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Secure Degrees of Freedom of One-hop Wireless Networks with No Eavesdropper CSIT

We consider three channel models: the wiretap channel with $M$ helpers, the $K$-user multiple access wiretap channel, and the $K$-user interference channel with an external eavesdropper, when no eavesdropper's channel state information (CSI) is available at the transmitters. In each case, we establish the optimal sum secure degrees of freedom (s.d.o.f.) by providing achievable schemes and matching converses. We show that the unavailability of the eavesdropper's CSIT does not reduce the s.d.o.f. of the wiretap channel with helpers. However, there is loss in s.d.o.f. for both the multiple access wiretap channel and the interference channel with an external eavesdropper. In particular, we show that in the absence of eavesdropper's CSIT, the $K$-user multiple access wiretap channel reduces to a wiretap channel with $(K-1)$ helpers from a sum s.d.o.f. perspective, and the optimal sum s.d.o.f. reduces from $\frac{K(K-1)}{K(K-1)+1}$ to $\frac{K-1}{K}$. For the interference channel with an external eavesdropper, the optimal sum s.d.o.f. decreases from $\frac{K(K-1)}{2K-1}$ to $\frac{K-1}{2}$ in the absence of the eavesdropper's CSIT. Our results show that the lack of eavesdropper's CSIT does not have a significant impact on the optimal s.d.o.f. for any of the three channel models, especially when the number of users is large. This implies that physical layer security can be made robust to the unavailability of eavesdropper CSIT at high signal to noise ratio (SNR) regimes by careful modification of the achievable schemes as demonstrated in this paper.