Physical layer security in light-fidelity systems

Abstract

Light-fidelity (LiFi) is a light-based wireless communication technology which can complement radio-frequency (RF) communication technologies for indoor applications. Although LiFi signals are spatially more contained than RF signals, the broadcasting nature of LiFi also makes it susceptible to eavesdropping. Therefore, it is important to secure the transmitted data against potential eavesdroppers. In this paper, an overview of the recent developments pertaining to LiFi physical layer security (PLS) is provided, and the main differences between LiFi PLS and RF PLS are explained. LiFi achievable secrecy rates and upper bounds are then investigated under practical channel models and transmission schemes. Beamforming and jamming, which received significant research attention recently as a means to achieve PLS in LiFi, are also investigated under indoor illumination constraints. Finally, future research directions of interest in LiFi PLS are identified and discussed.

This article is part of the theme issue ‘Optical wireless communication’.

1. Introduction

With the high and continuously increasing demand for data communications, the limited radio-frequency (RF) spectrum becomes a bottleneck which slows down communication speeds. The light spectrum, which consists of around 670 THz of unlicensed bandwidth, is a promising resource for aleviating this problem. In this context, light-fidelity (LiFi) [1–3] is a communication system that uses visible light communication (VLC) to provide indoor wireless communication. It is a promising technology to complement its RF-based counterpart, WiFi. Owing to its importance, several LiFi companies have recently emerged, including PureLiFi in England and Oledcom in France. Additionally, several existing companies have entered the LiFi market such as Apple, Light, Casio, Intel, LVX systems, Philips, Samsung and ZTE.

Similar to RF, a LiFi system sends signals in a broadcast fashion over a given location (its coverage), which makes the link vulnerable to attacks. In general, there are two forms of attacks in wireless communications that are studied in the literature [4]: passive attacks which do not disturb the system; and active attacks which do. Eavesdropping is a form of passive attack which can take place in an indoor LiFi system. Since LiFi may transmit private and personal information, securing LiFi systems against eavesdropping is essential. One direct approach is to adopt off-the-shelf security techniques developed for WiFi, such as the MAC-layer security mechanisms, which are based on cryptography, as defined in the IEEE 802.15.7 standard. Cryptography-based security techniques are efficient under the assumptions that the secret key is unknown to unintended receivers and the computational power of the unintended receivers is restricted. As described in [5,6], the shortage of these techniques includes inefficient key distribution, key exposure potential and decryption potential. Thus, cryptography-based methods are only conditionally secure. By contrast, physical layer security (PLS) provides theoretically provable security, and is a promising solution for securing LiFi systems [6].

PLS originated from information-theoretic studies on the wiretap channel [6,7], which makes use of physical properties of the transmission process to enhance security. As suggested by the name, PLS is a security technique for the physical layer and is independent of other layers of the OSI network model, and thus can provide either independent protection or cooperative protection with other layers [8]. In addition to being provably secure, advantages of PLS include [5] being key-free; assuming no limitations on the eavesdropper’s computational capabilities and knowledge of network parameters; and having a precisely quantified achieved security.

When studying PLS in LiFi, the studies on the Gaussian wiretap channel for RF systems can serve as a starting point, but do not apply directly. This is due to the constraints that the transmitted optical signals must satisfy, such as a peak constraints and/or an average constraint, in addition to a non-negativity constraint [9]. As LiFi is proposed as an add-on functional module to the indoor illumination system, the LiFi system should not disturb the indoor illumination level. Therefore, LiFi PLS should be designed for different illumination levels, such as dimming [10], where a strict average constraint should be imposed on the transmit signals of the LiFi system. Thus, PLS in LiFi must be investigated while paying careful attention to the optical signal constraints.

Various methods to realize secure transmissions have been studied in the literature. For instance, a security zone scheme was proposed in [11] to prevent eavesdropping within a certain region around the legitimate receiver. A MIMO-based VLC security zone was proposed based on light-emission pattern shaping in [12]. The effect of different light-fixture deployments on LiFi PLS was investigated in [13], where four types of deployments were tested, and an angle diversity receiver is shown to improve secrecy throughput significantly. Most of the related work only consider line-of-sight links, the secrecy performance under reflections and non-line-of-sight links was investigated in [14]. PLS for multi-user VLC networks was studied in [15]. A relay-aided secrecy communication is studied in [16], where different relaying schemes such as cooperative jamming, decode-and-forward, amplify-and-forward were compared. The works in [17–26] study the secrecy capacity and derive achievable secrecy rates for the LiFi wiretap channel, which is the focus of this paper.

In this paper, we focus on the recent theoretical studies of LiFi PLS, where we review bounds on the LiFi wiretap channel secrecy capacity, including single-input single-output (SISO) and multiple-input single-output (MISO) cases. For the LiFi SISO wiretap channel, we review and extend existing secrecy capacity bounds, while considering both peak and average constraints on the transmit signal. We comment on the applicability of existing results to a LiFi system which must satisfy a specific lighting requirement (average constraint with equality), which is important for light dimming. For the LiFi MISO wiretap channel, we review achievable secrecy rates using beamforming and friendly jamming under a peak constraint on the transmit signal. We then extend these results to LiFi MISO systems with both peak and average constraints on the transmit signal. We describe an enhanced beamforming scheme which makes better use of the emitting range of an LED, and we derive its achievable secrecy rate. Finally, we identify some gaps and future directions in LiFi PLS research.

Next, in §2, we review the theoretical background of the wiretap channel in general, and the LiFi wiretap channel in particular, and we review some LiFi P2P channel capacity bounds that are needed in the following sections. In §§3 and 4, we discuss secrecy capacity bounds for the LiFi SISO and MISO wiretap channels, respectively. We conclude the paper in §5 with some future research directions.

2. Preliminaries

In this section, we review PLS starting from Wyner’s discrete memoryless wiretap channel, and then introduce the LiFi wiretap channel. We also introduce some LiFi P2P capacity bounds, which are needed for bounding the secrecy capacity of the LiFi wiretap channel throughout the paper.

(a). A general review of physical layer security for the wiretap channel

(i). The wiretap channel and secrecy capacity

A discrete memoryless wiretap channel is depicted in figure 1. It consists of a transmitter (Alice), an intended receiver (Bob) and an eavesdropper (Eve). Alice wishes to transmit a message to Bob through a noisy channel while keeping it secret from Eve. This transmission can be described as follows [27]. Alice’s message is denoted by M, which is a random variable uniformly distributed on $\mathcal{M}=\{1,2,\dots ,2^{nR}\}$ for some $n\in {\mathbb{N}}_{+}$ and $R\in \mathbb{R}$. To realize this, Alice encodes the message M into a codeword $X^n={\{X^{[i]}\}}_{i=1}^n$ where $X^{[i]}\in \mathcal{X}$, for some alphabet $\mathcal{X}$. Then Alice sends this codeword symbol by symbol over n transmissions.¹

Figure 1.

The SISO wiretap channel model.

The signals received by Bob and Eve are denoted by $Y_1\in {\mathcal{Y}}_1$ and $Y_2\in {\mathcal{Y}}_2$, respectively. These are related to the transmitted signal X according to a conditional joint distribution p(y₁, y₂|x).

Bob collects Y₁ over n uses of the channel, i.e. $Y_1^n={\{Y_1^{[i]}\}}_{i=1}^n$, and then tries to decode M. If we denote the decoding result as $\hat{M}\in \mathcal{M}\cup \{\mathsf{e}\}$, where $\mathsf{e}$ is an error message, then the probability of error can be written as $P_{e,n}=\mathbb{P}\{\hat{M}\ne M\}$.

Similarly, Eve collects Y₂ over n uses of the channel, i.e. $Y_2^n={\{Y_2^{[i]}\}}_{i=1}^n$, and tries to infer as much information about M as possible. Clearly, Alice would like to reduce Eve’s abilities to infer information about M. The amount of information leaked to Alice can be expressed as $I(M;Y_2^n)$, where I( · ; · ) is the mutual information.

To communicate in a secret way, this transmission scheme should be designed so that

$$\underset{n\to \mathrm{\infty }}{lim}{P}_{e,n}=0\text{and}\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}I(M;Y_2^n)=0.$$ 2.1

The maximum rate R at which Alice can communicate with Bob while concealing the message from Eve (according to (2.1)) is called the secrecy capacity. The secrecy capacity of the wiretap channel is shown to be

$$C_s=\underset{p(u,x)}{max}I(U;Y_1)-I(U;Y_2),$$ 2.2

where $U\in \mathcal{U}$ is an auxiliary random and $|\mathcal{U}|\le |\mathcal{X}|$ [27,28]. The purpose of p(u, x) is to transform the end-to-end channel from p(y₁, y₂|x) to $\tilde{p}(y_1,y_2|u)$, which can be more favourable from a secrecy rate perspective. In particular, the achievable secrecy rate I(U;Y₁) − I(U;Y₂) for some p(x, u) can be larger than the achievable secrecy rate I(X;Y₁) − I(X;Y₂) achieved when X = U.

