Information Theoretic Foundations of Secret Sharing Systems
Summary
Secret sharing systems enable a sensitive datum to be split into distinct pieces or “shares” so that only authorised subsets of participants can reconstruct the original secret. The information theoretic approach to secret sharing employs entropy and mutual information measures to guarantee perfect secrecy against unauthorised coalitions and to quantify the efficiency of share distribution. Core principles include threshold schemes, where only groups of a minimum size can recover the secret, and general access structures, which specify arbitrary qualified subsets. The theoretical framework rests on the characterisation of entropic regions and information inequalities that delineate feasible distributions of entropy among shares. Fundamental metrics such as the information ratio and share size are derived from the intersection of linear and non-linear entropy inequalities, revealing optimal trade-offs between security, share size, and reconstruction thresholds. This mathematical foundation has underpinned advances in multiparty computation, cloud storage confidentiality and threshold cryptography, where rigorous guarantees of secrecy and resilience are essential.
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Technical terms
Entropic region: The set of all entropy vectors for a given number of random variables, subject to Shannon and non-Shannon information inequalities.
Access structure: A collection of authorised subsets of participants that are permitted to reconstruct the secret; the complement defines unauthorised coalitions.
Threshold scheme: A secret sharing design in which any subset of participants of size at least t can recover the secret, while smaller subsets learn nothing.
Information ratio: The ratio of the size (in bits) of the largest share to the size of the secret; a key measure of scheme efficiency.
Matroidal entropy function: An entropy function associated with a matroid, linking combinatorial designs and code constructions to entropic characterisations.
References
- Secret sharing in a special linear group. Informatics (2024).
- Recent Progresses in Characterising Information Inequalities. Entropy (2011).
- Reduced access structures with four minimal qualified subsets on six participants. Advances in Mathematics of Communications (2018).
- Matroidal Entropy Functions: A Quartet of Theories of Information, Matroid, Design, and Coding. Entropy (2021).
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