Physical Layer Security in LiFi Systemslampe/Preprints/2019-PhySec-VLC... · 2019-10-24 · Physical Layer Security in LiFi Systems Zhenyu Zhang1, Anas Chaaban1 and Lutz Lampe2 1School

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Research

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Subject Areas:

Optical wireless communications,

physical layer security

Keywords:

Intensity modulation, wiretap channel,

secrecy rate

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e-mail: [email protected]

Physical Layer Security in LiFiSystemsZhenyu Zhang1, Anas Chaaban1 and Lutz

Lampe2

1School of Engineering, University of British Columbia,

Kelowna, BC, Canada, V1V 1V7. Email:

[email protected], [email protected] of Electrical and Computer Engineering,

University of British Columbia, Vancouver, BC,

Canada, V6T 1Z4. Email: [email protected]

Light-Fidelity (LiFi) is a light-based wireless commun-ication technology which can complement radio-frequency (RF) communication technologies for indoorapplications. Although LiFi signals are spatially morecontained than RF signals, the broadcasting natureof LiFi also makes it susceptible to eavesdropping.Therefore, it is important to secure the transmitteddata against potential eavesdroppers. In this paper,an overview of the recent developments pertainingto LiFi physical layer security (PLS) is provided, andthe main differences between LiFi PLS and RF PLSare explained. LiFi achievable secrecy rates and upperbounds are then investigated under practical channelmodels and transmission schemes. Beamformingand jamming, which received significant researchattention recently as a means to achieve PLS inLiFi, are also investigated under indoor illuminationconstraints. Finally, future research directions ofinterest in LiFi PLS are identified and discussed.

1. IntroductionWith the high and continuously increasing demandfor data communications, the limited radio-frequency(RF) spectrum becomes a bottleneck which slows downcommunication speeds. The light spectrum, whichconsists of around 670 THz of unlicensed bandwidith, isa promising resource for aleviating this problem. In thiscontext, Light-Fidelity (LiFi) [1–3] is a communicationsystem that uses visible light communication (VLC) toprovide indoor wireless communication. It is a promisingtechnology to complement its RF-based counterpart,WiFi. Due to its importance, several LiFi companies have

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recently emerged, including PureLiFi in England and Oledcom in France. Additionally, severalexisting 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 (itscoverage), which makes the link vulnerable to attacks. In general, there are two forms of attacksin wireless communications that are studied in the literature [4]: Passive attacks which do notdisturb the system; and active attacks which do. Eavesdropping is a form of passive attack whichcan 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, whichare based on cryptography, as defined in the IEEE 802.15.7 standard. Cryptography-based securitytechniques are efficient under the assumptions that the secret key is unknown to unintendedreceivers 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. Incontrast, PLS provides theoretically-provable security, and is a promising solution for securingLiFi systems [6].

PLS originated from information-theoretic studies on the wiretap channel [6,7], which makesuse of physical properties of the transmission process to enhance security. As suggested by thename, PLS is a security technique for the physical layer and is independent of other layers ofthe OSI network model, and thus can provides either independent protection or cooperativeprotection 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 andknowledge of network parameters, and having a precisely quantified achieved security.

When studying PLS in LiFi, the studies on the Gaussian wiretap channel for RF systemscan serve as a starting point, but do not apply directly. This is due to the constraints that thetransmitted optical signals must satisfy, such as a peak constraints and/or an average constraint,in addition to a nonnegativity constraint [9]. As LiFi is proposed as an add-on functional moduleto the indoor illumination system, the LiFi system should not disturb the indoor illuminationlevel. 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 LiFisystem. Thus, PLS in LiFi must be investigated while paying careful attention to the optical signalconstraints.

Various methods to realize secure transmissions have been studied in the literature. Forinstance, a security zone scheme was proposed in [11] to prevent eavesdropping within a certainregion around the legitimate receiver. A MIMO-based VLC security zone was proposed basedon light-emission pattern shaping in [12]. The effect of different light-fixture deployments onLiFi PLS was investigated in [13], where four types of deployments were tested, and an anglediversity receiver is shown to improve secrecy throughput significantly. Most of the relatedwork 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 communicaton is studied in [16], where different relaying schemes suchas cooperative jamming, decode-and-forward, amplify-and-forward, were compared. The worksin [17–26] study the secrecy capacity and derive achievable secrecy rates for the LiFi wiretapchannel, which is the focus of this paper.

In this paper, we focus on the recent theoretical studies of LiFi PLS, where we review boundson the LiFi wiretap channel secrecy capacity, including single-input single-output (SISO) andmultiple-input single-output (MISO) cases. For the LiFi SISO wiretap channel, we review andextend existing secrecy capacity bounds, while considering both peak and average constraints onthe transmit signal. We comment on the applicability of existing results to a LiFi system whichmust satisfy a specific lighting requirement (average constraint with equality), which is importantfor light dimming. For the LiFi MISO wiretap channel, we review achievable secrecy rates using

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|Enc. Dec.

Eavesdropper

SISO Wiretap channel

Alice

Bob

Eve

Figure 1. The SISO wiretap channel model.

beamforming and friendly jamming under a peak constraint on the transmit signal. We thenextend these results to LiFi MISO systems with both peak and average constraints on the transmitsignal. We describe an enhanced beamforming scheme which makes better use of the emittingrange of an LED, and we derive its achievable secrecy rate. Finally, we identify some gaps andfuture directions in LiFi PLS research.

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

2. PreliminariesIn this section, we review PLS starting from Wyner’s discrete memoryless wiretap channel, andthen introduce the LiFi wiretap channel. We also introduce some LiFi P2P capacity bounds, whichare needed for bounding the secrecy capacity of the LiFi wiretap channel throughout the paper.

(a) A General Review of PLS for the Wiretap Channel

(i) The Wiretap Channel and Secrecy Capacity

A discrete memoryless wiretap channel is depicted in Fig. 1. It consists of a transmitter (Alice), anintended receiver (Bob), and an eavesdropper (Eve). Alice wishes to transmit a message to Bobthrough a noisy channel while keeping it secret from Eve. This transmission can be described asfollows [27]. Alice’s message is denoted by M , which is a random variable uniformly distributedon M= {1, 2, . . . , 2nR} for some n∈N+ and R ∈R. To realize this, Alice encodes the messageM into a codeword Xn = {X [i]}ni=1 where X [i] ∈X , for some alphabet X . Then Alice sends thiscodeword symbol by symbol over n transmissions.1

The signals received by Bob and Eve are denoted by Y1 ∈Y1 and Y2 ∈Y2, respectively. Theseare 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 n1 = {Y [i]1 }

ni=1, and then tries to decode M . If

we denote the decoding result as M ∈M∪ {e} where e is an error message, then the probabilityof error can be written as Pe,n = P{M 6=M}.

