Infinite probabilistic secret sharing

Laszlo Csirmaz

Kybernetika (2023)

  • Volume: 59, Issue: 2, page 179-197
  • ISSN: 0023-5954

Abstract

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A probabilistic secret sharing scheme is a joint probability distribution of the shares and the secret together with a collection of secret recovery functions. The study of schemes using arbitrary probability spaces and unbounded number of participants allows us to investigate their abstract properties, to connect the topic to other branches of mathematics, and to discover new design paradigms. A scheme is perfect if unqualified subsets have no information on the secret, that is, their total share is independent of the secret. By relaxing this security requirement, three other scheme types are defined. Our first result is that every (infinite) access structure can be realized by a perfect scheme where the recovery functions are non-measurable. The construction is based on a paradoxical pair of independent random variables which determine each other. Restricting the recovery functions to be measurable ones, we give a complete characterization of access structures realizable by each type of the schemes. In addition, either a vector-space or a Hilbert-space based scheme is constructed realizing the access structure. While the former one uses the traditional uniform distributions, the latter one uses Gaussian distributions, leading to a new design paradigm.

How to cite

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Csirmaz, Laszlo. "Infinite probabilistic secret sharing." Kybernetika 59.2 (2023): 179-197. <http://eudml.org/doc/299081>.

@article{Csirmaz2023,
abstract = {A probabilistic secret sharing scheme is a joint probability distribution of the shares and the secret together with a collection of secret recovery functions. The study of schemes using arbitrary probability spaces and unbounded number of participants allows us to investigate their abstract properties, to connect the topic to other branches of mathematics, and to discover new design paradigms. A scheme is perfect if unqualified subsets have no information on the secret, that is, their total share is independent of the secret. By relaxing this security requirement, three other scheme types are defined. Our first result is that every (infinite) access structure can be realized by a perfect scheme where the recovery functions are non-measurable. The construction is based on a paradoxical pair of independent random variables which determine each other. Restricting the recovery functions to be measurable ones, we give a complete characterization of access structures realizable by each type of the schemes. In addition, either a vector-space or a Hilbert-space based scheme is constructed realizing the access structure. While the former one uses the traditional uniform distributions, the latter one uses Gaussian distributions, leading to a new design paradigm.},
author = {Csirmaz, Laszlo},
journal = {Kybernetika},
keywords = {secret sharing; abstract probability space; Sierpinski topology; product measure; span program; Hilbert space program},
language = {eng},
number = {2},
pages = {179-197},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Infinite probabilistic secret sharing},
url = {http://eudml.org/doc/299081},
volume = {59},
year = {2023},
}

TY - JOUR
AU - Csirmaz, Laszlo
TI - Infinite probabilistic secret sharing
JO - Kybernetika
PY - 2023
PB - Institute of Information Theory and Automation AS CR
VL - 59
IS - 2
SP - 179
EP - 197
AB - A probabilistic secret sharing scheme is a joint probability distribution of the shares and the secret together with a collection of secret recovery functions. The study of schemes using arbitrary probability spaces and unbounded number of participants allows us to investigate their abstract properties, to connect the topic to other branches of mathematics, and to discover new design paradigms. A scheme is perfect if unqualified subsets have no information on the secret, that is, their total share is independent of the secret. By relaxing this security requirement, three other scheme types are defined. Our first result is that every (infinite) access structure can be realized by a perfect scheme where the recovery functions are non-measurable. The construction is based on a paradoxical pair of independent random variables which determine each other. Restricting the recovery functions to be measurable ones, we give a complete characterization of access structures realizable by each type of the schemes. In addition, either a vector-space or a Hilbert-space based scheme is constructed realizing the access structure. While the former one uses the traditional uniform distributions, the latter one uses Gaussian distributions, leading to a new design paradigm.
LA - eng
KW - secret sharing; abstract probability space; Sierpinski topology; product measure; span program; Hilbert space program
UR - http://eudml.org/doc/299081
ER -

References

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