Design of Energy-Efficient Artificial Noise for Physical Layer Security in Visible Light Communications

Indoor wireless activities generate roughly 80% of the total wireless Internet traffic [1]. In the case of indoor scenarios, a standard means of Internet access is Wireless Fidelity (Wi-Fi), which is currently imposed by increasingly stringent requirements for high data rates due to the tremendous growth in the number of mobile devices and data-intensive applications. A great deal of research effort has been devoted to satisfying this increasing demand by exploring new communications technologies, among which visible light communications (VLC) is a promising candidate. Compared to existing wireless systems, VLC offers the unique advantage of simultaneously providing illumination and communication, both essential for indoor activities. Furthermore, the promised high data rate comes at no license-fee spectrum and little infrastructural installation.

Over the years, numerous theoretical and experimental studies on VLC have focused on improving its practicality and data-rate performance (see [2, 3, 4] and references therein) for both indoor and outdoor scenarios. To realize practical VLC systems, one must also incorporate security and privacy in the design. Traditional approaches for acquiring secure transmissions over public communication media are based on well-established key-based cryptographic algorithms. The security of conventional cryptography relies on the extreme computational complexity of solving certain mathematical problems (e.g., the factoring problem). Given the current computing power of the classical computer, it is likely that breaking the secret keys is not feasible within a meaningful amount of time. However, quantum computers, which are expected to be fully functional in the foreseeable future, can threaten the security of current encryption schemes as they are exponentially faster than their classical counterparts [5]. The vulnerability of traditional cryptography to quantum computers has been the primary motivation for researching alternative security measures not based on computational difficulty. In this context, physical layer security (PLS) is a promising approach to complement and/or replace cryptographic techniques as the information confidentiality can be kept perfectly secure regardless of the computational power of the eavesdropper[6].

While the literature on PLS in VLC systems is considerably extensive (see [7, 8] and references therein), the existing studies mainly focused on two directions. The first focuses on analyzing different lower and upper bounds on the secrecy capacity of the single-input single-output (SISO) wiretap VLC channels [9, 10, 11]. Compared with the SISO, the multiple-input single-output (MISO) configuration is more relevant in practice, as deploying multiple LED luminaires is a norm in illumination. In the second direction, several studies, therefore, exploited the spatial degree of freedom at the transmitting side to investigate the application of precoding and artificial noise (AN) to improve secrecy performance. Specifically, in the case of precoding, Mostafa $et\leavevmode\nobreak\ al.$ studied the designs of zero-forcing (ZF) and general precoders to maximize the achievable secrecy rate for systems with a single legitimate user and a single eavesdropper considering both perfect and uncertain estimation of the channel state information (CSI) [12, 13]. Subsequent works then examined precoding designs for more general system configurations with different objectives, such as multi-eavesdropper with minimizing the total transmitted power [14], multi-legitimate users with maximizing the achievable secrecy sum-rate [15], and multi-user systems, in which each user treats others as eavesdroppers (hence, transmitted messages for users must be kept mutually confidential)[16]. ZF precoding strategies to deal with both active and passive eavesdroppers were also proposed in [17].

In addition to precoding, AN is another approach to enhancing PLS performance [18, 19, 20]. In essence, AN is a jamming signal which is purposely generated and transmitted by the transmitter so that it causes minimal or no interference to the legitimate user while potentially degrading the quality of the eavesdroppers’ channel. Note that AN should be deployed with precoding to fully exploit the available spatial degree of freedom. Hence, precoding can be considered a special case of AN-aided transmission where no power is allocated to AN generation. Initial research on the topic considered systems where a fixed LED luminaire transmits the information-bearing signal while the rest broadcast jamming signals [21, 22]. We term this scheme as AN-aided SISO transmission. Subsequent works then focused on the jamming scheme where the information-bearing signal and AN are combined and transmitted by all luminaires. This scheme is referred to as AN-aided MISO transmission. Joint precoding and AN designs were investigated under different performance criteria. For example, the authors in [23] studied the design that maximizes the signal-to-interference-plus-noise ratio (SINR) of the legitimate user’s channel while constraining that of the eavesdroppers’ channel to a predefined threshold. The study in [24] examined the same objective yet under the constraint on the average SINR of the eavesdroppers’ channel (assuming that eavesdroppers’ CSI is unknown at the transmitters). Considering a multi-Bob VLC system, AN designs to maximize the minimum SINR among Bobs’ channels were investigated in [25]. Given specific constraints on the SINRs of Bob’s and Eves’ channels, our study in [26] AN designs to minimize the total transmitted power considering both perfect and imperfect CSI estimations.

