C. Blundo, A. De Santis, D.R. Stinson, U. Vaccaro. Graph decompositions and secret sharing schemes. J. Cryptology 8 (1995) 39--64.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....potential conflicts of interest: boss, family member, etc. A secret sharing scheme based on a threshold may be inappropriate. Although general access structures have been defined by Ito, Saito, and Nishizeki [16] and have been heavily studied (see, e.g. Blundo, De Santis, Stinson, and Vaccaro [1]) no scientific way has been suggested to construct such an access structure. We suggest that a set of participants should only be authorized by the access structure if the participants forms a k independent set (note that this kind of access structures are not monotone) Whence the access ....

C. Blundo, A. De Santis, D. Stinson, and U. Vaccaro. Graph decompositions and secret sharing schemes. In: Advances in Cryptology, Proc. of Eurocrypt '92, LNCS 658, pages 1--24, Springer Verlag, 1992.

....denotes the size of the secret and jV i j denotes the size of the share of participant P i [Karnin et al. 82] Capocelli at el. 93] Kurosawa, Okada 96] More tight lower bounds of jV i j such that jV i j jSj which depend on the access structure have also been presented [Capocelli at el. 93] Blundo at el. 92b] Brickell, Stinson 92] Blundo at el. 92a] Stinson 92] This result means that every participants must hold very large information keeping secret. It will cost them much. If share size can be reduced then each participant save costs or obtains higher security with same cost. So, it is desired jV i ....

....holds for any PSS [Capocelli at el. 93] Kurosawa et al. proved that [Kurosawa, Okada 96] jV i j jSj (3) for any PSS. This is a more tight bound than eq. 2) because log 2 jSj H(S) For PSSs with certain Gamma s, more tight lower bounds on jV i j than eq. 3) is known [Capocelli at el. 93] Blundo at el. 92b] Brickell, Stinson 92] Blundo at el. 92a] Stinson 92] McEliece and Sarwate [McEliece, Sarwate 81] showed that Shamir s (k; n) threshold secret sharing scheme [Shamir 79] is closely related to Reed Solomon codes. A (d; k; n) ramp scheme is an NSS such that Gamma 1 = fA V j jAj kg; ....

Blundo, C., De Santis, A., Stinson, D.R., Vaccaro, U.: "Graph decomposition and secret sharing schemes"; Proc. of Eurocrypt'92, Lecture Notes on Comput. Sci., 658, Springer Verlag (1992) 1--20

....the ideal access structures, and to nd bounds on the optimal information rate. A necessary condition for an access structure to be ideal was given in [6] in terms of matroids. A sucient condition is obtained from the vector space construction [5] Several techniques have been introduced in [4, 7, 18] in order to construct secret sharing schemes for some families of access structures, which provide lower bounds on the optimal information rate. Upper bounds have been found for several particular access structures by using some tools from Information Theory [2, 3, 8] A general method to nd ....

....on the optimal information rate for most access structures. Due to the diculty of nding a general solution, those problems have been studied in several particular classes of access structures: access structures on sets of four [17] and ve [12] participants, access structures de ned by graphs [2, 3, 4, 6, 7, 8, 18], bipartite access structures [14] and access structures with three or four minimal quali ed subsets [13] The ideal access structures in all these families have been completely characterized. The optimal information rate of almost all access structures on a set of at most ve participants has ....

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C. Blundo, A. De Santis, D.R. Stinson, U. Vaccaro. Graph decompositions and secret sharing schemes. J. Cryptology. 8 (1995), 39-64.

....2. Since they are not, our assumption that G is not connected must be false and # 0 must be the edge set of a connected graph. We now prove that G is a complete multipartite graph by contradiction. We assume that G is not a complete multipartite graph. Blundo, De Santis, Stinson, and Vaccaro [4] proved that any such graph must contain as an induced subgraph which is isomorphic to H or P 3 , where V (H) V (P 3 ) 2, 3, 4 , E(H) 4 , and E(P 3 ) 4 . First we will show that if G contains an induced subgraph which is isomorphic to H, we have a contradiction. K 3 is an ....

