Secret sharing and non-Shannon information inequalities. (English) Zbl 1213.94150

Beimel, Amos; Orlov, Ilan

Reingold, Omer (ed.), Theory of cryptography. 6th theory of cryptography conference, TCC 2009, San Francisco, CA, USA, March 15–17, 2009. Proceedings. Berlin: Springer (ISBN 978-3-642-00456-8/pbk). Lecture Notes in Computer Science 5444, 539-557 (2009).

Summary: 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 \(2^{O(n)}\), 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 L. Csirmaz [“The size of a share must be large”, J. Cryptology 10, No. 4, 223–231 (1997; Zbl 0897.94012)], 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 \(\omega (n)\) on the share size. In the last decade, a sequence of non-Shannon information inequalities were discovered. This raises the hope that these inequalities can help in improving the lower bounds beyond \(n\). However, in this paper we show that all the inequalities known to date cannot prove a lower bound of \(\omega (n)\) on the share size.
For the entire collection see [Zbl 1156.94005].

MSC:

94A62

Authentication, digital signatures and secret sharing

Citations:

Zbl 0897.94012