  In many information-theoretic channel coding problems, adding an input cost
constraint to the operational setup amounts to restricting the optimization
domain in the capacity formula. This paper shows that, in contrast to common
belief, such a simple modification does not hold for the cost-constrained (CC)
wiretap channel (WTC). The secrecy-capacity of the discrete memoryless (DM) WTC
without cost constraints is described by a single auxiliary random variable.
For the CC DM-WTC, however, we show that two auxiliaries are necessary to
achieve capacity. Specifically, we first derive the secrecy-capacity formula,
proving the direct part via superposition coding. Then, we provide an example
of a CC DM-WTC whose secrecy-capacity cannot be achieved using a single
auxiliary. This establishes the fundamental role of superposition coding over
CC WTCs.
