Secure Transmission Design for Virtual Antenna Array-aided Device-to-device Multicast Communications

Device-to-Device Multicast (D2DM) communication which developed from Device-to-Device (D2D) unicast [1] communication has been regarded as one of the key technologies to improve the data rate and alleviate the workload of cellular networks [2, 3]. By allowing one transmitted information bearing signal to be received and decoded by multiple receivers, D2DM could achieve the large-scale parallel implementation of local contents sharing in high-density environments, thus can support massive connectivity and also reduce the redundancy transmission of the Base Station (BS), signaling overhead, and bandwidth waste. Due to the high spectrum utilization and transmission efficiency, D2DM is not only suitable for the data-rate-hungry mobile applications such as online conference, video streaming, mobile computing, and virtual reality, but also considered as a key enabling technique for various future cellular network services such as mobile edge computing (MEC)[4, 5, 6, 7, 8] and fog-radio access network (F-RAN) [9, 10].

Although D2DM has a wide range of applications and there have been many literatures on how to improve its network delivery efficiency[11, 12, 13, 3, 14, 15], on the other hand, as in other wireless communication systems, communication security is an important indicator to evaluate the performance of a D2DM system. Physical Layer Security (PLS), which is an essential strategy to protect the information by fully utilizing the basic characteristics of the wireless channel and noise, has been regarded as a highly promising security approach for wireless communications. Compared with the cryptography in the higher layer, PLS is independent of the computational complexity and thus more suitable for computation resource limited networks such as D2D communications. Motivated by this, many research efforts have been devoted to secure D2D communication from the physical layer security perspective [16, 17, 18, 19, 20, 21, 22, 23]. Ref. [16] attempted to increase the D2D link’s capacity while guaranteeing the security rate by controlling the transmit power of D2D pairs. Refs. [17, 18] considered that multiple D2D pairs can access the same resource block and one D2D pair is also permitted to access multiple RBs. In [19], aimed to secure CUE’s communication and increase energy efficiency of D2D communication, the condition whether D2D link could reuse a CUE’s RB was derived. Ref. [20] investigated both single-channel and multi-channel D2D communications. For providing security to CUEs and improving the spectral efficiency of the D2D links simultaneously, the authors jointly optimized the power allocation and channel assignment of the D2D links and CUEs. In [21], the model selection of a D2D link under the physical-layer security perspective was investigated. The PLS in UAV-enabled D2D communications was investigated in [22] where a UAV plays the role of flying BS. In [23], authors shown that the Non-Orthogonal Multiple Access (NOMA) technique could also be used to secure D2D communications. Note that, the above literatures are mainly focused on the D2D unicast scenarios. To the best of our knowledge, the PLS schemes in D2DM are rare [24]. No matter whether the design target is to increase network capacity or to provide PLS, the main difference between D2D unicast and D2DM is that the data rate in D2DM is limited by the worst Signal to Interference Plus Noise Ratio (SINR) UE and hence leading to a max-min optimization problem which is hard to deal with.

However, the above secure schemes in D2D communication are mainly based on cooperative jamming, in which the cellular communication signals and other D2D communication signals are utilized to interfere eavesdroppers for securing intended D2D communications, or vice versa. Constrained by the limited physical size, UEs usually have only one antenna or multiple antennas that are less distant than the coherent distance. Hence, the multiple-antenna based beamforming technologies for providing PLS [25, 26, 27] are not suitable for D2D communications. Thus, how to secure D2D communications is still an open issue and this paper makes attempt to deal with it by introducing the following Virtual Antenna Array (VAA) technique.

Recently, a novel implementation of Multiple-Input-Multiple-Output (MIMO) wireless system called VAA [28, 29, 30] was proposed as a promising approach to further increase the PLS performance and network delivery efficiency of the above D2DM scenario. The VAA system is realized by permitting adjacent single antenna mobile terminals to cooperate among each other and thus form the so called virtual antenna array. In the VAA system, as a virtual transmitter, the elements in VAA could perform cooperative beamforming to transmit information to receivers, while as a virtual receiver, the elements in VAA could exchange the received signals via some short-range communication techniques such as blue tooth and then perform some multiple-antenna receive techniques like maximum ratio combination to increase SINR. It is important to note that, compared with the Single-Input-Single-Output (SISO) channel, the spatial diversity of VAA together with cooperative beamforming can significantly improve the quality of the transmitted signal and then enhance the PLS performance[31, 32] and channel capacity [30, 33, 34, 35, 36].

