3 research outputs found
The Explicit Coding Rate Region of Symmetric Multilevel Diversity Coding
It is well known that {\em superposition coding}, namely separately encoding
the independent sources, is optimal for symmetric multilevel diversity coding
(SMDC) (Yeung-Zhang 1999). However, the characterization of the coding rate
region therein involves uncountably many linear inequalities and the constant
term (i.e., the lower bound) in each inequality is given in terms of the
solution of a linear optimization problem. Thus this implicit characterization
of the coding rate region does not enable the determination of the
achievability of a given rate tuple. In this paper, we first obtain closed-form
expressions of these uncountably many inequalities. Then we identify a finite
subset of inequalities that is sufficient for characterizing the coding rate
region. This gives an explicit characterization of the coding rate region. We
further show by the symmetry of the problem that only a much smaller subset of
this finite set of inequalities needs to be verified in determining the
achievability of a given rate tuple. Yet, the cardinality of this smaller set
grows at least exponentially fast with . We also present a subset entropy
inequality, which together with our explicit characterization of the coding
rate region, is sufficient for proving the optimality of superposition coding
Sliding Secure Symmetric Multilevel Diversity Coding
Symmetric multilevel diversity coding (SMDC) is a source coding problem where
the independent sources are ordered according to their importance. It was shown
that separately encoding independent sources (referred to as
``\textit{superposition coding}") is optimal. In this paper, we consider an
\textit{sliding secure} SMDC problem with security priority, where each
source is kept perfectly secure if no more
than encoders are accessible. The reconstruction requirements of the
sources are the same as classical SMDC. A special case of an
sliding secure SMDC problem that the first sources are constants is
called the \textit{multilevel secret sharing} problem. For , the
two problems coincide, and we show that superposition coding is optimal. The
rate regions for the problems are characterized. It is shown that
superposition coding is suboptimal for both problems. The main idea that joint
encoding can reduce coding rates is that we can use the previous source
as the secret key of . Based on this idea, we
propose a coding scheme that achieves the minimum sum rate of the general
multilevel secret sharing problem. Moreover, superposition coding of
the sets of sources , , , , achieves the minimum sum rate of the general sliding secure SMDC
problem
Weakly Secure Symmetric Multilevel Diversity Coding
Multilevel diversity coding is a classical coding model where multiple
mutually independent information messages are encoded, such that different
reliability requirements can be afforded to different messages. It is well
known that {\em superposition coding}, namely separately encoding the
independent messages, is optimal for symmetric multilevel diversity coding
(SMDC) (Yeung-Zhang 1999). In the current paper, we consider weakly secure SMDC
where security constraints are injected on each individual message, and provide
a complete characterization of the conditions under which superposition coding
is sum-rate optimal. Two joint coding strategies, which lead to rate savings
compared to superposition coding, are proposed, where some coding components
for one message can be used as the encryption key for another. By applying
different variants of Han's inequality, we show that the lack of opportunity to
apply these two coding strategies directly implies the optimality of
superposition coding. It is further shown that under a set of particular
security constraints, one of the proposed joint coding strategies can be used
to construct a code that achieves the optimal rate region.Comment: The paper has been accepted by IEEE Transactions on Information
Theor