Sum Secure Degrees Of Freedom Research Articles

Secure Degrees of Freedom of the Three-User MIMO Broadcast Channel With Delayed CSIT

This paper investigates the sum-secure degrees-of-freedom (SDoF) of three-user multiple-input multiple-output (MIMO) broadcast channel with confidential messages (BCCM) and delayed channel state information at the transmitter (CSIT). Specifically, we obtain non-trivial sum-SDoF upper and lower bounds. Firstly, we derive the sum-SDoF upper bound by means of statistical equivalence property, security constraints, and permutations. Then, for the sum-SDoF lower bound, we leverage the artificial noise transmission and interference re-transmission to design two transmission schemes, which have holistic and sequential higher-order symbol generation, respectively. For these two schemes, we propose the redundancy reduction approach for security analysis, by which the minimal duration of artificial noise transmission phase of the scheme is obtained. To eliminate the redundant equations in security analysis, this approach first identifies the constituent equations, and then analyzes the rank of assemble of them. As a result, both the proposed sum-SDoF upper and lower bounds are tighter than the existing sum-SDoF upper and lower bounds, respectively. Furthermore, the proposed lower bound showcases a three-user coding gain.

Full-Duplex Relay with Delayed CSI Elevates the SDoF of the MIMO X Channel

In this article, the sum secure degrees-of-freedom (SDoF) of the multiple-input multiple-output (MIMO) X channel with confidential messages (XCCM) and arbitrary antenna configurations is studied, where there is no channel state information (CSI) at two transmitters and only delayed CSI at a multiple-antenna, full-duplex, and decode-and-forward relay. We aim at establishing the sum-SDoF lower and upper bounds. For the sum-SDoF lower bound, we design three relay-aided transmission schemes, namely, the relay-aided jamming scheme, the relay-aided jamming and one-receiver interference alignment scheme, and the relay-aided jamming and two-receiver interference alignment scheme, each corresponding to one case of antenna configurations. Moreover, the security and decoding of each scheme are analyzed. The sum-SDoF upper bound is proposed by means of the existing SDoF region of two-user MIMO broadcast channel with confidential messages (BCCM) and delayed channel state information at the transmitter (CSIT). As a result, the sum-SDoF lower and upper bounds are derived, and the sum-SDoF is characterized when the relay has sufficiently large antennas. Furthermore, even assuming no CSI at two transmitters, our results show that a multiple-antenna full-duplex relay with delayed CSI can elevate the sum-SDoF of the MIMO XCCM. This is corroborated by the fact that the derived sum-SDoF lower bound can be greater than the sum-SDoF of the MIMO XCCM with output feedback and delayed CSIT.

On Secure Degrees of Freedom of the MIMO Interference Channel With Local Output Feedback

This article studies the problem of Sum-secure Degrees of Freedom (SDoF) of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(M,M,N,N)$ </tex-math></inline-formula> multiple-input–multiple-output (MIMO) interference channel with local output feedback, so as to build an information-theoretic foundation and provide practical transmission schemes for 6G-enabled Vehicles-to-Vehicles (V2V). For this problem, we propose two novel transmission schemes, i.e., the interference decoding scheme and the interference alignment scheme, and thus establish a sum-SDoF lower bound. In particular, to optimize the phase duration, we analyze the security and decoding constraints and formulate a linear-fractional optimization problem. Furthermore, we show that the derived sum-SDoF lower bound is the sum-SDoF for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M \le N/2$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N=M$ </tex-math></inline-formula> , and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2N \le M$ </tex-math></inline-formula> antenna configurations, and reveal that for a fixed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> , the optimal <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> to maximize the sum-SDoF is not less than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2N$ </tex-math></inline-formula> . Through simulations, we examine the secure sum-rate performance of proposed transmission schemes and reveal that using local output feedback can lead to a higher secure sum-rate than that by using delayed channel state information at the transmitter (CSIT).

