Wen-Ai Jackson

Geometric secret sharing schemes and their duals

Given a set of participants we wish to distribute information relating to a secret in such a way that only specified groups of participants can reconstruct the secret. We consider here a special class of such schemes that can be described in terms of finite geometries as first proposed by Simmons. We formalize the Simmons model and show that given a geometric scheme for a particular access structure it is possible to find another geometric scheme whose access structure is the dual of the original scheme, and which has the same average and worst-case information rates as the original scheme. In particular this shows that if an ideal geometric scheme exists then an ideal geometric scheme exists for the dual access structure.

Perfect Secret Sharing Schemes on Five Participants

A perfect secret sharing scheme is a system for the protection of a secret among a number of participants in such a way that only certain subsets of these participants can reconstruct the secret, and the remaining subsets can obtain no additional information about the secret. The efficiency of a perfect secret sharing scheme can be assessed in terms of its information rates. In this paper we discuss techniques for obtaining bounds on the information rates of perfect secret sharing schemes and illustrate these techniques using the set of monotone access structures on five participants. We give a full listing of the known informtion rate bounds for all the monotone access structures on five participants.

Multisecret Threshold Schemes

A threshold scheme is a system that protects a secret (key) among a group of participants in such a way that it can only be reconstructed from the joint information held by some predetermined number of these participants. In this paper we extend this problem to one where there is more than one secret that participants can reconstruct using the information that they hold. In particular we consider the situation where there is a secret sK associated with each k-subset K of participants and sK can be reconstructed by any group of t participants in K (t ? k). We establish bounds on the minimum amount of information that participants must hold in order to ensure that up to w participants (0 ? w ? n - k + t - 1) cannot obtain any information about a secret with which they are not associated. We also discuss examples of systems that satisfy this bound.

On sharing many secrets

We consider secret sharing schemes which, through an initial issuing of shares to a group of participants, permit a number of different secrets to be protected. Each secret is associated with a (potentially different) access structure and a particular secret can be reconstructed by any group of participants from its associated access structure without the need for further broadcast information. Two distinct problems are addressed. Firstly we consider ideal secret sharing schemes in this more general environment. In particular, we classify the collections of access structures that can be combined in such an ideal secret sharing scheme and we provide a general method of construction for such schemes. We also explore the extent to which the results that connect ideal secret sharing schemes to matroids can be appropriately generalised. Secondly we consider secret sharing schemes that can be used more than once. This problem can be considered as a type of secret sharing scheme wi! th different secrets but with the same access structure for each of the secrets.

Ideal secret sharing schemes with multiple secrets

We consider secret sharing schemes which, through an initial issuing of shares to a group of participants, permit a number of different secrets to be protected. Each secret is associated with a (potentially different) access structure and a particular secret can be reconstructed by any group of participants from its associated access structure without the need for further broadcast information. We consider ideal secret sharing schemes in this more general environment. In particular, we classify the collections of access structures that can be combined in such an ideal secret sharing scheme and we provide a general method of construction for such schemes. We also explore the extent to which the results that connect ideal secret sharing schemes to matroids can be appropriately generalized.

Updating the parameters of a threshold scheme by minimal broadcast

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.

Cumulative Arrays and Geometric Secret Sharing Schemes

Cumulative secret sharing schemes were introduced by Simmons et al (1991) based on the generalised secret sharing scheme of Ito et al (1987). A given monotone access structure together with a security level is associated with a unique cumulative scheme. Geometric secret sharing schemes form a wide class of secret sharing schemes which have many desirable properties including good information rates. We show that every non-degenerate geometric secret sharing scheme is ‘contained’ in the corresponding cumulative scheme. As there is no known practical algorithm for constructing efficient secret sharing schemes, the significance of this result is that, at least theoretically, a geometric scheme can be constructed from the corresponding cumulative scheme.

A Construction for Multisecret Threshold Schemes

AbstractA multisecret threshold scheme is a system that protects a number of secrets (or keys) among a group of participants, as follows. Given a set of n participants, there is a secret sK associated with each k–subset K of these participants. The scheme ensures that sK can be reconstructed by any group of t participants in K (

Mutually Trusted Authority-Free Secret Sharing Schemes

1 < t < k