  In coding schemes for the wire-tap channel or the broadcast channels with
confidential messages, it is well known that the sender needs to use a
stochastic encoding to avoid the information about the transmitted confidential
message to be leaked to an eavesdropper. In this paper, it is investigated that
the trade-off between the rate of the random number to realize the stochastic
encoding and the rates of the common, private, and confidential messages. For
the direct theorem, the superposition coding scheme for the wire-tap channel
recently proposed by Chia and El Gamal is employed, and its strong security is
proved. The matching converse theorem is also established. Our result clarifies
that a combination of the ordinary stochastic encoding and the channel
prefixing by the channel simulation is suboptimal.
