Cryptology ePrint Archive: Report 2016/726

Bounds on the Information Ratios of Secret Sharing Schemes for Close Access Structures

Oriol Farrąs and Jordi Ribes-Gonzįlez and Sara Ricci

Abstract: The information ratio of a secret sharing scheme $\Sigma$ measures the size of the largest share of the scheme, and is denoted by $\sigma(\Sigma)$. The optimal information ratio of an access structure $\Gamma$ is the infimum of $\sigma(\Sigma)$ among all schemes $\Sigma$ for $\Gamma$, and is denoted by $\sigma(\Gamma)$. The main result of this work is that for every two access structures $\Gamma$ and $\Gamma'$, $|\sigma(\Gamma)- \sigma(\Gamma')|\leq|\Gamma\cup\Gamma'|-|\Gamma\cap\Gamma'|$. As a consequence of this result, we see that close access structures admit secret sharing schemes with similar information ratio. We show that this property is also true for particular families of secret sharing schemes and models of computation, like the family of linear secret sharing schemes, span programs, Boolean formulas and circuits. In order to understand this property, we also study the limitations of the techniques for finding lower bounds on the information ratio and other complexity measures. We analyze the behavior of these bounds when we add or delete subsets from an access structure

Category / Keywords: foundations / secret sharing

Date: received 22 Jul 2016, last revised 22 Jul 2016

Contact author: oriol farras at urv cat

Available format(s): PDF | BibTeX Citation

Version: 20160727:174800 (All versions of this report)

Short URL: ia.cr/2016/726

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