Similarly, Eve collects Y2 over n uses of the channel, i.e., Y n2 = {Y [i]2 }

ni=1, and tries to infer as

much information about M as possible. Clearly, Alice would like to reduce Eve’s abilities to inferinformation about M . The amount of information leaked to Alice can be expressed as I(M ;Y n2 ),where I(·; ·) is the mutual information.

1The time index iwill only be indicated when needed.

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

limn→∞

Pe,n = 0, limn→∞

1

nI(M ;Y n2 ) = 0. (2.1)

The maximum rateR at which Alice can communicate with Bob while concealing the messagefrom Eve (according to (2.1)) is called the secrecy capacity. The secrecy capacity of the wiretapchannel is shown to be

Cs = maxp(u,x)

I(U ;Y1)− I(U ;Y2), (2.2)

where U ∈ U is an auxiliary random and |U| ≤ |X | [27,28]. The purpose of p(u, x) is to transformthe end-to-end channel from p(y1, y2|x) to p(y1, y2|u), which can be more favourable from asecrecy rate perspective. In particular, the achievable secrecy rate I(U ;Y1)− I(U ;Y2) for somep(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 randomizedencoding scheme (Fig. 2). In this coding scheme, Alice first generates a subcodebook of codewords

un(`) for each message m∈M, with `∈Lm , {(m− 1)2n(R−R) + 1, . . . ,m2n(R−R)} for someR≥R. The reason for generating many codewords for each m is to increase the randomness ofthe transmission of m so as to confuse Eve. The symbols of un(`) are independent and identicallydistributed according to p(u). Consequently, each message m is represented by many codewordsun(`). To send m, Alice randomly chooses a un(`) with `∈Lm, and then generates the transmitsignal xn whose symbols are independent and identically distributed according to p(x|u).

Note that the total number of generated codewords un(`) is 2nR. If R < I(U ;Y1), then by thechannel coding theorem [27], Bob will be able to discern the sent codewords if n is large enough.It then decodes `, from which it knows m.

The maximum number of codewords that Eve can discern is given by 2nI(U ;Y2) asymptoticallyfor large n, which can be proved using the channel coding theorem [27, Chap. 3.1]. If wechoose R−R> I(U ;Y2), then Eve will have ambiguity between multiple codewords un(`)

corresponding to m= 1, multiple codewords un(`) corresponding to m= 2, etc. Moreover, thisambiguity will show no preference to a specific m, since the number of ‘likely’ codewords un(`)will be roughly the same for all m. This confuses Eve and establishes secrecy. When optimizedwith respect to p(u, x), this coding scheme achieves the secrecy capacity in (2.2). Note that thiscoding 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 wiretapchannel satisfies the condition p(y1, y2|x) = p(y1|x)p(y2|y1), i.e, X→ Y1→ Y2 forms a Markovchain, then the wiretap channel is said to be physically degraded.2 In this case, Eve receives adegraded version of Bob’s received signal (as if Bob’s received signal is passed to Eve throughanother noisy channel p(y2|y1)). If there exists Y2 so that Y2|X has the same distribution as Y |Xand X→ Y1→ Y2 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]

Cs =maxp(x)

I(X;Y1)− I(X;Y2). (2.3)

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

As we shall see next, the single-input single-output (SISO) LiFi wiretap channel where Alicehas a single transmitter and each of Bob and Eve have a single receiver is a degraded wiretapchannel if h1 >h2. The LiFi wiretap channel is not degraded in general if Alice is equipped withmultiple transmitters that can transmit non-identical signals. We call this a multiple-input single-output (MISO) wiretap channel. We introduce the LiFi wiretap channel next.2A channel satisfying the Markov chain X→ Y2→ Y1 is also physically degraded, but has zero secrecy capacity, and henceis not of interest in this paper.

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m

1

2

...

2nR

`1

2...2n(R−R)

2n(R−R) + 1

2n(R−R) + 2...(2)2n(R−R)

...

(2nR − 1)2n(R−R) + 1

(2nR − 1)2n(R−R) + 2...2nR2n(R−R)

un(`)

.

.

.

••◦•••◦

.

.

.

••◦••◦•...

.

.

.

•••◦••◦

p(x|u) xn

to channel

Figure 2. The secrecy capacity achieving scheme encodes the message m into many codewordsun(`). Alice selects a random codeword un(`) corresponding to the message m to be sent, andthen sends a random signal xn distributed according to p(x|u). Bob is able to decode the correctmessage by decoding the correct un(`) (marked with a square), while Eve will have ambiguitybetween many un(`) corresponding to different messages m (white).

(b) LiFi Wiretap ChannelIn a LiFi system, the transmitter is a light source such as an LED (or a group thereof as in a lightfixture), and the receiver is a photodetector (PD) consisting of a photodiode (or a group thereof).Light intensity of the LED is modulated by modulating the driving current, and the PD transformslight intensity to current.

Let us denote the driving current by X . Due to the use of LEDs, the driving current can onlybe positive. Therefore, X ∈R+. In addition to this constraint, the current X must satisfy someadditional 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 tobe satisfied, which is given by X ≤A. Additionally, for eye safety X has to satisfy an averageconstraint E[X]≤E , which arises due to a (roughly) linear relation between current and lightintensity in LiFi solid-state lighting devices [30]. Moreover, To meet a desired illuminationrequirement, we might also require E[X] = E . Some studies on LiFi ignore the average constraintand only focus on a peak constraint. In this paper, we will consider both average and peakconstraints.

In general, if Alice is equipped with multiple transmitters (light-fixtures e.g.), then the transmitsignal will be a vector X∈RK+ where K is the number of transmitters. In this case, eachcomponent of X is constrained by a peak constraint A, and the average constraint can applyto the sum as

∑Kk=1 E[Xk]≤E or individually as E[Xk]≤Ek for all k= 1, 2, . . . ,K.