The secrecy capacity achieving coding scheme [27] is a multi-coding and two-step randomized encoding scheme (figure 2). In this coding scheme, Alice first generates a subcodebook of codewords uⁿ(ℓ) for each message $m\in \mathcal{M}$, with $\ell \in {\mathcal{L}}_m\triangleq \{(m-1)2^{n(\tilde{R}-R)}+1,\dots ,m{2}^{n(\tilde{R}-R)}\}$ for some $\tilde{R}\ge R$. The reason for generating many codewords for each m is to increase the randomness of the transmission of m so as to confuse Eve. The symbols of uⁿ(ℓ) are independent and identically distributed according to p(u). Consequently, each message m is represented by many codewords uⁿ(ℓ). To send m, Alice randomly chooses a uⁿ(ℓ) with $\ell \in {\mathcal{L}}_m$, and then generates the transmit signal xⁿ whose symbols are independent and identically distributed according to p(x|u).

Figure 2.

The secrecy capacity achieving scheme encodes the message m into many codewords uⁿ(ℓ). Alice selects a random codeword uⁿ(ℓ) corresponding to the message m to be sent, and then sends a random signal xⁿ distributed according to p(x|u). Bob is able to decode the correct message by decoding the correct uⁿ(ℓ) (marked with a square), while Eve will have ambiguity between many uⁿ(ℓ) corresponding to different messages m (white).

Note that the total number of generated codewords uⁿ(ℓ) is $2^{n\tilde{R}}$. If $\tilde{R}<I(U;Y_1)$, then by the channel coding theorem [27], Bob will be able to discern the sent codewords if n is large enough. It then decodes ℓ, from which it knows m.

The maximum number of codewords that Eve can discern is given by $2^{nI(U;Y_2)}$ asymptotically for large n, which can be proved using the channel coding theorem [27, ch. 3.1]. If we choose $\tilde{R}-R>I(U;Y_2)$, then Eve will have ambiguity between multiple codewords uⁿ(ℓ) corresponding to m = 1, multiple codewords uⁿ(ℓ) corresponding to m = 2, etc. Moreover, this ambiguity will show no preference to a specific m, since the number of ‘likely’ codewords uⁿ(ℓ) will be roughly the same for all m. This confuses Eve and establishes secrecy. When optimized with respect to p(u, x), this coding scheme achieves the secrecy capacity in (2.2). Note that this coding scheme is also known as ‘binning’ [29].

(ii). The degraded wiretap channel

The above description applies to a general discrete-memoryless wiretap channel. If the wiretap channel satisfies the condition p(y₁, y₂|x) = p(y₁|x) p(y₂|y₁), i.e. X → Y₁ → Y₂ forms a Markov chain, then the wiretap channel is said to be physically degraded.² In this case, Eve receives a degraded version of Bob’s received signal (as if Bob’s received signal is passed to Eve through another noisy channel p(y₂|y₁)). If there exists ${\tilde{Y}}_2$ so that ${\tilde{Y}}_2|X$ has the same distribution as Y|X and $X\to Y_1\to {\tilde{Y}}_2$ forms a Markov chain, then the channel is said to be stochastically degraded [27].

Under both types of degradedness, the secrecy capacity simplifies to [27]

$$C_s=\underset{p(x)}{max}I(X;Y_1)-I(X;Y_2).$$ 2.3

In other words, in this case, it is optimal to choose X = U. Note that this expression also applies to the family of continuous-memoryless wiretap channels (cf. [29, ch. 5]).

As we shall see next, the SISO LiFi wiretap channel where Alice has a single transmitter and each of Bob and Eve have a single receiver is a degraded wiretap channel if h₁ > h₂. The LiFi wiretap channel is not degraded in general if Alice is equipped with multiple transmitters that can transmit non-identical signals. We call this a MISO wiretap channel. We introduce the LiFi wiretap channel next.

(b). Light-fidelity wiretap channel

In a LiFi system, the transmitter is a light source such as an LED (or a group thereof as in a light fixture), and the receiver is a photodetector (PD) consisting of a photodiode (or a group thereof). The light intensity of the LED is modulated by modulating the driving current, and the PD transforms light intensity to current.

Let us denote the driving current by X. Owing to the use of LEDs, the driving current can only be positive. Therefore, $X\in {\mathbb{R}}_{+}$. In addition to this constraint, the current X must satisfy some additional constraints due to safety and practical considerations. For instance, to avoid over-driving the LED, or operating it beyond its saturation current, we require a peak constraint to be satisfied, which is given by $X\le \mathcal{A}$. Additionally, for eye safety X has to satisfy an average constraint $\mathbb{E}[X]\le \mathcal{E}$, which arises due to a (roughly) linear relation between current and light intensity in LiFi solid-state lighting devices [30]. Moreover, to meet a desired illumination requirement, we might also require $\mathbb{E}[X]=\mathcal{E}$. Some studies on LiFi ignore the average constraint and only focus on a peak constraint. In this paper, we will consider both average and peak constraints.

In general, if Alice is equipped with multiple transmitters (light-fixtures for example), then the transmit signal will be a vector $\mathbf{\text{X}}\in {\mathbb{R}}_{+}^K$ where K is the number of transmitters. In this case, each component of X is constrained by a peak constraint $\mathcal{A}$, and the average constraint can apply to the sum as $\sum _{k=1}^K\mathbb{E}[X_k]\le \mathcal{E}$ or individually as $\mathbb{E}[X_k]\le {\mathcal{E}}_k$ for all k = 1, 2, …, K.

Upon transmitting X, Bob and Eve receive Y₁ and Y₂ (representing currents), which can be described by the following input–output relations:

$$Y_r={\mathbf{\text{h}}}_r^T\mathbf{\text{X}}+Z_r,r=1,2,$$ 2.4

where h₁, ${\mathbf{\text{h}}}_2\in {\mathbb{R}}_{+}^K$ are the time-invariant channel coefficient vectors from Alice to Bob and Eve, respectively, and Z_i-s are independent Gaussian noises which have zero mean and variance σ². This noise combines thermal noise and noise from ambient light. Throughout the paper, we assume that the channels from the transmitters to the receivers are known globally.

Remark 2.1. —

Note the following main differences between the RF wiretap channel with a power constraint. First, the transmit signal and channel gains are real positive, whereas they are complex-valued in RF. Second, the transmit signal is subject to peak and average constraints, whereas in RF it is subject to power (second moment) and possibly peak constraints. These make up the main differences between the two cases.

(i). Light-fidelity single-input single-output secrecy capacity

If K = 1, then we have a LiFi SISO wiretap channel. If h₁ > h₂, this LiFi SISO wiretap channel is stochastically degraded. In particular, ${\tilde{Y}}_2=(h_2/h_1)Y_1+\tilde{Z}$ where $\tilde{Z}\sim \mathcal{N}(0,{\sigma }^2(1-h_2^2{h}_1^{-2}))$ has the same conditional distribution ${\tilde{Y}}_2|X$ as Y₂|X, and satisfies the Markov chain $X\to Y_1\to {\tilde{Y}}_2$. Consequently, in this case, the secrecy capacity is given by

$$C_{s,i}^{\text{SISO}}=\underset{p(x)\in {\mathcal{P}}^{(t)}}{max}I(X;Y_1)-I(X;Y_2),$$ 2.5

where t = 1, 2, ${\mathcal{P}}^{(1)}$ is the collection of distributions of $X\in [0,\mathcal{A}]$ (peak constraint), and ${\mathcal{P}}^{(2)}$ is the collection of distributions of $X\in [0,\mathcal{A}]$ with $\mathbb{E}[X]\le \mathcal{E}$ (peak and average constraint).

If h₁ ≤ h₂, then it can be shown that the secrecy capacity is zero.

Now the question is how to maximize (2.5). The optimal solution of this problem is not obvious, and requires numerical methods. Moreover, it was shown in [26] that the optimal p(x) in (2.5) (under both i = 1, 2) is discrete with a finite number of mass points. At low signal-to-noise ratio (SNR) ($(\mathcal{A}/\sigma )\to 0$), Soltani & Rezki [26] show that the secrecy capacity achieving distribution has two mass points. It is also shown that at high SNR ($(\mathcal{A}/\sigma )\to \mathrm{\infty }$), a large number of mass points is needed. However, the optimal distribution p(x) remains unknown.

(ii). Light-fidelity multiple-input single-output secrecy capacity

If K > 1, then we have a LiFi MISO wiretap channel. The LiFi MISO wiretap channel is not degraded, unless h₁ = ηh₂ for some $\eta \in \mathbb{R}$ (aligned),³ which is unlikely in practice. Thus, the secrecy capacity of the MISO wiretap channel is given by [22]

$$C_{s,i}^{\text{MISO}}=\underset{p(\mathbf{\text{u}},\mathbf{\text{x}}):p(x_i)\in {\mathcal{P}}^{(t)}}{max}I(\mathbf{\text{U}};Y_1)-I(\mathbf{\text{U}};Y_2),$$ 2.6

i = 1, 2, where U is an auxiliary vector [19,22]. The optimal input distribution which maximizes (2.6) is unknown, and works in the literature on LiFi MISO wiretap channels generally focus on achievable rates based on beamforming and friendly jamming.

It is often convenient to express capacity in general, and secrecy capacity in our case in particular, using simple analytical expressions. Using a discrete p(x) in (2.5) or (2.6) prohibits this due to the integral form of the mutual information, which does not simplify easily in this case. To circumvent this issue, it is useful to develop bounds on the secrecy capacity and asymptotic capacity expressions at high and low SNR. To develop such bounds, we first introduce capacity lower and upper bounds for the SISO LiFi channel without an eavesdropper.

