Ideal Hierarchical Secret Sharing Schemes

Hierarchical secret sharing is among the most natural generalizations of threshold secret sharing, and it has attracted a lot of attention since the invention of secret sharing until nowadays. Several constructions of ideal hierarchical secret sharing schemes have been proposed, but it was not known...

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Published in:	IEEE transactions on information theory Vol. 58; no. 5; pp. 3273 - 3286
Main Authors:	Farras, O., Padro, C.
Format:	Journal Article Publication
Language:	English
Published:	New York, NY IEEE 01.05.2012 Institute of Electrical and Electronics Engineers The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:	90 Operations research, mathematical programming 90B Operations research and management science 90C Mathematical programming Applied sciences Bismuth Boolean polymatroids Classificació AMS Code Division Multiple Access Construction Cryptography Electronic mail Exact sciences and technology Government hierarchical secret sharing ideal secret sharing schemes Information sharing Information theory Information, signal and communications theory integer polymatroids Integers Inventions Investigació operativa Joints Matemàtiques i estadística Mathematical analysis multi partite secret sharing multipartite secret sharing Operations research Periodic structures Ports Programació (Matemàtica) Programming (Mathematics) secret sharing Telecommunications and information theory Thresholds Vectors weighted secret sharing schemes weighted threshold secret sharing Àrees temàtiques de la UPC Finite field integer polymatroids hierarchical secret sharing weighted threshold secret sharing Matroid multipartite secret sharing ideal secret sharing schemes Secret sharing Hierarchized structure Boolean polymatroids
ISSN:	0018-9448, 1557-9654
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Abstract	Hierarchical secret sharing is among the most natural generalizations of threshold secret sharing, and it has attracted a lot of attention since the invention of secret sharing until nowadays. Several constructions of ideal hierarchical secret sharing schemes have been proposed, but it was not known what access structures admit such a scheme. We solve this problem by providing a natural definition for the family of the hierarchical access structures and, more importantly, by presenting a complete characterization of the ideal hierarchical access structures, that is, the ones admitting an ideal secret sharing scheme. Our characterization is based on the well-known connection between ideal secret sharing schemes and matroids and, more specifically, on the connection between ideal multipartite secret sharing schemes and integer polymatroids. In particular, we prove that every hierarchical matroid port admits an ideal linear secret sharing scheme over every large enough finite field. Finally, we use our results to present a new proof for the existing characterization of the ideal weighted threshold access structures.
AbstractList	Hierarchical secret sharing is among the most natural generalizations of threshold secret sharing, and it has attracted a lot of attention since the invention of secret sharing until nowadays. Several constructions of ideal hierarchical secret sharing schemes have been proposed, but it was not known what access structures admit such a scheme. We solve this problem by providing a natural definition for the family of the hierarchical access structures and, more importantly, by presenting a complete characterization of the ideal hierarchical access structures, that is, the ones admitting an ideal secret sharing scheme. Our characterization is based on the well-known connection between ideal secret sharing schemes and matroids and, more specifically, on the connection between ideal multipartite secret sharing schemes and integer polymatroids. In particular, we prove that every hierarchical matroid port admits an ideal linear secret sharing scheme over every large enough finite field. Finally, we use our results to present a new proof for the existing characterization of the ideal weighted threshold access structures. Hierarchical secret sharing is among the most natural generalizations of threshold secret sharing, and it has attracted a lot of attention since the invention of secret sharing until nowadays. Several constructions of ideal hierarchical secret sharing schemes have been proposed, but it was not known what access structures admit such a scheme. We solve this problem by providing a natural definition for the family of the hierarchical access structures and, more importantly, by presenting a complete characterization of the ideal hierarchical access structures, that is, the ones admitting an ideal secret sharing scheme. Our characterization is based on the well-known connection between ideal secret sharing schemes and matroids and, more specifically, on the connection between ideal multipartite secret sharing schemes and integer polymatroids. In particular, we prove that every hierarchical matroid port admits an ideal linear secret sharing scheme over every large enough finite field. Finally, we use our results to present a new proof for the existing characterization of the ideal weighted threshold access structures. [PUBLICATION ABSTRACT] Hierarchical secret sharing is among the most natural generalizations of threshold secret sharing, and it has attracted a lot of attention since the invention of secret sharing until nowadays. Several constructions of ideal hierarchical secret sharing schemes have been proposed, but it was not known what access structures admit such a scheme. We solve this problem by providing a natural definition for the family of the hierarchical access structures and, more importantly, by presenting a complete characterization of the ideal hierarchical access structures, that is, the ones admitting an ideal secret sharing scheme. Our characterization is based on the well known connection between ideal secret sharing schemes and matroids and, more specifically, on the connection between ideal multipartite secret sharing schemes and integer polymatroids. In particular, we prove that every hierarchical matroid port admits an ideal linear secret sharing scheme over every large enough finite field. Finally, we use our results to present a new proof for the existing characterization of the ideal weighted threshold access structures. Peer Reviewed
Author	Farras, O. Padro, C.
Author_xml	– sequence: 1 givenname: O. surname: Farras fullname: Farras, O. email: oriol.farras@urv.cat organization: Univ. Rovira i Virgili, Tarragona, Spain – sequence: 2 givenname: C. surname: Padro fullname: Padro, C. email: carlespl@ntu.edu.sg organization: Nanyang Technol. Univ., Singapore, Singapore
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References	ref35 ref13 beimel (ref1) 2011; lncs 6639 ref34 ref12 ref15 ref30 ref33 ref32 welsh (ref36) 1976 ref10 ito (ref17) 1987 fujishige (ref14) 1991; 47 ref2 edmonds (ref11) 2003; 2570 ref16 ref19 ref18 ref24 ref23 ng (ref26) 2003; 30 ref20 oxley (ref29) 1992 ref22 blakley (ref5) 1979 ref21 murota (ref25) 2003 ref28 ref27 ref8 ref7 schrijver (ref31) 2003 ref4 brickell (ref6) 1989; 6 ref3 collins (ref9) 0
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SubjectTerms	90 Operations research, mathematical programming 90B Operations research and management science 90C Mathematical programming Applied sciences Bismuth Boolean polymatroids Classificació AMS Code Division Multiple Access Construction Cryptography Electronic mail Exact sciences and technology Government hierarchical secret sharing ideal secret sharing schemes Information sharing Information theory Information, signal and communications theory integer polymatroids Integers Inventions Investigació operativa Joints Matemàtiques i estadística Mathematical analysis multi partite secret sharing multipartite secret sharing Operations research Periodic structures Ports Programació (Matemàtica) Programming (Mathematics) secret sharing Telecommunications and information theory Thresholds Vectors weighted secret sharing schemes weighted threshold secret sharing Àrees temàtiques de la UPC
Title	Ideal Hierarchical Secret Sharing Schemes
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