Upon transmitting X, Bob and Eve receive Y1 and Y2 (representing currents), which can bedescribed by the following input-output relations

Yr = hTr X+ Zr, r= 1, 2, (2.4)

where h1,h2 ∈RK+ are the time-invariant channel coefficient vectors from Alice to Bob and Eve,respectively, and Zi-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, weassume that the channels from the transmitters to the receivers are known globally.

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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) LiFi SISO Secrecy Capacity

If K = 1, then we have a LiFi SISO wiretap channel. If h1 >h2, this LiFi SISO wiretap channelis stochastically degraded. In particular, Y2 = h2

h1Y1 + Z where Z ∼N (0, σ2(1− h22h−21 )) has

the same conditional distribution Y2|X as Y2|X , and satisfies the Markov chain X→ Y1→ Y2.Consequently, in this case the secrecy capacity is given by

CSISOs,i = max

p(x)∈P(t)I(X;Y1)− I(X;Y2), (2.5)

where t= 1, 2, P(1) is the collection of distributions of X ∈ [0,A] (peak constraint), and P(2) isthe collection of distributions of X ∈ [0,A] with E[X]≤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-noiseratio (SNR) (Aσ → 0), [26] shows that the secrecy capacity achieving distribution has two masspoints. It is also shown that at high SNR (Aσ →∞), a large number of mass points is needed.However, the optimal distribution p(x) remains unknown.

(ii) LiFi MISO Secrecy Capacity

If K > 1, then we have a LiFi MISO wiretap channel. The LiFi MISO wiretap channel is notdegraded, unless h1 = ηh2 for some η ∈R (aligned),3 which is unlikely in practice. Thus, thesecrecy capacity of the MISO wiretap channel is given by [22]

CMISOs,i = max

p(u,x):p(xi)∈P(t)I(U;Y1)− I(U;Y2), (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 onachievable rates based on beamforming and friendly jamming.

It is often convenient to express capacity in general, and secrecy capacity in our case inparticular, using simple analytical expressions. Using a discrete p(x) in (2.5) or (2.6) prohibits thisdue 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 asymptoticcapacity expressions at high and low SNR. To develop such bounds, we first introduce capacitylower and upper bounds for the SISO LiFi channel without an eavesdropper.

(c) LiFi P2P Channel Capacity BoundsSeveral works in the literature studied the LiFi P2P channel capacity [9,31,32] and developedupper and lower bounds. These bounds have been used intensively in the literature to studyother types of LiFi channels’ [33–35], including the LiFi wiretap channel [22,25].

Consider a LiFi P2P channel with input X satisfying X ∈ [0,A] and E[X]≤ αA for someα∈ [0, 1] (average-to-peak ratio), and output Y =X + Z, with Z Gaussian with zero mean andvariance σ2. Denote the capacity of this channel by Cα(A, σ) =maxp(X) I(Y ;X). We reviewbounds on Cα(A, σ) in the following lemmas. We start with α≥ 1

2 , in which case the averageconstraint is inactive, and the peak constraint dominates.3 In this case, the channel is equivalent to a SISO channel with input X = hT2 X and outputs Y1 = ηX + Z1 and Y2 =

X + Z2, which is degraded.

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Lemma 2.1 ( [9]). If 12 ≤ α< 1 (inactive average constraint) or α= 1 (peak constraint only), then the

capacity of the LiFi P2P channel Cα(A, σ) is bounded as follows

Cα(A, σ)≥C[1]α (A, σ), 1

2log

(1 +

A2

2πeσ2

), (2.7)

Cα(A, σ)≤C[1]α (A, σ), 1

2log

(1 +

A2

4σ2

), (2.8)

Cα(A, σ)≤ C[1]α (A, σ), (2.9)

where

C[1]α (A, σ),min

δ>0

(1− 2Q

(δ + A2σ

))log

(A+ 2δ

σ√2π(1− 2Q( δσ ))

)

+ log(e)

(−1

2+Q

(δ

σ

)+

δ√2πσ

e−δ2

2σ2

). (2.10)

It can be seen that α does not affect the bounds in Lemma 2.1. This is because if α≥ 12 , then

E ≥ A2 , and the capacity-achieving input distribution has mean A2 in this case [9]. If α< 1

2 , thenthe average constraint is active, and capacity is bounded as given next.

Lemma 2.2 ( [9]). If 0<α< 12 , then the capacity of the LiFi P2P channelCα(A, σ) is bounded as follows

Cα(A, σ)≥C[2]α (A, σ), 1

2log

(1 +A2 e

2αµα

2πeσ2

(1− e−µα

µα

)), (2.11)

Cα(A, σ)≤C[2]α (A, σ), 1

2log

(1 + α(1− α)A

2

σ2

), (2.12)

Cα(A, σ)≤ C[2]α (A, σ), (2.13)

where µα is the unique positive solution of α= 1µα− e−µα

1−e−µα , and

C[2]α (A, σ), min

µ,δ>0

(1−Q

(δ + αAσ

)−Q

(δ + (1− α)A

σ

))· log

(Aσ

eµδA − e−µ(1+

δA )

√2πµ(1− 2Q( δσ ))

)

+ log(e)

[− 1

2+Q

(δ

σ

)+

δ√2πσ

e−δ2

2σ2 +σ

Aµ√2π

(e−

δ2

2σ2 − e−(A+δ)2

2σ2

)

+ µα

(1− 2Q

(δ + A2σ

))], (2.14)

The upper bounds C[i]α (A, σ), i= 1, 2, have been shown to be asymptotically tight at high SNR

where they coincide with the lower bounds C[i]α (A, σ), i= 1, 2, respectively, while the bounds

C[i]α (A, σ), 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 2.2 and 2.1, in addition to a new bound which we developto fill some gaps in the literature.

3. LiFi SISO Secrecy Capacity BoundsIn this section, we present LiFi SISO secrecy capacity results from the literature, and develop somenew bounds. Clearly, we focus on h1 >h2, since the secrecy capacity is zero otherwise. We willsplit the review into two parts depending on the constraint on X .

