Secret sharing schemes for ports of matroids of rank 3

Oriol Farràs

Kybernetika (2020)

  • Volume: 56, Issue: 5, page 903-915
  • ISSN: 0023-5954

Abstract

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A secret sharing scheme is ideal if the size of each share is equal to the size of the secret. Brickell and Davenport showed that the access structure of an ideal secret sharing scheme is determined by a matroid. Namely, the minimal authorized subsets of an ideal secret sharing scheme are in correspondence with the circuits of a matroid containing a fixed point. In this case, we say that the access structure is a matroid port. It is known that, for an access structure, being a matroid port is not a sufficient condition to admit an ideal secret sharing scheme. In this work we present a linear secret sharing scheme construction for ports of matroids of rank 3 in which the size of each share is at most n times the size of the secret. Using the previously known secret sharing constructions, the size of each share was O ( n 2 / log n ) the size of the secret. Our construction is extended to ports of matroids of any rank k ≥ 2 , obtaining secret sharing schemes in which the size of each share is at most n k - 2 times the size of the secret. This work is complemented by presenting lower bounds: There exist matroid ports that require ( 𝔽 q , ℓ ) -linear secret schemes with total information ratio Ω ( 2 n / 2 / ℓ n 3 / 4 log q ) .

How to cite

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Farràs, Oriol. "Secret sharing schemes for ports of matroids of rank 3." Kybernetika 56.5 (2020): 903-915. <http://eudml.org/doc/297215>.

@article{Farràs2020,
abstract = {A secret sharing scheme is ideal if the size of each share is equal to the size of the secret. Brickell and Davenport showed that the access structure of an ideal secret sharing scheme is determined by a matroid. Namely, the minimal authorized subsets of an ideal secret sharing scheme are in correspondence with the circuits of a matroid containing a fixed point. In this case, we say that the access structure is a matroid port. It is known that, for an access structure, being a matroid port is not a sufficient condition to admit an ideal secret sharing scheme. In this work we present a linear secret sharing scheme construction for ports of matroids of rank 3 in which the size of each share is at most $n$ times the size of the secret. Using the previously known secret sharing constructions, the size of each share was $O(n^2/\log n)$ the size of the secret. Our construction is extended to ports of matroids of any rank $k\ge 2$, obtaining secret sharing schemes in which the size of each share is at most $n^\{k-2\}$ times the size of the secret. This work is complemented by presenting lower bounds: There exist matroid ports that require $(\mathbb \{F\}_q,\ell )$-linear secret schemes with total information ratio $\Omega (2^\{n/2\}/\ell n^\{3/4\}\sqrt\{\log q\})$.},
author = {Farràs, Oriol},
journal = {Kybernetika},
keywords = {secret sharing schemes; matroids; matroid ports},
language = {eng},
number = {5},
pages = {903-915},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Secret sharing schemes for ports of matroids of rank 3},
url = {http://eudml.org/doc/297215},
volume = {56},
year = {2020},
}

TY - JOUR
AU - Farràs, Oriol
TI - Secret sharing schemes for ports of matroids of rank 3
JO - Kybernetika
PY - 2020
PB - Institute of Information Theory and Automation AS CR
VL - 56
IS - 5
SP - 903
EP - 915
AB - A secret sharing scheme is ideal if the size of each share is equal to the size of the secret. Brickell and Davenport showed that the access structure of an ideal secret sharing scheme is determined by a matroid. Namely, the minimal authorized subsets of an ideal secret sharing scheme are in correspondence with the circuits of a matroid containing a fixed point. In this case, we say that the access structure is a matroid port. It is known that, for an access structure, being a matroid port is not a sufficient condition to admit an ideal secret sharing scheme. In this work we present a linear secret sharing scheme construction for ports of matroids of rank 3 in which the size of each share is at most $n$ times the size of the secret. Using the previously known secret sharing constructions, the size of each share was $O(n^2/\log n)$ the size of the secret. Our construction is extended to ports of matroids of any rank $k\ge 2$, obtaining secret sharing schemes in which the size of each share is at most $n^{k-2}$ times the size of the secret. This work is complemented by presenting lower bounds: There exist matroid ports that require $(\mathbb {F}_q,\ell )$-linear secret schemes with total information ratio $\Omega (2^{n/2}/\ell n^{3/4}\sqrt{\log q})$.
LA - eng
KW - secret sharing schemes; matroids; matroid ports
UR - http://eudml.org/doc/297215
ER -

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