In this papar, we propose a VAA-aided D2DM (VD2DM) scheme for securing D2DM communication. In the VD2DM, UEs who cache the common contents cooperatively form a VAA system and each UE could be treated as one of the antennas in the VAA. By multiplying the signals each UE sends itself by a beamforming coefficient (i.e., the corresponding element in the beamforming vector), VAA could jointly conduct cooperative beamforming to multicast the common contents to receivers while prevent eavesdroppers from overhearing. Further, unlike the traditional D2D communications which only allow one UE as the transmitter, the VD2DM could take full use of the same the common contents cached on different UEs. The main contributions of this paper are summarized as follows:

• 1)

  We propose a novel secure VD2DM transmission scheme, which integrates the VAA into the D2DM underlying cellular network. In the proposed scheme, all UEs are equipped with one antenna, and the BS is equipped with multiple antennas. A group of UEs (i.e., D2D receivers) needs a common confidential information data and in the D2D transmitter group, each UE (i.e., D2D transmitters) has cached the common confidential information desired by the D2D receivers. Under the coordination of the BS, D2D transmitters form a VAA, where each UE multiplies the signals by a complex beamforming coefficient¹¹1Amplitude control can be accomplished by power control and phase control can be accomplished by delay.to perform cooperative beamforming for multicasting the confidential information to D2D receivers. Meanwhile, a group of baleful UEs (i.e., eavesdroppers) desire to overhear the information delivered by the VAA. Since the VAA transmission reuses the downlink subcarrier resources of a CUE and hence the downlink signal transmitted by the BS could be treated as a cooperative jamming to secure VD2DM. To further limit the overhearing capacity of eavesdroppers, the BS splits a part of power to transmit an Artificial Noise (AN) to block eavesdroppers.

• 2)

  We explore the difference between the VD2DM and the traditional D2D unicast, and then formulate the weighted sum rate maximization problem (WRMP) to increase the achievable rate of CUE and the achievable secrecy rate of VD2DM simultaneously by jointly optimizing the transmit beamforming of the BS, the cooperative beamforming vector of the VAA and the covariance matrix of AN. The formulated problem is non-convex and non-smooth since both max-min and min-max problem are included, i.e., the achievable performance of the multicast group is generally limited by the receiver with the lowest SINR and the secrecy performance is limited by the eavesdropper with the highest SINR. We then introduce a series of auxiliary variables to transfer the non-convex objective function and constraints into a series of convex objective function and constraints. As last, an effective iteration algorithm is designed to solve it.

• 3)

  Moving beyond that, we investigate an extreme case (i.e., the worst case) in which the eavesdroppers also form a VAA and gang up together to jointly overhearing the communication of VD2DM. In our analysis, we treat eavesdroppers as one equivalent one multi-antenna eavesdropper called Eve. In this case, with the help of semidefinite relaxation and slack operation, we can decompose the WRMP into a two-level optimization problem, in which the outer level is a single-variable optimization, and the inner level is a non-convex problem, which can be recast to a SemiDefinite Problem (SDP) based on Arithmetic Geometry Mean (AGM) inequality.

The remainder of the paper is organized as follows. Section II introduces our considered system model. Section III gives our formulated WRMP problem and the detailed solution to the formulated problem. Section IV extends the considered scenario to the worst case and gives the corresponding solution. Simulation results are shown in Section V and finally Section VI concludes the paper.

$Notations$: $\mathbb{C}$ and $\mathbb{R}$ denote the sets of complex and real numbers, respectively; $\rm Rank(\bullet)$, $\rm Tr(\bullet)$, ${\left[{\bullet}\right]^{-1}}$, ${\left[{\bullet}\right]^{T}}$, and ${\left[{\bullet}\right]^{H}}$ denote the rank, trace, inverse, transpose, and conjugate-transpose operations, respectively; ${\left\|\bullet\right\|_{2}}$ and ${\left[{\bullet,\bullet}\right]^{+}}$ denote the Frobenius norm and the maximum value of its argument, respectively. Except for $\bf e$, bold symbols in capital letter and small letter denote matrices and vectors, respectively and $\bf e$ is the natural constant. $\bf{A}\succeq\bf{0}$ means $\bf{A}$ is positive semidefinite matrix. ${\bf{a}}[n]$ and ${\bf{A}}[n]$ are the $n^{th}$ element of the vector ${\bf{a}}$ and the $n^{th}$ element of the diagonal entry of the matrix ${\bf{A}}$, respectively. $\log(\bullet)$ denotes the natural logarithm of its argument; ${{\cal C}{\cal N}}\left({a,b}\right)$ denotes the complex Gaussian distribution with mean $a$ and variance $b$; ${{\bf{I}}_{L}}$ is an $L\times L$ identity matrix.

