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The impact of quantized channel direction information (CDI) on the achievable secrecy rate is studied for multiple antenna wiretap channels. By assuming that the eavesdropper's channel is unknown at the transmitter, we adopt the transmission scheme where artificial noise (AN) is imposed in the null space of the legitimate receiver's channel to disrupt the eavesdropper's reception. It has been shown that, in the ideal case where perfect CDI is available at the transmitter, the achievable secrecy rate can be made arbitrarily large by increasing the transmission power. However, when only quantized CDI is available, the AN that was originally intended to jam the eavesdropper may now leak into the legitimate receiver's channel, causing significant secrecy rate loss. For a given number of feedback bits B and transmission power P, we derive the optimal power allocation among the message-bearing signal and the AN to maximize the secrecy rate under AN leakage. We show that, when B is sufficiently large, one should allocate power evenly among the message-bearing signal and the AN; whereas when B is small, one should be more conservative in allocating power to the AN. Moreover, by showing that the achievable secrecy rate under quantized CDI is bounded by a constant, we derive a scaling law between B and P that is necessary to maintain a constant secrecy rate loss compared to the perfect CDI case. The scaling of B is shown to be logarithmic of P. These results are first derived for the multiple-input single-output single-antenna-eavesdropper scenario and are later extended to the multiple-input multiple-output multiple-antenna-eavesdropper case. Numerical simulations are provided to verify our theoretical claims.