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Vide Leaf, Hyderabad eBooks

Security Notions of Secret Sharing Schemes via Different Information Measures

Chenxi Liu, Songsong Dai, Donghui Guo

Published Jan 1, 2020• 0 citations
Publisher
DOI
10.37247/etni.1.2020.14

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The following standard relaxations of perfect security for secret sharing schemes (SSSs) exist in the literature:quasi-perfect,almost-perfect, andstatistical. Understanding the power of these relaxations on the efficiency of SSSs, measured via a parameter calledinformation ratio, is a long-standing open problem. In this article, we introduce and study an extremely relaxed security notion, calledpartial security, for which it is only required that any qualified set gains strictly more information about the secret than any unqualified one. To get a meaningful efficiency measure, we normalize the (standard) information ratio of such schemes by an appropriate parameter and refer to the new measure aspartial information ratio. We present three main results in this paper.First, we prove that partial and perfect information ratios coincide for the class of linear SSSs.Second, we prove that for the general (i.e., non-linear) class of SSSs, partial and statistical information ratios are equal.Third, we show that partial and almost-perfect information ratios do not coincide for the class of mixed-linear schemes (i.e., schemes constructed by combining linear schemes with different underlying finite fields). We also use the notion of partial secret sharing to strengthen and unify the previousdecompositiontheorems for constructing SSSs.

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Secret sharing is a cryptographic technique to share a secret among participants in such a way that only authorized subsets are able to recover the secret. Ramp secret sharing schemes can achieve better information ratio than perfect schemes while some partial information on a secret which is composed of several sub-secrets leaks out. The notion of strong security has been introduced to control the amount of information on every subset of the sub-secrets unauthorized sets can obtain. In this paper, we reduce the construction of strongly secure ramp secret sharing for general access structures to a linear algebraic problem. As a result, we show that previous results on strongly secure network coding imply two constructions of a linear transformation which makes a given linear ramp scheme strongly secure. They are explicit or provide a deterministic algorithm while the previous method which works for any linear ramp scheme is probabilistic.

International Symposium on Information Theory and its Applications•Oct 24, 2020•0 citations
Different security notions of secret sharing schemes have been proposed by different information measures. Entropies, such as Shannon entropy and min entropy, are frequently used in the setting security notions for secret sharing schemes. Different to the entropies, Kolmogorov complexity was also defined and used in study the security of individual instances for secret sharing schemes. This paper is concerned with these security notions for secret sharing schemes defined by the variational measures, including Shannon entropy, guessing probability, min entropy and Kolmogorov complexity.
The following standard relaxations of perfect security for secret sharing schemes (SSSs) exist in the literature:quasi-perfect,almost-perfect, andstatistical. Understanding the power of these relaxations on the efficiency of SSSs, measured via a parameter calledinformation ratio, is a long-standing open problem. In this article, we introduce and study an extremely relaxed security notion, calledpartial security, for which it is only required that any qualified set gains strictly more information about the secret than any unqualified one. To get a meaningful efficiency measure, we normalize the (standard) information ratio of such schemes by an appropriate parameter and refer to the new measure aspartial information ratio. We present three main results in this paper.First, we prove that partial and perfect information ratios coincide for the class of linear SSSs.Second, we prove that for the general (i.e., non-linear) class of SSSs, partial and statistical information ratios are equal.Third, we show that partial and almost-perfect information ratios do not coincide for the class of mixed-linear schemes (i.e., schemes constructed by combining linear schemes with different underlying finite fields). We also use the notion of partial secret sharing to strengthen and unify the previousdecompositiontheorems for constructing SSSs.
Secret sharing is a cryptographic technique to share a secret among participants in such a way that only authorized subsets are able to recover the secret. Ramp secret sharing schemes can achieve better information ratio than perfect schemes while some partial information on a secret which is composed of several sub-secrets leaks out. The notion of strong security has been introduced to control the amount of information on every subset of the sub-secrets unauthorized sets can obtain. In this paper, we reduce the construction of strongly secure ramp secret sharing for general access structures to a linear algebraic problem. As a result, we show that previous results on strongly secure network coding imply two constructions of a linear transformation which makes a given linear ramp scheme strongly secure. They are explicit or provide a deterministic algorithm while the previous method which works for any linear ramp scheme is probabilistic.