K. M. Martin. Discrete Structures in the Theory of Secret Sharing. PhD thesis, University of London, 1991.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....that, when the general access structure considered in the above construction is a threshold structure, a secret sharing scheme with information rate ae = 1 (called ideal [42] i.e. where each share has the same size of the secret, realizing the access structure does exist. As shown by K. Martin [32] and, independently by Dawson et al. 20] if we represent such a secret sharing scheme by means of a distribution table, this table is exactly an orthogonal array. For the general case, a proof that the indexing structure satisfies the same properties of an orthogonal array can be achieved ....

K. Martin, Discrete Structures in the Theory of Secret Sharing, PhD Thesis, University of London, 1991.

....the t Gamma 1 given shares and such that M(w 1; r) K 0 . It can be proved easily that jSj jKj in any (t; w) threshold scheme. The construction described above provides schemes in which this bound is met with equality, i.e. in which the shares are as small as possible. A result of Martin [67] shows that the converse also holds: Combinatorial Designs in Communications 21 Theorem 5.1 There exists a (t; w) threshold scheme in which jSj = jKj = v if and only if there exists an OA 1 (t; w 1; v) We stated above that jSj jKj in any (t; w) threshold scheme. This result can be easily ....

K.M. Martin, Discrete Structures in the Theory of Secret Sharing, Ph. D. Thesis, University of London, 1991.

....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 ....

....ae (G) 2=3. 2 A Decomposition Construction Our main recursive construction uses small schemes as building blocks in the construction of larger schemes. We call this the decomposition construction. Note that various versions of this construction have been described in several papers, such as [6, 2, 16, 11, 10, 17]. We now describe a new, more general version of the technique. Suppose Gamma is an access structure having basis Gamma 0 . Let 1 be an integer. A Gammadecomposition of Gamma 0 consists of a collection (i.e. a multiset) f Gamma 1 ; Gamma n g such that the following properties are ....

K. M. Martin. Discrete Structures in the Theory of Secret Sharing. PhD thesis, University of London, 1991.

.... scheme for any P in P we have, H(P ) n 1)H(S) Moreover, the broadcast message b k;P 0 that enables the access structure A (k;P 0 ) has entropy equal to H(B k;P 0 ) n Gamma k 1) n 1)H(S) With a slight modification of the previous scheme (using techniques described in [24] and [18]) we can obtain a geometric scheme in which H(P ) H(S) and H(B k;P 0 ) n Gamma k 1)H(S) The following algorithms describe a secret sharing scheme with broadcast message such that for all P in P , H(P ) H(S) We suppose that S = GF (q) where q maxf2n; mg 1 is a prime power. ....

K. M. Martin, Discrete Structures in the Theory of Secret Sharing, PhD Thesis, University of London, 1991.

....satisfy, and these properties do not necessarily hold for an arbitrary access structure. Therefore not every access structure has an appropriate matroid. But if a connected matroid is appropriate for an access structure, then it is the only matroid with this property (see [18] Theorem 5.4. 1, and [13, 9]) Brickell and Davenport [6] have found relations between the two notions when A is an ideal access structure. The next two theorems almost characterize m Gammaideal access structures. The formulation of Theorem 2.7 is implicit in [6] and explicit in the works of Jackson and Martin [9, 13] ....

....and [13, 9] Brickell and Davenport [6] have found relations between the two notions when A is an ideal access structure. The next two theorems almost characterize m Gammaideal access structures. The formulation of Theorem 2. 7 is implicit in [6] and explicit in the works of Jackson and Martin [9, 13]. Theorem 2.7 : 6, 9, 13] necessary condition) If a non degenerate access structure A is m Gammaideal for some positive integer m, then there exists a connected matroid T that is appropriate for A . Theorem 2.8: 6] sufficient condition) 1 Let q be a prime power, and A be a non degenerate ....

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K. M. Martin. Discrete Structures in the Theory of Secret Sharing. PhD thesis, University of London, 1991.

....appropriate measure. We define the average information rate as follows. Given a set of secrets S, a non trivial probability distribution Pi S on S, and a fixed secret sharing scheme Sigma for A, we define e ae(A; Pi S ; Sigma) H(S) P P2P H(P ) jP j : This measure was introduced in [5] [33], and [34] when an uniform probability distribution on the set of secrets is assumed. In such a case the average information rate reduces to jP j log jSj= P P2P log jK(P )j. Blundo, De Santis, Stinson, and Vaccaro [9] analyzed secret sharing schemes by means of this measure, when the ....

K. M. Martin, Discrete Structures in the Theory of Secret Sharing, PhD Thesis, University of London, 1991.

....appropriate measure. We define the average information rate as follows. Given a set of secrets S, a non trivial probability distribution Pi S on S, and a fixed secret sharing scheme Sigma for A, we define e ae(A; Pi S ; Sigma) H(S) P P2P H(P ) jP j : This measure was introduced in [5] [31], and [32] when an uniform probability distribution on the set of secrets is assumed. In such a case the average information rate reduces to jP j log jSj= P P2P log jK(P )j. Blundo, De Santis, Stinson, and Vaccaro [9] analyzed secret sharing schemes by means of this measure, when the probability ....

K. M. Martin, Discrete Structures in the Theory of Secret Sharing, PhD Thesis, University of London, 1991.

....they showed that the bound is tight. Ideal secret sharing schemes, that is, schemes where the shares are taken from the same domain as that of the secret, were characterized by Brickell and Davenport [11] in terms of matroids. The uniqueness of the associated matroid is established by Martin in [26]. Brickell constructed some classes of ideal schemes in [10] and an interesting non existence result was proved by Seymour [30] Beimel and Chor [1] investigate the access structures for which an ideal scheme can be constructed for every possible size of the set of secrets. Finally, equivalence ....

....size of the shares, we can use the information rate [12] which is the ratio between the secret size and the maximum size of the shares. When we are interested in the total size of all the shares (and not just the maximum one) it is preferable to use as a measure the average information rate [6, 26, 27], which is the ratio between the secret size and the arithmetic mean of the size of all the shares. In this paper, we study secret sharing schemes in the case where the access structure consists of the closure of a (connected) graph. We consider all 30 connected graphs on at most five vertices, ....

K. M. Martin. Discrete Structures in the Theory of Secret Sharing. PhD Thesis, University of London, 1991.

....of the H(X) X 2 P , is a more appropriate measure. We define the average information rate as follows ae(A; P S ; Sigma) H(S) P X2P H(X) jP j for a given secret sharing scheme Sigma and non trivial probability distribution P S on the set of secrets S. This measure was introduced in [3] [13], and [14] when an uniform probability distribution on the set of secrets and the set of shares is assumed. In such a case the average information rate e ae(A) reduces to e ae(A) jP j log jSj= P X fflP log jX j. The optimal average information rate of the access structure A is defined as ....

K. M. Martin, Discrete Structures in the Theory of Secret Sharing, PhD Thesis, University of London, 1991.

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K. M. Martin. Discrete Structures in the Theory of Secret Sharing. PhD thesis, University of London, 1991.

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K. Martin, Discrete Structures in the Theory of Secret Sharing, PhD thesis, Royal Holloway and Bedford New College, 1993.

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K. M. Martin. Discrete Structures in the Theory of Secret Sharing. PhD Thesis, University of London, 1991.

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