(c). LiFi P2P channel capacity bounds

Several works in the literature studied the LiFi P2P channel capacity [9,31,32] and developed upper and lower bounds. These bounds have been used intensively in the literature to study other types of LiFi channels’ [33–35], including the LiFi wiretap channel [22,25].

Consider a LiFi P2P channel with input X satisfying $X\in [0,\mathcal{A}]$ and $\mathbb{E}[X]\le \alpha \mathcal{A}$ for some α ∈ [0, 1] (average-to-peak ratio), and output Y = X + Z, with Z Gaussian with zero mean and variance σ². Denote the capacity of this channel by $C_{\alpha }(\mathcal{A},\sigma )=\underset{p(X)}{max}I(Y;X)$. We review bounds on $C_{\alpha }(\mathcal{A},\sigma )$ in the following lemmas. We start with α ≥ 1/2, in which case the average constraint is inactive, and the peak constraint dominates.

Lemma 2.2 ([9]). —

If 1/2 ≤ α < 1 (inactive average constraint) or α = 1 (peak constraint only), then the capacity of the LiFi P2P channel $C_{\alpha }(\mathcal{A},\sigma )$ is bounded as follows

$$\begin{array}{r}{C}_{\alpha }(\mathcal{A},\sigma )\ge {\underset{\_}{C}}_{\alpha }^{[1]}(\mathcal{A},\sigma )\triangleq \frac{1}{2}\log (1+\frac{{\mathcal{A}}^2}{2\pi e{\sigma }^2}),\end{array}$$ 2.7

$$\begin{array}{r}{C}_{\alpha }(\mathcal{A},\sigma )\le {\overline{C}}_{\alpha }^{[1]}(\mathcal{A},\sigma )\triangleq \frac{1}{2}\log (1+\frac{{\mathcal{A}}^2}{4{\sigma }^2})\end{array}$$ 2.8

$$\begin{array}{r}\text{and}{C}_{\alpha }(\mathcal{A},\sigma )\le {\dot{C}}_{\alpha }^{[1]}(\mathcal{A},\sigma ),\end{array}$$ 2.9

where

$$\begin{array}{rl}{\dot{C}}_{\alpha }^{[1]}(\mathcal{A},\sigma )& \triangleq \underset{\delta >0}{min}(1-2\mathcal{Q}\left(\frac{\delta +\frac{\mathcal{A}}{2}}{\sigma }\right))\log \left(\frac{\mathcal{A}+2\delta }{\sigma \sqrt{2\pi }(1-2\mathcal{Q}((\delta /\sigma )))}\right)\\ & +\log (e)(-\frac{1}{2}+\mathcal{Q}\left(\frac{\delta }{\sigma }\right)+\frac{\delta }{\sqrt{2\pi }\sigma }\hspace{0.17em}{\text{e}}^{-({\delta }^2/2{\sigma }^2)}).\end{array}$$ 2.10

It can be seen that α does not affect the bounds in lemma 2.2. This is because if α ≥ 1/2, then $\mathcal{E}\ge (\mathcal{A}/2)$, and the capacity-achieving input distribution has mean $\frac{A}{2}$ in this case [9]. If α < (1/2), then the average constraint is active, and capacity is bounded as given next.

Lemma 2.3 ([9]). —

If 0 < α < 1/2, then the capacity of the LiFi P2P channel $C_{\alpha }(\mathcal{A},\sigma )$ is bounded as follows

$$\begin{array}{r}{C}_{\alpha }(\mathcal{A},\sigma )\ge {\underset{\_}{C}}_{\alpha }^{[2]}(\mathcal{A},\sigma )\triangleq \frac{1}{2}\log (1+{\mathcal{A}}^2\frac{e^{2\alpha {\mu }_{\alpha }}}{2\pi e{\sigma }^2}\left(\frac{1-{\text{e}}^{-{\mu }_{\alpha }}}{{\mu }_{\alpha }}\right)),\end{array}$$ 2.11

$$\begin{array}{r}{C}_{\alpha }(\mathcal{A},\sigma )\le {\overline{C}}_{\alpha }^{[2]}(\mathcal{A},\sigma )\triangleq \frac{1}{2}\log (1+\alpha (1-\alpha )\frac{{\mathcal{A}}^2}{{\sigma }^2})\end{array}$$ 2.12

$$\begin{array}{r}\text{and}{C}_{\alpha }(\mathcal{A},\sigma )\le {\dot{C}}_{\alpha }^{[2]}(\mathcal{A},\sigma ),\end{array}$$ 2.13

where ${\mu }_{\alpha }$ is the unique positive solution of $\alpha =(1/{\mu }_{\alpha })-({\text{e}}^{-{\mu }_{\alpha }})/(1-{\text{e}}^{-{\mu }_{\alpha }})$, and

$$\begin{array}{rl}{\dot{C}}_{\alpha }^{[2]}(\mathcal{A},\sigma )& \triangleq \underset{\mu ,\delta >0}{min}(1-\mathcal{Q}\left(\frac{\delta +\alpha \mathcal{A}}{\sigma }\right)-\mathcal{Q}\left(\frac{\delta +(1-\alpha )\mathcal{A}}{\sigma }\right))\cdot \log \left(\frac{\mathcal{A}}{\sigma }\frac{{\text{e}}^{(\mu \delta /\mathcal{A})}-{\text{e}}^{-\mu (1+(\delta /\mathcal{A}))}}{\sqrt{2\pi }\mu (1-2\mathcal{Q}((\delta /\sigma )))}\right)\\ & +\log (e)[-\frac{1}{2}+\mathcal{Q}\left(\frac{\delta }{\sigma }\right)+\frac{\delta }{\sqrt{2\pi }\sigma }\hspace{0.17em}{\text{e}}^{-({\delta }^2/2{\sigma }^2)}+\frac{\sigma }{\mathcal{A}}\frac{\mu }{\sqrt{2\pi }}({\text{e}}^{-({\delta }^2/2{\sigma }^2)}-{\text{e}}^{-({(\mathcal{A}+\delta )}^2)/(2{\sigma }^2)})\\ & +\mu \alpha (1-2\mathcal{Q}\left(\frac{\delta +\mathcal{A}/2}{\sigma }\right))].\end{array}$$ 2.14

The upper bounds ${\dot{C}}_{\alpha }^{[i]}(\mathcal{A},\sigma )$, i = 1, 2, have been shown to be asymptotically tight at high SNR where they coincide with the lower bounds ${\underset{\_}{C}}_{\alpha }^{[i]}(\mathcal{A},\sigma )$, i = 1, 2, respectively, while the bounds ${\overline{C}}_{\alpha }^{[i]}(\mathcal{A},\sigma )$, i = 1, 2, have been shown to be asymptotically tight at low SNR [9]. Needless to say, all the lower bounds above are achievable rates.

Now, we are ready to present LiFi SISO secrecy capacity bounds that have been derived in the recent literature, which rely on lemmas 2.3 and 2.2, in addition to a new bound which we develop to fill some gaps in the literature.

3. LiFi SISO secrecy capacity bounds

In this section, we present LiFi SISO secrecy capacity results from the literature, and develop some new bounds. Clearly, we focus on h₁ > h₂, since the secrecy capacity is zero otherwise. We will split the review into two parts depending on the constraint on X.

(a). Secrecy capacity bounds under a peak constraint

When only a peak constraint is imposed, the LiFi SISO wiretap channel capacity is given by

$$C_{s,1}^{\text{SISO}}=\underset{p(x)\in {\mathcal{P}}^{(1)}}{max}I(X;Y_1)-I(X;Y_2).$$ 3.1

In what follows, we use (x)⁺ to denote max{0, x} and log ⁺(x) to denote max{0, log (x)}. The following theorem presents a secrecy capacity lower bound which relies on lemma 2.2.

Theorem 3.1 (Lower bound [22]). —

Under a peak constraint, the LiFi SISO wiretap channel secrecy capacity is lower bounded by $C_{s,1}^{\text{SISO}}\ge {\underset{\_}{C}}_{s,1}^{\text{SISO}}\triangleq {(1/2\log (1+(h_1^2{\mathcal{A}}^2)/(2\pi e{\sigma }^2))-{\dot{C}}_1^{[1]}(h_2\mathcal{A},\sigma ))}^{+}$.

Proof. —

Since $C_{s,1}^{\text{SISO}}=\underset{p(x)\in {\mathcal{P}}^{(1)}}{max}I(X;Y_1)-I(X;Y_2)$, we can write

$$\begin{array}{rl}{C}_{s,1}^{\text{SISO}}& \ge \underset{p(x)\in {\mathcal{P}}^{(1)}}{max}I(X;Y_1)-\underset{p(x)\in {\mathcal{P}}^{(1)}}{max}I(X;Y_2)\\ & \ge {({\underset{\_}{C}}_1^{[1]}(h_1\mathcal{A},\sigma )-{\dot{C}}_1^{[1]}(h_2\mathcal{A},\sigma ))}^{+}.\end{array}$$ 3.2

where the last step follows using lemma 2.2. ▪

Another lower bound that is derived using a different method, and leads to a simple expression is given next.