Secure Degrees of Freedom of the Multiple Access Wiretap Channel With Multiple Antennas

We consider a two-user multiple-input multiple-output 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 achievable schemes based on vector space alignment and real alignment techniques for fixed and fading channel gains. To prove the optimality of the achievable schemes, we provide 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.

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.

Secrecy in MIMO Networks with No Eavesdropper CSIT

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 $K\leq N$ , and show that in this regime, the MAC-WT channel reduces to the WTH in the absence of eavesdropper CSIT. For the regime $N\leq K\leq 2N$ , 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 $N\leq K\leq 2N$ . 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 $K\geq N$ .

Secure Degrees of Freedom Regions of Multiple Access and Interference Channels: The Polytope Structure

The sum secure degrees of freedom (s.d.o.f.) of two fundamental multi-user network structures, the K-user Gaussian multiple access (MAC) wiretap channel and the K-user interference channel (IC) with secrecy constraints, have been determined recently as K(K-1)/(K(K-1)+1) [1,2] and K(K-1)/(2K-1) [3,4], respectively. In this paper, we determine the entire s.d.o.f. regions of these two channel models. The converse for the MAC follows from a middle step in the converse of [1,2]. The converse for the IC 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. In both MAC and IC cases, we develop explicit schemes that achieve the extreme points of the polytope region given by the converse. Specifically, 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.

Secure Degrees of Freedom of &lt;inline-formula&gt; &lt;tex-math notation="LaTeX"&gt;$K$ &lt;/tex-math&gt;&lt;/inline-formula&gt;-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.

Secure Degrees of Freedom of MIMO X-Channels With Output Feedback and Delayed CSIT

We investigate the problem of secure transmission over a two-user multi-input multi-output (MIMO) X-channel in which channel state information is provided with one-unit delay to both transmitters (CSIT), and each receiver feeds back its channel output to a different transmitter. We refer to this model as MIMO X-channel with asymmetric output feedback and delayed CSIT. The transmitters are equipped with M-antennas each, and the receivers are equipped with N-antennas each. For this model, accounting for both messages at each receiver, we characterize the optimal sum secure degrees of freedom (SDoF) region. We show that, in presence of asymmetric output feedback and delayed CSIT, the sum SDoF region of the MIMO X-channel is same as the SDoF region of a two-user MIMO BC with 2M-antennas at the transmitter, N-antennas at each receiver and delayed CSIT. This result shows that, upon availability of asymmetric output feedback and delayed CSIT, there is no performance loss in terms of sum SDoF due to the distributed nature of the transmitters. Next, we show that this result also holds if only output feedback is conveyed to the transmitters, but in a symmetric manner, i.e., each receiver feeds back its output to both transmitters and no CSIT. We also study the case in which only asymmetric output feedback is provided to the transmitters, i.e., without CSIT, and derive a lower bound on the sum SDoF for this model. Furthermore, we specialize our results to the case in which there are no security constraints. In particular, similar to the setting with security constraints, we show that the optimal sum DoF region of the (M,M,N,N)--MIMO X-channel with asymmetric output feedback and delayed CSIT is same as the DoF region of a two-user MIMO BC with 2M-antennas at the transmitter, N-antennas at each receiver, and delayed CSIT. We illustrate our results with some numerical examples.

Sum Secure Degrees of Freedom of Two-Unicast Layered Wireless Networks

In this paper, we study the sum secure degrees of freedom (d.o.f.) of two-unicast layered wireless networks. Without any secrecy constraints, the sum d.o.f. of this class of networks was studied by and shown to take only one of three possible values: 1, 3/2 and 2, for all network configurations. We consider the setting where, in addition to being reliably transmitted, each message is required to be kept information-theoretically secure from the unintended receiver. We show that the sum secure d.o.f. can only take one of five possible values: 0, 2/3, 1, 3/2, 2, for all network configurations. To determine the sum secure d.o.f., we divide the class of two-unicast layered networks into several sub-classes, and propose an achievable scheme based on the specific structure of the networks in each sub-class. Our achievable schemes are based on real interference alignment, cooperative jamming, interference neutralization and cooperative jamming neutralization techniques.