Consider a VD2DM communication scenario which underlays over the downlink spectrum of a cellular communication network as shown in Fig. 1. A group of D2D receivers want to obtain identical confidential information, and a group of UEs who pre-cache the identical confidential information will form the transmit VAA (i.e., the D2D transmitter group) to multicast the information under the coordination of the BS. A CUE shares its downlink resource (e.g., subcarriers in OFDMA system) for the D2D link to reuse. Also, in the system, a group of baleful UEs (i.e., eavesdroppers group) try to overhear the communication between D2D transceivers²²2For example, in commercial corporation or research institution, colleagues could share the internal information via VD2DM in a timely and convenient manner, meanwhile the VD2DM may be under the risk of eavesdrop attack from other UEs in the cellular system., thus the physical layer security is taken into consideration. Except for the BS, all nodes are equipped with one antenna, and the BS has $K$ antennas for its downlink transmission.

Let ${{\cal M}}\buildrel\Delta\over{=}{\rm{\{1,2,}}\ldots{\rm{,}}M{\rm{\}}}$ denote the set of D2D transmitters, ${{\cal N}}\buildrel\Delta\over{=}{\rm{\{1,2,}}\ldots{\rm{,}}N{\rm{\}}}$ denote the set of D2D receivers and ${{\cal L}}\buildrel\Delta\over{=}{\rm{\{1,2,}}\ldots{\rm{,}}L{\rm{\}}}$ denote the set of eavesdroppers. Define ${\bf{h}}_{D,n}$ as the channel vector between D2D transmitters and D2D receiver $n$, ${\bf{h}}_{l}$ as the channel vector between D2D transmitters and eavesdropper $l$, ${{\bf{h}}_{C}}$ as the channel vector between D2D transmitters and the CUE, ${{\bf{h}}_{B,n}}$ as the channel vector between BS and D2D receiver $n$, ${{\bf{h}}_{B,l}}$ as the channel vector between BS and eavesdropper $l$, and ${{\bf{h}}_{BC}}$ as the channel vector between BS and CUE, respectively. The Rayleigh block-fading channel model is assumed in this paper. Note that, the cellular network is considered as a kind of centralized network architecture and the eavesdroppers are also be considered as the registered active user in the cellular network, hence the BS can collect the channel state information (CSI) of all links in the network³³3In LTE system, the BS and D2D transmitters could perform channel quality indication estimation from the received sounding reference signal transmitted by eavesdroppers then obtain $\bf{h}_{C}$ and $\bf{h}_{l}$, respectively. Further, D2D transmitters could report periodic or aperiodic report $\bf{h}_{l}$ to the BS from the physical uplink control channel. Interested readers can read [37] for the details. [1]. Also, since the channels are quasi-static over each time block, the BS could precalculate the the optimal beamforming vectors at the beginning of each time block, then forwards the beamforming vector of the VAA to each D2D transmitters via physical downlink control channel.

We first consider the scenario in which each eavesdropper overhears the confidential information independently, and there has no extra information exchanges between eavesdroppers. The transmitted signal of the BS is

in which ${s_{B}}\sim{{\cal C}{\cal N}}\left({0,1}\right)$ is the desired signal of CUE with beamforming vector ${{\bf{w}}_{B}}$, ${{\bf{n}}_{B}}\sim{{\cal C}{\cal N}}\left({{\bf{0}},{\bf{Q}}}\right)$ is the AN vector to interfere eavesdroppers from overhearing with covariance matrix ${\bf{Q}}\in\mathbb{C}{{}^{K\times K}}$. The D2D transmitters form a VAA and employ the cooperative beamforming strategy to multicast the confidential information ${s_{D}}\sim{{\cal C}{\cal N}}\left({0,1}\right)$. Specifically, the $m^{th}$ D2D transmitter multiplies the identical confidential information bearing signal $s_{D}$ with a complex weight ${w_{m}}$. Denote the cooperative beamforming vector by ${\bf{w}}\buildrel\Delta\over{=}{[{w_{1}},{w_{2}},\ldots,{w_{M}}]^{T}}$, then the received signals of D2D receiver $n$, eavesdropper $l$ and CUE can be respectively expressed as