Theorem 3.2 (Lower bound [22]). —

Under a peak constraint, the LiFi SISO wiretap channel secrecy capacity is lower bounded by $C_{s,1}^{\text{SISO}}\ge \hspace{0.17em}{\underset{\dot{}}{C}}_{s,1}^{\text{SISO}}\triangleq 1/2{\log }^{+}((6h_1^2{\mathcal{A}}^2+12\pi e{\sigma }^2)/(\pi e{h}_2^2{\mathcal{A}}^2+12\pi e{\sigma }^2))$.

Proof. —

We have $C_{s,1}^{\text{SISO}}=\underset{p(x)\in {\mathcal{P}}^{(1)}}{max}I(X;Y_1)-I(X;Y_2)$ which we can write as $C_{s,1}^{\text{SISO}}=\underset{p(x)\in {\mathcal{P}}^{(1)}}{max}h(Y_1)-h(Y_1|X)-h(Y_2)+h(Y_2|X)$ where h( · ) and h( · | · ) are the differential and conditional differential entropy, respectively. We proceed as follows:

$$\begin{array}{rl}{C}_{s,1}^{\text{SISO}}& =\underset{p(x)\in {\mathcal{P}}^{(1)}}{max}h(Y_1)-h(Z_1)-h(Y_2)+h(Z_2)\\ & \stackrel{(a)}{\ge }\underset{p(x)\in {\mathcal{P}}^{(1)}}{max}{(\frac{1}{2}\log (2^{2h(h_{1}X)}+2^{2h(Z_1)})-\frac{1}{2}\log (2\pi e\mathbb{V}\mathbb{A}\mathbb{R}\{Y_2\}))}^{+}\\ & \stackrel{(b)}{\ge }{(\frac{1}{2}\log (h_1^2{\mathcal{A}}^2+2\pi e{\sigma }^2)-\frac{1}{2}\log \left(2\pi e(\frac{h_2^2{\mathcal{A}}^2}{12}+{\sigma }^2)\right))}^{+}\\ & =\frac{1}{2}{\log }^{+}\left(\frac{6h_1^2{\mathcal{A}}^2+12\pi e{\sigma }^2}{\pi e{h}_2^2{\mathcal{A}}^2+12\pi e{\sigma }^2}\right),\end{array}$$ 3.3

where (a) follows by using h(Z₁) = h(Z₂), lower bounding h(Y₁) using the entropy-power inequality (EPI), and upper bounding h(Y₂) by the differential entropy of a Gaussian random variable with variance $\mathbb{V}\mathbb{A}\mathbb{R}\{Y_2\}$; and (b) follows by choosing X to be uniformly distributed on $[0,\mathcal{A}]$. ▪

The advantage of this lower bound compared to the one in theorem 3.1 is that it has a closed-form expression. Next, we present a secrecy capacity upper bound.

Theorem 3.3 (Upper bound [22]). —

Under a peak constraint, the LiFi SISO wiretap channel secrecy capacity is upper bounded by $C_{s,1}^{\text{SISO}}\le {\overline{C}}_{s,1}^{\text{SISO}}\triangleq 1/2\log ((h_1^2{\mathcal{A}}^2+4{\sigma }^2)/(h_2^2{\mathcal{A}}^2+4{\sigma }^2))$.

This statement was proved in [22, Prop. 2.3] using a dual secrecy capacity expression (for degraded channels), which is an extension of the dual capacity expression in [36]. This statement can also be proved by converting the LiFi SISO wiretap channel into a Gaussian wiretap channel with a power constraint $\mathbb{E}[X^2]\le \frac{{\mathcal{A}}^2}{4}$. Namely, we can consider an equivalent degraded wiretap channel with $Y_i=h_i(X-(\mathcal{A}/2))+Z_i$. Then, we note that $X-(\mathcal{A}/2)$ is in $[-(\mathcal{A}/2),(\mathcal{A}/2)]$ and has maximum variance $({\mathcal{A}}^2/4)$. By dropping the peak constraints and allowing X to be in $\mathbb{R}$ instead, we obtain a Gaussian wiretap channel with a power constraint $\mathbb{E}[X^2]\le ({\mathcal{A}}^2/4)$, whose secrecy capacity is given as in theorem 3.3 [27]. This bound can be also obtained from [25].

(b). Secrecy capacity bounds under peak and average constraints

When both peak and average constraints are imposed, the secrecy capacity is given by

$$C_{s,2}^{\text{SISO}}=\underset{p(x)\in {\mathcal{P}}^{(2)}}{max}I(X;Y_1)-I(X;Y_2).$$ 3.4

In this case, the secrecy capacity can be bounded using the LiFi P2P channel capacity bounds in lemmas 2.2 and 2.3 as shown next.

Theorem 3.4 (Lower bound [25]). —

Under both peak and average constraints, the LiFi SISO wiretap channel secrecy capacity is lower bounded by

$$C_{s,2}^{\text{SISO}}\ge {\underset{\_}{C}}_{s,2}^{\text{SISO}}(\alpha )\triangleq {({\underset{\_}{C}}_{\alpha }^{[i]}(h_1\mathcal{A},\sigma )-min\{{\overline{C}}_{\alpha }^{[i]}(h_2\mathcal{A},\sigma ),{\dot{C}}_{\alpha }^{[i]}(h_2\mathcal{A},\sigma )\})}^{+},$$ 3.5

where i = 1 if $\alpha \ge \frac{1}{2}$ and i = 2 if α < 1/2.

The proof of this theorem is similar to the proof of ${\underset{\_}{C}}_{s,1}^{\text{SISO}}$ in theorem 3.1, but uses both lemmas 2.2 and 2.3.

Similar to theorem 3.2, we can develop a lower bound under both peak and average constraints. Instead of the uniform distribution used in theorem 3.2, we use a truncated-exponential distribution similar to [9]. The following theorem describes this new lower bound.

Theorem 3.5 (Lower bound). —

Under both peak and average constraints with α < 1/2, the LiFi SISO wiretap channel secrecy capacity is lower bounded by

$$C_{s,2}^{\text{SISO}}\ge \hspace{0.17em}{\underset{\dot{}}{C}}_{s,2}^{\text{SISO}}(\alpha )\triangleq \frac{1}{2}{\log }^{+}\left(\frac{\rho h_1^2{\mathcal{A}}^2+2\pi e{\sigma }^2}{2\pi e\varphi h_2^2{\mathcal{A}}^2+2\pi e{\sigma }^2}\right),$$ 3.6

where $\rho ={\text{e}}^{2(1-({\mu }_{\alpha })/({\text{e}}^{{\mu }_{\alpha }}-1))}{((1-{\text{e}}^{-{\mu }_{\alpha }})/({\mu }_{\alpha }))}^2$, $\varphi =(1/{\mu }_{\alpha }^2)+(1/4)-((1/{\text{e}}^{{\mu }_{\alpha }}-1)+(1/2))$ and ${\mu }_{\alpha }$ is defined in lemma 2.3.

Proof. —

The proof is similar to that of theorem 3.2 until (b), where instead of a uniform distribution, we choose X according to a truncated-exponential distribution which maximizes the differential entropy h(X), where $X\in [0,\mathcal{A}]$ and $\mathbb{E}[X]=\mathcal{E}=\alpha \mathcal{A}$ [9, (42)], given by

$$p(x)=\frac{1}{\mathcal{A}}\frac{\mu }{1-{\text{e}}^{-\mu }}{\text{e}}^{-(\mu x/\mathcal{A})}.$$ 3.7

Thus, we have

$$C_{s,2}^{\text{SISO}}\ge {(\frac{1}{2}\log (2^{2h(h_{1}X)}+2^{2h(Z_1)})-\frac{1}{2}\log (2\pi e\mathbb{V}\mathbb{A}\mathbb{R}[Y_2]))}^{+},$$ 3.8

where X follows the distribution p(x) above, and $\mathbb{V}\mathbb{A}\mathbb{R}\{Y_2\}=h_2^2\mathbb{V}\mathbb{A}\mathbb{R}\{X\}+{\sigma }^2$. Then we have $\mathbb{V}\mathbb{A}\mathbb{R}[X]\triangleq \mathbb{E}[X^2]-{(\mathbb{E}[X])}^2=\varphi h_2^2{A}^2$, and $h(h_{1}X)=h(X)+\log (h_1)=-\log ((1h_1\mathcal{A})({\text{e}}^{-\mu })/(1-{\text{e}}^{-\mu }))+\log (e)(1-({\mu }_{\alpha })/({\text{e}}^{{\mu }_{\alpha }}-1))$. Then $2^{h(h_{1}X)}=\rho h_1^2{A}^2$. Substituting in (3.8) concludes the proof. ▪

Next, we present a secrecy capacity upper bound derived using the EPI and the bounds in lemmas 2.2 and 2.3.

Theorem 3.6 (Upper bound [25]). —

Under both peak and average constraints, the LiFi SISO wiretap channel secrecy capacity is upper bounded by

$$C_{s,2}^{\text{SISO}}\le {\overline{C}}_{s,2}^{\text{SISO}}(\alpha )\triangleq \frac{1}{2}\log \left(\frac{h_1^2{2}^{2{\overline{C}}_{\alpha }^{[i]}(h_1\mathcal{A},\sigma )}}{h_2^2-h_1^2+h_2^2{2}^{2{\overline{C}}_{\alpha }^{[i]}(h_1\mathcal{A},\sigma )}}\right)$$ 3.9

and

$$C_{s,2}^{\text{SISO}}\le {\dot{C}}_{s,2}^{\text{SISO}}(\alpha )\triangleq \frac{1}{2}\log \left(\frac{h_1^2{2}^{2{\dot{C}}_{\alpha }^{[i]}(h_1\mathcal{A},\sigma )}}{h_1^2-h_2^2+h_2^2{2}^{2{\dot{C}}_{\alpha }^{[i]}(h_1\mathcal{A},\sigma )}}\right),$$ 3.10

where i = 1 if α ≥ 1/2 and i = 2 if α < 1/2.

