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IEEE Transactions on Information Theory

Ideal Hierarchical Secret Sharing Schemes

Oriol Farràs, Carles Padró

Published Jan 25, 2012• 69 citations
Publisher

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.

DOI
10.1109/tit.2011.2182034

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This thesis is dedicated to the study of secret sharing schemes, which are cryptographic methods to share information in a secure way. The topics that are considered in the thesis are two of the main open problems in secret sharing: the characterization of the ideal access structures and the optimization of the length of the shares for general access structures. These open problems are studied for multipartite secret sharing schemes. In these schemes the set of participants is divided into parts and the participants in each part have the same rights to obtain the secret. The results of the thesis are based on a new combinatorial property of secret sharing schemes, which is a connection between ideal multipartite secret sharing schemes and integer polymatroids. It provides new sufficient conditions and necessary conditions for an access structure to be ideal. Moreover, this connection is also used in the construction ideal linear multipartite secret sharing schemes. These results are useful for the study of multipartite access structures in which the number of parts is small in relation to the number of participants, and multipartite access structures in which the parts are related in a special way. This is the case of the family of hierarchical access structures, which are the ones in which the participants can be hierarchically ordered, and the family of tripartite access structures. Applying these results, the ideal access structures in these families are completely characterized. All the ideal multipartite secret sharing schemes presented in the literature are related to a particular family of integer polymatroids, the boolean ones. The analysis of these polymatroids leds to the find of new ideal multipartite secret sharing schemes. The optimization of the length of the shares is also studied for multipartite secret sharing schemes, in particular for the bipartite ones. The main results are a new method to find bound on the length of the shares that combines linear El tema d'aquesta tesi són els esquemes de compartició de secrets, uns mètodes criptogràfics que permeten fragmentar missatges de manera segura. En particular s'aborden dos dels principals problemes oberts en aquest camp: la caracterització de les estructures d'accés que admeten esquemes de compartició de secrets ideals i l'optimització de la llargada dels fragments en construccions per estructures d'accés generals. Aquests problemes oberts s'estudien per estructures d'accés multipartides, que són estructures en què el conjunt de participants està dividit en diferents parts i a cada part tots els participants tenen el mateix rol. Aquest plantejament resulta útil per estudiar famílies d'estructures d'accés en què el nombre de parts és petit i estructures en què hi ha alguna relació especial entre les parts, com les jeràrquiques i les compartmentades. L'eina principal d'aquest estudi és una nova propietat de les estructures ideals que es presenta en aquesta tesi, una connexió entre les estructures ideals multipartides i els polimatroides enters. A partir d'aquesta connexió s'obté una condició suficient i una de necessària perquè una estructura d'accés multipartida sigui ideal. Així, aquesta connexió permet discutir la idealitat de les estructures d'accés i també, en el cas que el polimatoride sigui representable, construir esquemes ideals a partir de les representacions. Aplicant aquests resultats s'obté un nou marc per descriure i analitzar mètodes per construir esquemes multipartits ideals. Com a resultat d'aquest estudi s'obté una caracterització completa de les estructures d'accés tripartides ideals i de les jeràrquiques ideals. Una estructura és jeràrquica si el conjunt de participants es pot ordenar de tal manera que si en un conjunt autoritzat es canvia un participant per un de jeràrquicament superior, el nou conjunt també és autoritzat. A partir d'aquest resultat s'obté una nova caracterització de les estructures de llindar amb pesos ideals. Després d'analitzar tots els esquemes de compartició de secrets ideals multipartits presentats fins l'actualitat, es mostra que tots ells estan relacionats amb una família molt senzilla de polimatroides enters, els booleans. A partir d'aquests polimatroides s'obté una nova família d'estructures ideals, les compartmentades, que inclouen diverses famílies estudiades prèviament. L'altre problema obert considerat a la tesi, l'estudi de l'optimització de la llargada dels fragments, es centra en les estructures bipartides. Combinant tècniques de programació lineal i polimatroides, es presenta un mètode per calcular fites en la llargada dels fragments per estructures d'accés no ideals que és vàlid per qualsevol estructura multipartida. A partir de tècniques algebraiques es presenten noves fites i una família d'esquemes bipartits òptims.

