D.R. Stinson. Decomposition constructions for secret-sharing schemes. IEEE Trans. Inform. Theory 40 (1994) 118--125.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....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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D.R. Stinson. Decomposition constructions for secret-sharing schemes. IEEE Trans. on Information Theory. 40 (1994), 118-125. 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 ....

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D.R. Stinson. Decomposition constructions for secret-sharing schemes. IEEE Trans. on Information Theory. 40 (1994), 118-125.

....and combining secret sharing schemes on these substructures in order to obtain a secret sharing scheme for Gamma. For instance, one of the most powerful decomposition techniques to construct secret sharing schemes with good information rate is the decomposition construction due to Stinson [27]. A linear secret sharing scheme is obtained when combining linear schemes in a decomposition construction. 7 4.2 The Scheme In this section, we present a method to construct a metering scheme realizing an access structure Gamma from any linear secret sharing scheme realizing the same access ....

D.R. Stinson, Decomposition Constructions for Secret-Sharing Schemes, IEEE Transactions on Information Theory, Vol. 40, pp. 118--125, 1994. 14

....to realize secret sharing schemes for any given monotone access structure. In both constructions, the information rate decreases exponentially as a function of n, the number of participants. There are several performance and efficiency measures proposed for analyzing secret sharing schemes [5,11 14]. Their goal is to maximize the information rate of a secret sharing scheme. Brickell and Stinson [5] studied a perfect secret sharing scheme for graph based access structure G where the monotone increasing access structure G contains the pairs of participants corresponding to edges (the ....

.... of subsets of participants corresponding to any independent set of the graph) They proved that, for any graph G with n vertices having maximum degree d, there exists a perfect secret sharing scheme for the access structure based on G in which the information rate is at least 2 (d 3) Stinson [14] improved the general result that there exists a perfect secret sharing scheme realizing access structure based on G in which the information rate is at least 2 (d 1) After that, van Dijk [12] showed that Stinson s lower bound is tight because he proved that there exists graphs having maximum ....

D.R. Stinson, "Decomposition constructions for secret sharing schemes," IEEE Trans. Inform. Theory 40(1) 118-125 (1994).

....as corollaries of our result. Moreover, we prove that for any integer d there exists a d regular graph for which any secret sharing scheme has information rate upper bounded by 2= d 1) This improves on van Dijk s result [14] and matches the corresponding lower bound proved by Stinson in [22]. Index terms : Secret Sharing, Data Security, Entropy, Information Rate, and Cryptography. 1 Introduction A secret sharing scheme is a protocol by means of which a dealer distributes a secret s among a set of participants P in such a way that only qualified subsets of P can reconstruct the ....

....of the basic issue in the area of secret sharing schemes is that of estimating the information rate of the scheme, that is, the ratio between the size of the secret and that of the largest share given to any participant. This problem has received considerable attention in the last few years (e.g. [1, 5, 4, 10, 11, 13, 14, 22]) The practical relevance of this issue is based on the following observations: Firstly, the security of any system tends to degrade as the amount of information that must be kept secret, i.e. the shares of the participants, increases. Secondly, if the shares given to participants are too long, ....

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D. R. Stinson, Decomposition Constructions for Secret Sharing Schemes, IEEE Trans. on Inform. Theory, vol. IT-40, pp.118-125, Jan. 1994.

....Each vertex gets one bit from each of its neighbor, plus two for its own, a total two more than its degree. This is for two bits of secret, which gives the average rate 2= d 2) where d is the average degree. By a more sophisticated argument this still can be improved: Theorem 2. 3 (Stinson [10]) For any graph G with average degree d, G) 2= d 1) The following theorem, which we also quote without proof, shows that for dense graphs, i.e. graphs with average degree near to n, the previous bound is not the best possible. Theorem 2.4 (Erd os and Pyber [7] There is a constant c 0 ....

....is also c log n n . Unfortunately the entropy method, presently the only available method for proving upper bounds, cannot give better estimate than 1= 1 (G) where (G) is the size of the maximal independent set, and this is log n for random graphs. From the other side Stinson showed in [10] that the average information rate for any cycle of length 5 is exactly 2=3. Capocelli et al. [3] constructed access structures with information rate below 1=2 . For each 0 and d 2 van Dijk in [6] constructed a d regular graph with information rate below 2= d 1) In this paper we ....