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(a) Secrecy Capacity Bounds under a Peak ConstraintWhen only a peak constraint is imposed, the LiFi SISO wiretap channel capacity is given by

CSISOs,1 = max

p(x)∈P(1)I(X;Y1)− I(X;Y2). (3.1)

In what follows, we use (x)+ to denote max{0, x} and log+(x) to denote max{0, log(x)}. Thefollowing theorem presents a secrecy capacity lower bound which relies on Lemma 2.1.

Theorem 3.1 (Lower bound [22]). Under a peak constraint, the LiFi SISO wiretap channel secrecy

capacity is lower bounded by CSISOs,1 ≥CSISO

s,1 ,(12 log

(1 +

h21A

2

2πeσ2

)− C[1]

1 (h2A, σ))+

.

Proof. Since CSISOs,1 =maxp(x)∈P(1) I(X;Y1)− I(X;Y2), we can write

CSISOs,1 ≥ max

p(x)∈P(1)I(X;Y1)− max

p(x)∈P(1)I(X;Y2)

≥(C

[1]1 (h1A, σ)− C

[1]1 (h2A, σ)

)+.

(3.2)

where the last step follows using Lemma 2.1.

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

Theorem 3.2 (Lower bound [22]). Under a peak constraint, the LiFi SISO wiretap channel secrecy

capacity is lower bounded by CSISOs,1 ≥C.

SISOs,1 , 1

2 log+(

6h21A

2+12πeσ2

πeh22A2+12πeσ2

).

Proof. We have CSISOs,1 =maxp(x)∈P(1) I(X;Y1)− I(X;Y2) which we can write as CSISO

s,1 =

maxp(x)∈P(1) h(Y1)− h(Y1|X)− h(Y2) + h(Y2|X) where h(·) and h(·|·) are the differential andconditional differential entropy, respectively. We proceed as follows

CSISOs,1 = max

p(x)∈P(1)h(Y1)− h(Z1)− h(Y2) + h(Z2)

(a)≥ max

p(x)∈P(1)

(1

2log(22h(h1X) + 22h(Z1)

)− 1

2log (2πeVAR{Y2})

)+

(b)≥(1

2log(h21A2 + 2πeσ2

)− 1

2log

(2πe

(h22A2

12+ σ2

)))+

=1

2log+

(6h21A2 + 12πeσ2

πeh22A2 + 12πeσ2

),

(3.3)

where (a) follows by using h(Z1) = h(Z2), lower bounding h(Y1) using the entropy-powerinequality (EPI), and upper bounding h(Y2) by the differential entropy of a Gaussian randomvariable with variance VAR{Y2}; and (b) follows by choosing X to be uniformly distributed on[0,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 CSISOs,1 ≤C

SISOs,1 , 1

2 log(h21A

2+4σ2

h22A2+4σ2

).

This statement was proved in [22, Prop. 2.3] using a dual secrecy capacity expression (fordegraded channels), which is an extension of the dual capacity expression in [36]. This statementcan also be proved by converting the LiFi SISO wiretap channel into a Gaussian wiretap channel

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with a power constraint E[X2]≤ A2

4 . Namely, we can consider an equivalent degraded wiretapchannel with Yi = hi(X − A2 ) + Zi. Then, we note that X − A2 is in [−A2 ,

A2 ] and has maximum

variance A2

4 . By dropping the peak constraints and allowing X to be in R instead, we obtain a

Gaussian wiretap channel with a power constraint E[X2]≤ A2

4 , whose secrecy capacity is givenas in Theorem 3.3 [27]. This bound can be also obtained from [25].

(b) Secrecy Capacity Bounds under Peak and Average ConstraintsWhen both peak and average constraints are imposed, the secrecy capacity is given by

CSISOs,2 = max

p(x)∈P(2)I(X;Y1)− I(X;Y2). (3.4)

In this case, the secrecy capacity can be bounded using the LiFi P2P channel capacity boundsin Lemma 2.1 and 2.2 as shown next.

Theorem 3.4 (Lower bound [25]). Under both peak and average constraints, the LiFi SISO wiretapchannel secrecy capacity is lower bounded by

CSISOs,2 ≥CSISO

s,2 (α),(C[i]α (h1A, σ)−min

{C

[i]α (h2A, σ), C

[i]α (h2A, σ)

})+, (3.5)

where i= 1 if α≥ 12 and i= 2 if α< 1

2 .

The proof of this theorem is similar to the proof ofCSISOs,1 in Theorem 3.1, but uses both Lemmas

2.1 and 2.2.Similar to Theorem 3.2, we can develop a lower bound under both peak and average

constraints. Instead of the uniform distribuion 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< 12 , the LiFi SISO

wiretap channel secrecy capacity is lower bounded by

CSISOs,2 ≥C.

SISOs,2 (α),

1

2log+

(ρh21A2 + 2πeσ2

2πeφh22A2 + 2πeσ2

), (3.6)

where ρ= e2(1−µα

eµα−1 )(1−e−µαµα

)2, φ= 1

µ2α+ 1

4 − ( 1eµα−1 + 1

2 ), and µα is defined in Lemma. 2.2.

Proof. The proof is similar to that of Theorem (3.2) until (b), where instead of a uniformdistribution, we choose X according to a truncated-exponential distribution which maximizesthe differential entropy h(X), where X ∈ [0,A] and E[X] = E = αA [9, (42)], given by

p(x) =1

Aµ

1− e−µ e−µxA . (3.7)

Thus, we have

CSISOs,2 ≥

(1

2log(22h(h1X) + 22h(Z1)

)− 1

2log (2πeVAR[Y2])

)+

, (3.8)

where X follows the distribution p(x) above, and VAR{Y2}= h22VAR{X}+ σ2. Then we haveVAR[X],E[X2]− (E[X])2 = φh22A

2, and h(h1X) = h(X) + log(h1) =− log(

1h1A

e−µ

1−e−µ)+

log(e)(1− µαeµα−1 ). Then 2h(h1X) = ρh21A

2. Substituting in (3.8) concludes the proof.

Next, we present a secrecy capacity upper bound derived using the EPI and the bounds inLemmas 2.1 and 2.2.