Carlo Blundo, Alfredo De Santis, Douglas R. Stinson, Ugo Vaccaro. Graph decomposition and secret sharing schemes. Journal of Cryptology, 8, pp. 3964, 1995.

....of K to each participant using the matrix M , i.e. participant P j receives the entry M r;j as his share. The general requirements of a perfect secret scheme translate into the following combinatorial conditions in the matrix model, cf. Stinson [84] and Blundo, De Santis, Stinson and Vaccaro [15]. Suppose that Gamma is an access structure. 1. If B 2 Gamma and M(r; P ) M(r ; P ) for all P 2 B, then M(r; D) M(r ; D) 2. If B 62 Gamma, then for every possible assignment f of shares to the participants in B, say f = f P : P 2 B) a nonnegative integer (f; B) exists such that ....

....such decompositions of access structures, Stinson [84] derives a lower bound: ae ( Gamma) R, where and R are two quantities defined in terms of the ideal decomposition of Gamma 0 . The decomposition construction and its precursor, the graph decomposition construction (cf. Blundo et al. [15]) can be formulated as linear programming problems in order to derive the best possible information rates that are obtainable using these constructions. Other ways of realizing schemes with optimal or close to optimal information rates are considered by Charnes and Pieprzyk [30] Their method ....

C. Blundo, A. De Santis, D. R. Stinson and U. Vaccaro. Graph decompositions and secret sharing schemes. J. of Cryptology, 8(1):39-64, 1995.

....where ISl denotes the size of the secret and IV I denotes the size of the share of participant Pi [Karnin et al. 82] Capocelli at el. 93] Kurosawa, Okada 96] More tight lower bounds of IV I such that IV I which depend on the access structure have also been presented [Capocelli at el. 93] Blundo at el. 92b] Brickell, Stinson 92] Blundo at el. 92a] Stinson 92] This re suit means that every participants must hold very large information keeping secret. It will cost them much. If share size can be reduced then each participant save costs or obtains higher security with same cost. So, it is desired ....

....showed that the above bound holds for any PSS [Capocelli at el. 93] Kurosawa et al. proved that [Kurosawa, Okada 96] Il I l (3) for any PSS. This is a more tight bound than eq. 2) because log For PSSs with certain Fs, more tight lower bounds on Il than eq. 3) is known [Capocelli at el. 93] Blundo at el. 92b] Brickell, Stinson 92] Blundo at el. 92a] Stinson 92] McEliece and Sarwate [McEliece, Sarwate 81] showed that Shamir s (k, n) threshold secret sharing scheme [Shamir 79] is closely related to Reed Solomon codes. A (d, k, n) ramp scheme is an NSS such that rl: far V IIAI r2: B c V I ....

Blundo, C., De Santis, A., Stinson, D.R., Vaccaro, U.: "Graph decomposition and secret sharing schemes"; Proc. of Eurocrypt'92, Lecture Notes on Cornput. Sci., 658, Springer Verlag (1992) 1-20

....for an access structure to be ideal was given in [6] in terms of matroids. A sucient condition is obtained from the vector space construction [5] which is a method to construct ideal secret sharing schemes for a wide family of access structures. Several techniques have been introduced in [4, 7, 16] in order to construct secret sharing schemes for some families of access structures, which provide lower bounds on the optimal information rate. Upper bounds have been found for several particular access structures by using some tools from Information Theory [2, 3, 8] 2 A general method to nd ....

....access structures. Due to the diculty of nding a general solution, those problems have been studied in several particular classes of access structures: access structures on a set of four participants [15] access structures on a set of ve participants [11] access structures de ned by graphs [2, 3, 4, 6, 7, 8, 16]; and bipartite access structures [12] The ideal access structures in all these families have been completely characterized. The optimal information rate of almost all access structures on a set of at most ve participants has been determined. Bounds on the optimal information rate, which are ....

C. Blundo, A. De Santis, D.R. Stinson, U. Vaccaro. Graph decompositions and secret sharing schemes. J. Cryptology. 8 (1995), 39-64.

