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.
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.
In this paper, we determine the entire secure degrees of freedom (s.d.o.f.) regions of the K-user Gaussian multiple access (MAC) wiretap channel and the K-user interference channel (IC) with secrecy constraints. For the IC, we consider three secrecy constraints: K-user IC with an external eavesdropper (ICEE), K-user IC with confidential messages (IC-CM), and their combination Kuser IC with confidential messages and external eavesdropper (IC-CM-EE). The converse for the IC includes constraints both due to secrecy as well as due to interference. For the IC, although the portion of the region close to the optimum sum s.d.o.f. point is governed by the upper bounds due to secrecy constraints, the other portions of the region are governed by the upper bounds due to interference constraints. Different from the existing literature, in order to fully understand the characterization of the s.d.o.f. region of the IC, one has to study the four-user case, i.e., the twoor three-user cases do not illustrate the full generality of the problem. In order to prove the achievability, we use the polytope structure of the converse region. In both MAC and IC cases, we develop explicit schemes that achieve the extreme points of the polytope region given by the converse. In particular, the extreme points of the MAC region are 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. The extreme points of the IC region are achieved by a (K - m)-user IC with confidential messages, m helpers, and N external eavesdroppers, for m ≥ 1 and a finite N. A byproduct of our results in this paper is that the sum s.d.o.f. is achieved only at one extreme point of the s.d.o.f. region, which is the symmetric-rate extreme point, for both MAC and IC channel models.
We show that the sum secure degrees of freedom (d.o.f.) of the K-user Gaussian multiple access (MAC) wiretap channel is K(K-1)/K(K-1)+1. Our achievability is based on real interference alignment and structured cooperative jamming. Each user divides its message into K - 1 sub-messages, and sends a linear combination of signals carrying these sub-messages together with a structured cooperative jamming signal. All cooperative jamming signals are aligned in a single dimension at the legitimate receiver allowing for reliable decoding of the message carrying signals by the legitimate receiver. Each cooperative jamming signal is aligned with K-1 message signals at the eavesdropper limiting the information leakage rate to the eavesdropper. We provide a matching converse establishing the exact sum secure d.o.f. of the Gaussian MAC wiretap channel as K(K-1)/K(K-1)+1.
The sum secure degrees of freedom (s.d.o.f.) of the K-user interference channel (IC) with secrecy constraints has been determined recently as equation [1], [2]. In this paper, we determine the entire s.d.o.f. region of this channel model. The converse includes constraints both due to secrecy as well as due to interference. Although the portion of the region close to the optimum sum s.d.o.f. point is governed by the upper bounds due to secrecy constraints, the other portions of the region are governed by the upper bounds due to interference constraints. Different from the existing literature, in order to fully understand the characterization of the s.d.o.f. region of the IC, one has to study the 4-user case, i.e., the 2 or 3-user cases do not illustrate the generality of the problem. In order to prove the achievability, we use the polytope structure of the converse region. The extreme points of the converse region are achieved by a (K - m)-user IC with confidential messages, m helpers, and N external eavesdroppers, for m ≥ 1 and a finite N. A byproduct of our results in this paper is that the sum s.d.o.f. is achieved only at one extreme point of the s.d.o.f. region, which is the symmetric-rate extreme point.
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 study the secure degrees of freedom (d.o.f.) of one-hop wireless networks by considering four fundamental wireless network structures: 1) Gaussian wiretap channel; 2) Gaussian broadcast channel with confidential messages; 3) Gaussian interference channel with confidential messages; and 4) Gaussian multiple access wiretap channel. The secrecy capacity of the canonical Gaussian wiretap channel does not scale with the transmit power, and hence, the secure d.o.f. of the Gaussian wiretap channel with no helpers is zero. It has been known that a strictly positive secure d.o.f. can be obtained in the Gaussian wiretap channel by using a helper, which sends structured cooperative signals. We show that the exact secure d.o.f. of the Gaussian wiretap channel with a helper is 1/2. Our achievable scheme is based on real interference alignment and cooperative jamming, which renders the message signal and the cooperative jamming signal separable at the legitimate receiver, but aligns them perfectly at the eavesdropper preventing any reliable decoding of the message signal. Our converse is based on two key lemmas. The first lemma quantifies the secrecy penalty by showing that the net effect of an eavesdropper on the system is that it eliminates one of the independent channel inputs. The second lemma quantifies the role of a helper by developing a direct relationship between the cooperative jamming signal of a helper and the message rate. We extend this result to the case of M helpers, and show that the exact secure d.o.f. in this case is M/M+1.We then generalize this approach to more general network structures with multiple messages. We show that the sum secure d.o.f. of the Gaussian broadcast channel with confidential messages and M helpers is 1, the sum secure d.o.f. of the twouser interference channel with confidential messages is 2/3, the sum secure d.o.f. of the two-user interference channel with confidential messages and M helpers is 1, and the sum secure d.o.f. of the K-user multiple access wiretap channel is K(K-1)/K(K-1)+1.
We study deterministic approximations of the Gaussian two-user multiple access wiretap channel (G-MAC-WT) and the Gaussian wiretap channel with a helper (G-WT-H). These approximations enable results beyond the recently shown 2/3 and 1/2 secure degrees of freedom (s.d.o.f.) for the G-MAC-WT and the G-WT-H, respectively. While the s.d.o.f. were obtained by real interference alignment, our approach uses signal-scale alignment. We show achievable schemes which are independent of the rationality of the channel gains. Moreover, our results can differentiate between channel strengths, in particular, between both users, and will establish secrecy rates dependent on this difference. We can show that the resulting achievable secrecy rates tend to the s.d.o.f. for vanishing channel gain differences. Moreover, we extend previous and develop new techniques to prove generalized s.d.o.f. bounds for varying channel strengths and show that our achievable schemes reach the bounds for certain channel gain parameters. We believe that our analysis is the next step toward a constant-gap analysis of the G-MAC-WT and the G-WT-H.