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Theorem 3.6 (Upper bound [25]). Under both peak and average constraints, the LiFi SISO wiretapchannel secrecy capacity is upper bounded by

CSISOs,2 ≤C

SISOs,2 (α),

1

2log

h2122C

[i]α (h1A,σ)

h22 − h21 + h2222C

[i]α (h1A,σ)

, (3.9)

CSISOs,2 ≤ CSISO

s,2 (α),1

2log

(h212

2C[i]α (h1A,σ)

h21 − h22 + h2222C

[i]α (h1A,σ)

), (3.10)

where i= 1 if α≥ 12 and i= 2 if α< 1

2 .

Proof. Since I(X;Y1)− I(X;Y2) = I(X;h1X + Z1)− I(X;h2X + Z2), we can write

I(X;Y1)− I(X;Y2) = I(X;h1X + Z1)− h(h2X + Z2) + h(Z2)

(a)= I(X;h1X + Z1)− h(h1X + Z1 + Z′)− log(h2h

−11 ) + h(Z2)

(b)≤ I(X;h1X + Z1)−

1

2log(22h(h1X+Z1) + 22h(Z

′))− log(h2h

−11 ) + h(Z2)

(c)=

1

2log

(h212

2I(X;h1X+Z1)+2h(Z2)

h2222I(X;h1X+Z1)+2h(Z1) + h222

2h(Z′)

)(3.11)

where (a) follows by using h(X) = h(aX)− log(a) for a> 0 and writing h1h2Z2 =Z1 + Z′ with Z′

being Gaussian with zero mean and variance(h21

h22− 1)σ2 (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;h1X + Z1)≤C[i]α (h1A, σ) and I(X;h1X + Z1)≤ C

[i]α (h1A, σ) by Lemmas 2.1 and 2.2.

Furthermore, 22h(Z1) = 22h(Z2) = 2πeσ2 and 22h(Z′) = 2πe

(h21

h22− 1)σ2. Substituting these

identities and inequalities in the last line in (3.11) concludes the proof.

(c) Extension to Systems with Specific Lighting RequirementsThe above discussion considered peak, or both peak and average constraints, where the averageconstraint is an inequality constraint given by E[X]≤E . As discussed earlier, in LiFi, we are alsointerested in a practical average constraint which is an equality constraint E[X] = E , which reflectsa specific lighting requirement. So it is also relevant to develop secrecy capacity bounds under thisequality constraint.

Towards this end, we discuss below how the above bounds can be extended to LiFi systemswith E[X] = E . Some results above apply immediately to this case which are discussed in thefollowing remarks. The only missing piece is when α> 1

2 , which is discussed afterwards.

Remark 3.1. Given α≤ 12 , the secrecy capacity lower bounds in Theorems 3.4 and 3.5 (peak and average

constraint E[X]≤E) are also secrecy capacity lower bounds for a LiFi SISO wiretap channel with a peakconstraint and an average constraint E[X] = E . The reason is that the schemes achieving the secrecy ratesin these theorems satisfy the constraint E[X]≤E with equality.

Remark 3.2. The secrecy capacity upper bounds in Theorem 3.6 (peak and average constraint E[X]≤E)are also secrecy capacity upper bounds for a LiFi SISO wiretap channel with a peak constraint and anaverage constraint E[X] = E . The reason is that the constraint E[X]≤E is a relaxed version of E[X] = E ,and hence the former has larger secrecy capacity than the latter. We improve this upper bound further next.

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−20 0 20 40 60 800

2

4

6h1h2

= 102

h1h2

= 10

SNR (dB)

Secr

ecy

Rat

e(b

it/s

/Hz)

CSISOs,1 (Thm. 3.1)

C.SISOs,1 (Thm. 3.2)

CSISOs,1 (Thm. 3.3)

(a) Peak constraint only.

−20 0 20 40 60 800

2

4

6

h1h2

= 103

h1h2

= 102

h1h2

= 10

SNR (dB)

Secr

ecy

Rat

e(b

it/s

/Hz)

CSISOs,2 (α) (Thm. 3.4)

C.SISOs,2 (α) (Thm. 3.5)


(b) Peak and average constraints: α= 0.3.

Figure 3. Secrecy capacity bounds for the LiFi SISO wiretap channel.

Based on these remarks, for a LiFi SISO system with a constraint E[X] = E , it remains todevelop secrecy capacity lower bounds for the case α> 1

2 , and to improve the upper bound fromRemark 3.2. This task is simple after observing the following.

The secrecy capacity for a LiFi SISO wiretap channel with α> 12 is the same as for a

channel with outputs Y ′1 = h1A− Y1 = h1(A−X) + Z′1 and Y ′2 = h2A− Y2 = h2(A−X) + Z′2by symmetry of the noises’ Gaussian distribution. But the signal A−X has lies in [0,A] andsatisfies E[A−X] =A− E = (1− α)A if E[X] = E = αA. Moreover, if α> 1

2 , then 1− α, α′ < 12 .

Hence, the developed lower bounds in Theorems 3.4 and 3.5 and the upper bound in Theorem 3.6for the case α< 1

2 also apply to this new channel with α′ < 12 . Thus, we can state the following.

Corollary 3.1. Under a peak constraint and an average constraints E[X] = E = αA with α> 12 , the LiFi

SISO wiretap channel secrecy capacity is upper bounded by CSISOs,2 (α′) and lower bounded by CSISO

s,2 (α′)

and C.SISOs,2 (α′), where

α′ =

{α α≤ 1

2

1− α α> 12 .

(3.12)

Next, we present some numerical evaluations.

(d) Numerical resultsFirst, we show a comparison of the secrecy capacity bounds for a LiFi SISO wiretap channel witha peak constraint only, under different ratios h1

h2. Fig. 3a compares lower bounds CSISO

s,1 and C.SISOs,1

from Theorems 3.1 and 3.2, respectively, with upper bound CSISOs,1 from Theorem 3.3, versus SNR

defined as Aσ . It can be seen that all bounds increase with h1h2

, which supports the intuition thata larger disparity between Bob’s and Eve’s channels lead to higher secrecy capacity. The figureshows that C.

SISOs,1 outperforms CSISO

s,1 when SNR is not very large. On the other hand, as SNR

increases, CSISOs,1 converges to C

SISOs,1 . In particular, the two converge to the high-SNR secrecy

capacity characterized in [25] as CSISOs,1 → 1

2 log(h21

h22

)as SNR→∞.

