Citations: the Classification of Ideal Secret Sharing Schemes

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E. F. Brickell and D. M. Davenport. On the classification of ideal secret sharing schemes. J. of Cryptology, 4(73):123--134, 1991.

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On the Power of Nonlinear Secret-Sharing - Beimel, Ishai (2001) (2 citations) (Correct)

....the above state of affairs. To this end, we take two different approaches. By default, we ignore the computational complexity of the scheme. However, most of our efficient constructions are also computationally efficient. We explicitly indicate when this is not the case. A construction of [20] has been shown to be incorrect by [ Specific candidates. The main contribution of this work is the construction of specific efficient nonlinear secretsharing schemes, whose access structures are conjectured to be hard. We present two main schemes, whose access structures are related to two ....

E. F. Brickell and D. M. Davenport. On the classification of ideal secret sharing schemes. J. of Cryptology, 4(73):123--134, 1991.

Distributing the Encryption and Decryption of a Block.. - Martin, Safavi-Naini.. (2003) (Correct)

....about the key by pooling their shares. In perfect secret sharing schemes it can be seen that any share must be at least the size of the secret, in other words H(S i ) H(K) for i = 1, n. If equality in this bound is met for each share set then the secret sharing scheme is called ideal [6]. Secret sharing schemes defined on n participants, whose access structure consists of all sets of size at least t are referred to as (t, n) threshold schemes. The well known (t, n) threshold schemes defined by Shamir [26] are examples of ideal threshold schemes. 3.3 Block Cipher Sharing Block ....

E.F. Brickell and D.M. Davenport. On the classification of ideal secret sharing schemes. J. of Cryptology, 4 (1991)123--134.

Some Basic Properties of General Nonperfect Secret Sharing.. - OGATA, KUROSAWA (1998) (1 citation) (Correct)

....protocols to cope with faulty players. Franklin and Yung used a (d; k; n) ramp scheme to parallelize a multi party protocol d times [Franklin, Yung 92] Their method can reduce the communication complexity although only k Gamma d 1 faulty players can be allowed. Brickell and Davenport [Brickell, Davenport 91] characterized ideal PSS in terms of a matroid. Kurosawa et al. generalized this result to NSSs as follows [Kurosawa et al. 93] Definition 7. Kurosawa et al. 93] Suppose that S = S 1 ffi S 2 ffi Delta Delta Delta ffi S d and jS i j = jSj=d for all i (ffi means concatenation) Let W = fS ....

Brickell, E.F., Davenport, D.M.: "On the classification of ideal secret sharing schemes"; Journal of Cryptology, 4, 2 (1991) 123--134

Crypto Topics And Applications II - Seberry, Charnes, Pieprzyk.. (Correct)

....schemes by this construction. In particular the access structure Gamma(G) where G = V; E) is a complete multigraph can be realized as an ideal scheme. A proof of this is given by Stinson [84] A relation between ideal secret sharing schemes and matroids was established by Brickell and Davenport [19]. The matroid theory counterpart of a minimal linearly dependent set of vectors in a vector space is called a circuit. A coordinatizable matroid is one which can be mapped into a vector space over a field in a way that preserves linear independence. Brickell and Davenport [19] prove the following ....

....and Davenport [19] The matroid theory counterpart of a minimal linearly dependent set of vectors in a vector space is called a circuit. A coordinatizable matroid is one which can be mapped into a vector space over a field in a way that preserves linear independence. Brickell and Davenport [19] prove the following theorem about coordinatizable matroids. Theorem 3 ( 19] Suppose the connected matroid M = X; I) is coordinatizable over a finite field. Let x 2 X and let P = X nfxg. Then there exists an ideal scheme for the connected access structure having basis Gamma 0 = fC n fxg : x 2 ....

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E. F. Brickell and D. M. Davenport. On the classification of ideal secret sharing schemes. Journal of Cryptology, 4:123-134, 1991.

Some Basic Properties of General Nonperfect Secret Sharing.. - Ogata, Kurosawa (1998) (1 citation) (Correct)

....multi party protocols to cope with faulty players. Franklin and Yung used a (d, k, n) ramp scheme to parallelize a multi party protocol d times [Franklin, Yung 92] Their method can reduce the communication complexity although only k d 1 faulty players can be allowed. Brickell and Davenport [Brickell, Davenport 91] characterized ideal PSS in terms of a matroid. Kurosawa et al. generalized this result to NSSs as follows [Kurosawa et al. 93] Definition7. Kurosawa et al. 93] Suppose that S = S1 o S2 o . o Sa and IS I ISl d for all i (o means concatenation) Let W S1, Sa, V1, Vn . We say that ....

