Secure Degrees Of Freedom Research Articles

Secure Degrees-of-Freedom of the MIMO X Channel With Delayed CSIT

In this letter, we study the secure degrees-of-freedom (SDoF) characterization for the multiple-input multiple-output (MIMO) X channel with confidential messages and delayed channel state information at the transmitter (CSIT). In particular, we propose a transmission scheme, which can be regarded as a generalization of the state-of-the-art scheme without security and with delayed CSIT. The key of this generalization is performing the security analysis, by which we derive the optimal duration of the artificial noise transmission phase. As a result, we derive the sum-SDoF lower bound. Furthermore, we reveal that if the number of receive antennas, denoted by <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> , is fixed, the minimum number of transmit antennas achieving the maximum of the lower bound is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\frac {7+\sqrt {33}}{8}N$ </tex-math></inline-formula> .

Achievable Secure Degrees of Freedom of MIMO Interference Channel With a Cooperative Jammer

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.

Secure Degrees of Freedom of MIMO Two-Way Wiretap Channel With no CSI Anywhere

This article considers a two-way multiple-input multiple-output (MIMO) Rayleigh block fading wiretap channel with two full-duplex nodes (Alice and Bob) exchanging messages simultaneously and wiretapped by a $ {N}_{ {e}}$ -antennas eavesdropper (Eve). The channel state remains unchanged over a coherence interval T and is unknown to all terminals, including Alice, Bob, and Eve, at the beginning of the coherence interval. We first show the expression of the optimal signal waveform for Alice and Bob, which maximizes the sum secrecy rate, should be the products of independent random diagonal matrices and isotropically distributed unitary matrices. Then, conditioned on large T and small $ {N}_{ {e}}$ , the upper-bound for the secure degree of freedom (s.d.o.f.) of the two-way wiretap channel is derived, and a constant-norm-signaling (CNS) scheme is presented to achieve this theoretical bound. Finally, we formulate an optimization problem of the s.d.o.f. achieved by the CNS scheme for general cases with arbitrary T and $ {N}_{ {e}}$ , and solve this problem by exploring the monotonicity of the achievable s.d.o.f.. Our result shows that the optimal s.d.o.f. can be independent of coherence time T and becomes the product of the numbers of legitimate nodes’s transmitter antennas (NLNTA) for large $ {N}_{ {e}}$ and small NLNTA.

Secure Degrees of Freedom of Rank-Deficient BCCM and Rank-Deficient ICCM

The secure degrees of freedom (SDoF) of two-user rank-deficient multiple-input multiple-output (MIMO) broadcast channel with confidential messages (BCCM) and two-user rank- deficient MIMO interference channel with confidential messages (ICCM) are studied respectively in this paper. In the systems, each transmitter has $M$ antennas, and each receiver has $N$ antennas. We consider that each transmitter intends to send secure messages to the corresponding receiver, and another receiver can receive the messages but cannot decode them, which ensures that the messages are confidential. Comparing with previous studies on two-user broadcast channel and interference channel, we explore the case that channel matrices are rank-deficient, and the rank of channel matrices can take arbitrary value, $D \leq \text{min} (M,N)$ . For the two rank-deficient channels, BCCM and ICCM, we divide the problem into several regimes according to the different configurations of the rank of channel matrices and the number of antennas at each node. The outer bound of rank-deficient BCCM is inferred from information theoretic methods and the achievable schemes are given by zero forcing techniques. We derive the outer bound of rank-deficient ICCM by taking the minimum region of the three outer bounds, where the first outer bound is the result of rank-deficient interference channel model without secrecy constraints, the second outer bound is obtained by considering secrecy constraints, and the third outer bound is achieved by transmitters cooperation. Then we present the achievable schemes by combining zero-forcing, real interference alignment, and spatial alignment. Finally, we determine the exact SDoF of the rank-deficient BCCM and the rank-deficient ICCM.

Secure Retrospective Interference Alignment

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 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, SDoF is achievable, and for the K-user IC-CM-EE, 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 , square-root of number of users.

On the Secure Degrees of Freedom of Two-Way 2 × 2 × 2 MIMO Interference Channel

We investigate the secure degrees of freedom (SDoF) for the two-way $2\times 2\times 2$ MIMO interference channel (IC) under three wiretap models, i.e., the confidential messages (CM) model, the untrusted relays (UR) model, and the combined CM and UR (CM-UR) model. For the general case of arbitrary antenna configuration under each wiretap model, we derive the upper bound on SDoF with Markov chain and secrecy constraints, and obtain the achievability schemes with designed interference neutralization, cooperative jamming, and interference alignment schemes. To gain insight on these bounds, we further consider the special case where each user node has $M$ antennas and each relay node has $N$ antennas, and highlight the modification process when achieving the maximum SDoF. For such special case, we show that the optimum SDoF of the CM model is achieved in the regimes $M\ge N$ and $M ; and for the UR model and CM-UR model, the optimum SDoF is achieved in the regimes $N\le ({M}/{2})$ , $N > 2M$ , and $N=M$ .

