  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
$(L,s)$ \textit{sliding secure} SMDC problem with security priority, where each
source $X_{\alpha}~(s\leq \alpha\leq L)$ is kept perfectly secure if no more
than $\alpha-s$ encoders are accessible. The reconstruction requirements of the
$L$ sources are the same as classical SMDC. A special case of an $(L,s)$
sliding secure SMDC problem that the first $s-1$ sources are constants is
called the $(L,s)$ \textit{multilevel secret sharing} problem. For $s=1$, the
two problems coincide, and we show that superposition coding is optimal. The
rate regions for the $(3,2)$ 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
$X_{\alpha-1}$ as the secret key of $X_{\alpha}$. Based on this idea, we
propose a coding scheme that achieves the minimum sum rate of the general
$(L,s)$ multilevel secret sharing problem. Moreover, superposition coding of
the $s$ sets of sources $X_1$, $X_2$, $\cdots$, $X_{s-1}$, $(X_s, X_{s+1},
\cdots, X_L)$ achieves the minimum sum rate of the general sliding secure SMDC
problem.