Proof. —

Since I(X;Y₁) − I(X;Y₂) = I(X;h₁ X + Z₁) − I(X;h₂ X + Z₂), we can write

$$\begin{array}{rl}I(X;Y_1)-I(X;Y_2)& =I(X;h_{1}X+Z_1)-h(h_{2}X+Z_2)+h(Z_2)\\ & \stackrel{(a)}{=}I(X;h_{1}X+Z_1)-h(h_{1}X+Z_1+Z^{\prime })-\log (h_2{h}_1^{-1})+h(Z_2)\\ & \stackrel{(b)}{\le }I(X;h_{1}X+Z_1)-\frac{1}{2}\log (2^{2h(h_{1}X+Z_1)}+2^{2h(Z^{\prime })})-\log (h_2{h}_1^{-1})+h(Z_2)\\ & \stackrel{(c)}{=}\frac{1}{2}\log \left(\frac{h_1^2{2}^{2I(X;h_{1}X+Z_1)+2h(Z_2)}}{h_2^2{2}^{2I(X;h_{1}X+Z_1)+2h(Z_1)}+h_2^2{2}^{2h(Z^{\prime })}}\right),\end{array}$$ 3.11

where (a) follows by using h(X) = h(aX) − log (a) for a > 0 and writing (h₁/h₂)Z₂ = Z₁ + Z′ with Z′ being Gaussian with zero mean and variance $((h_1^2/h_2^2)-1){\sigma }^2$ (note that h₂ > h₁ > 0 for a degraded LiFi wiretap channel), (b) follows by using the EPI, and (c) follows since I(X;h₁X + Z₁) = h(h₁X + Z₁) − h(Z₁).

Note that the last line in (3.11) is monotonically increasing in I(X;h₁X + Z₁). Moreover, $I(X;h_{1}X+Z_1)\le {\overline{C}}_{\alpha }^{[i]}(h_1\mathcal{A},\sigma )$ and $I(X;h_{1}X+Z_1)\le {\dot{C}}_{\alpha }^{[i]}(h_1\mathcal{A},\sigma )$ by lemmas 2.2 and 2.3. Furthermore, $2^{2h(Z_1)}=2^{2h(Z_2)}=2\pi e{\sigma }^2$ and $2^{2h(Z^{\prime })}=2\pi e((h_1^2/h_2^2)-1){\sigma }^2$. Substituting these identities and inequalities in the last line in (3.11) concludes the proof. ▪

(c). Extension to systems with specific lighting requirements

The above discussion considered peak, or both peak and average constraints, where the average constraint is an inequality constraint given by $\mathbb{E}[X]\le \mathcal{E}$. As discussed earlier, in LiFi, we are also interested in a practical average constraint which is an equality constraint $\mathbb{E}[X]=\mathcal{E}$, which reflects a specific lighting requirement. So it is also relevant to develop secrecy capacity bounds under this equality constraint.

Towards this end, we discuss below how the above bounds can be extended to LiFi systems with $\mathbb{E}[X]=\mathcal{E}$. Some results above apply immediately to this case which are discussed in the following remarks. The only missing piece is when α > 1/2, which is discussed afterwards.

Remark 3.7. —

Given α ≤ 1/2, the secrecy capacity lower bounds in theorems 3.4 and 3.5 (peak and average constraint $\mathbb{E}[X]\le \mathcal{E}$) are also secrecy capacity lower bounds for a LiFi SISO wiretap channel with a peak constraint and an average constraint $\mathbb{E}[X]=\mathcal{E}$. The reason is that the schemes achieving the secrecy rates in these theorems satisfy the constraint $\mathbb{E}[X]\le \mathcal{E}$ with equality.

Remark 3.8. —

The secrecy capacity upper bounds in theorem 3.6 (peak and average constraint $\mathbb{E}[X]\le \mathcal{E}$) are also secrecy capacity upper bounds for a LiFi SISO wiretap channel with a peak constraint and an average constraint $\mathbb{E}[X]=\mathcal{E}$. The reason is that the constraint $\mathbb{E}[X]\le \mathcal{E}$ is a relaxed version of $\mathbb{E}[X]=\mathcal{E}$, and hence the former has larger secrecy capacity than the latter. We improve this upper bound further next.

Based on these remarks, for a LiFi SISO system with a constraint $\mathbb{E}[X]=\mathcal{E}$, it remains to develop secrecy capacity lower bounds for the case α > 1/2, and to improve the upper bound from remark 3.8. This task is simple after observing the following.

The secrecy capacity for a LiFi SISO wiretap channel with α > 1/2 is the same as for a channel with outputs $Y_1^{\prime }=h_1\mathcal{A}-Y_1=h_1(\mathcal{A}-X)+Z_1^{\prime }$ and $Y_2^{\prime }=h_2\mathcal{A}-Y_2=h_2(\mathcal{A}-X)+Z_2^{\prime }$ by symmetry of the noises’ Gaussian distribution. But the signal $\mathcal{A}-X$ has lies in $[0,\mathcal{A}]$ and satisfies $\mathbb{E}[\mathcal{A}-X]=\mathcal{A}-\mathcal{E}=(1-\alpha )\mathcal{A}$ if $\mathbb{E}[X]=\mathcal{E}=\alpha \mathcal{A}$. Moreover, if α > 1/2, then 1 − α≜α′ < 1/2. Hence, the developed lower bounds in theorems 3.4 and 3.5 and the upper bound in theorem 3.6 for the case α < 1/2 also apply to this new channel with α′ < 1/2. Thus, we can state the following.

Corollary 3.9. —

Under a peak constraint and an average constraints $\mathbb{E}[X]=\mathcal{E}=\alpha \mathcal{A}$ with α > 1/2, the LiFi SISO wiretap channel secrecy capacity is upper bounded by ${\overline{C}}_{s,2}^{\text{SISO}}({\alpha }^{\prime })$ and lower bounded by ${\underset{\_}{C}}_{s,2}^{\text{SISO}}({\alpha }^{\prime })$ and $\hspace{0.17em}{\underset{\dot{}}{C}}_{s,2}^{\text{SISO}}({\alpha }^{\prime }),$ where

$${\alpha }^{\prime }=\{\begin{array}{ll}\alpha & \alpha \le \frac{1}{2}\\ 1-\alpha & \alpha >\frac{1}{2}.\end{array}$$ 3.12

Next, we present some numerical evaluations.

(d). Numerical results

First, we show a comparison of the secrecy capacity bounds for a LiFi SISO wiretap channel with a peak constraint only, under different ratios (h₁/h₂). Figure 3a compares lower bounds ${\underset{\_}{C}}_{s,1}^{\text{SISO}}$ and $\hspace{0.17em}{\underset{\dot{}}{C}}_{s,1}^{\text{SISO}}$ from theorems 3.1 and 3.2, respectively, with upper bound ${\overline{C}}_{s,1}^{\text{SISO}}$ from theorem 3.3, versus SNR defined as $(\mathcal{A}/\sigma )$. It can be seen that all bounds increase with (h₁/h₂), which supports the intuition that a larger disparity between Bob’s and Eve’s channels lead to higher secrecy capacity. The figure shows that $\hspace{0.17em}{\underset{\dot{}}{C}}_{s,1}^{\text{SISO}}$ outperforms ${\underset{\_}{C}}_{s,1}^{\text{SISO}}$ when SNR is not very large. On the other hand, as SNR increases, ${\underset{\_}{C}}_{s,1}^{\text{SISO}}$ converges to ${\overline{C}}_{s,1}^{\text{SISO}}$. In particular, the two converge to the high-SNR secrecy capacity characterized in [25] as $C_{s,1}^{\text{SISO}}\to 1/2\log ((h_1^2/h_2^2))$ as SNR → ∞.

Figure 3.

Secrecy capacity bounds for the LiFi SISO wiretap channel. (a) Peak constraint only, (b) peak and average constraints: α = 0.3. (Online version in colour.)

Figure 3b compares lower bounds ${\underset{\_}{C}}_{s,2}^{\text{SISO}}$ and $\hspace{0.17em}{\underset{\dot{}}{C}}_{s,2}^{\text{SISO}}$ from theorems 3.4 and 3.5, respectively, with upper bound ${\overline{C}}_{s,2}^{\text{SISO}}$ from theorem 3.6, versus SNR defined as $(\mathcal{A}/\sigma )$. The impact of increasing (h₁/h₂) is similar to that in figure 3a and the newly derived lower bound, $\hspace{0.17em}{\underset{\dot{}}{C}}_{s,2}^{\text{SISO}}$, is tighter than ${\underset{\_}{C}}_{s,2}^{\text{SISO}}$ when SNR is relatively low. It can also be seen that ${\underset{\_}{C}}_{s,2}^{\text{SISO}}$ and ${\overline{C}}_{s,2}^{\text{SISO}}$ converge to the asymptotic high-SNR secrecy capacity $(1/2)\log ((h_1^2/h_2^2))$ [25], which is the same as for the case under a peak constraint, i.e. the high-SNR secrecy capacity is independent of α. This conclusion is examined next.

