  Conditional disclosure of secrets (CDS) is the problem of disclosing as
efficiently as possible, one secret from Alice and Bob to Carol if and only if
the inputs at Alice and Bob satisfy some function $f$. The information
theoretic capacity of CDS is the maximum number of bits of the secret that can
be securely disclosed per bit of total communication. All CDS instances, where
the capacity is the highest and is equal to $1/2$, are recently characterized
through a noise and signal alignment approach and are described using a graph
representation of the function $f$. In this work, we go beyond the best case
scenarios and further develop the alignment approach to characterize the linear
capacity of a class of CDS instances to be $(\rho-1)/(2\rho)$, where $\rho$ is
a covering parameter of the graph representation of $f$.