Brickell, E.F., Davenport, D.M.: "On the classification of ideal secret sharing schemes"; Journal of Cryptology, 4, 2 (1991) 123-134

Lower Bounds for Monotone Span Programs - Beimel, Gál, Paterson (1994) (9 citations) (Correct)

....The question whether there exist more efficient schemes, or if there exists a Boolean function that does not have (space ) efficient schemes is open. This problem is one of the most important open problems concerning secret sharing. Some weak lower bounds on the length of the shares were proved in [15, 2, 7, 9, 6, 20, 10]. The best lower bound was proved by Csirmaz [11] His proof shows that for every m there exists a Boolean function with m variables for which, in every secret sharing scheme, the sum of the lengths of the shares is Omega Gamma m = log m) times the length of the secret (for every finite set of ....

.... function with m variables for which, in every secret sharing scheme, the sum of the lengths of the shares is Omega Gamma m = log m) times the length of the secret (for every finite set of possible secrets) Small monotone span programs give rise to efficient linear secret sharing schemes (see [7, 14, 4]) We call these schemes linear, since the shares are a linear combination of the secret and some random inputs. Karchmer and Wigderson [14] proved that if there is a monotone span program of size s for some function then there exists a scheme for the corresponding secret sharing problem in which ....

[Article contains additional citation context not shown here]

E. F. Brickell and D. M. Davenport. On the Classification of Ideal Secret Sharing Schemes. Journal of Cryptology, 4(73)(1991) 123--134.

On the Power of Nonlinear Secret-Sharing - Beimel, Ishai (2001) (2 citations) (Correct)

....in this class [9, 2, 35] As opposed to linear secret sharing schemes, nearly nothing is known for general (i.e. possibly nonlinear) schemes. Several constructions of nonlinear secret sharing schemes have been suggested, both for the threshold case [55, 29, 47] and for general access structures [19, 33]. The question of basing verifiable secret sharing and secure multi party computation on nonlinear secret sharing has been recently studied in [24] However, none of these works provides evidence that nonlinear schemes are significantly more powerful than their linear counterparts. The relation ....

E. F. Brickell and D. M. Davenport. On the classification of ideal secret sharing schemes. J. of Cryptology, 4(73):123-- 134, 1991.

General Short Computational Secret Sharing Schemes - Béguin, Cresti (Correct)

....is of size at least n log n times the size of the secret k. We have better upper bounds when the access structure is based on graphs. If, for example, the graph on which the access structure is based is complete multipartite, then there exists an ideal perfect secret sharing scheme for A (see [3]) and the size of the shares becomes log jSj= max log jKj. Otherwise, using bounds found in [17] we can say that log jV i j log jKj( Delta 1) 2; where Delta is the maximum degree of the graph. Moreover, better bounds on V i can be obtained if the graph is acyclic. 5 Conclusions We have ....

E. F. Brickell and D. M. Davenport, On the Classification of Ideal Secret Sharing Schemes, J. Cryptology, Vol. 4, No. 2, pp. 123--124, 1991.

Decomposition Constructions for Secret Sharing Schemes - Stinson (1998) (15 citations) (Correct)

....scheme realizing cl( Gamma) It is easy to prove that ae e ae 1 for any Gamma, and that ae = 1 if and only if e ae = 1. Since ae = e ae = 1 is the optimal situation, we refer to such 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 ....

....graph is a vertex disjoint union of cliques. Note that the complete graph K n can be thought of as a complete multipartite graph with n parts of size 1. We now briefly mention some results we will need later. Ideal schemes for connected graphs were characterized by Brickell and Davenport [5], as follows: Theorem 1.1 Suppose G is a connected graph. Then ae (G) 1 if and only if G is a complete multipartite graph. The following result from [6] specifies some values of q for which these ideal schemes can be constructed. Corollary 1.2 Suppose q t is a prime power. Then there is a ....

E. F. Brickell and D. M. Davenport. On the classification of ideal secret sharing schemes. J. Cryptology 4 (1991), 123--134.

Some New Results on Nonperfect Secret Sharing Schemes - Ogata, Kurosawa (1995) (Correct)

....[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) jV i j jSj in a PSS. Therefore, if we desire that jV i j jSj, the secret sharing scheme must be an NSS. ....

E.F.Brickell, D.M.Davenport, On the classification of ideal secret sharing schemes, Journal of Cryptology, vol.4, No.2 (1991) 123--134

Characterizing Ideal Weighted Threshold Secret - Sharing Amos Beimel (2005) (4 citations) (Correct)

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E. F. Brickell and D. M. Davenport. On the classification of ideal secret sharing schemes. J. of Cryptology, 4(73):123--134, 1991.

Ideal Multipartite Secret Sharing Schemes - Oriol Farras Jaume (2006) (Correct)

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E.F. Brickell, D.M. Davenport. On the classification of ideal secret sharing schemes. J. Cryptology 4 (1991) 123--134.

On Ideal Non-Perfect Secret Sharing Schemes - Published In Christianson (Correct)