and

where ${n_{i}}\sim{{\cal C}{\cal N}}\left({0,1}\right),i\in\left\{{{{\cal N}},{{\cal L}},C}\right\}$ is the Additive White Gaussian Noise (AWGN).

From (2), (3) and (4), the channel capacities of D2D receiver $n$, the eavesdropper $l$ to overhearing $s_{D}$, the eavesdropper $l$ to decoding $s_{B}$ and the CUE are respectively

and

in which where ${{\bf{H}}_{B,n}}\buildrel\Delta\over{=}{\bf{h}}_{B,n}{\bf{h}}_{B,n}^{H}$, ${{\bf{H}}_{B,l}}\buildrel\Delta\over{=}{\bf{h}}_{B,l}{\bf{h}}_{B,l}^{H}$ and ${{\bf{H}}_{BC}}\buildrel\Delta\over{=}{\bf{h}}_{BC}{\bf{h}}_{BC}^{H}$

Let ${R_{S}}$ be the achievable secrecy rate of the D2D multicast communication and ${R_{C}}$ be the achievable rate of the CUE. Then an achievable rate region is given as the set of nonnegative rate pairs $\left({{R_{S}},{R_{C}}}\right)$ satisfying

and

in which $\mathop{\min}\limits_{n\in{{\cal N}}}{C_{n}}$ means the achievable rate of D2D multicast is determined by the receiver who has the minimum channel capacity. Also, $\mathop{\max}\limits_{l\in{{\cal L}}}{C_{l}}$ means the eavesdropping rate is determind by the eavesdropper who has the maximum channel capacity, since the confidential information cannot be overheard by any eavesdropper. ${C^{\prime}_{l}}<{R_{C}}$ guarantees that the eavesdroppers could not perform non-orthogonal multiple access (NOMA) to decode and eliminate the $s_{B}$ first, hence the $s_{B}$ could jointly interfere with the eavesdroppers.

As can be seen in (5), (6), (7) and (8), the beamforming vector ${{\bf{w}}_{B}}$, cooperative beamforming vector ${\bf{w}}$ and the covariance matrix of AN, $\bf{Q}$ directly determine the achievable data rates, thus our work focuses on the design of ${{\bf{w}}_{B}}$, ${\bf{w}}$ and $\bf{Q}$, under an achievable rate region maximization (RRM) formulation with power constraint. The formulated RRM is a vector maximization problem, i.e. (12), in which $\chi$ is an arbitrarily number greater than zero. In (12), $P_{max}$ and $P_{B}$ are the total power budgets of the VAA and the BS, respectively, the constraints C1 and C2 are the maximal power constraint of each D2D transmitter and the BS, C3, C4 and C5 are corresponding to (9), (10) and (11), respectively. The RRM problem in (12) is a non-convex vector optimization and thus difficult to solve. In the next section, we elaborate our approaches to tackle (12).

A common approach for dealing with the vector optimization problem (12) is referred to as scalarization, whose basic idea is to transform the vector optimization into the weighted sum maximization optimization. In the rest of this section, we transform the RRM into a WRMP and then deal with it.

In Section III above, we only considered the scenario in which multiple eavesdroppers overhear the confidential information independently. However, in the worst case that the eavesdroppers may collude and form another VAA to jointly process the eavesdropped information (such as maximum ratio combination and MMSE receiver) to increase the overhearing capacity. We next make efforts to investigate the PLS scheme in such a scenario.

Since the eavesdroppers are ganged up together and form an eavesdrop VAA, the group of eavesdroppers i.e., the eavesdrop VAA could be treated as a virtual multi-antenna eavesdropper which is equipped with $L$ antennas. In the rest of this paper, unless particularly stated, we use Eve to represent the so-called eavesdrop VAA, i.e., the equivalent $L$-antenna eavesdropper.