Aside from improving the secrecy rate, there is also a need to optimize energy efficiency (EE), which measures how much energy is consumed per a transmitted information bit [27]. This has been of particular interest in recent years due to the increased effort in reducing energy consumption to combat climate change [28]. Although several studies have been conducted on EE in VLC and hybrid VLC/RF systems [29, 30, 31, 32, 33, 34], little attention has been paid to that in the context of PLS. In fact, to the best of the authors’ knowledge, our previous studies in [35, 36] are the first works concerning the PLS in VLC from the EE perspective. Defining secrecy energy efficiency (SEE) as the consumed energy per a transmitted confidential bit, we designed precoding schemes to maximize the sum SEE for multi-user VLC systems where messages among users are kept mutually confidential.

Against the above background, this paper aims to study energy-efficient AN designs for PLS in VLC systems. We consider the systems with a legitimate user (i.e., Bob) and multiple eavesdroppers (i.e., Eves) whose CSI can be known or unknown on the transmitting side. The AN-aided MISO and a refined AN-aided SISO transmission scheme, which we term as selective AN-aided SISO, are investigated and compared in terms of the achievable SEE. Contrary to the original AN-aided SISO, where the information-bearing signal is transmitted by a fixed LED luminaire, in the proposed scheme, that is done by the luminaire, which has the highest channel gain to Bob. The rest of the luminaires act as jammers transmitting AN. By doing so, Bob can enjoy the strongest information-bearing signal while suffering less from the AN. Hence, given the same amount of consumed energy, the proposed scheme should achieve a better achievable secrecy rate, resulting in a higher SEE compared with the original counterpart. In the conference version of this paper [37], the design of the selective AN-aided SISO scheme considering the unavailability of Eves’ CSI was studied. Extending from this, the contributions of this paper are summarized as follows.

• •

  In addition to the presented design in [37], which was solved using the Dinkelbach algorithm and the convex-concave procedure (CCP), a simple and low-complexity ZF AN-aided SISO design is investigated. Simulation results show that this design approach achieves comperable SEE that of the design in [37] with significantly reduced computational time.

• •

  We study an AN design when Eves’ CSI is available at the transmitting side. The presence of multiple Eves results in multiple sub-wiretap channels, each characterized by a different achievable secrecy rate. To ensure that the transmitted information to Bob is secure against all Eves, the system’s achievable secrecy rate is defined as the minimum among all sub-wiretap channels. Then, an AN design is cast as a max-min SEE problem, which requires a different approach to solve. Specifically, we transform the original problem into a feasibility problem, which can then be solved using the bisection method.

• •

  Comprehensive simulations are conducted to compare the complexity and performance of the original AN-aided SISO, the proposed selective AN-aided SISO, and the AN-aided MISO transmissions under various parameter settings. Our main finding is that the selective AN-aided SISO scheme achieves a better SEE than the AN-aided MISO does in the case of unknown Eves’ CSI, while it is the opposite in the case of known Eves’ CSI as the AN-aided MISO scheme is superior.

The rest of the paper is organized as follows. The system models, including the channel model, signal model, and energy consumption, are described in Section II. Section III presents different AN-aided designs from the perspective of SEE maximization. Numerical results with related discussions are given in Section IV. Finally, Section V concludes the paper.