Secure Degrees of Freedom of Rank-Deficient MIMO Interference Channel

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, $D\le M$ . According to different configurations of ranks and number of antennas, the problem is divided into two regimes, and the exact SDoF is given by $\text {min}\left({2D,\frac {4M-2D}{3}}\right)$ . 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 in Networks with User Misbehavior

We investigate the secure degrees of freedom (s.d.o.f.) of three new channel models: broadcast channel with combating helpers, interference channel with selfish users, and multiple access wiretap channel with deviating users. The goal of introducing these channel models is to investigate various malicious interactions that arise in networks, including active adversaries. That is in contrast with the common assumption in the literature that the users follow a certain protocol altruistically and transmit both message-carrying and cooperative jamming signals in an optimum manner. In the first model, over a classical broadcast channel with confidential messages (BCCM), there are two helpers, each associated with one of the receivers. In the second model, over a classical interference channel with confidential messages (ICCM), there is a helper and users are selfish. By casting each problem as an extensive-form game and applying recursive real interference alignment, we show that, for the first model, the combating intentions of the helpers are neutralized and the full s.d.o.f. is retained; for the second model, selfishness precludes secure communication and no s.d.o.f. is achieved. In the third model, we consider the multiple access wiretap channel (MAC-WTC), where multiple legitimate users wish to have secure communication with a legitimate receiver in the presence of an eavesdropper. We consider the case when a subset of users deviate from the optimum protocol that attains the exact s.d.o.f. of this channel. We consider two kinds of deviation: when some of the users stop transmitting cooperative jamming signals, and when a user starts sending intentional jamming signals. For the first scenario, we investigate possible responses of the remaining users to counteract such deviation. For the second scenario, we use an extensive-form game formulation for the interactions of the deviating and well-behaving users. We prove that a deviating user can drive the s.d.o.f. to zero; however, the remaining users can exploit its intentional jamming signals as cooperative jamming signals against the eavesdropper and achieve an optimum s.d.o.f.

On the Secure Degrees of Freedom for Two-User MIMO Interference Channel With a Cooperative Jammer

We investigate the secure degrees of freedom (SDoF) for the two-user multiple-input multiple-output (MIMO) interference channel with a cooperative jammer, where each transmitter is equipped with $M$ antennas, each receiver is equipped with $N$ antennas, and the jammer is equipped with $K$ antennas. For each one of the four regions of parameters $(M,N)$ , i.e., $1\leq \frac {N}{M} , $\frac {N}{M} \geq 2$ , $\frac {1}{2}\le \frac {N}{M} , and $\frac {N}{M} , we provide both upper and lower bounds on the SDoF of the system which depend on the value of $K$ . In particular, the upper bound is the union of three upper bounds that are based on the DoF of the MIMO interference channel, the SDoF of the MIMO wiretap channel with a jammer, and the Secrecy Penalty Lemma and the Role of A Helper Lemma for the interference channel, respectively. For the lower bound, we provide achievability schemes that make use of signal alignment, zero forcing, cooperative jamming, and multi-slot transmission. Our results indicate that the SDoF of the system under consideration improves with respect to the no jammer case in all regimes, especially when $N>2M$ . Moreover, we quantify the SDoF gap of the network and reveal it as a function of the number of the antennas. For large $K$ , the upper and lower bounds coincide in all regimes, and the exact SDoF can be achieved.

Blind MIMO Cooperative Jamming: Secrecy via ISI Heterogeneity Without CSIT

We investigate the secure degrees of freedom (SDoF) of the multiple-input multiple-output (MIMO) wiretap channel with intersymbol interference (ISI) in the presence of a multi-antenna cooperative jammer. We focus on the practically relevant setting with no channel state information at the transmitters (CSIT). More specifically, the legitimate transmitter and the cooperative jammer only have statistical knowledge of the channel in terms of the effective number of ISI channel taps, i.e., channel impulse response (CIR) lengths, toward the legitimate receiver and the eavesdropper. Our main contribution is to show that in the absence of CSIT, positive SDoF can be achieved by carefully exploiting: 1) the heterogeneity of the CIR lengths toward both receivers and 2) the relative number of antennas at the four terminals. To achieve secrecy, we propose a scheme in which the transmitter strategically sends a mixture of information and artificial noise symbols in a way that exploits the heterogeneity of CIR lengths. Additionally, the cooperative jammer transmits artificial noise symbols in a way that completely masks all the information symbols that are received at the eavesdropping node. Under the proposed scheme, positive SDoF can be achieved, even when the number of antennas at the legitimate receiver is less than the number of antennas at the eavesdropper.