Figure 4 compares the bounds for a wiretap channel with both peak and average constraints, for different values of α, under fixed (h₁/h₂) = 10. It can be seen that for each α, the corresponding lower bound converges to the upper bound when SNR is high enough. It can also be seen that the lower bound increases with α, which depicts the impact of dimming in LiFi, since decrease α decreases the average light intensity.

Figure 4.

Secrecy capacity bounds under both peak and average constraints with (h₁/h₂) = 10 and different values of α. (Online version in colour.)

4. Light-fidelity multiple-input single-output achievable secrecy rates

An indoor illumination system usually has multiple light fixtures to cover a large area. LiFi transmission through these multiple light fixtures can be modelled as a MISO channel. In this section, we review achievable secrecy rate results for the LiFi MISO wiretap channel. We split the discussion into two parts: (i) beamforming and (ii) friendly jamming; and we conclude with numerical evaluations.

(a). Light-fidelity multiple-input single-output secrecy by beamforming

In LiFi applications, one can use beamforming to improve transmission or secrecy, however, subject to the condition that this beamforming does not affect illumination—the primary function of light fixtures. The light intensity is controlled by means of the driving current, i.e. its DC component. As such, it is desirable to separate the DC component from the modulated signal in order to be able to control each of them separately, and ensure that communication does not affect illumination. As such, the transmit signal can be written as

$$\mathbf{\text{X}}=\mathbf{\text{w}}S+d\mathbf{\text{1}},$$

where $\mathbf{\text{w}}={[w_k]}_{k=1}^K\in {\mathbb{R}}^K$ is the beamforming vector, S is the codeword symbol to be transmitted, d is the DC offset, and 1 is the all-ones vector of length K. This construction has been studied in [21] with $d=(\mathcal{A}/2)$. Next, we generalize this to any $d\in [0,\mathcal{A}]$, and then we present some achievable secrecy rate results.

The illumination is set by choosing $d\in [0,\mathcal{A}]$. To preserve this illumination level, we force S to have zero mean, and to vary between $[-d,\mathcal{A}-d]$, and we constrain w to be in ${[-(d)/(\mathcal{A}-d),1]}^K$. This guarantees that $\mathbb{E}[X_i]=d$ and $X_i\in [0,\mathcal{A}]$. Note that d must satisfy $d\le \mathcal{E}$ if the system is constrained by an average constraint $\mathbb{E}[X_i]\le \mathcal{E}$, and $d=\mathcal{E}$, if the system is constrained by an average constraint $\mathbb{E}[X_i]=\mathcal{E}$.

As a result, the input–output relationship becomes [21]

$$Y_r={\mathbf{\text{h}}}_r^T\mathbf{\text{X}}+Z_r=({\mathbf{\text{h}}}_r^T\mathbf{\text{w}})S+d{\mathbf{\text{h}}}_r^T\mathbf{\text{1}}+Z_r,r=1,2.$$ 4.1

The DC component $d{\mathbf{\text{h}}}_r^T\mathbf{\text{1}}$ can be subtracted at the receiver. This converts the LiFi MISO wiretap channel into a SISO one, where bounds from §3 can be applied. However, the choice of w remains to be optimized.

In the following, we present an achievable secrecy rate derived in [21] based on (4.1), under the peak constraint $X_i\in [0,\mathcal{A}]$ and illumination constraint $\mathbb{E}[X_i]=(\mathcal{A}/2)$. Then, we discuss the cases $\mathbb{E}[X_i]\le (\mathcal{A}/2)$, $\mathbb{E}[X_i]=\mathcal{E}$, and $\mathbb{E}[X_i]\le \mathcal{E}$.

Theorem 4.1 (Beamforming lower bound [21]). —

Under the constraints $X_i\in [0,\mathcal{A}]$ and $\mathbb{E}[X_i]=(\mathcal{A}/2)$, the secrecy capacity of the LiFi MISO wiretap channel is lower bounded as $C_{s,1}^{\text{MISO}}\ge {\underset{\_}{C}}_{s,1}^{\text{MISO}}$, where this achievable lower bound is given by

$${\underset{\_}{C}}_{s,1}^{\text{MISO}}=\underset{\mathbf{\text{w}}\in {[-1,1]}^K}{max}\frac{1}{2}{\log }^{+}\left(\frac{6{({\mathbf{\text{h}}}_1^T\mathbf{\text{w}})}^2{\mathcal{A}}^2+12\pi e{\sigma }^2}{\pi e{({\mathbf{\text{h}}}_2^T\mathbf{\text{w}})}^2{\mathcal{A}}^2+12\pi e{\sigma }^2}\right).$$ 4.2

Proof. —

We start by setting U = X in (2.6) to obtain $C_{s,1}^{\text{MISO}}\ge \underset{p(\mathbf{\text{x}})}{max}I(\mathbf{\text{X}};Y_1)-I(\mathbf{\text{X}};Y_2)$. Then we choose X = wS + d1, $d=(\mathcal{A}/2)$, and S to be uniformly distributed on $[-(\mathcal{A}/2),(\mathcal{A}/2)]$, to obtain $C_{s,1}^{\text{MISO}}\ge \underset{\mathbf{\text{w}}}{max}I(\mathbf{\text{w}}S;Y_1)-I(\mathbf{\text{w}}S;Y_2)=\underset{\mathbf{\text{w}}}{max}I(S;Y_1)-I(S;Y_2)$. Then we use the lower bound $\hspace{0.17em}{\underset{\dot{}}{C}}_{s,1}^{\text{SISO}}$ in theorem 3.2 to bound I(S;Y₁) − I(S;Y₂) which yields $C_{s,1}^{\text{MISO}}\ge {\underset{\_}{C}}_{s,1}^{\text{MISO}}$ as defined in (4.2). This completes the proof. ▪

The optimization problem in theorem 4.1 is non-convex. However, it can be transformed into a quasi-convex line search problem and solved using bisection as described in [21].

Note that ${\underset{\_}{C}}_{s,1}^{\text{MISO}}$ can be achieved if a peak constraint $X_i\in [0,\mathcal{A}]$ is imposed without an average constraint, or with an average constraint $\mathbb{E}\{X_i\}=(\mathcal{A}/2)$ since this is automatically satisfied by choosing S to follow a zero-mean uniform distribution. If an inequality average constraint is imposed such that $\mathbb{E}\{X_i\}\le \mathcal{E}\le (\mathcal{A}/2)$, or an equality average constraint $\mathbb{E}\{X_i\}=\mathcal{E}\le (\mathcal{A}/2)$, the same scheme of theorem 4.1 can also be applied, as shown in the following corollary.

Corollary 4.2. —

Under the constraints $X_i\in [0,\mathcal{A}]$ and either $\mathbb{E}[X_i]\le \mathcal{E}$ or $\mathbb{E}[X_i]=\mathcal{E}$, where $\mathcal{E}\le \frac{\mathcal{A}}{2}$, the secrecy capacity of the LiFi MISO wiretap channel is lower bounded as $C_{s,2}^{\text{MISO}}\ge {\underset{\_}{C}}_{s,2}^{\text{MISO}}$, where

$${\underset{\_}{C}}_{s,2}^{\text{MISO}}=\underset{\mathbf{\text{w}}\in {[-1,1]}^K}{max}\frac{1}{2}{\log }^{+}\left(\frac{6{({\mathbf{\text{h}}}_1^T\mathbf{\text{w}})}^2{\mathcal{E}}^2+3\pi e{\sigma }^2}{\pi e{({\mathbf{\text{h}}}_2^T\mathbf{\text{w}})}^2{\mathcal{E}}^2+3\pi e{\sigma }^2}\right)$$ 4.3

The proof is similar to theorem 4.1, with $\mathcal{A}$ replaced with $2\mathcal{E}$. This forces $\mathbb{E}[X_i]$ to be equal to $\mathcal{E}$. Note that one may also replace $\mathcal{A}$ in the proof of theorem 4.1 by $d\le \mathcal{E}$ if the average constraint is $\mathbb{E}[X_i]\le \mathcal{E}$, and then maximize with respect to d. However, the maximum can be shown to be achieved when $d=\mathcal{E}$.

The advantage of the scheme used in corollary 4.2 is that we can still choose w ∈ [ − 1, 1]^K, which provides flexibility in beamforming. The disadvantage is that we limit the range of X_i to $[0,2\mathcal{E}]$, which is a subset of $[0,\mathcal{A}]$. Another way to develop an achievable secrecy rate for the case $\mathbb{E}[X_i]\le \mathcal{E}$ is by choosing S to be distributed according to a truncated-exponential distribution (similar to [9, (42)]) between $[-\mathcal{E},\mathcal{A}-\mathcal{E}]$ with zero mean. However, since we construct the transmit signal as $\mathbf{\text{X}}=\mathbf{\text{w}}S+\mathcal{E}\mathbf{\text{1}}$, this imposes the restriction that $\mathbf{\text{w}}\in {[\frac{\alpha }{\alpha -1},1]}^K$ where $\alpha =(\mathcal{E}/\mathcal{A})$, since otherwise, X_i may become negative. Despite this disadvantage, the advantage is that X_i can occupy the whole range $[0,\mathcal{A}]$ when w_i (the corresponding component of w) is positive. It still occupies a portion of $[0,\mathcal{A}]$ when w_i is negative. We present a new lower bound which applies this scheme next.