Notation: The following notations are used in the paper. $\mathbb{R}$ and $\mathbb{R}_{\geq 0}$ denote the set of real and nonnegative real numbers, respectively. Uppercase and lowercase bold letters (e.g., $\mathbf{A}$ and $\mathbf{a}$) represent matrices and column vectors, respectively. $\mathbf{A}^{T}$ is the transpose of $\mathbf{A}$. In addition, $|\cdot|$, $\lVert\cdot\rVert_{2}$, $\lVert\cdot\rVert_{\infty}$, and $\text{tr}(\cdot)$ are the absolute value, Euclidean norm, maximum norm, and trace operators, respectively. Finally, $\mathbb{E}[x]$ is the expected value of $x$, $[\mathbf{a}]_{n}$ denotes the $n$-th element of $\mathbf{a}$ and $\mathbf{1}_{N}$ is the all-one column vector of size $N$.

In indoor environments, the transmitted signal propagates to a user via a direct light-of-sight (LoS) and multiple non-light-of-sight paths (NLoS) (due to reflections of walls and ceiling). Nevertheless, it was experimentally verified that the total received optical power at the receiver is predominantly contributed by the LoS signal. On the other hand, the NLoS channel gain, which accounts for less than 5% of the total received optical power, Therefore, only the LoS propagation path is considered in this work to simplify the analysis.

The LoS channel coefficient between an LED luminaire and a user denoted as $h$ is given by [38]

where $\mathds{1}_{[x,y]}(\cdot)$ denotes the indicator function, $A_{r}$ is the active area of the photodiode (PD), $l$ is the link length, $\phi$ is the angle of irradiance, $\Psi$ is the optical field of view (FOV) of the PD, and $\psi$ is the angle of incidence. Also, $m=-\frac{\ln(2)}{\ln(\Theta_{0.5})}$ is the order of the Lambertian emission where $\Theta_{0.5}$ is the LED’s semi-angle for half illuminance. $T_{s}(\psi)$ is the gain of the optical filter, and $g(\psi)=\frac{\kappa^{2}}{\sin^{2}(\Psi)}\mathds{1}_{[0,\Psi]}(\psi)$ is the gain of the optical concentrator, with $\kappa$ being the refractive index of the optical concentrator.

In this section, only the signal model of the selective AN-aided transmission scheme is presented for the sake of conciseness. Without otherwise noted, analysis in the case of the AN-aided MISO transmission scheme follows the same manner. In the following, signal models for the case of unknown and known Eves’s CSI are respectively described.

In most practical cases, Eves are passive eavesdroppers who do not feed their CSI back to the transmitters, leading to the unavailability of Eves’ CSI on the transmitting side. Without the knowledge of Eves’ CSI, direct optimization of the SEE may not be possible. Instead, an indirect design approach is to maximize the EE of Bob’s channel. The obtained solution $(v,\mathbf{w})$ is then used to calculate the SEE. Due to the proportionality between the capacity of Bob’s channel and the secrecy capacity, this design approach possibly results in good SEE performance as well.

With constraints on the precoders $v$ and $\mathbf{w}$ given in (4) and (5), a naive way to formulate an EE maximization problem for Bob’s channel is as follows

Proposition 1 shows that in order to achieve the maximal $\Phi_{\text{B}}(v,\mathbf{w})$, jammers should be inactive. However, this may not lead to an improved SEE because without being interfered with by the AN, Eves can enjoy high-quality communication channels (thus, lowering the achievable secrecy rate). To enable the use of AN, one can set a constraint to ensure that a certain amount of power is allocated to generating the AN signal, for example, $\lVert\mathbf{w}\rVert^{2}_{2}\geq P_{\text{th}}$ where $P_{\text{th}}>0$ is the minimal allocated power threshold. The choice of $P_{\text{th}}$ is critical as due to (21c), this power allocation renders the design problem infeasible if $P_{\text{th}}>(N_{T}-1)\Delta_{\text{DC}}^{2}$. Moreover, although increasing $P_{\text{th}}$ decreases Eves’ SINRs, it potentially degrades the quality of Bob’s channel as well and increases the total consumption power. Consequently, the overall SEE might be reduced. To prevent too much power from being allocated to the AN and guarantee that the quality of Bob’s channel is above a desirable limit, a constraint on Bob’s SINR is taken into consideration. Specifically, we study a modification for $\bf{\mathcal{P}1}$ as given below

