We consider the K-user interference channel with an external eavesdropper, with no eavesdropper's channel state information at the transmitters (CSIT). We determine the exact sum secure degrees of freedom (s.d.o.f.) for this channel by providing a new alignment based achievable scheme and a matching converse. Our results show that the lack of eavesdropper's CSIT does not have a significant impact on the optimal s.d.o.f. of the interference channel with an external eavesdropper, especially when the number of users is large.
We consider the K-user interference channel with an external eavesdropper, with no eavesdropper's channel state information at the transmitters (CSIT). We determine the exact sum secure degrees of freedom (s.d.o.f.) for this channel by providing a new alignment based achievable scheme and a matching converse. Our results show that the lack of eavesdropper's CSIT does not have a significant impact on the optimal s.d.o.f. of the interference channel with an external eavesdropper, especially when the number of users is large.
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 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
This paper studies the K-user Gaussian interference channel with secrecy constraints. Two distinct network models, namely the interference channel with confidential messages and the one with an external eavesdropper, are analyzed. Using interference alignment along with secrecy pre-coding at each transmitter, it is shown that each user in the network can achieve non-zero secure Degrees of Freedoms (DoFs) in both scenarios. In particular, the proposed coding scheme achieves K−2/2K−2 secure DoFs for each user in the interference channel with confidential messages model, and K−2/2K secure DoFs in the case of an external eavesdropper. The fundamental difference between the two scenarios stems from the lack of channel state information (CSI) about the external eavesdropper. Remarkably, the results establish the positive impact of interference on the secrecy capacity of wireless networks.
Secure Degrees of Freedom of K -User Gaussian Interference Channels: A Unified View
We determine the exact sum secure degrees of freedom (d.o.f.) of the K-user Gaussian interference channel. We consider three different secrecy constraints: 1) K-user interference channel with one external eavesdropper (IC-EE); 2) K-user interference channel with confidential messages (IC-CM); and 3) K-user interference channel with confidential messages and one external eavesdropper (IC-CM-EE). We show that for all of these three cases, the exact sum secure d.o.f. is K(K - 1)/(2K - 1). We show converses for IC-EE and IC-CM, which imply a converse for IC-CM-EE. We show achievability for IC-CM-EE, which implies achievability for IC-EE and IC-CM. Our converse is based on developing a direct relationship between the differential entropies of the channel inputs and the rates of the users, and quantifying the effect of eavesdropping on the rates in terms of the differential entropies of the eavesdroppers' observations. Our achievability is based on structured signaling, structured cooperative jamming, channel prefixing, and asymptotic real interference alignment. While the traditional interference alignment provides some amount of secrecy by mixing unintended signals in a smaller subspace at every receiver, in order to attain the optimum sum secure d.o.f., we incorporate structured cooperative jamming into the achievable scheme, and intricately design the structure of all of the transmitted signals jointly.
In this paper, the K-user interference channel with secrecy constraints is considered with delayed channel state information at transmitters (CSIT). We propose a novel secure retrospective interference alignment scheme in which the transmitters carefully mix information symbols with artificial noises to ensure confidentiality. Achieving positive secure degrees of freedom (SDoF) is challenging due to the delayed nature of CSIT, and the distributed nature of the transmitters. Our scheme works over two phases: Phase one, in which each transmitter sends information symbols mixed with artificial noises, and repeats such transmission over multiple rounds. In the next phase, each transmitter uses the delayed CSIT of the previous phase and sends a function of the net interference and artificial noises (generated in previous phase), which is simultaneously useful for all receivers. These phases are designed to ensure the decodability of the desired messages while satisfying the secrecy constraints. We present our achievable scheme for three models, namely: (1) K-user interference channel with confidential messages (IC-CM), and we show that 1 2 ( K − 6 ) SDoF is achievable; (2) K-user interference channel with an external eavesdropper (IC-EE); and (3) K-user IC with confidential messages and an external eavesdropper (IC-CM-EE). We show that for the K-user IC-EE, 1 2 ( K − 3 ) SDoF is achievable, and for the K-user IC-CM-EE, 1 2 ( K − 6 ) is achievable. To the best of our knowledge, this is the first result on the K-user interference channel with secrecy constrained models and delayed CSIT that achieves an SDoF which scales with K , square-root of number of users.
We consider a two-user multiple-input multiple- output (MIMO) multiple access wiretap channel with N antennas at each transmitter, N antennas at the legitimate receiver, and K antennas at the eavesdropper. We determine the optimal sum secure degrees of freedom (s.d.o.f.) for this model for all values of N and K. We subdivide our problem into several regimes based on the values of N and K, and provide alignment based achievable schemes and matching converses for each regime. Our results show how the number of eavesdropper antennas affects the optimal sum s.d.o.f. of the multiple access wiretap channel.