....must remember the more reliable the scheme is. Theoretically, the known upper and lower bounds are very far from each other, and closing the gap even in some special cases is also an intriguing task. For a more complete description of the problem as well as a detailed list of references, see e.g. [1]. This paper is organized as follows. First we give the necessary de nitions, and then state the theorems to be proved. Section 3 gives the proofs, and in Section 4 we conclude the paper. For unde ned notions see [1] for secret sharing schemes, and [5] for those in information theory. 2 De ....

....description of the problem as well as a detailed list of references, see e.g. 1] This paper is organized as follows. First we give the necessary de nitions, and then state the theorems to be proved. Section 3 gives the proofs, and in Section 4 we conclude the paper. For unde ned notions see [1] for secret sharing schemes, and [5] for those in information theory. 2 De nitions In this section we give a rough de nition of the notions we shall use later. First we de ne formally what a perfect secret sharing scheme is, then connect it to certain submodular function. 1 Let G be a graph, we ....

C. Blundo, A. De Santis, D. R. Stinson, U. Vaccaro, Graph Decomposition and Secret Sharing Schemes Journal of Cryptology, Vol 8(1995) pp. 39-64. 7

....bound of the size of V i . Let jXj denotes the bit length of X. For simplicity, assume that S is uniformly distributed. Then, 10] showed that jV i j jSj for any PSS by extending the technique of [13] More tight lower bounds of jV i j which depend on the access structure have also been shown [10, 6, 9, 5, 15]. 3) The third question is what happens if jV i j = jSj. A PSS is called ideal if jV i j = jSj for all i. 8] characterized ideal PSSs in terms of a matroid. It is important that jV i j is as small as possible because, otherwise, the security of the system degrades. However, as mentioned in (2) ....

C.Blundo, A.De Santis, D.R.Stinson, U.Vaccaro, Graph decomposition and secret sharing schemes, Proceedings of Eurorypt'92, Lecture Notes on Comput. Sci., 658 (1992) 1--20

....opening a bank vault or even opening a safety deposit box. Secret sharing schemes are also used in management of cryptographic keys and multi party secure protocols. 7.2. 1 Size of Shares The question of how much information one has to distribute to participants is addressed in [137] 138] [102], 99] and [100] New techniques of constructing schemes by suitable decomposition of the access structure are presented. Moreover, it is proved that there are access structures for which there is a participant whose share of information has to be at least twice the secret size. The security of ....

C. Blundo, A. De Santis, D. R. Stinson, and U. Vaccaro. Graph decompositions and secret sharing schemes. In R. A. Rueppel, editor, Advances in Cryptology -- EUROCRYPT 92, volume 658 of Lecture Notes in Computer Science, pages 1--24. Springer-Verlag. To appear in Journal of Cryptology.

....denotes the size of the secret and jV i j denotes the size of the share of participant P i [Karnin et al. 82] Capocelli at el. 93] Kurosawa, Okada 96] More tight lower bounds of jV i j such that jV i j jSj which depend on the access structure have also been presented [Capocelli at el. 93] Blundo at el. 92b] Brickell, Stinson 92] Blundo at el. 92a] Stinson 92] This result means that every participants must hold very large information keeping secret. It will cost them much. If share size can be reduced then each participant save costs or obtains higher security with same cost. So, it is desired jV i ....

....holds for any PSS [Capocelli at el. 93] Kurosawa et al. proved that [Kurosawa, Okada 96] jV i j jSj (3) for any PSS. This is a more tight bound than eq. 2) because log 2 jSj H(S) For PSSs with certain Gamma s, more tight lower bounds on jV i j than eq. 3) is known [Capocelli at el. 93] Blundo at el. 92b] Brickell, Stinson 92] Blundo at el. 92a] Stinson 92] McEliece and Sarwate [McEliece, Sarwate 81] showed that Shamir s (k; n) threshold secret sharing scheme [Shamir 79] is closely related to Reed Solomon codes. A (d; k; n) ramp scheme is an NSS such that Gamma 1 = fA V j jAj kg; Gamma ....