Fig. 3b compares lower bounds CSISOs,2 and C.

SISOs,2 from Theorems 3.4 and 3.5, respectively, with

upper bound CSISOs,2 from Theorem 3.6, versus SNR defined as Aσ . The impact of increasing h1

h2is

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−20 0 20 40 60 800

1

2

3


CSISOs,2 (α)

(Thm. 3.6)

SNR

Secr

ecy

Rat

e(b

it/s

/Hz)

α= 0.1

α= 0.3

α= 0.5

Figure 4. Secrecy capacity bounds under both peak and average constraints with h1h2

= 10 anddifferent values of α.

similar to that in Fig. 3a and the newly derived lower bound, C.SISOs,2 , is tighter than CSISO

s,2 when

SNR is relatively low. It can also be seen that CSISOs,2 and C

SISOs,2 converge to the asymptotic high-

SNR secrecy capacity 12 log

(h21

h22

)[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.Fig. 4 compares the bounds for a wiretap channel with both peak and average constraints, for

different values of α, under fixed h1h2

= 10. It can be seen that for each α, the corresponding lowerbound converges to the upper bound when SNR is high enough. It can also be seen that the lowerbound increases with α, which depicts the impact of dimming in LiFi, since decrease α decreasesthe average light intensity.

4. LiFi MISO Achievable Secrecy RatesAn indoor illumination system usually has multiple light fixtures to cover a large area. LiFitransmission through these multiple light fixtures can be modelled as a MISO channel. In thissection, we review achievable secrecy rate results for the LiFi MISO wiretap channel. We splitthe discussion into two parts: (i) Beamforming and (ii) friendly jamming; and we conclude withnumerical evaluations.

(a) LiFi MISO Secrecy by BeamformingIn 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 functionof light fixtures. The light intensity is controlled by means of the driving current, i.e., its DCcomponent. As such, it is desirable to separate the DC component from the modulated signal inorder to be able to control each of them separately, and ensure that communication does not affectillumination. As such, the transmit signal can be written as

X=wS + d1,

where w= [wk]Kk=1 ∈RK 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 studiedin [21] with d= A2 . Next, we generalize this to any d∈ [0,A], and then we present some achievablesecrecy rate results.

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The illumination is set by choosing d∈ [0,A]. To preserve this illumination level, we force Sto have zero mean, and to vary between [−d,A− d], and we constrain w to be in [− d

A−d , 1]K .

This guarantees that E[Xi] = d and Xi ∈ [0,A]. Note that d must satisfy d≤E if the system isconstrained by an average constraint E[Xi]≤E , and d= E , if the system is constrained by anaverage constraint E[Xi] = E .

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

Yr = hTr X+ Zr = (hTr w)S + dhTr 1+ Zr, r= 1, 2. (4.1)

The DC component dhTr 1 can be subtracted at the receiver. This converts the LiFi MISO wiretapchannel into a SISO one, where bounds from Sec. 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 Xi ∈ [0,A] and illumination constraint E[Xi] = A2 . Then, we discuss the casesE[Xi]≤ A2 , E[Xi] = E , and E[Xi]≤E .

Theorem 4.1 (Beamforming lower bound [21]). Under the constraints Xi ∈ [0,A] and E[Xi] = A2 ,the secrecy capacity of the LiFi MISO wiretap channel is lower bounded as CMISO

s,1 ≥CMISOs,1 , where this

achievable lower bound is given by

CMISOs,1 = max

w∈[−1,1]K1

2log+

(6(hT1 w)2A2 + 12πeσ2

πe(hT2 w)2A2 + 12πeσ2

). (4.2)

Proof. We start by setting U=X in (2.6) to obtain CMISOs,1 ≥maxp(x) I(X;Y1)− I(X;Y2). Then we

choose X=wS + d1, d= A2 , and S to be uniformly distributed on [−A2 ,A2 ], to obtain CMISO

s,1 ≥maxw I(wS;Y1)− I(wS;Y2) =maxw I(S;Y1)− I(S;Y2). Then we use the lower bound C.

SISOs,1 in

Theorem 3.2 to bound I(S;Y1)− I(S;Y2) which yields CMISOs,1 ≥CMISO

s,1 as defined in (4.2). Thiscompletes the proof.

The optimization problem in Theorem 4.1 is non-convex. However, it can be transformed intoa quasiconvex line search problem and solved using bisection as described in [21].

Note thatCMISOs,1 can be achieved if a peak constraintXi ∈ [0,A] is imposed without an average

constraint, or with an average constraint E{Xi}= A2 since this is automatically satisfied bychoosing S to follow a zero-mean uniform distribution. If an inequality average constraint isimposed such that E{Xi} ≤ E ≤ A2 , or an equality average constraint E{Xi}= E ≤ A2 , the samescheme of Theorem 4.1 can also be applied, as shown in the following corollary.

Corollary 4.1. Under the constraints Xi ∈ [0,A] and either E[Xi]≤E or E[Xi] = E , where E ≤ A2 , thesecrecy capacity of the LiFi MISO wiretap channel is lower bounded as CMISO

s,2 ≥CMISOs,2 , where

CMISOs,2 = max

w∈[−1,1]K1

2log+

(6(hT1 w)2E2 + 3πeσ2

πe(hT2 w)2E2 + 3πeσ2

)(4.3)

The proof is similar to Theorem 4.1, with A replaced with 2E . This forces E[Xi] to be equal toE . Note that one may also replaceA in the proof of Theorem 4.1 by d≤E if the average constraintis E[Xi]≤E , and then maximize with respect to d. However, the maximum can be shown to beachieved when d= E .

The advantage of the scheme used in Corollary 4.1 is that we can still choose w ∈ [−1, 1]K ,which provides flexibility in beamforming. The disadvantage is that we limit the range of Xi to[0, 2E ] which is a subset of [0,A]. Another way to develop an achievable secrecy rate for the caseE[Xi]≤E is by choosing S to be distributed according to a truncated-exponential distribution(similar to [9, (42)]) between [−E ,A− E ] with zero mean. However, since we construct thetransmit signal as X=wS + E1, this imposes the restriction that w ∈ [ α

α−1 , 1]K where α= EA ,

since otherwise, Xi may become negative. Despite this disadvantage, the advantage is that Xi

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can occupy the whole range [0,A] whenwi (the corresponding component of w) is positive. It stilloccupies a portion of [0,A] when wi is negative. We present a new lower bound which appliesthis scheme next.

