TY - GEN

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.

PY - 2012/9/3

Y1 - 2012/9/3

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, we study very dense graphs, that is, graphs whose complement contains few edges. We show that if a graph with n vertices contains ( n 2 - n 1+β edges for some constant 0 ≤ β < 1, then there is a scheme realizing the graph with total share size of Õ(n 5/4+3β/4). This should be compared to O(n 2/logn) - the best upper bound known for 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 k-homogeneous access structures for a constant k. To complement our results, we prove lower bounds for secret-sharing schemes realizing very dense graphs, e.g., for linear secret-sharing schemes we prove a lower bound of Ω(n 1 + β/2).

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, we study very dense graphs, that is, graphs whose complement contains few edges. We show that if a graph with n vertices contains ( n 2 - n 1+β edges for some constant 0 ≤ β < 1, then there is a scheme realizing the graph with total share size of Õ(n 5/4+3β/4). This should be compared to O(n 2/logn) - the best upper bound known for 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 k-homogeneous access structures for a constant k. To complement our results, we prove lower bounds for secret-sharing schemes realizing very dense graphs, e.g., for linear secret-sharing schemes we prove a lower bound of Ω(n 1 + β/2).

UR - http://www.scopus.com/inward/record.url?scp=84865527639&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-32009-5_10

DO - 10.1007/978-3-642-32009-5_10

M3 - Conference contribution

AN - SCOPUS:84865527639

SN - 9783642320088

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 144

EP - 161

BT - Advances in Cryptology, CRYPTO 2012 - 32nd Annual Cryptology Conference, Proceedings

T2 - 32nd Annual International Cryptology Conference, CRYPTO 2012

Y2 - 19 August 2012 through 23 August 2012

ER -