Blundo, C., De Santis, A., Stinson, D.R., Vaccaro, U.: "Graph decomposition and secret sharing schemes"; Proc. of Eurocrypt'92, Lecture Notes on Comput. Sci., 658, Springer Verlag (1992) 1--20

....secret (see [12] for a simple information theoretic proof) Moreover, there are access structures for which any corresponding secret sharing scheme must give to some participant a share of size strictly bigger than the secret size. Lower bounds on the size of shares can be found in [12] 5] and [6]. The best lower bound on the size of shares that has been proved is dlog jSj 2 Gammaffl e, where jSj is the number of possible secrets and ffl is any constant 0 [5] Upper bounds on the size of shares have been given in [10] and [27] We also briefly mention some extended capabilities of ....

C. Blundo, A. De Santis, D. R. Stinson, and U. Vaccaro, Graph Decomposition and Secret Sharing Schemes, in "Advances in Cryptology -- EUROCRYPT 92", Lecture Notes in Computer Science, Vol. 658, R. Rueppel (Ed.), Springer-Verlag, pp. 1--24, 1993. Also to appear in: Journal of Cryptology.

....of our result. We also consider secret sharing schemes for access structures based on graphs, that is, access structures whose elements corresponds to subsets of vertices of a graph joined by at least one edge. Such a class of access structures has been extensively studied (see for example [6, 10, 22] and references therein quoted) One of the main result in the area is a theorem by Stinson [22] that proves the existence of a secret sharing scheme with information rate lower bounded by 2= d 1) where d is the maximum degree of the graph. We prove that this result is best possible, in the ....

C. Blundo, A. De Santis, D. R. Stinson, and U. Vaccaro, Graph Decomposition and Secret Sharing Schemes, Journal of Cryptology, vol. 8, pp. 39--64, 1995.

....a scheme an ideal scheme. Ideal schemes have been studied extensively; see for example [4, 5, 11, 9, 12] In the cases where ideal schemes do not exist, the objective is to construct a scheme with (average) information rate as close to one as possible. Research in this direction can be found in [6, 7, 2, 16, 10, 3, 17]. 1.2 Graph Access Structures The situation that has been studied the most is when the basis consists of the edges of a graph (i.e. the access structure has rank two) see [6, 2, 7, 3, 17] for example. If G is a graph, then we will denote the vertex set of G by V (G) and the edge set by E(G) ....

....(average) information rate as close to one as possible. Research in this direction can be found in [6, 7, 2, 16, 10, 3, 17] 1. 2 Graph Access Structures The situation that has been studied the most is when the basis consists of the edges of a graph (i.e. the access structure has rank two) see [6, 2, 7, 3, 17], for example. If G is a graph, then we will denote the vertex set of G by V (G) and the edge set by E(G) For a vertex v 2 V (G) the neighbourhood of v, denoted by N(v) consists of all vertices w such tath vw 2 E(G) If V 1 V (G) then the induced subgraph G[V 1 ] is defined to have vertex ....

[Article contains additional citation context not shown here]

C. Blundo, A. De Santis, D. R. Stinson, and U. Vaccaro. Graph decompositions and secret sharing schemes. Presented at EUROCRYPT '92, submitted to Journal of Cryptology.

No context found.

C. Blundo, A. De Santis, D. R. Stinson, and U. Vaccaro, Graph Decomposition and Secret Sharing Schemes, in "Advances in Cryptology -- Eurocrypt '92", Lecture Notes in Computer Science, Vol. 658, R. Rueppel Ed., Springer-Verlag, pp. 1--24, 1993. To appear in Journal of Cryptology.

....can be identified with the vertex set V (G) of a graph G = V (G) E(G) and the subsets of participants qualified to reconstruct the secret are only those containing an edge of G. Secret sharing schemes based on graph access structures have been extensively studied in several papers, such as [11, 12, 14, 6, 5, 34, 36]. We give both lower and upper bounds for infinite classes of access structures. Lower bounds are obtained using entropy arguments. We prove a general lower bound on the dealer s randomness for access structures based on graphs. As a result we obtain a general bound when the graph is the cycle Cn ....