Since the eavesdroppers form the eavesdrop VAA, the channel vector from D2D transmitter group to the $l^{th}$ eavesdropper, ${\bf{h}}_{l}$ could be treated as the channel vector from D2D transmitter group to the $l^{th}$ antenna of Eve. We use ${{\bf{G}}_{E}}\buildrel\Delta\over{=}\left[{{\bf{h}}_{1},{\bf{h}}_{2},\ldots,{\bf{h}}_{L}}\right]$ represent the channel matrix between D2D transmitter group and Eve. Similarly, we use ${{\bf{G}}_{B}}\buildrel\Delta\over{=}\left[{{{\bf{h}}_{B,1}}{{\bf{h}}_{B,2}},...,{{\bf{h}}_{B,L}}}\right]$ represent the channel matrix between BS and Eve. Then the received signals of D2D receiver $n$, Eve and CUE can be expressed as

and

respectively, where ${{\bf{n}}_{E}}\sim CN\left({{\bf{0}},{{\bf{I}}_{L}}}\right)$ is the AWGN vector at Eve. Similar with the normal case, from (30), (31) and (32), the channel capacities of D2D receiver $n$, the eavesdropper Eve to overhearing $s_{D}$⁵⁵5 The channel capacity of Eve is obtained by calculating the mean mutual information which is the upper bound of the achievable rate of Eve, regardless of which receiving filter Eve used. Hence, the achievable secrecy rate of the worst case is a lower bound., the eavesdropper $l$ to decoding $s_{B}$ and the CUE are

and

respectively, where ${\bf{H}}_{C}\buildrel\Delta\over{=}{\bf{h}}_{C}{\bf{h}}_{C}^{H}$. It is important to note that since the eavesdroppers form another VAA, the eavesdrop channel has transformed from MISO to MIMO, and the corresponding channel capacity becomes (34) and (35).

The achievable rate region is then defined as the set of nonnegative rate pairs $\left({R_{S}^{worst},R_{C}^{worst}}\right)$ satisfying

and

In the worst case, to reduce computational complexity, we discuss the WRMP in two subcases and formulate corresponding optimization problems. Note that, different from the WRMP in the normal case which include constraints C5 to guarantee that the eavesdroppers could not perform NOMA to decode and eliminate the $s_{B}$ first. In both subcases, we do not consider (39) into the optimization problems, the detailed reasons please see Remark 3 at last of this section.

In subcase 1, we consider that Eve could not decode and eliminate the $s_{B}$ first and formulate the corresponding sub-WRMP, ${\rm{P4\_1}}$. On the contrary, in subcase 2, we consider that Eve could decode and eliminate the $s_{B}$ first and formulate the corresponding sub-WRMP, ${\rm{P4\_2}}$.

We first consider subcase 1 and formulate the following ${\rm{P4\_1}}$ to maximize the weighted sum of $R_{S}^{worst}$ and $R_{C}^{worst}$

where $R_{S}^{worst}=\varphi^{worst}-{\beta^{worst}}$ and $\alpha$ is the weighted coefficient. ${\rm{C}}4^{\prime}$, ${\rm{C}}6^{\prime}$ and ${\rm{C}}7^{\prime}$ are evolved from C4, C5 and C7 in (14) respectively, because of the different expressions of the channel capacity of CUE, D2D receivers and Eve.

Note that, ${\rm{P4\_2}}$ is basically the same as ${\rm{P4\_1}}$, by replacing ${\rm{C7^{\prime}}}$ with

${\rm{P4\_2}}$ can be obtained. In ${\rm{C7^{\prime}\_2:}}$, only the AN could interfere with Eve since Eve could decode and eliminate $s_{B}$ first.

We first make effort to solve ${\rm{P4\_1}}$, it is easy to see that ${\rm{C}}4^{\prime}$ and ${\rm{C}}6^{\prime}$ have the same structure as C4 and C6, respectively, and the technique used to transform ${\rm{C}}4$ and ${\rm{C}}6$ to series of convex constraints in Section III.B could still be used to deal with ${\rm{C}}4^{\prime}$ and ${\rm{C}}6^{\prime}$. However, due to the determinant operation, ${\rm{C}}7^{\prime}$ in (40) is a non-convex constraint and non-trivial to deal with.