  The secrecy capacity of a discrete memoryless Gaussian Wire-Tap Channel when
the input is from a finite complex constellation is studied. It is shown that
the secrecy capacity curves of a finite constellation plotted against the SNR,
for a fixed noise variance of the eavesdropper's channel has a global maximum
at an internal point. This is in contrast to what is known in the case of
Gaussian codebook input where the secrecy capacity curve is a bounded,
monotonically increasing function of SNR. Secrecy capacity curves for some well
known constellations like BPSK, 4-QAM, 16-QAM and 8-PSK are plotted and the SNR
at which the maximum occurs is found through simulation. It is conjectured that
the secrecy capacity curves for finite constellations have a single maximum.