....t (v; k; designs) A decomposition construction can be considered as a recursive technique that uses small schemes to build schemes for larger access 3 structures. The decomposition of a given access structure into smaller ones has been accomplished in several ways; we refer the reader to [12, 6, 35, 28]. Also, we study the access structures on at most five participants, obtaining exact values for the dealer s randomness for all access structures on at most four participants, and for all connected graphs on five vertices. Finally, we analyze the dealer s randomness of anonymous threshold schemes, ....

[Article contains additional citation context not shown here]

C. Blundo, A. De Santis, D. R. Stinson, and U. Vaccaro, Graph Decompositions and Secret Sharing Schemes, Journal of Cryptology, Vol. 8 (1995), pp. 39--64.

....rate of access structures based on graphs. Stinson in [43] presented new lower bounds on general access structures. Capocelli, De Santis, Gargano, and Vaccaro [17] gave the first example of access structures with information rate bounded away from 1. Blundo, De Santis, Stinson, and Vaccaro [9] analyzed the information rate and the average information rate of secret sharing schemes based on graphs. The average information rate is the ratio between the secret size and the arithmetic mean of the size of the shares for such schemes. They proved the existence of a gap in the values of ....

....and are the best possible for the considered structures since we exhibit secret sharing schemes that meet the bounds. In particular, we give the first proof of the existence of access structures with information rate and average information rate strictly less that 2=3. This solves a problem of [9]. In Section 3.1 we also consider the problem of efficiently testing if one of these low information rate access structures is a sub structure of an arbitrary access structure. This is important since it would immediately give an efficient way to get upper bounds on the information rate for ....

[Article contains additional citation context not shown here]

C. Blundo, A. De Santis, D. R. Stinson, and U. Vaccaro, Graph Decomposition and Secret Sharing Schemes, in "Advances in Cryptology -- EUROCRYPT '92", R. Rueppel Ed., "Lecture Notes in Computer Science", Vol. 658, Springer-Verlag, Berlin, pp. 1--24, 1993. To appear in Journal of Cryptology.

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C. Blundo, A. De Santis, D. R. Stinson, and U. Vaccaro, Graph Decomposition and Secret Sharing Schemes, Journal of Cryptology, Vol. 8, (1995), pp. 39-64.

....secret (see [12] for a simple information theoretic proof) Moreover, there are access structures for which any corresponding secret sharing scheme must give to some participant a share of size strictly bigger than the secret size. Lower bounds on the size of shares can be found in [12] 5] and [6]. The best lower bound on the size of shares that has been proved is dlog jSj 2 Gammaffl e, where jSj is the number of possible secrets and ffl is any constant 0 [5] Upper bounds on the size of shares have been given in [10] and [27] We also briefly mention some extended capabilities of ....

C. Blundo, A. De Santis, D. R. Stinson, and U. Vaccaro, Graph Decomposition and Secret Sharing Schemes, in "Advances in Cryptology -- EUROCRYPT 92", Lecture Notes in Computer Science, Vol. 658, R. Rueppel (Ed.), Springer-Verlag, pp. 1--24, 1993. Also to appear in: Journal of Cryptology.

No context found.

C. Blundo, A. De Santis, D.R. Stinson, U. Vaccaro. Graph decompositions and secret sharing schemes. J. Cryptology 8 (1995) 39--64.

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C. Blundo, A. De Santis, D.R. Stinson and U. Vaccaro. Graph decompositions and secret sharing schemes. Journal of Cryptology, Vol. 8, No. 1, pp. 39-64 (1995).

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C. Blundo, A. De Santis, D.R. Stinson, U. Vaccaro. Graph decompositions and secret sharing schemes. J. Cryptology 8 (1995) 39--64.

No context found.

C. Blundo, A. De Santis, D. R. Stinson, U. Vaccaro, Graph Decomposition and Secret Sharing Schemes Journal of Cryptology, Vol 8(1995) pp. 39--64.

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Blundo, C., De Santis, A., Stinson, D.R., Vaccaro, U., Graph Decompositions and Secret Sharing Schemes Lecture Notes in Computer Science vol. 658 pp 1-24

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C. Blund, A. De Santis, D. R. Stinson, U. Vaccaro, "Graph decomposition and secret sharing schemes", Journal of Cryptology, vol. 8, No. 1, pp. 39--64, 1995.

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