Achievable Secure Degrees of Freedom of ${K}$ -User MISO Broadcast Channel With Imperfect CSIT

In this letter, we derive the achievable secure degrees of freedom (SDoF) of ${K}$ -user multiple-input single-output broadcast channel with imperfect channel state information (CSI) at the transmitter. With the help of an $N_{j}$ -antenna external jammer, we propose a jamming rate splitting (J-RS) scheme to ensure the security of two wiretap models: 1) broadcast channel with confidential messages and 2) broadcast channel with an $N_{e}$ -antenna external eavesdropper, where each legal node has no CSI of wiretap channels. Our converse is based on the degrees of freedom (DoF) of the network with secrecy constraints, and the achievability is based on J-RS. By numerical analysis, it indicates that the proposed scheme can achieve more SDoF than the traditional rate splitting scheme, zero forcing scheme, and jamming scheme, which can reach to the DoF of the network.

On the Gaussian Multiple Access Wiretap Channel and the Gaussian Wiretap Channel With a Helper: Achievable Schemes and Upper Bounds

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.

Secure degrees of freedom of two‐user X‐channel with synergistic alternating channel state information at transmitters

In this study, a two-user single-input single-output X-channel with confidential messages is addressed. In this model, the authors assume that the transmitters have access to synergistic alternating channel state information. During different time slots, the channel state information at transmitters (CSIT) alternate between three states including perfect CSIT, delayed CSIT, and no CSIT. By using the eminent synergistic benefits of the CSIT pattern, some schemes capable of attaining the maximum achievable secure degrees of freedom (SDoF) are presented. Additionally, in devising the schemes, the minimal CSIT patterns required to achieve optimal SDoF are introduced. It is shown that for CSIT patterns which are weaker than minimal ones, using a half-duplex relay can assist the network in obtaining the optimal SDoF. Indeed, the relay alleviates the effects of the lack of knowledge at transmitters on achievable SDoF.

Physical-Layer Security in TDD Massive MIMO

We consider a single-cell downlink time-division duplex-based massive MIMO communication in the presence of an adversary capable of jamming and eavesdropping simultaneously. We show that the massive MIMO communication is naturally resilient to no training-phase jamming attack in which the adversary jams only the data communication and eavesdrops both the data communication and the training. Specifically, we show that the secure degrees of freedom (SDoF) attained in the presence of such an attack are identical to the maximum DoF attainable under no attack. Furthermore, we evaluate the number of base station (BS) antennas necessary in order to establish information theoretic security without even a need for Wyner encoding for a given rate of information leakage to the attacker. Next, we show that things are completely different once the adversary starts jamming the training phase. Specifically, we consider the pilot contamination attack, called training-phase jamming in which the adversary jams and eavesdrops both the training and the data communication. We show that under such an attack, the maximum achieved SDoF is identical to zero. Furthermore, the maximum achievable secure rates of users also vanish, even in the asymptotic regime in the number of the BS antennas. We finally address this attack and show that, under training-phase jamming , if the number of pilot signals is scaled in a certain way and the pilot signal assignments can be hidden from the adversary, the users achieve an SDoF identical to the maximum achievable DoF under no attack.

Secrecy DoF of Blind MIMOME Wiretap Channel With Delayed CSIT

We study the Gaussian multiple-input multiple-output multiple-eavesdropper wiretap channel where a transmitter wishes to communicate a confidential message to a legitimate receiver in the presence of eavesdroppers. Each node in the network is equipped with an arbitrary number of antennas. Furthermore, channels are time varying, and there is no channel state information available at the transmitter (CSIT) with respect to eavesdroppers' channels; and transmitter only has access to delayed CSIT of the channel to the legitimate receiver. The secure degrees of freedom (SDoF) for such a network has only been characterized for special cases, and is unknown in general. We completely characterize the SDoF of this network for all antenna configurations. In particular, we strictly improve the state-of-the-art achievable scheme for this network by proposing more efficient artificial noise alignment at the receivers. Furthermore, we develop a tight upper bound by utilizing three important inequalities that provide lower bounds on the received signal dimensions at receivers which supply delayed CSIT or no CSIT. These inequalities together allow for analysis of signal dimensions in networks with heterogeneous CSIT; and as a result, we present a converse proof that leads to characterization of SDoF for all possible antenna configurations.

Secure Degrees of Freedom of the Asymmetric Gaussian MIMO Interference Channel

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.