Theorem 4.2 (Beamforming lower bound). Under the constraints Xi ∈ [0,A] and either E[Xi]≤Eor E[Xi] = E , where E = αA and α< 1

2 , the secrecy capacity of the LiFi MISO wiretap channel is lowerbounded as CMISO

s,1 ≥C.MISOs,2 , where

C.MISOs,2 (α) = max

w∈[ αα−1 ,1]

K

1

2log+

(ρ(hT1 w)2A2 + 2πeσ2

2πeφ(hT2 w)2A2 + 2πeσ2

), (4.4)

where ρ= e2(1−µα

eµα−1 )(1−e−µαµα

)2, φ= 1

µ2α+ 1

4 − ( 1eµα−1 + 1

2 ), and µα is defined in Lemma. 2.2.

Proof. The proof is similar to the proof of Theorem 4.1 except for the last inequality which isobtained by using the bounds in Theorem 3.5. In this case, Alice uses a truncated-exponentialdistribution for S ∈ [−E ,A− E ] with zero mean (a shifted version of [9, (42)]), and sends X=

wS + E1 where w ∈ [ αα−1 , 1]

K .

The optimization problem in Theorem 4.2 can be solved using the same algorithm asin Theorem 4.1 by replacing the constraint w ∈ [−1, 1]K in Theorem 4.1 by two constraints,maxk wk ≤ 1 and mink wk ≥ α

α−1 .Finally, the achievable secrecy rate using this scheme under α≥ 1

2 can be obtained by replacingα and E by α′ = 1− α and E ′ =A− E , respectively, in Corollary 4.1 and Theorem 4.2.

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

(b) LiFi MISO Secrecy by Friendly JammingSuppose the room is illuminated by K identical light fixtures. Let K =KJ +KA. Alice uses KAlight fixtures to transmit a message. On the other hand, a friendly jammer uses the remainingKJ light fixtures to support Alice by sending noise. When Eve is closer to Alice than Bob, securecommunication using the beamforming method described in the previous subsection becomesless promising. To worsen the channel quality from Alice to Eve and maintain a good channelquality from Alice to Bob, the jammer sends a jamming signal which is friendly to Bob but hostileto Eve.

While KA can be generally any number between 1 and K, in [18,20], KA is chosen to be 1. Westick to this case in the current description. In this case, the input-output relation under friendlyjamming can be expressed as follows

Yr = hrX + hTJrXJ + Zr, 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, hJ1 and hJ2 are the channels from the jammer to Bob and Eve, respectively, and XJis a jamming noise signal.

The design of friendly jamming can be achieved as follows. First, the jamming signal isbeamformed in a way that maximizes the jamming effect at Eve, so that XJ =wJSJ, where

wJ = [wJ,k]KJk=1 is the jammer’s beamforming vector, and SJ is the jamming noise signal. Second,

the jamming signal is made friendly to Bob by choosing wJ in a way that decreases the impact ofSJ at Bob, i.e., ideally hTJ1wJ = 0.

In [18], an achievable secrecy rate was also developed for this jamming scheme, where Xand SJ were chosen to be uniformly distributed. An improved secrecy rate was derived in

[20], by using truncated-Gaussian distributions instead. Define φ(x) = 1√2πe−

x2

2 as the standard

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Gaussian distribution, and Φ(x) to be the corresponding cumulative distribution. We define atruncated-Gaussian distribution pA,σt(x) as follows

pA,σt(x) =

1σtφ(xσt

)Φ(β)−Φ(−β) , x∈ [−A2 ,

A2 ],

0, |x|> A2 .(4.6)

where β = A2σt

. Also, define ζ =Φ (β)− Φ (−β). Using this distribution, we can derive thefollowing lower bound on secrecy capacity of the LiFi MISO wiretap channel.

Theorem 4.3 (Jamming lower bound [20]). Under the constraints Xi ∈ [0,A] and E[Xi] = A2 , thesecrecy capacity of the LiFi MISO wiretap channel is lower bounded by CMISO

s,1 ≥C.MISOs,1 where

C.MISOs,1 ,

1

2log

(1 +

σ2t h21e

2η

σ2

)− h(V0) +

1

2log(2πeσ2t

)+ max

hTJ1w=0

w∈[−1,1]K−1

log(|hTJ2wJ|

)+ η, (4.7)

where V0 = h2Xt + hTJ2wJSt, Xt and St follow a truncated-Gaussian distribution fσt(x), and η=

log (ζ) +−βφ(−β)−βφ(β)

2ζ .

The optimal wJ for maximizing C.MISOs,1 can be found by solving the following optimization

problem [20]

w∗J = arg maxwJ∈[−1,1]K−1

hTJ2wJ

s.t. hTJ1wJ = 0, (4.8)

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

Extension to the inequality or equality average constraint, i.e., E[Xi]≤E or E[Xi] = E , whereE ≤ A2 is not considered in this paper, but is a relevant research direction. Now we present somenumerical evaluations which evaluate the presented bounds.

(c) Numerical ResultsIn the following numerical evaluations, we consider a room of size 6× 6× 3 m3 which isilluminated by 4 lighting fixtures located at the corners of a 3× 3m square centered at theceiling of the room, as shown in Fig. 5a. Each of the fixtures consists of an array of LEDs, andassume identical PDs for Bob and Eve. The channel gains between the light fixtures and PDs, i.e.,h1, h2, hJ1, hJ2, can be estimated using the following model [21, (6a)] [37]:

h=(m+ 1)gA

2πd2(cosφ)2 cosψ I(ψ <Ψc), (4.9)

where m= − ln 2ln(cosΦh)

is the Lambertian emission order, Φh is the half-intensity semi-angle of thelight 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, Ψc is thefield of view (FOV) of the PD, and I(·) is a logic function which returns 1 if its argument is true,and zero otherwise. In the simulations, we assume that gA= 10, Φh = 60◦, Ψc = 70◦, and we alsoassume that the PD is always placed upright, so that φ=ψ.

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

In Fig. 5b, we compare the achieved secrecy rate by beamforming CMISOs,1 in (4.2) and by

friendly jamming C.MISOs,1 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

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-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

x

y

(a) The room layout, where the dots represent the

light fixture, the dashed circles identify the half-

intensity regions.

