17,103 research outputs found
Secret Sharing Schemes with General Access Structures (Full version)
Secret sharing schemes with general monotone access structures have been widely discussed in the literature. But in some scenarios, non-monotone access structures may have more practical significance. In this paper, we shed a new light on secret sharing schemes realizing general (not necessarily monotone) access structures. Based on an attack model for secret sharing schemes with general access structures, we redefine perfect secret sharing schemes, which is a generalization of the known concept of perfect secret sharing schemes with monotone access structures. Then, we provide for the first time two constructions of perfect secret sharing schemes with general access structures. The first construction can be seen as a democratic scheme in the sense that the shares are generated by the players themselves.
Our second construction significantly enhance the efficiency of the system, where the shares are distributed by the trusted center (TC)
An Epitome of Multi Secret Sharing Schemes for General Access Structure
Secret sharing schemes are widely used now a days in various applications,
which need more security, trust and reliability. In secret sharing scheme, the
secret is divided among the participants and only authorized set of
participants can recover the secret by combining their shares. The authorized
set of participants are called access structure of the scheme. In Multi-Secret
Sharing Scheme (MSSS), k different secrets are distributed among the
participants, each one according to an access structure. Multi-secret sharing
schemes have been studied extensively by the cryptographic community. Number of
schemes are proposed for the threshold multi-secret sharing and multi-secret
sharing according to generalized access structure with various features. In
this survey we explore the important constructions of multi-secret sharing for
the generalized access structure with their merits and demerits. The features
like whether shares can be reused, participants can be enrolled or dis-enrolled
efficiently, whether shares have to modified in the renewal phase etc., are
considered for the evaluation
Probabilistic Infinite Secret Sharing
The study of probabilistic secret sharing schemes using arbitrary probability
spaces and possibly infinite number of participants lets us investigate
abstract properties of such schemes. It highlights important properties,
explains why certain definitions work better than others, connects this topic
to other branches of mathematics, and might yield new design paradigms.
A probabilistic secret sharing scheme is a joint probability distribution of
the shares and the secret together with a collection of secret recovery
functions for qualified subsets. The scheme is measurable if the recovery
functions are measurable. Depending on how much information an unqualified
subset might have, we define four scheme types: perfect, almost perfect, ramp,
and almost ramp. Our main results characterize the access structures which can
be realized by schemes of these types.
We show that every access structure can be realized by a non-measurable
perfect probabilistic scheme. The construction is based on a paradoxical pair
of independent random variables which determine each other.
For measurable schemes we have the following complete characterization. An
access structure can be realized by a (measurable) perfect, or almost perfect
scheme if and only if the access structure, as a subset of the Sierpi\'nski
space , is open, if and only if it can be realized by a span
program. The access structure can be realized by a (measurable) ramp or almost
ramp scheme if and only if the access structure is a set
(intersection of countably many open sets) in the Sierpi\'nski topology, if and
only if it can be realized by a Hilbert-space program
Optimal non-perfect uniform secret sharing schemes
A secret sharing scheme is non-perfect if some subsets of participants that cannot recover the secret value have partial information about it. The information ratio of a secret sharing scheme is the ratio between the maximum length of the shares and the length of the secret. This work is dedicated to the search of bounds on the information ratio of non-perfect secret sharing schemes. To this end, we extend the known connections between polymatroids and perfect secret sharing schemes to the non-perfect case. In order to study non-perfect secret sharing schemes in all generality, we describe their structure through their access function, a real function that measures the amount of information that every subset of participants obtains about the secret value. We prove that there exists a secret sharing scheme for every access function. Uniform access functions, that is, the ones whose values depend only on the number of participants, generalize the threshold access structures. Our main result is to determine the optimal information ratio of the uniform access functions. Moreover, we present a construction of linear secret sharing schemes with optimal information ratio for the rational uniform access functions.Peer ReviewedPostprint (author's final draft
Infinite Secret Sharing -- Examples
The motivation for extending secret sharing schemes to cases when either the
set of players is infinite or the domain from which the secret and/or the
shares are drawn is infinite or both, is similar to the case when switching to
abstract probability spaces from classical combinatorial probability. It might
shed new light on old problems, could connect seemingly unrelated problems, and
unify diverse phenomena.
Definitions equivalent in the finitary case could be very much different when
switching to infinity, signifying their difference. The standard requirement
that qualified subsets should be able to determine the secret has different
interpretations in spite of the fact that, by assumption, all participants have
infinite computing power. The requirement that unqualified subsets should have
no, or limited information on the secret suggests that we also need some
probability distribution. In the infinite case events with zero probability are
not necessarily impossible, and we should decide whether bad events with zero
probability are allowed or not.
In this paper, rather than giving precise definitions, we enlist an abundance
of hopefully interesting infinite secret sharing schemes. These schemes touch
quite diverse areas of mathematics such as projective geometry, stochastic
processes and Hilbert spaces. Nevertheless our main tools are from probability
theory. The examples discussed here serve as foundation and illustration to the
more theory oriented companion paper
- …