Physical Layer Security Based on Interference Alignment in K-User MIMO Y Wiretap Channels

This paper studies the secure degree of freedom (SDOF) of the multiway relay wiretap system $K$ -user MIMO Y wiretap channel, where each legitimate user equipped with $M$ antennas intends to convey independent signals via an intermediate relay with $N$ antennas. There exists an eavesdropper which is equipped with $N_{e}$ antennas close to the legitimate users. In order to eliminate the multiuser interference and keep the system security, interference alignment is mainly utilized in the proposed broadcast wiretap channel (BWC) and multi-access BWC (MBWC), and cooperative jamming is adopted as a supplementary approach in the MBWC model. The general feasibility conditions of the interference alignment are deduced as $M \ge K-1, 2M > N$ and $N \ge (({K(K-1)})/2)$ . In the BWC model, we have deduced the secure degrees of freedom (SDOF) as $K\min \{M,N\}-\min \{N_{e},K(K-1)/2\}$ , which is always a positive value. While in the MBWC model, the SDOF is given by $K\min \{M,N\}$ . Finally, since the relay transmits the synthesized signals of the legal signal and the jamming signal in the MBWC model, we propose a power allocation scheme to maximize the secrecy rate. Simulation results demonstrate that our proposed power allocation scheme can improve secrecy rate under various antenna configurations.

Signal Alignment for Secure Underwater Coordinated Multipoint Transmissions

Underwater acoustic networked systems often consist of geographically distributed antenna elements (DAEs) that are connected via cables or high-rate radio links (e.g., bottom-anchored nodes or surface buoys). This paper investigates countermeasures against eavesdropping attack in the coordinated multipoint (CoMP) transmission of DAEs to an underwater legitimate user. Exploiting the low sound speed in water and the spatial diversity of DAEs, we propose signal alignment for transmission secrecy, where a transmission strategy will be judiciously designed such that useful signals will collide at the eavesdropper while stay collision-free at the legitimate user. Specifically, the transmit DAE set, and the transmission schedule and transmission power of each active DAE are jointly optimized with a goal of minimizing the maximal received signal-to-interference-and-noise ratio (SINR) of useful signals at the eavesdropper, under a lower bound constraint of the received signal-to-noise ratio (SNR) at the legitimate user. Taking the orthogonal frequency-division multiplexing (OFDM) as the modulation technique, simulation and emulated experimental results demonstrate that the proposed method significantly degrades the eavesdropper's interception capability. We further investigate the secrecy capacity and the secure degrees of freedom (d.o.f.) of the signal alignment method from an information-theoretic perspective, which reveals that without external helpers, secure d.o.f. greater than 1/2 can be achieved.

Secure degrees of freedom in cooperative [formula omitted]-helper MIMO cognitive radio systems

We study the achievable secure degrees of freedom (DoF) in a cooperative MIMO cognitive radio system comprised of one primary source–destination pair, multiple secondary source–destination pairs and an eavesdropper against whom the primary user intends to secure its data. In this system, multiple secondary user pairs help to secure primary user’s data against eavesdropping. In return, these secondary users are allowed to access primary user’s spectrum. All users, including primary user pair, K≥2 secondary user pairs and the eavesdropper are equipped with M antennas. We investigate the secure DoF without the knowledge of eavesdropper’s channel state information (CSI), using the interference alignment concept combined with a zero inter-user interference constraint. A beamforming design is proposed to achieve secure DoF d for the primary user and DoF d for all secondary users if Kd≤M. The case of untrusted secondary users is also investigated where the secondary users may potentially eavesdrop. Simulation examples corroborating the theoretical results are presented.

On SDoF of Multi-Receiver Wiretap Channel With Alternating CSIT

We study the problem of secure transmission over a Gaussian multi-input single-output (MISO) two receiver channel with an external eavesdropper, under the assumption that the state of the channel which is available to each receiver is conveyed either perfectly ( $P$ ) or with delay ( $D$ ) to the transmitter. Denoting by $S_{1}$ , $S_{2}$ , and $S_{3}$ , the channel state information at the transmitter (CSIT) of user 1, user 2, and eavesdropper, respectively, the overall CSIT can then alternate between eight possible states, i.e., $(S_{1},S_{2},S_{3}) \in \{P,D\}^{3}$ . We denote by $\lambda _{S_{1} S_{2} S_{3}}$ the fraction of time during which the state $S_{1}S_{2}S_{3}$ occurs. Under these assumptions, we first consider the Gaussian MISO wiretap channel and characterize the secure degrees of freedom (SDoF). Next, we consider the general multi-receiver setup and characterize the SDoF region of fixed hybrid states $PPD$ , $PDP$ , and $DDP$ . We then focus our attention on the symmetric case in which $\lambda _{PDD}=\lambda _{DPD}$ . For this case, we establish bounds on SDoF region. The analysis reveals that alternating CSIT allows synergistic gains in terms of SDoF, and shows that by opposition to encoding separately over different states, joint encoding across the states strictly enables better secure rates. Furthermore, we specialize our results for the two receivers channel with an external eavesdropper to the two-user broadcast channel. We show that the synergistic gains in terms of SDoF by alternating CSIT are not restricted to multi-receiver wiretap channels, and can also be harnessed under broadcast setting.