0 10 20 30 400

1

2

3

4

SNR(dB)

Secr

ecy

Rat

e(b

it/s

/Hz)

CMISOs,1 (Thm. 4.1): (p1, p0)

CMISOs,2 (Thm. 4.3): (p1, p0)

CMISOs,1 (Thm. 4.1): (p2, p0)

CMISOs,2 (Thm. 4.3): (p2, p0)

(b) Secrecy rate as a function of SNR.

Figure 5. Room layout used in the simulations and secrecy capacity bounds.

use σt = 9 in (4.2) [20]. It can be seen that when Bob is at p1, the beamforming scheme alwaysoutperforms the friendly jamming scheme. However, when Bob is at p2, friendly jamming canachieve higher secrecy rates when SNR is lower than 20dB. This suggests that a scheme whichcombines beamforming and friendly jamming can achieve a better performance. Developing sucha scheme is a relevant research direction.

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

In Fig. 7, we compare CMISOs,2 (α) and C.

MISOs,2 (α) for SNR between 0 and 40dB, where the

locations of Bob and Eve are fixed at p1 and p0, respectively. We choose α to be 0.1 or 0.3. Itcan be seen that, for α= 0.1, although C. s,2(α) uses a larger signal range, it is a little better thanCs,2(α) only when SNR≤ 30dB and becomes much worse than Cs,2(α) when SNR= 40dB, butfor α= 0.3, C. s,2(α) outperforms Cs,2(α) for all the tested SNR region.

In Fig. 8, we compare the beamforming lower boundsCMISOs,2 (α) in Corollary 4.1 and C.

MISOs,2 (α)

in Theorem 4.2, under an average constraint E[Xi] = αA with α= 0.3. In this evaluation, Eve’slocation is fixed at p0 but Bob can move around the room. It can be seen that because of a largersignal range (range of S), C.

MISOs,2 (α) outperforms CMISO

s,2 (α) in general.

5. Conclusion and Future Research DirectionsIn this paper, we reviewed theoretical results on LiFi PLS focusing on the LiFi SISO and MISOwiretap channel models and their secrecy capacity bounds. We considered systems with a peakconstraint only and also systems with both peak and average constraints. As LiFi should notdisturb illumination levels, an equality average constraint has been discussed, and new bounds orextensions of existing bounds are developed for this case. We studied beamforming and jamming

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(a) Beamforming. (b) Friendly Jamming.

Figure 6. Secrecy rate as a function of Bob’s location when Eve is fixed at position p0 (room center).

0 10 20 30 400

1

2

3

α= 0.1

α= 0.3

SNR (dB)

Secr

ecy

Rat

e(b

it/s

/Hz)

CMISOs,2 (α)

C.MISOs,2 (α)

Figure 7. Secrecy rate constraint E[Xi] = αA as a function of SNR, where Bob and Eve are locatedat p1 and p0, respectively.

schemes, and evaluated their performance numerically. We have also derived new achievablesecrecy rates under beamforming, which improve upon existing results. Existing results and newresults 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, signallingschemes, 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 torealize secrecy, but they only consider the peak constraint and average total power constraint,i.e., Xk ≤A and

∑Kk=1 E[Xk]≤E . In a LiFi system, each light fixture should provide a constant

average light intensity, and thus X has to be constrained by E[Xk] = Ek, k= 1, . . . ,K. In thespecial case of LiFi MISO, the current studies adopt beamforming and friendly jamming. As wehave seen above, if the equality average constraint is imposed, the transmitted signal may not beable to vary over the whole permissible range [0,A] – the linear range of the LED. For example,in Theorem 4.2, if wk < 0, then Xk varies over a subset of [0, αA1−α ] (α∈ [0, 0.5]), which is itselfa subset of the whole range [0,A]. As seen in Fig. 7, a larger emitting range can increase the

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(a) CMISOs,2 (α) (b) C.

MISOs,2 (α)

Figure 8. Secrecy rate under constraint E[Xi] = 0.3A as a function of Bob’s location, where Eve isfixed at p0 and SNR= 20dB.

secrecy performance. Therefore, it is important to rethink the current design to make full use ofthe emitting range of the LED. Moreover, in such a design, Ek does not have to be the same for allk. For instance, in some large indoor scenarios, not all light have to be turned on or provide thesame level of illumination. For example, light fixtures in a large library may only need to turn onwhen people walk in the nearby region, leading to non-equal values of Ek.

Another promising direction is designing signaling schemes for LiFi PLS. As shown in [26],the optimal input distribution for the LiFi SISO wiretap channel studied in this paper is discretewith a finite number of mass points. But the optimal distribution is unknown, and even iffound numerically, it is not clear how to design the transmitted signal to match the theoreticallyoptimal input distribution. On the other hand, existing modulation schemes produce a uniformdistribution (realized by PPM and PAM) and truncated Gaussian distribution (realized by varioustypes of unipolar OFDM-based schemes [38]). It is important to develop modulation schemes thatproduce a truncated-exponential distribution, which has superior performance compared with auniform distribution as shown in Fig. 7. The work in [39] proposes a method for generating anon-uniform discrete distribution of the SISO LiFi channel without an eavesdropper, which mightalso be useful for the wiretap channel. This research direction is relevant and requires additionalinvestigation.

It is also important to note that LiFi is vulnerable to physical blockage, and RF accesspoints (such as WiFi) can be used to support LiFi connection in such cases. Hybrid VLC/RFcommunication has been demonstrated in the literature [40]. The current studies on the secrecyof hybrid VLC/RF systems mostly use zero-forcing beamforming and try to minimize the systempower consumption [41–43]. These works consider VLC and RF as independent componentswhich perform secure communication independently without cooperation, and they do notconsider an specific lighting requirement in the form of an average constraint E[Xk] = Ek.This calls upon designing and studying more cooperative PLS schemes for hybrid LiFi/RF orLiFi/WiFi systems.

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Physical Layer Security in LiFi Systemslampe/Preprints/2019-PhySec-VLC... · 2019-10-24 · Physical Layer Security in LiFi Systems Zhenyu Zhang1, Anas Chaaban1 and Lutz Lampe2 1School

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