TY - JOUR

T1 - Secret sharing and non-Shannon information inequalities

AU - Beimel, Amos

AU - Orlov, Ilan

N1 - Funding Information:
Manuscript received February 28, 2010; revised February 14, 2011; accepted February 14, 2011. Date of current version August 31, 2011. This work was supported in part by an ISF grant 938/09 and in part by the Frankel Center for Computer Science. The material in this paper was presented at the 6th Theory of Cryptography Conference (TCC), San Francisco, CA, March 2009.

PY - 2011/9/1

Y1 - 2011/9/1

N2 - The known secret-sharing schemes for most access structures are not efficient; even for a one-bit secret the length of the shares in the schemes is 2O(n) , where n is the number of participants in the access structure. It is a long standing open problem to improve these schemes or prove that they cannot be improved. The best known lower bound is by Csirmaz, who proved that there exist access structures with n participants such that the size of the share of at least one party is n/log n times the secret size. Csirmaz's proof uses Shannon information inequalities, which were the only information inequalities known when Csirmaz published his result. On the negative side, Csirmaz proved that by only using Shannon information inequalities one cannot prove a lower bound of ω(n) on the share size. In the last decade, a sequence of non-Shannon information inequalities were discovered. In fact, it was proved that there are infinity many independent information inequalities even in four variables. This raises the hope that these inequalities can help in improving the lower bounds beyond n. However, we show that any information inequality with four or five variables cannot prove a lower bound of ω(n) on the share size. In addition, we show that the same negative result holds for all information inequalities with more than five variables that are known to date.

AB - The known secret-sharing schemes for most access structures are not efficient; even for a one-bit secret the length of the shares in the schemes is 2O(n) , where n is the number of participants in the access structure. It is a long standing open problem to improve these schemes or prove that they cannot be improved. The best known lower bound is by Csirmaz, who proved that there exist access structures with n participants such that the size of the share of at least one party is n/log n times the secret size. Csirmaz's proof uses Shannon information inequalities, which were the only information inequalities known when Csirmaz published his result. On the negative side, Csirmaz proved that by only using Shannon information inequalities one cannot prove a lower bound of ω(n) on the share size. In the last decade, a sequence of non-Shannon information inequalities were discovered. In fact, it was proved that there are infinity many independent information inequalities even in four variables. This raises the hope that these inequalities can help in improving the lower bounds beyond n. However, we show that any information inequality with four or five variables cannot prove a lower bound of ω(n) on the share size. In addition, we show that the same negative result holds for all information inequalities with more than five variables that are known to date.

KW - Linear programs

KW - lower bounds

KW - monotone span programs

KW - non-Shannon information inequalities

KW - rank inequalities

KW - secret-sharing

UR - http://www.scopus.com/inward/record.url?scp=79959324084&partnerID=8YFLogxK

U2 - 10.1109/TIT.2011.2162183

DO - 10.1109/TIT.2011.2162183

M3 - Article

AN - SCOPUS:79959324084

SN - 0018-9448

VL - 57

SP - 5634

EP - 5649

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

IS - 9

M1 - 6006590

ER -