Local bounds for the optimal information ratio of secret sharing schemes
| Autor: | Jordi Ribes-González, Oriol Farràs, Sara Ricci |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Discrete mathematics
Applied Mathematics Prove it Sigma Order (ring theory) 020206 networking & telecommunications 0102 computer and information sciences 02 engineering and technology 01 natural sciences Secret sharing Infimum and supremum Computer Science Applications Information ratio 010201 computation theory & mathematics Scheme (mathematics) 0202 electrical engineering electronic engineering information engineering Access structure Mathematics |
| Zdroj: | Designs, Codes and Cryptography. 87:1323-1344 |
| ISSN: | 1573-7586 0925-1022 |
| DOI: | 10.1007/s10623-018-0529-7 |
| Popis: | The information ratio of a secret sharing scheme $$\varSigma $$ is the ratio between the length of the largest share and the length of the secret, and it is denoted by $$\sigma (\varSigma )$$ . The optimal information ratio of an access structure $$\varGamma $$ is the infimum of $$\sigma (\varSigma )$$ among all schemes $$\varSigma $$ with access structure $$\varGamma $$ , and it is denoted by $$\sigma (\varGamma )$$ . The main result of this work is that for every two access structures $$\varGamma $$ and $$\varGamma '$$ , $$|\sigma (\varGamma )-\sigma (\varGamma ')|\le |\varGamma \cup \varGamma '|-|\varGamma \cap \varGamma '|$$ . We prove it constructively. Given any secret sharing scheme $$\varSigma $$ for $$\varGamma $$ , we present a method to construct a secret sharing scheme $$\varSigma '$$ for $$\varGamma '$$ that satisfies that $$\sigma (\varSigma ')\le \sigma (\varSigma )+|\varGamma \cup \varGamma '|-|\varGamma \cap \varGamma '|$$ . 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 classes 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. |
| Databáze: | OpenAIRE |
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