TY - GEN
T1 - Secret sharing schemes for dense forbidden graphs
AU - Beimel, Amos
AU - Farràs, Oriol
AU - Peter, Naty
N1 - Publisher Copyright:
© Springer International Publishing Switzerland 2016.
PY - 2016/1/1
Y1 - 2016/1/1
N2 - A secret-sharing scheme realizes a given graph if every two vertices connected by an edge can reconstruct the secret and every independent set in the graph does not get any information about the secret. A secret-sharing scheme realizes a forbidden graph if every two vertices connected by an edge can reconstruct the secret and every two vertices which are not connected by an edge do not get any information about the secret. Similar to secret-sharing schemes for general access structures, there are gaps between the known lower bounds and upper bounds on the total share size for graphs and for forbidden graphs. Following [Beimel et al. CRYPTO 2012], our goal in this paper is to understand how the total share size increases by removing few edges from a graph that can be realized by an efficient secret-sharing scheme. We show that if a graph with n vertices contains at least (Formula Presented) edges for some (Formula Presented), i.e., it is obtained by removing few edges from the complete graph, then there is a scheme realizing its forbidden graph in which the total share size is O(n7/6+2β/3). This should be compared to O(n3/2), the best known upper bound for the total share size in general forbidden graphs. Additionally, we show that a forbidden graph access structure obtained by removing few edges from an arbitrary graph G can be realized by a secret-sharing scheme with total share size of O(m+ n7/6+2β/3), where m is the total size of the shares in a secretsharing scheme realizing G and n1+β is the number of the removed edges. We also show that for a graph obtained by removing few edges from an arbitrary graph G with n vertices, if the chromatic number of the graph that contains the removed edges is small, then there is a fairly efficient scheme realizing the resulting graph; specifically, we construct a secret-sharing scheme with total share size of Õ(m2/3n2/3+2β/3c1/3), where m is the total size of the shares in a secret-sharing scheme realizing G, the value n1+β is an upper bound on the number of the removed edges, and c is the chromatic number of the graph of the removed edges. This should be compared to O(n2/ log(n)), the best known upper bound for the total share size for general graphs.
AB - A secret-sharing scheme realizes a given graph if every two vertices connected by an edge can reconstruct the secret and every independent set in the graph does not get any information about the secret. A secret-sharing scheme realizes a forbidden graph if every two vertices connected by an edge can reconstruct the secret and every two vertices which are not connected by an edge do not get any information about the secret. Similar to secret-sharing schemes for general access structures, there are gaps between the known lower bounds and upper bounds on the total share size for graphs and for forbidden graphs. Following [Beimel et al. CRYPTO 2012], our goal in this paper is to understand how the total share size increases by removing few edges from a graph that can be realized by an efficient secret-sharing scheme. We show that if a graph with n vertices contains at least (Formula Presented) edges for some (Formula Presented), i.e., it is obtained by removing few edges from the complete graph, then there is a scheme realizing its forbidden graph in which the total share size is O(n7/6+2β/3). This should be compared to O(n3/2), the best known upper bound for the total share size in general forbidden graphs. Additionally, we show that a forbidden graph access structure obtained by removing few edges from an arbitrary graph G can be realized by a secret-sharing scheme with total share size of O(m+ n7/6+2β/3), where m is the total size of the shares in a secretsharing scheme realizing G and n1+β is the number of the removed edges. We also show that for a graph obtained by removing few edges from an arbitrary graph G with n vertices, if the chromatic number of the graph that contains the removed edges is small, then there is a fairly efficient scheme realizing the resulting graph; specifically, we construct a secret-sharing scheme with total share size of Õ(m2/3n2/3+2β/3c1/3), where m is the total size of the shares in a secret-sharing scheme realizing G, the value n1+β is an upper bound on the number of the removed edges, and c is the chromatic number of the graph of the removed edges. This should be compared to O(n2/ log(n)), the best known upper bound for the total share size for general graphs.
KW - Avoiding covers
KW - Covers by graphs
KW - Secret sharing
UR - https://www.scopus.com/pages/publications/84984794059
U2 - 10.1007/978-3-319-44618-9_27
DO - 10.1007/978-3-319-44618-9_27
M3 - Conference contribution
AN - SCOPUS:84984794059
SN - 9783319446172
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 509
EP - 528
BT - Security and Cryptography for Networks - 10th International Conference, SCN 2016, Proceedings
A2 - De Prisco, Roberto
A2 - Zikas, Vassilis
PB - Springer Verlag
T2 - 10th International Conference on Security and Cryptography for Networks, SCN 2016
Y2 - 31 August 2016 through 2 September 2016
ER -