Geometrical contributions to secret sharing theory

Jackson, Wen-Ai; Martin, Keith M.; O'Keefe, Christine M.

doi:10.1007/s00022-003-1633-y

“…From (13) it follows that and so (14) Also by (13), and so (see Section II-A). Since we have and thus…”

Section: B Proof For Theorem 14mentioning

confidence: 88%

“…They thus have no information about the secret . To see how to extract the distribution rules of an ideal -threshold scheme from this configuration of points see, for example, [13] or [20].…”

Section: Definitionmentioning

confidence: 99%

See 1 more Smart Citation

Updating the Parameters of a Threshold Scheme by Minimal Broadcast

Barwick

¹

,

Jackson

²

,

Martin

³

2005

IEEE Trans. Inform. Theory

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Abstract-Threshold schemes allow secret data to be protected among a set of participants in such a way that only a prespecified threshold of participants can reconstruct the secret from private information (shares) distributed to them on a system setup using secure channels. We consider the general problem of designing unconditionally secure threshold schemes whose defining parameters (the threshold and the number of participants) can later be changed by using only public channel broadcast messages. In this paper, we are interested in the efficiency of such threshold schemes, and seek to minimize storage costs (size of shares) as well as optimize performance in low-bandwidth environments by minimizing the size of necessary broadcast messages. We prove a number of lower bounds on the smallest size of broadcast message necessary to make general changes to the parameters of a threshold scheme in which each participant already holds shares of minimal size. We establish the tightness of these bounds by demonstrating optimal schemes.

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“…Starting with P G(2, q), the mapping ζ : P G(2, q) → P G(5, q) with ζ([x 0 , x 1 , x 2 ]) = [x 2 0 , x 2 1 , x 2 2 , x 0 x 1 , x 0 x 2 , x 1 x 2 ] defines the quadratic Veronesean V 4 2 . If P = [a, b, c], the planes D(P ) defined above have the equation…”

Section: Examplementioning

confidence: 99%

Generalised Veroneseans

Klein¹,

Schillewaert²,

Storme³

2012

Advances in Geometry

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In [8], a characterization of the finite quadric Veronesean V 2 n n by means of properties of the set of its tangent spaces is proved. These tangent spaces form a regular generalised dual arc. We prove an extension result for regular generalised dual arcs. To motivate our research, we show how they are used to construct a large class of secret sharing schemes.A typical problem in (finite) geometry is the study of highly symmetrical substructures. For example, arcs are configurations of points in PG(n, q) such that each n + 1 of them are in general position, while n 1 -dimensional dual arcs are sets of n 1 -spaces such that each two intersect in a point and any three of them are skew. These two structures appear naturally in cryptographical applications.In this article, we define objects, called (generalised) dual arcs; a class of objects that contain classical arcs and n 1 -dimensional dual arcs as special cases. These (generalised) dual arcs have applications in cryptography as well.We give a construction method for a wide class of parameters and prove an extension result for regular generalised dual arcs of order d = 1.In Sections 1 and 2, we give the necessary definitions, constructions, and examples of applications in cryptography. Section 3 refers to known classification results, and Section 4 states our main characterization theorem (Theorem 13). We now start with the required definitions to make this article self-contained.

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“…For an overview of secret sharing and the links with geometry we refer to [4]. A recent overview of different adversary models in secret sharing can be found in [6].…”

Section: Secret Sharingmentioning

confidence: 99%

Generalised Veroneseans

Klein¹,

Schillewaert²,

Storme³

2011

Advances in Geometry

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In [8], a characterization of the finite quadric Veronesean V 2 n n by means of properties of the set of its tangent spaces is proved. These tangent spaces form a regular generalised dual arc. We prove an extension result for regular generalised dual arcs. To motivate our research, we show how they are used to construct a large class of secret sharing schemes.A typical problem in (finite) geometry is the study of highly symmetrical substructures. For example, arcs are configurations of points in PG(n, q) such that each n + 1 of them are in general position, while n 1 -dimensional dual arcs are sets of n 1 -spaces such that each two intersect in a point and any three of them are skew. These two structures appear naturally in cryptographical applications.In this article, we define objects, called (generalised) dual arcs; a class of objects that contain classical arcs and n 1 -dimensional dual arcs as special cases. These (generalised) dual arcs have applications in cryptography as well.We give a construction method for a wide class of parameters and prove an extension result for regular generalised dual arcs of order d = 1.In Sections 1 and 2, we give the necessary definitions, constructions, and examples of applications in cryptography. Section 3 refers to known classification results, and Section 4 states our main characterization theorem (Theorem 13). We now start with the required definitions to make this article self-contained.

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Geometrical contributions to secret sharing theory

Cited by 11 publications

References 60 publications

Updating the Parameters of a Threshold Scheme by Minimal Broadcast

Updating the Parameters of a Threshold Scheme by Minimal Broadcast

Generalised Veroneseans

Generalised Veroneseans

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