Secret Sharing and Shared Information

Abstract

Secret sharing is a cryptographic discipline in which the goal is to distribute information about a secret over a set of participants in such a way that only specific authorized combinations of participants together can reconstruct the secret. Thus, secret sharing schemes are systems of variables in which it is very clearly specified which subsets have information about the secret. As such, they provide perfect model systems for information decompositions. However, following this intuition too far leads to an information decomposition with negative partial information terms, which are difficult to interpret. One possible explanation is that the partial information lattice proposed by Williams and Beer is incomplete and has to be extended to incorporate terms corresponding to higher order redundancy. These results put bounds on information decompositions that follow the partial information framework, and they hint at where the partial information lattice needs to be improved.

9 pages, 1 figure. The material was presented at a Workshop on information decompositions at FIAS, Frankfurt, in 12/2016. The revision includes changes in the definition of combinations of secret sharing schemes. Section 3 and Appendix now discusses in how far existing measures satisfy the proposed properties. The concluding section is considerably revised

References in corpus (9)

• Quantifying unique information
• Nonnegative Decomposition of Multivariate Information
• Exploration of synergistic and redundant information sharing in static and dynamical Gaussian systems
• A Bivariate Measure of Redundant Information
• Measuring multivariate redundant information with pointwise common change in surprisal
• Bits from Biology for Computational Intelligence
• Redundancy and synergy in dual decompositions of mutual information gain and information loss
• Reconsidering unique information: Towards a multivariate information decomposition
• The Partial Entropy Decomposition: Decomposing multivariate entropy and mutual information via pointwise common surprisal

Cited by in corpus (15)

• Measuring multivariate redundant information with pointwise common change in surprisal
• Bits and Pieces: Understanding Information Decomposition from Part-whole Relationships and Formal Logic
• Pointwise Partial Information Decomposition using the Specificity and Ambiguity Lattices
• A Novel Approach to the Partial Information Decomposition
• Unique Information and Secret Key Agreement
• Generalised Measures of Multivariate Information Content
• Exact partial information decompositions for Gaussian systems based on dependency constraints
• Unique Informations and Deficiencies
• The identity of information: how deterministic dependencies constrain information synergy and redundancy
• Pooling Probability Distributions and the Partial Information Decomposition
• Orders between channels and implications for partial information decomposition
• MAXENT3D_PID: An Estimator for the Maximum-entropy Trivariate Partial Information Decomposition
• A Measure of Synergy based on Union Information
• Modes of Information Flow
• Quantifying multivariate redundancy with maximum entropy decompositions of mutual information