Here, $\delta_{\text{B}}$ is the minimum SINR required for Bob’s channel.

One should emphasize that the introduction of constraints in (22b) and (22c) may render $\bf{\mathcal{P}2}$ infeasible. Indeed, it can be easily verified that the problem is infeasible when $\delta_{\text{B}}>\frac{\left(h_{\text{B}}\Delta_{\text{DC}}\right)^{2}}{\overline{\sigma}^{2}_{\text{B}}}$. Therefore, we assume that $\delta_{\text{B}}$ and $P_{\text{th}}$ are explicitly chosen so that $\bf{\mathcal{P}2}$ is feasible. It is seen that $\bf{\mathcal{P}2}$ is a fractional non-convex optimization problem. Hence, we employ the Dinkelbach algorithm, which is efficient in solving fractional programming to tackle the problem. The principle of the Dinkelbach algorithm is to get rid of the fractional objective function by introducing an equivalent non-fractional parametric objective function, which can be handled more conveniently. In particular to our problem, let $D(v,\mathbf{w})=P_{\text{circuit}}+\sum_{N_{T}}U_{\text{LEDs}}I_{\text{DC}}+\zeta\left(v^{2}+\lVert\mathbf{w}\rVert_{2}^{2}\right)$ and $\lambda\geq 0$ be the value of the objective function. The algorithm then iteratively solves the following parametric problem

In each iteration, say the $i$th iteration, a new $\lambda^{(i)}=\frac{C_{\text{B}}\left(v^{{(i)}^{*}},\mathbf{w}^{{(i)}^{*}}\right)}{D\left(v^{{(i)}^{*}},\mathbf{w}^{{(i)}^{*}}\right)}$ is updated, where $v^{{(i)^{*}}}$ and $\mathbf{w}^{{(i)^{*}}}$ are the iteration’s optimal solutions to $v$ and $\mathbf{w}$, respectively. The algorithm terminates when $C_{\text{B}}\left(v^{{(i)^{*}}},\mathbf{w}^{{(i)^{*}}}\right)-\lambda^{(i)}D\left(v^{{(i)^{*}}},\mathbf{w}^{{(i)^{*}}}\right)$ converges to a predefined sufficiently low value. The algorithm is summarized as follows.

Recall that when Eves’ CSI is known at the transmitter, we consider a direct design to maximize the minimum SEE. Following the expression in (20), the AN design problem, in this case, is formulated as

It is seen that $\mathcal{P}\textbf{8}$ is a nonconvex maximin optimization problem. A standard technique to handle the problem is to introduce a slack variable $t=\widetilde{\Phi}^{\text{min}}_{s}(v,\mathbf{W})$ and then transform $\mathcal{P}\textbf{8}$ to

The use of the slack variable $t$ results in additional constraints $\widetilde{\Phi}_{s}(v,\mathbf{W})\geq t,\hskip 2.0pt\forall k=1,2,...,K$, which are equivalently represented in (38b), which is non-convex and difficult to handle due to the product between optimizing variables on the right-hand side. It is, therefore, challenging to directly solve ${\bf{\mathcal{P}9}}$. However, one can transform ${\bf{\mathcal{P}9}}$ into a feasibility problem by fixing the value of $t$. Then, a solution to the original problem can be found using the bisection method. Specifically, given a fixed value of $t$,

Similar to (25)-(28), we make use of the following variable transformations