Theorem 4.3 (Beamforming lower bound). —

Under the constraints $X_i\in [0,\mathcal{A}]$ and either $\mathbb{E}[X_i]\le \mathcal{E}$ or $\mathbb{E}[X_i]=\mathcal{E}$, where $\mathcal{E}=\alpha \mathcal{A}$ and α < (1/2), the secrecy capacity of the LiFi MISO wiretap channel is lower bounded as $C_{s,1}^{\text{MISO}}\ge \hspace{0.17em}{\underset{\dot{}}{C}}_{s,2}^{\text{MISO}}$, where

$$\hspace{0.17em}{\underset{\dot{}}{C}}_{s,2}^{\text{MISO}}(\alpha )=\underset{\mathbf{\text{w}}\in {[(\alpha /\alpha -1),1]}^K}{max}\frac{1}{2}{\log }^{+}\left(\frac{\rho {({\mathbf{\text{h}}}_1^T\mathbf{\text{w}})}^2{\mathcal{A}}^2+2\pi e{\sigma }^2}{2\pi e\varphi {({\mathbf{\text{h}}}_2^T\mathbf{\text{w}})}^2{\mathcal{A}}^2+2\pi e{\sigma }^2}\right),$$ 4.4

where $\rho ={\text{e}}^{2(1-({\mu }_{\alpha }/{\text{e}}^{{\mu }_{\alpha }}-1))}({(1-{\text{e}}^{-{\mu }_{\alpha }})/({\mu }_{\alpha }))}^2$, $\varphi =(1/{\mu }_{\alpha }^2)+1/4-((1/e^{{\mu }_{\alpha }}-1)+(1/2))$ and ${\mu }_{\alpha }$ is defined in lemma 2.3.

Proof. —

The proof is similar to the proof of theorem 4.1 except for the last inequality which is obtained by using the bounds in theorem 3.5. In this case, Alice uses a truncated-exponential distribution for $S\in [-\mathcal{E},\mathcal{A}-\mathcal{E}]$ with zero mean (a shifted version of [9, (42)]), and sends $\mathbf{\text{X}}=\mathbf{\text{w}}S+\mathcal{E}\mathbf{\text{1}}$ where w ∈ [(α)/(α − 1), 1]^K. ▪

The optimization problem in theorem 4.3 can be solved using the same algorithm as in theorem 4.1 by replacing the constraint w ∈ [ − 1, 1]^K in theorem 4.1 by two constraints, max _kw_k ≤ 1 and min _kw_k ≥ (α)/(α − 1).

Finally, the achievable secrecy rate using this scheme under α ≥ (1/2) can be obtained by replacing α and $\mathcal{E}$ by α′ = 1 − α and ${\mathcal{E}}^{\prime }=A-\mathcal{E}$, respectively, in corollary 4.2 and theorem 4.3.

In addition to beamforming, another technique that can be used to enhance secrecy is jamming. This is discussed next.

(b). Light-fidelity multiple-input single-output secrecy by friendly jamming

Suppose the room is illuminated by K identical light fixtures. Let K = K_J + K_A. Alice uses K_A light fixtures to transmit a message. On the other hand, a friendly jammer uses the remaining K_J light fixtures to support Alice by sending noise. When Eve is closer to Alice than Bob, secure communication using the beamforming method described in the previous subsection becomes less promising. To worsen the channel quality from Alice to Eve and maintain a good channel quality from Alice to Bob, the jammer sends a jamming signal which is friendly to Bob but hostile to Eve.

While K_A can be generally any number between 1 and K, in [18,20], K_A is chosen to be 1. We stick to this case in the current description. In this case, the input–output relation under friendly jamming can be expressed as follows:

$$Y_r=h_{r}X+{\mathbf{\text{h}}}_{\text{J}r}^T{\mathbf{\text{X}}}_{\text{J}}+Z_r,r=1,2,$$ 4.5

Here, h₁ and h₂ are the channels from Alice to Bob and Eve, respectively, X is the information-bearing signal, h_J1 and h_J2 are the channels from the jammer to Bob and Eve, respectively, and X_J is a jamming noise signal.

The design of friendly jamming can be achieved as follows. First, the jamming signal is beamformed in a way that maximizes the jamming effect at Eve, so that X_J = w_JS_J, where ${\mathbf{\text{w}}}_{\text{J}}={[w_{\text{J},k}]}_{k=1}^{K_{\text{J}}}$ is the jammer’s beamforming vector, and S_J is the jamming noise signal. Second, the jamming signal is made friendly to Bob by choosing w_J in a way that decreases the impact of S_J at Bob, i.e., ideally ${\mathbf{\text{h}}}_{\text{J}1}^T{\mathbf{\text{w}}}_{\text{J}}=0$.

In [18], an achievable secrecy rate was also developed for this jamming scheme, where X and S_J were chosen to be uniformly distributed. An improved secrecy rate was derived in [20], by using truncated-Gaussian distributions instead. Define $\varphi (x)=(1/\sqrt{2\pi })\hspace{0.17em}{\text{e}}^{-(x^2/2)}$ as the standard Gaussian distribution, and Φ(x) to be the corresponding cumulative distribution. We define a truncated-Gaussian distribution $p_{\mathcal{A},{\sigma }_t}(x)$ as follows:

$$p_{\mathcal{A},{\sigma }_t}(x)=\{\begin{array}{ll}\frac{(1/{\sigma }_t)\varphi ((x/{\sigma }_t))}{\mathit{\Phi }(\beta )-\mathit{\Phi }(-\beta )},& x\in [-\frac{\mathcal{A}}{2},\frac{\mathcal{A}}{2}],\\ 0,& |x|>\frac{\mathcal{A}}{2}\end{array}$$ 4.6

where $\beta =(\mathcal{A})/(2{\sigma }_t)$. Also, define ζ = Φ(β) − Φ( − β). Using this distribution, we can derive the following lower bound on secrecy capacity of the LiFi MISO wiretap channel.

Theorem 4.4 (Jamming lower bound [20]). —

Under the constraints $X_i\in [0,\mathcal{A}]$ and $\mathbb{E}[X_i]=(\mathcal{A}/2)$, the secrecy capacity of the LiFi MISO wiretap channel is lower bounded by $C_{s,1}^{\text{MISO}}\ge \hspace{0.17em}{\underset{\dot{}}{C}}_{s,1}^{\text{MISO}}$ where

$$\hspace{0.17em}{\underset{\dot{}}{C}}_{s,1}^{\text{MISO}}\triangleq \frac{1}{2}\log (1+\frac{{\sigma }_t^2{h}_1^2{e}^{2\eta }}{{\sigma }^2})-h(V_0)+\frac{1}{2}\log (2\pi e{\sigma }_t^2)+\underset{\begin{array}{c}{\mathbf{\text{h}}}_{\text{J}1}^T\mathbf{\text{w}}=0\\ \mathbf{\text{w}}\in {[-1,1]}^{K-1}\end{array}}{max}\log (|{\mathbf{\text{h}}}_{\text{J}2}^T{\mathbf{\text{w}}}_{\text{J}}|)+\eta ,$$ 4.7

where $V_0=h_2{X}_t+{\mathbf{\text{h}}}_{\text{J}2}^T{\mathbf{\text{w}}}_{\text{J}}{S}_t$, X_t and S_t follow a truncated-Gaussian distribution $f_{{\sigma }_t}(x)$, and η = log (ζ) + ( − βϕ( − β) − βϕ(β))/(2ζ).

The optimal w_J for maximizing $\hspace{0.17em}{\underset{\dot{}}{C}}_{s,1}^{\text{MISO}}$ can be found by solving the following optimization problem [20]:

$$\begin{array}{rl}& {\mathbf{\text{w}}}_{\text{J}}^{\ast }=\arg \underset{{\mathbf{\text{w}}}_{\text{J}}\in {[-1,1]}^{K-1}}{max}{\mathbf{\text{h}}}_{\text{J}2}^T{\mathbf{\text{w}}}_{\text{J}}\\ & \text{s.t.}{\mathbf{\text{h}}}_{\text{J}1}^T{\mathbf{\text{w}}}_{\text{J}}=0,\end{array}\}$$ 4.8

which is a zero-forcing beamforming problem and can be solved using a linear programming algorithm. To maximize the achievable secrecy rate further, one can optimize the selection of one transmitter and K − 1 jammers from among the K light fixtures.

Extension to the inequality or equality average constraint, i.e. $\mathbb{E}[X_i]\le \mathcal{E}$ or $\mathbb{E}[X_i]=\mathcal{E}$, where $\mathcal{E}\le (\mathcal{A}/2)$ is not considered in this paper, but is a relevant research direction. Now, we present some numerical evaluations which evaluate the presented bounds.

