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IEEE Transactions on Vehicular Technology

Secure Degrees of Freedom of the Asymmetric Gaussian MIMO Interference Channel

Qiang Wang, Yi Wang, Haojin Wang

Published Mar 22, 2017• 11 citations
Publisher

Abstract

We study the secure degrees of freedom (s.d.o.f.) for a two-user multiple-input multiple-output (MIMO) Gaussian interference channel with confidential messages, each transmitter–receiver pair of which intends to keep secret from the other pair. Different from existing works on symmetric scenario, in this paper, an asymmetric system is considered, where transmitters have$M_1$,$M_2$antennas and receivers have$N_1$,$N_2$antennas, respectively. The channel is assumed to be fast fading and channel matrices are fixed and full rank. We subdivide the problem into several regimes based on the configuration of antennas at each node and present the exact secure degrees of freedom of corresponding regime. To prove a converse, the outer bound is developed by combining information theoretic derivation and the results of fundamental models. We also provide the achievable schemes based on zero forcing, real interference alignment, and cooperative jamming. One part of private signal is sent in the null space, while the other part is sent in a way that it aligns with cooperative jamming signal at unintended receiver, but leaves the secret message intact at intended receiver. As a consequence of the scheme, the objective s.d.o.f. can be achieved and the system ensures that the confidential messages cannot be eavesdropped by unintended receivers.

DOI
10.1109/tvt.2017.2685526

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We study the secure degrees of freedom (s.d.o.f.) for a two-user multiple-input multiple-output (MIMO) Gaussian interference channel with confidential messages, each transmitter–receiver pair of which intends to keep secret from the other pair. Different from existing works on symmetric scenario, in this paper, an asymmetric system is considered, where transmitters haveM1M_1M1​,M2M_2M2​antennas and receivers haveN1N_1N1​,N2N_2N2​antennas, respectively. The channel is assumed to be fast fading and channel matrices are fixed and full rank. We subdivide the problem into several regimes based on the configuration of antennas at each node and present the exact secure degrees of freedom of corresponding regime. To prove a converse, the outer bound is developed by combining information theoretic derivation and the results of fundamental models. We also provide the achievable schemes based on zero forcing, real interference alignment, and cooperative jamming. One part of private signal is sent in the null space, while the other part is sent in a way that it aligns with cooperative jamming signal at unintended receiver, but leaves the secret message intact at intended receiver. As a consequence of the scheme, the objective s.d.o.f. can be achieved and the system ensures that the confidential messages cannot be eavesdropped by unintended receivers.
We consider the two-user multiple-input multiple- output (MIMO) interference channel with confidential messages (ICCM). We determine the exact sum secure degrees of freedom (s.d.o.f.) for the symmetric case of M antennas at both transmitters and N antennas at both receivers. We develop the converse by combining the broadcast channel with confidential messages (BCCM) cooperative upper bound, decodability upper bound for the IC with no secrecy constraints, and extensions of the secrecy penalty and role of a helper lemmas. We propose a novel achievable scheme for the 2 × 2 ICCM, which combines asymptotic real interference alignment with spatial interference alignment. Using this scheme, we provide achievable schemes for any M and N by proper vector space operations.
In this letter, the secure degrees of freedom (SDoF) of 2-user multiple-input multiple-output (MIMO) interference channel with an external N-antenna cooperative jammer (CJ) is considered. In this model, each transmitter intends to send confidential messages to the corresponding receiver and keep secret with unintended receiver. Different from the previous studies on interference channels, our model has an external cooperative jammer, which can improve the SDoF by transmitting interference signals. We derive the converse and give the corresponding achievable schemes to determine the exact SDoF, and study the influence of CJ on the SDoF of this system. We find that adding one CJ antenna has the same effect as adding one transmitting antenna, and has a better effect than adding one receiving antenna. The converse is given by using vector extensions of the role of a helper and secrecy penalty lemmas. For arbitrary values of N and M, we provide different achievable schemes for the corresponding regimes by making use of the symbol extension and spatial alignment techniques.
We consider the two-user multiple-input multipleoutput (MIMO) interference channel with confidential messages (ICCM). We determine the exact secure degrees of freedom (s.d.o.f.) region for the symmetric case of M antennas at both transmitters and N antennas at both receivers. We develop the converse by combining the broadcast channel with confidential messages (BCCM) cooperative upper bound, decodability upper bound for the interference channel with no secrecy constraints, and vector extensions of the secrecy penalty and role of a helper lemmas. For the achievability, we first show that the s.d.o.f. region is a four-vertex polytope. For the sum s.d.o.f. point, we propose a novel achievable scheme for the 2 × 2 ICCM, which combines asymptotic real interference alignment with spatial interference alignment. Using this scheme, we provide achievable schemes for any M and N by proper vector space operations. We achieve the other non-trivial extreme polytope points by employing one of the transmitters as a deaf helper for assisting the secure transmission of the other user. We present simplified achievable schemes for the special case of time-varying MIMO ICCM. The achievable schemes, in this case, make use of the time-varying nature of the channel to construct vector-space alignment counterpart of the real interference alignment used in the static channel case.
We investigate the secure degrees of freedom (SDoF) of a two-transmitter Gaussian multiple access channel with multiple antennas at the transmitters, a legitimate receiver and an unknown number of eavesdroppers each with a number of antennas less than or equal to a known value NE. The channel matrices between the legitimate transmitters and the receiver are available everywhere, while the legitimate pair have no information about the fading eavesdroppers’ channels. We provide the exact sum SDoF for the considered system. We show that it is equal to min(M1 + M2 − NE, 1/2(max(M1;N2 + max(M2;N) − NE,N). A new comprehensive upperbound is deduced and a new achievable scheme based on utilizing jamming is exploited. We prove that Cooperative Jamming is SDoF optimal even without the eavesdropper CSI available at the transmitters.
This letter investigates the secure degrees of freedom (SDoF) for a 2-user rank-deficient (M, D) interference channel with confidential messages (ICCM) in which each node is equipped with M antennas. In this model, each transmitter intends to send a confidential message to the corresponding receiver while ensuring that the messages are kept secret against the unintended receiver. Unlike the existing studies on 2-user multiple-input multiple-output (MIMO) interference channel models, channel matrices in this letter do not need to be full rank, and they can take arbitrary rank, D ≤ M. According to different configurations of ranks and number of antennas, the problem is divided into two regimes, and the exact SDoF is given by min(2D, 4M - 2D/3). Then, we propose a converse proof, in which the outer bound is proved by using information theoretic methods. Finally, we propose different achievable schemes for the corresponding regimes by utilizing the spatial alignment, symbol extension, and zero-forcing techniques. It can be obviously seen that the full rank situation on ICCM can be regarded as a special case when D = M.
Secure Degrees of Freedom of KKK -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.