23,416 research outputs found
Linear codes with few weights from non-weakly regular plateaued functions
Linear codes with few weights have significant applications in secret sharing
schemes, authentication codes, association schemes, and strongly regular
graphs. There are a number of methods to construct linear codes, one of which
is based on functions. Furthermore, two generic constructions of linear codes
from functions called the first and the second generic constructions, have
aroused the research interest of scholars. Recently, in \cite{Nian}, Li and
Mesnager proposed two open problems: Based on the first and the second generic
constructions, respectively, construct linear codes from non-weakly regular
plateaued functions and determine their weight distributions.
Motivated by these two open problems, in this paper, firstly, based on the
first generic construction, we construct some three-weight and at most
five-weight linear codes from non-weakly regular plateaued functions and
determine the weight distributions of the constructed codes. Next, based on the
second generic construction, we construct some three-weight and at most
five-weight linear codes from non-weakly regular plateaued functions belonging
to (defined in this paper) and determine the weight
distributions of the constructed codes. We also give the punctured codes of
these codes obtained based on the second generic construction and determine
their weight distributions. Meanwhile, we obtain some optimal and almost
optimal linear codes. Besides, by the Ashikhmin-Barg condition, we have that
the constructed codes are minimal for almost all cases and obtain some secret
sharing schemes with nice access structures based on their dual codes.Comment: 52 pages, 34 table
Fourier-based Function Secret Sharing with General Access Structure
Function secret sharing (FSS) scheme is a mechanism that calculates a
function f(x) for x in {0,1}^n which is shared among p parties, by using
distributed functions f_i:{0,1}^n -> G, where G is an Abelian group, while the
function f:{0,1}^n -> G is kept secret to the parties. Ohsawa et al. in 2017
observed that any function f can be described as a linear combination of the
basis functions by regarding the function space as a vector space of dimension
2^n and gave new FSS schemes based on the Fourier basis. All existing FSS
schemes are of (p,p)-threshold type. That is, to compute f(x), we have to
collect f_i(x) for all the distributed functions. In this paper, as in the
secret sharing schemes, we consider FSS schemes with any general access
structure. To do this, we observe that Fourier-based FSS schemes by Ohsawa et
al. are compatible with linear secret sharing scheme. By incorporating the
techniques of linear secret sharing with any general access structure into the
Fourier-based FSS schemes, we show Fourier-based FSS schemes with any general
access structure.Comment: 12 page
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
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