D. R. Stinson, Decomposition construction for secret sharing schemes, IEEE Trans. Inform. Theory Vol 40(1994) pp. 118-125. 8

....from K(P ) chosen according to some, non necessarily uniform, probability distribution. Given a set of participants A = fP i 1 ; P i r g P , where i 1 i 2 Delta Delta Delta i r , denote by K(A) K(P i 1 ) Theta Delta Delta Delta Theta K(P i r ) We represent, as in [25], a perfect secret sharing scheme, or simply a secret sharing scheme, Sigma by a collection of distribution rules. A distribution rule is a function f : P [ fDg K(P) S which satisfies the conditions f(D) 2 S and f(P i ) 2 K(P i ) for i = 1; 2; n. A distribution rule f represents a ....

D. R. Stinson, Decomposition Constructions for Secret Sharing Schemes, IEEE Trans. on Inform. Theory, Vol. IT-40, pp. 118--125, 1994.

....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 ....

....) chosen according to some specified (not necessarily uniform) probability distribution, which is public knowledge. Given a set of participants A = fP i 1 ; P i r g P , with i 1 : i r , denote K(A) K(P i 1 ) Theta Delta Delta Delta Theta K(P i r ) We represent, as in [36], a secret sharing scheme Sigma by a pair (F ; f Pi s g s2S ) where F is a collection of distribution rules and f Pi s g s2S is a family of probability distributions. A distribution rule is a function f : P [ fDg [P2P K(P ) S which satisfies the conditions that f(D) 2 S and f(P i ) 2 K(P i ....

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D. R. Stinson, Decomposition Constructions for Secret Sharing Schemes, IEEE Trans. Inform. Theory, Vol. 40 (1994), pp. 118-125.

....does this by giving each participant P 2 P a share from K(P ) chosen according to some, non necessarily uniform, probability distribution. Given a set of participants A = fP i 1 ; P i r g ae P , denote by K(A) K(P i 1 ) Theta Delta Delta Delta Theta K(P i r ) We represent, as in [44], a secret sharing scheme Sigma by a collection of distribution rules. A distribution rule is a function f : P [ fDg K(P) S which satisfies the conditions f(D) 2 S and f(P i ) 2 K(P i ) for i = 1; 2; n. A distribution rule f represents a possible distribution of shares to the ....

....result for trees. Lemma 2.1 Let G be a tree. Then for any set of secrets S of cardinality q 2, there exists a secret sharing scheme Sigma with information rate ae(G; U S ; Sigma) 1=2. In Section 3 we will show how to improve this bound for any tree. The following results, proved in [9] and [44] will be used to obtain good secret sharing schemes for graphs with maximum degree 3: Theorem 2.2 Let C n be a cycle of length n; n 5: For any set of secrets S of cardinality q 2 , with q n, a secret sharing scheme Sigma for C n exists with information rate ae(C n ; U S ; Sigma) 2=3. ....

D. R. Stinson, Decomposition Constructions for Secret Sharing Schemes, IEEE Trans. Inform. Theory, Vol. 40, pp. 118--125, 1994.

....shares distributed to participants since the security of a system degrades as the amount of the information that must be kept secret increases. Recently, several papers studied this topic and both upper bounds and lower bounds on the size of the shares have been provided [3] 4] 5] 6] 8] [19]. In this paper we study the amount of secret information that must be given to participants in terms of the probability that the previously described attack be successful. Our motivations are based on the observation that the Tompa and Woll secret sharing scheme requires that each participant ....

D. R. Stinson, Decomposition Constructions for Secret Sharing Schemes, Technical Report UNL-CSE-92-020, Department of Computer Science and Engineering, University of Nebraska, September 1992.

....with the basic terminology in graph theory. a share from K(P ) chosen according to some, non necessarily uniform, probability distribution. Given a set of participants A = fP i 1 ; P i r g ae P , denote by K(A) K(P i 1 ) Theta Delta Delta Delta Theta K(P i r ) We represent, as in [42], a secret sharing scheme Sigma by a collection of distribution rules. A distribution rule is a function f : P [ fDg K(P) S which satisfies the conditions f(D) 2 S and f(P i ) 2 K(P i ) for i = 1; 2; n. A distribution rule f represents a possible distribution of shares to the ....

