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IEEE International Symposium on Information Theory (ISIT)

Secure degrees of freedom of the multiple access wiretap channel with no eavesdropper CSI

Pritam Mukherjee, Şennur Ulukuş

Published Jun 1, 2015• 10 citations
Publisher

Abstract

We consider the K-user Gaussian multiple access wiretap channel (MAC-WT), where no eavesdropper channel state information (CSI) is available at the transmitters. We show that the exact sum secure degrees of freedom (s.d.o.f.) of this channel model is K-1/K . This result shows that, under the condition of no eavesdropper CSI, the MAC-WT acts like a single-transmitter K - 1 helper wiretap channel. We further show that, when a subset of the transmitters have eavesdropper CSI, then higher sum s.d.o.f. can be achieved, and the system can be operated as a MAC-WT for the users with eavesdropper CSI, with the remaining users acting as helpers. In particular, if m of the K transmitters have eavesdropper CSI, we show that m(K-1)/m(K-1)+1 sum s.d.o.f. can be achieved, showing the benefits of having the eavesdropper CSI at the transmitters.

DOI
10.1109/isit.2015.7282868

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We consider the K-user Gaussian multiple access wiretap channel (MAC-WT), where no eavesdropper channel state information (CSI) is available at the transmitters. We show that the exact sum secure degrees of freedom (s.d.o.f.) of this channel model is K-1/K . This result shows that, under the condition of no eavesdropper CSI, the MAC-WT acts like a single-transmitter K - 1 helper wiretap channel. We further show that, when a subset of the transmitters have eavesdropper CSI, then higher sum s.d.o.f. can be achieved, and the system can be operated as a MAC-WT for the users with eavesdropper CSI, with the remaining users acting as helpers. In particular, if m of the K transmitters have eavesdropper CSI, we show that m(K-1)/m(K-1)+1 sum s.d.o.f. can be achieved, showing the benefits of having the eavesdropper CSI at the transmitters.
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 channel state information at the transmitter (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 K(K-1) /K(K-1)+1 to K-1 K . For the interference channel with an external eavesdropper, the optimal sum s.d.o.f. decreases from K(K-1) /2K-1 to 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 regimes by the careful modification of the achievable schemes as demonstrated in this paper.
We consider two fundamental multi-user channel models: the multiple-input multiple-output (MIMO) wiretap channel with one helper (WTH) and the MIMO multiple access wiretap (MAC-WT) channel. In each case, the eavesdropper has K antennas while the remaining terminals have N antennas each. We consider a fast fading channel where the channel state information (CSI) of the legitimate receiver is available at the transmitters but no CSI at the transmitters (CSIT) is available for the eavesdropper's channel. We determine the optimal sum secure degrees of freedom (s.d.o.f.) for each channel model for the regime K ≤ N, and show that in this regime, the MAC-WT channel reduces to the WTH in the absence of eavesdropper CSIT. For the regime N ≤ K ≤ 2N, we obtain the optimal linear s.d.o.f., and show that the MAC-WT channel and the WTH have the same optimal s.d.o.f. when restricted to linear encoding strategies. In the absence of any such restrictions, we provide an upper bound for the sum s.d.o.f. of the MAC-WT channel in the regime N ≤ K ≤ 2N. Our results show that unlike in the single-input single-output case, there is loss of s.d.o.f. for even the WTH due to lack of eavesdropper CSIT when K ≥ N.
We consider three channel models: the wiretap channel withM 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 free-dom (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 K(K−1)K(K−1)+1 to K−1 K. For the interference channel with an external eavesdropper, the optimal sum s.d.o.f. de-creases from K(K−1)2K−1 to 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 signicant 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 modication of the achievable schemes as demonstrated in this paper. 1
We derive an upper bound for the sum secure Degrees of Freedom (sDoF) for the symmetric K-user MIMO Wiretap Multiple Access channel with full channel state information at transmitters (CSIT) but without the knowledge of eavesdroppers channel state information (CSI). We consider the symmetric case with K transmitters each equipped with M antennas, a base station with N antennas and an unknown number of eavesdroppers. Without assuming knowledge of the number (or location) of eavesdroppers, we assume that each eavesdropper has a maximum of Neantennas. The sum sDoF is shown to be upper bounded by Ds= min(max(M, N) - Ne/K, K M - Ne, N). Achievable schemes utilizing cooperative artificial noise transmission and zero-forcing decorrelators are shown to achieve the upper bound for a wide range of values of N/M.
Two fundamental multi-user channel models: the multiple-input multiple-output (MIMO) wiretap channel with one helper (WTH) and the MIMO multiple access wiretap channel (MAC-WT) are considered. In each case, the eavesdropper has K antennas while the remaining terminals have N antennas. We consider a fast fading channel where the channel state information (CSI) of the legitimate receiver is available at the transmitters but no channel state information at the transmitters (CSIT) is available for the eavesdropper's channel. The optimal sum secure degrees of freedom (s.d.o.f.) for each channel model is determined for the regime K ≤ N, and we show that in this regime, the MAC-WT channel reduces to the WTH in the absence of eavesdropper CSIT. For the regime N ≤ K ≤ 2N, we obtain the optimal linear s.d.o.f., and show that the MAC-WT channel and the WTH have the same optimal s.d.o.f. when restricted to linear encoding strategies. In the absence of any such restrictions, we provide an upper bound for the sum s.d.o.f. of the MAC-WT chanel in the regime N ≤ K ≤ 2N. Our results show that unlike in the single-input single-output (SISO) case, there is loss of s.d.o.f. for even the WTH due to lack of eavesdropper CSIT, when K ≥ N.
The paper showed that the sum secure degrees of freedom (s.d.o.f.) of the K-user Gaussian multiple access (MAC) wiretap channel is K(K-1)/K(K-1);1. In this paper, we determine the entire s.d.o.f. region of the K-user Gaussian MAC wiretap channel. The converse follows from a middle step in the converse of [1]. The achievability follows from exploring the polytope structure of the converse region, determining its extreme points, and then showing that each extreme point can be achieved by an m-user MAC wiretap channel with K-m helpers, i.e., by setting K-m users' secure rates to zero and utilizing them as pure (structured) cooperative jammers. A byproduct of our result is that the sum s.d.o.f. is achieved only at one corner point of the s.d.o.f. region.