  In the conditional disclosure of secrets (CDS) problem, Alice and Bob (each
holds an input and a common secret) wish to disclose, as efficiently as
possible, the secret to Carol if and only if their inputs satisfy some
function. The capacity of CDS is the maximum number of bits of the secret that
can be securely disclosed per bit of total communication. We characterize the
necessary and sufficient condition for the extreme case where the capacity of
CDS is the highest and is equal to 1/2. For the simplest instance where the
capacity is smaller than 1/2, we show that the linear capacity is 2/5.