....result for trees. Lemma 2.1 Let G be a tree. Then for any set of secrets S of size q 2, there exists a secret sharing scheme Sigma with information rate ae(G; U S ; Sigma) 1=2. In Section 3 we will show how to improve this bound for any tree. The following results, proved in [9] and [42] will be used to obtain good secret sharing schemes for graphs with maximum degree 3: Theorem 2.2 Let C n be a cycle of length n; n 5: For any set of secrets S of size q 2 , with q n, a secret sharing scheme for C n exists with optimal information rate 2=3: The following lemmas have been ....

D. R. Stinson, Decomposition Constructions for Secret Sharing Schemes, IEEE Trans. Inform. Theory, 40:118--125, 1994.

....shares to the participants. The family of distribution rules F can also be depicted as a matrix M , each row of which corresponds to one distribution rule. One column of M will be indexed by D, and the remaining columns are indexed by the members of P . 2 A similar model has been proposed in [38] where, for any s 2 S, the probability distribution Pi s is the uniform one. Any secret sharing scheme for secrets in S and a probability distribution fp S (s)g s2S naturally induce a probability distribution on K(A) for any A P . Denote such probability distribution by fp K(A) a)g a2K(A) ....

....set of participants qualified to reconstruct the secret are only those containing an edge of G. Equivalently, these are access structures that coincides with the closure of E(G) for some graph G = V (G) E(G) Secret sharing schemes for such access structures have been extensively studied in [8, 10, 15, 16, 17, 38]. For an access structure A which consists of the closure of the edge set of a graph G we denote the dealer s randomness by (G; jSj) A; jSj) The next theorem proves a gap on the values that (G; jSj) can take, that is, the dealer s randomness is either log jSj or 2 log jSj, depending on ....

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Stinson D. R., Decomposition Constructions for Secret Sharing Schemes, IEEE Trans. Inform. Theory, 40, 1994, pp. 118--125.

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D. R. Stinson. Decomposition constructions for secret sharing schemes. To appear in IEEE Trans. Inform. Theory.

....c copies of S 0 and S 1 to get a ( Gamma Qual ; Gamma Forb ; m Delta c) VCS with contrast c. In this section we describe a general technique to construct VCS having any higher contrast, which provides better schemes with respect to the value of m. This technique was introduced by Stinson [10] in the context of secret sharing schemes and it is referred to as a (w; decomposition. For the rest of this section, we confine our attention to strong access structures. Let ( Gamma Qual ; Gamma Forb ) be a strong access structure having basis Gamma 0 and let ; w 1 be integers. A (w; ....

D. R. Stinson, Decomposition Constructions for Secret Sharing Schemes, IEEE Trans. Inform. Theory, Vol. 40, N. 1, pp. 118--125, 1994.

....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. Non anonymous secret sharing schemes for graph access structures have been extensively studied in several papers, such as [5, 6, 8, 4, 3, 24, 25]. 2 Definitions and Notation A perfect secret sharing scheme permits a secret to be shared among a set P of w participants in such a way that a qualified subset of P can recover the secret, but any non qualified subset has absolutely no information on the secret. An access structure Gamma is ....

D. R. Stinson, Decomposition Constructions for Secret Sharing Schemes, IEEE Trans. Inform. Theory, 40 (1994), 118--125.

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D. R. Stinson. Decomposition constructions for secret sharing schemes. To appear in IEEE Trans. Inform. Theory.

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D.R. Stinson. Decomposition constructions for secret-sharing schemes. IEEE Trans. Inform. Theory 40 (1994) 118--125.

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D. R. Stinson, Decomposition construction for secret sharing schemes, IEEE Trans. Inform. Theory Vol 40(1994) pp. 118-125. 11

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D.R. Stinson, Decomposition Constructions for Secret Sharing Schemes, IEEE Trans. Inform. Theory, vol. IT-40 (1994), 118-125.

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