(c). Numerical results

In the following numerical evaluations, we consider a room of size 6 × 6 × 3 m³ which is illuminated by four lighting fixtures located at the corners of a 3 × 3 m square centred at the ceiling of the room, as shown in figure 5a. Each of the fixtures consists of an array of LEDs, and assume identical PDs for Bob and Eve. The channel gains between the light fixtures and PDs, i.e. h₁, h₂, h_J1, h_J2, can be estimated using the following model [21, (6a)][37]:

$$h=\frac{(m+1)gA}{2\pi d^2}{(\cos \varphi )}^2\cos \psi \hspace{0.17em}\mathsf{I}(\psi <{\mathit{\Psi }}_c),$$ 4.9

where m = ( − ln2)/(ln(cosΦ_h)) is the Lambertian emission order, Φ_h is the half-intensity semi-angle of the light fixture, d is the distance of the light propagation, g is the PD receiving gain and A is its area, ϕ and ψ are the emitting and receiving angle of the transmitter and PD, respectively, Ψ_c is the field of view (FOV) of the PD, and $\mathsf{I}(\cdot )$ is a logic function which returns 1 if its argument is true, and zero otherwise. In the simulations, we assume that gA = 10, Φ_h = 60°, Ψ_c = 70°, and we also assume that the PD is always placed upright, so that ϕ = ψ.

Figure 5.

Room layout used in the simulations and secrecy capacity bounds. (a) The room layout, where the dots represent the light fixture, the dashed circles identify the half-intensity regions, (b) secrecy rate as a function of SNR. (Online version in colour.)

We note that when evaluating the friendly jamming scheme, the selection of the light fixtures into a transmitter and K − 1 jammers is optimized to increase the secrecy rate.

In figure 5b, we compare the achieved secrecy rate by beamforming ${\underset{\_}{C}}_{s,1}^{\text{MISO}}$ in (4.2) and by friendly jamming $\hspace{0.17em}{\underset{\dot{}}{C}}_{s,1}^{\text{MISO}}$ in (4.7), under different SNRs, where we fix Eve’s location at coordinates p₀ = (0, 0, 0.8), and fix Bob’s location at coordinates p₁ = (1.5, 1.5, 0.8) or p₂ = (0, 1.5, 0.8). We use σ_t = 9 in (4.2) [20]. It can be seen that when Bob is at p₁, the beamforming scheme always outperforms the friendly jamming scheme. However, when Bob is at p₂, friendly jamming can achieve higher secrecy rates when SNR is lower than 20 dB. This suggests that a scheme which combines beamforming and friendly jamming can achieve a better performance. Developing such a scheme is a relevant research direction.

In figure 6, we fix SNR = 20 dB and compare the achieved secrecy rate by the beamforming and the friendly jamming schemes, where Eve’s location is fixed at p₀ and Bob can move around the room. It can be seen the beamforming and the friendly jamming schemes are suitable for different regions. For example, when Bob is very close to one of the light fixtures, the beamforming scheme will be a better choice. However, when Bob is between two nearby light fixtures, the friendly jamming scheme can achieve much higher secrecy rate. In conclusion, the friendly jamming scheme can complement the beamforming, since it can achieve superior performance in some ranges of SNR, α, and relative locations of Bob and Eve.

Figure 6.

Secrecy rate as a function of Bob’s location when Eve is fixed at position p₀ (room centre). (a) Beamforming, (b) friendly jamming. (Online version in colour.)

In figure 7, we compare ${\underset{\_}{C}}_{s,2}^{\text{MISO}}(\alpha )$ and $\hspace{0.17em}{\underset{\dot{}}{C}}_{s,2}^{\text{MISO}}(\alpha )$ for SNR between 0 and 40 dB, where the locations of Bob and Eve are fixed at p₁ and p₀, respectively. We choose α to be 0.1 or 0.3. It can be seen that, for α = 0.1, although  C̣_s,2(α) uses a larger signal range, it is a little better than C_s,2(α) only when SNR ≤ 30 dB, and becomes much worse than C_s,2(α) when SNR = 40 dB, but for α = 0.3,  C̣_s,2(α) outperforms C_s,2(α) for all the tested SNR region.

Figure 7.

Secrecy rate constraint $\mathbb{E}[X_i]=\alpha \mathcal{A}$ as a function of SNR, where Bob and Eve are located at p₁ and p₀, respectively. (Online version in colour.)

In figure 8, we compare the beamforming lower bounds ${\underset{\_}{C}}_{s,2}^{\text{MISO}}(\alpha )$ in corollary 4.2 and $\hspace{0.17em}{\underset{\dot{}}{C}}_{s,2}^{\text{MISO}}(\alpha )$ in theorem 4.3, under an average constraint $\mathbb{E}[X_i]=\alpha \mathcal{A}$ with α = 0.3. In this evaluation, Eve’s location is fixed at p₀ but Bob can move around the room. It can be seen that because of a larger signal range (range of S), $\hspace{0.17em}{\underset{\dot{}}{C}}_{s,2}^{\text{MISO}}(\alpha )$ outperforms ${\underset{\_}{C}}_{s,2}^{\text{MISO}}(\alpha )$ in general.

Figure 8.

Secrecy rate under constraint $\mathbb{E}[X_i]=0.3\mathcal{A}$ as a function of Bob’s location, where Eve is fixed at p₀ and SNR = 20 dB. (a) ${\underset{\_}{C}}_{s,2}^{\text{MISO}}(\alpha )$, (b) $\hspace{0.17em}{\underset{\dot{}}{C}}_{s,2}^{\text{MISO}}(\alpha )$. (Online version in colour.)

5. Conclusion and future research directions

In this paper, we reviewed theoretical results on LiFi PLS focusing on the LiFi SISO and MISO wiretap channel models and their secrecy capacity bounds. We considered systems with a peak constraint only and also systems with both peak and average constraints. As LiFi should not disturb illumination levels, an equality average constraint has been discussed, and new bounds or extensions of existing bounds are developed for this case. We studied beamforming and jamming schemes, and evaluated their performance numerically. We have also derived new achievable secrecy rates under beamforming, which improve upon existing results. Existing results and new results presented in this paper can be further improved and extended along the following lines: Transmission under a specific illumination requirement for MISO and MIMO LiFi, signalling schemes and hybrid LiFi/WiFi systems. These three directions are discussed next.

Current studies on LiFi MIMO wiretap channels in the literature [23,24] adopt precoding to realize secrecy, but they only consider the peak constraint and average total power constraint, i.e. $X_k\le \mathcal{A}$ and $\sum _{k=1}^K\mathbb{E}[X_k]\le \mathcal{E}$. In a LiFi system, each light fixture should provide a constant average light intensity, and thus X has to be constrained by $\mathbb{E}[X_k]={\mathcal{E}}_k$, k = 1, …, K. In the special case of LiFi MISO, the current studies adopt beamforming and friendly jamming. As we have seen above, if the equality average constraint is imposed, the transmitted signal may not be able to vary over the whole permissible range $[0,\mathcal{A}]$—the linear range of the LED. For example, in theorem 4.3, if w_k < 0, then X_k varies over a subset of $[0,(\alpha \mathcal{A})/(1-\alpha )]$ (α ∈ [0, 0.5]), which is itself a subset of the whole range $[0,\mathcal{A}]$. As seen in figure 7, a larger emitting range can increase the secrecy performance. Therefore, it is important to rethink the current design to make full use of the emitting range of the LED. Moreover, in such a design, ${\mathcal{E}}_k$ does not have to be the same for all k. For instance, in some large indoor scenarios, not all lights have to be turned on or provide the same level of illumination. For example, light fixtures in a large library may only need to turn on when people walk in the nearby region, leading to non-equal values of ${\mathcal{E}}_k$.

Another promising direction is designing signalling schemes for LiFi PLS. As shown in [26], the optimal input distribution for the LiFi SISO wiretap channel studied in this paper is discrete with a finite number of mass points. But the optimal distribution is unknown, and even if found numerically, it is not clear how to design the transmitted signal to match the theoretically optimal input distribution. On the other hand, existing modulation schemes produce a uniform distribution (realized by PPM and PAM) and truncated Gaussian distribution (realized by various types of unipolar OFDM-based schemes [38]). It is important to develop modulation schemes that produce a truncated-exponential distribution, which has superior performance compared with a uniform distribution as shown in figure 7. The work in [39] proposes a method for generating a non-uniform discrete distribution of the SISO LiFi channel without an eavesdropper, which might also be useful for the wiretap channel. This research direction is relevant and requires additional investigation.

It is also important to note that LiFi is vulnerable to physical blockage, and RF access points (such as WiFi) can be used to support LiFi connection in such cases. Hybrid VLC/RF communication has been demonstrated in the literature [40]. The current studies on the secrecy of hybrid VLC/RF systems mostly use zero-forcing beamforming and try to minimize the system power consumption [41–43]. These works consider VLC and RF as independent components which perform secure communication independently without cooperation, and they do not consider a specific lighting requirement in the form of an average constraint $\mathbb{E}[X_k]={\mathcal{E}}_k$. This calls upon designing and studying more cooperative PLS schemes for hybrid LiFi/RF or LiFi/WiFi systems.

Data accessibility

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Authors' contributions

This is mostly a review article. All three authors participated equally in the design of the manuscript scope and structure. Z.Z. carried out all the numerical evaluations and the drafting of the manuscript. A.C. and L.L. performed revisions and provided comments.

Competing interests

We declare we have no competing interest.

Funding

This work has been supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).