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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.
This thesis is dedicated to the study of secret sharing schemes, which are cryptographic methods to share information in a secure way. The topics that are considered in the thesis are two of the main open problems in secret sharing: the characterization of the ideal access structures and the optimization of the length of the shares for general access structures. These open problems are studied for multipartite secret sharing schemes. In these schemes the set of participants is divided into parts and the participants in each part have the same rights to obtain the secret. The results of the thesis are based on a new combinatorial property of secret sharing schemes, which is a connection between ideal multipartite secret sharing schemes and integer polymatroids. It provides new sufficient conditions and necessary conditions for an access structure to be ideal. Moreover, this connection is also used in the construction ideal linear multipartite secret sharing schemes. These results are useful for the study of multipartite access structures in which the number of parts is small in relation to the number of participants, and multipartite access structures in which the parts are related in a special way. This is the case of the family of hierarchical access structures, which are the ones in which the participants can be hierarchically ordered, and the family of tripartite access structures. Applying these results, the ideal access structures in these families are completely characterized. All the ideal multipartite secret sharing schemes presented in the literature are related to a particular family of integer polymatroids, the boolean ones. The analysis of these polymatroids leds to the find of new ideal multipartite secret sharing schemes. The optimization of the length of the shares is also studied for multipartite secret sharing schemes, in particular for the bipartite ones. The main results are a new method to find bound on the length of the shares that combines linear El tema d'aquesta tesi són els esquemes de compartició de secrets, uns mètodes criptogràfics que permeten fragmentar missatges de manera segura. En particular s'aborden dos dels principals problemes oberts en aquest camp: la caracterització de les estructures d'accés que admeten esquemes de compartició de secrets ideals i l'optimització de la llargada dels fragments en construccions per estructures d'accés generals. Aquests problemes oberts s'estudien per estructures d'accés multipartides, que són estructures en què el conjunt de participants està dividit en diferents parts i a cada part tots els participants tenen el mateix rol. Aquest plantejament resulta útil per estudiar famílies d'estructures d'accés en què el nombre de parts és petit i estructures en què hi ha alguna relació especial entre les parts, com les jeràrquiques i les compartmentades. L'eina principal d'aquest estudi és una nova propietat de les estructures ideals que es presenta en aquesta tesi, una connexió entre les estructures ideals multipartides i els polimatroides enters. A partir d'aquesta connexió s'obté una condició suficient i una de necessària perquè una estructura d'accés multipartida sigui ideal. Així, aquesta connexió permet discutir la idealitat de les estructures d'accés i també, en el cas que el polimatoride sigui representable, construir esquemes ideals a partir de les representacions. Aplicant aquests resultats s'obté un nou marc per descriure i analitzar mètodes per construir esquemes multipartits ideals. Com a resultat d'aquest estudi s'obté una caracterització completa de les estructures d'accés tripartides ideals i de les jeràrquiques ideals. Una estructura és jeràrquica si el conjunt de participants es pot ordenar de tal manera que si en un conjunt autoritzat es canvia un participant per un de jeràrquicament superior, el nou conjunt també és autoritzat. A partir d'aquest resultat s'obté una nova caracterització de les estructures de llindar amb pesos ideals. Després d'analitzar tots els esquemes de compartició de secrets ideals multipartits presentats fins l'actualitat, es mostra que tots ells estan relacionats amb una família molt senzilla de polimatroides enters, els booleans. A partir d'aquests polimatroides s'obté una nova família d'estructures ideals, les compartmentades, que inclouen diverses famílies estudiades prèviament. L'altre problema obert considerat a la tesi, l'estudi de l'optimització de la llargada dels fragments, es centra en les estructures bipartides. Combinant tècniques de programació lineal i polimatroides, es presenta un mètode per calcular fites en la llargada dels fragments per estructures d'accés no ideals que és vàlid per qualsevol estructura multipartida. A partir de tècniques algebraiques es presenten noves fites i una família d'esquemes bipartits òptims.
