in general graphs. Thus, if a graph is “hard,” then the graph and its complement should have many edges. We generalize these results to nearly complete (Formula presented.) -homogeneous access structures for a constant (Formula presented.). To complement our results, we prove lower bounds on the total share size for secret-sharing schemes realizing very dense graphs, e.g., for linear secret-sharing schemes, we prove a lower bound of (Formula presented.) for a graph with (Formula presented.) edges.

AB - A secret-sharing scheme realizes a graph if every two vertices connected by an edge can reconstruct the secret while every independent set in the graph does not get any information on the secret. Similar to secret-sharing schemes for general access structures, there are gaps between the known lower bounds and upper bounds on the share size for graphs. Motivated by the question of what makes a graph “hard” for secret-sharing schemes (that is, they require large shares), we study very dense graphs, that is, graphs whose complement contains few edges. We show that if a graph with (Formula presented.) vertices contains (Formula presented.) edges for some constant (Formula presented.) , then there is a scheme realizing the graph with total share size of (Formula presented.). This should be compared to (Formula presented.) , the best upper bound known for the total share size in general graphs. Thus, if a graph is “hard,” then the graph and its complement should have many edges. We generalize these results to nearly complete (Formula presented.) -homogeneous access structures for a constant (Formula presented.). To complement our results, we prove lower bounds on the total share size for secret-sharing schemes realizing very dense graphs, e.g., for linear secret-sharing schemes, we prove a lower bound of (Formula presented.) for a graph with (Formula presented.) edges.

KW - Complete bipartite covers

KW - Equivalence covers

KW - Graph access structures

KW - Secret-sharing

KW - Share size

UR - http://www.scopus.com/inward/record.url?scp=84959158082&partnerID=8YFLogxK

U2 - 10.1007/s00145-014-9195-8

DO - 10.1007/s00145-014-9195-8

M3 - Article

AN - SCOPUS:84959158082

VL - 29

SP - 336

EP - 362

JO - Journal of Cryptology

JF - Journal of Cryptology

SN - 0933-2790

IS - 2

ER -