Secret-sharing schemes are a tool used in many cryptographic protocols. In these schemes, a dealer holding a secret string distributes shares to the parties such that only authorized subsets of participants can reconstruct the secret from their shares. The collection of authorized sets is called an access structure. An access structure is ideal if there is a secret-sharing scheme realizing it such that the shares are taken from the same domain as the secrets. Brickell and Davenport (Journal of Cryptology, 1991) have shown that ideal access structures are closely related to matroids. They give a necessary condition for an access structure to be ideal-the access structure must be induced by a matroid. Seymour (Journal of Combinatorial Theory B, 1992) has proved that the necessary condition is not sufficient: There exists an access structure induced by a matroid that does not have an ideal scheme. The research on access structures induced by matroids is continued in this work. The main result in this paper is strengthening the result of Seymour. It is shown that in any secret-sharing scheme realizing the access structure induced by the Vamos matroid with domain of the secrets of size k, the size of the domain of the shares is at least k + Omega(radic(k)). The second result considers nonideal secret-sharing schemes realizing access structures induced by matroids. It is proved that the fact that an access structure is induced by a matroid implies lower and upper bounds on the size of the domain of shares of subsets of participants even in nonideal schemes (as long as the shares are still relatively short). This generalized results of Brickell and Davenport for ideal schemes. Finally, an example of a nonideal access structure that is nearly ideal is presented.
A secret-sharing scheme is a cryptographic protocol to distribute a secret state in an encoded form among a group of players such that only authorized subsets of the players can reconstruct the secret. Classically, efficient secret-sharing schemes have been shown to be induced by matroids. Furthermore, access structures of such schemes can be characterized by an excluded minor relation. No such relations are known for quantum secret-sharing schemes. In this paper we take the first steps toward a matroidal characterization of quantum-secret-sharing schemes. In addition to providing a new perspective on quantum-secret-sharing schemes, this characterization has important benefits. While previous work has shown how to construct quantum-secret-sharing schemes for general access structures, these schemes are not claimed to be efficient. In this context the present results prove to be useful; they enable us to construct efficient quantum-secret-sharing schemes for many general access structures. More precisely, we show that an identically self-dual matroid that is representable over a finite field induces a pure-state quantum-secret-sharing scheme with information rate 111.
Optimizing the maximum, or average, length of the shares in relation to the length of the secret for every given access structure is a difficult and long-standing open problem in cryptology. Most of the known lower bounds on these parameters have been obtained by implicitly or explicitly using that every secret sharing scheme defines a polymatroid related to the access structure. The best bounds that can be obtained by this combinatorial method can be determined by using linear programming, and this can be effectively done for access structures on a small number of participants. By applying this linear programming approach, we improve some of the known lower bounds for the access structures on five participants and the graph access structures on six participants for which these parameters were still undetermined. Nevertheless, the lower bounds that are obtained by this combinatorial method are not tight in general. For some access structures, they can be improved by adding to the linear program non-Shannon information inequalities as new constraints. We obtain in this way new separation results for some graph access structures on eight participants and for some ports of non-representable matroids. Finally, we prove that, for two access structures on five participants, the combinatorial lower bound cannot be attained by any linear secret sharing scheme.
Beimel, Tassa and Weinreb (2008) and Farras and Padro (2010) partially characterized access structures of ideal weighted threshold secret sharing schemes in terms of the operation of composition. They classified indecomposable ideal weighted threshold access structures, and proved that any other ideal weighted threshold access structure is a composition of indecomposable ones. It remained unclear which compositions of indecomposable weighted threshold access structures are weighted. In this paper we fill the gap. Using game-theoretic techniques we determine which compositions of indecomposable ideal access structures are weighted, and obtain an if and only if characterization of ideal weighted threshold secret sharing schemes.
In a bipartite secret sharing scheme, the set of participants is divided into two parts, and all participants in each part play an equivalent role. The ideal bipartite access structures were characterized by Padr´o and S´aez, but it is not known which is the optimal information rate of non ideal bipartite access structures. Determining the optimal information rate of general access structures is one of the major problems in secret sharing. We study this open problem restricted to the bipartite access structures, obtaining a new method to compute bounds on the optimal information rate. Namely, by using the connection between secret sharing schemes and polymatroids, we show a linear programming problem that determines, for each access structure, an upper bound on the information rate. Moreover, we show new optimal constructions for certain bipartite access structures.