I Introduction
With the widespread use of lightemitting diodes (LEDs) for commercial lighting applications, visible light communication (VLC) has attracted increasing attention in recent years. Due to the combination of communication and illumination, VLC is regarded as one of the most important wireless communication technologies for future indoor access [1].
In the last decade, the pointtopoint (P2P) VLC has achieved rapid development in many fields, especially in channel modelling [2], modulation [3], coding [4], equalization [5]
, channel estimation
[6], indoor positioning [7], channel capacity analysis [8, 9, 10, 11] and transceiver design [12]. At present, the research focus of VLC is being changed from P2P communications to network aspects. In VLC networks, data privacy is becoming a main concern for users. Although it is propagated via the lineofsight path, the VLC signal is broadcasted to all users illuminated by the LEDs. Such a broadcast feature provides convenience for data transmission, but it also offers an opportunity for unintended users to eavesdrop the information, which imposes a security risk to legitimate users. Therefore, information security becomes an urgent issue to be addressed. Traditional security schemes are performed at upperlayers of the network stack by using access control, password protection and endtoend encryption [13]. The safety of traditional security schemes is built on the limited storage capacity and computational power of the eavesdroppers. Recently, physicallayer (PHY) security, which exploits the channel characteristics to hide information from eavesdroppers and does not rely on the upperlayer encryption, has been proposed as an efficient supplement to traditional security schemes.Secure transmission is important for radio frequency wireless communications (RFWC). The PHY security was first investigated in 1949 by Shannon, who proposed the concept of perfect secrecy over noiseless channels [14]. Under the noisy channels, Wyner analyzed the secrecy capacity via the wiretap channel [15]. In [16], the secrecy capacity of the singleinput singleoutput (SISO) Gaussian wiretap channel was derived. Under the nondegraded wiretap channel, a singleletter characterization of the secrecy capacity was derived in [17]. Recently, the secrecy performance analysis over the SISO scenario was extended to that over the multiinput multioutput (MIMO) scenario. For MIMO wiretap channels with confidential messages, the authors in [18] analyzed the secrecy capacity region. For artificial noisy MIMO channels, the secrecy capacity was studied in [19]
by using the ordered eigenvalues of Wishart matrices. The authors in
[20] obtained the secrecy capacity for MIMO channels with finite memory. With the help of a cooperative jammer, a lower bound of the secrecy capacity for the MIMO channels was derived in [21].Although much work has been done to investigate the secrecy capacity for RFWC, the developed theory is not directly applicable to VLC. The main differences between RFWC and VLC are highlighted as follows. First, the transmit signal in RFWC can be bipolar or unipolar, while the signal in VLC must be unipolar because the optical intensity is typically used to carry information. Moreover, the average power in RFWC is the mean square value of a signal, but the average power in VLC is the mean value of the signal [22]. Also, a lower average power is usually preferred for RFWC, but VLC has a predefined average intensity according to the dimming target, which is not an objective function but a constraint [22]. Therefore, the aforementioned features should be considered for practical VLC. In [23], the secrecy capacity was analyzed for direct current biased VLC, where a uniform input distribution is used to derive the lower bound of secrecy capacity. In [24], the secrecy capacity of multipleinput singleoutput (MISO) VLC channels was investigated, where the input distribution was chosen as a truncated generalized normal (TGN) distribution. To obtain the optimal and robust beamforming, the authors in [25] also employed the TGN distribution for the input. Due to the constraints of the input signal in VLC, the uniform and TGN input distributions are generally not optimal [8, 10]. By using the variational method, an improved input distribution can be obtained [8, 10]. By employing a discrete input distribution, the authors in [26] derived an upper bound on the secrecy capacity for the MISO VLC channels. However, the theoretical expression of the secrecy capacity is not obtained. In [27]
, the secrecy outage probability (SOP) was analyzed for a hybrid VLCRFWC system with energy harvesting. In
[28], the SOP and the average secrecy capacity were discussed for VLC with spatially random terminals. Note that the dimming requirement for indoor VLC was not considered in [27, 28]. In our previous work [29], three lower bounds on the secrecy capacity were obtained. However, no upper bound has been obtained. Moreover, the peak optical intensity constraint is not considered. To the best of the authors’ knowledge, the secrecy capacity for VLC has not been systematically investigated.In this paper, the secrecy capacity for an indoor VLC system with a transmitter, a legitimate receiver, and an eavesdropper is investigated. The main contributions are summarized as follows:

The secrecy capacity for indoor VLC with only an average optical intensity constraint is analyzed. By using the existing channel capacity results, the entropypower inequality (EPI) and the variational method, two lower bounds on secrecy capacity are obtained. Applying the dual expression of the secrecy capacity, the upper bound on the secrecy capacity is obtained. Numerical results validate the derived closedform expressions.

The secrecy capacity for indoor VLC with both average and peak optical intensity constraints is investigated. In practical VLC, the peak optical intensity of the LED is also limited. By adding a peak optical intensity constraint on the input signal, the lower and upper bounds on the secrecy capacity are further derived, which are in closed forms. The accuracy of the derived closedform expressions are confirmed by numerical results.

The asymptotic behaviors at high signaltonoise ratio (SNR) are analyzed. Through theoretical analysis, it is shown that the asymptotic lower and upper bounds do not coincide but with a small gap when only considering the average optical intensity constraint. When considering both average and peak optical intensity constraints, the asymptotic lower and upper bounds coincide, and thus the secrecy capacity can be obtained precisely.
The reminder of this paper is organized as follows. Section II describes the system model. With different constraints, the secrecy capacity bounds and the asymptotic behavior are analyzed in Section III and Section IV, respectively. Numerical results are presented in Section V. Conclusions and future directions are presented in Section VI.
Notations: Throughout this paper, italicized symbols denote scalar values;
stands for a Gaussian distribution with zero mean and variance
; denotes the expectation operator; denotes the variance of a variable;denotes the probability density function (PDF) of
. We use for the entropy, for the relative entropy, and for the mutual information. We use for the natural logarithm and for the Gaussian Qfunction.Ii System Model
As shown in Fig. 1, we consider an indoor VLC network consisting of a transmitter (i.e., Alice), a legitimate receiver (i.e., Bob), and an eavesdropper (i.e., Eve). Alice is deployed on the ceiling, while Bob and Eve are placed on the floor. When Alice transmits data bits to Bob, Eve as a passive eavesdropper can also receive the signals intended for Bob. In the network, Alice is equipped with a single LED to transmit optical intensity signals, while Bob and Eve are equipped with one photodiode (PD) individually to perform the opticaltoelectrical conversions. The received signals at Bob and Eve can be expressed, respectively, as
(1) 
where is the transmit optical intensity signal; and denote the channel gains of the main channel and the eavesdropping channel, respectively; and stand for the additive white Gaussian noises at Bob and Eve, where and denote the variances of the noises at Bob and Eve, respectively.
Because the intensity modulation and direct detection is employed for VLC, is restricted to be nonnegative such that
(2) 
In practical VLC systems, the peak optical intensity of the LED is also limited. Therefore, the peak optical intensity constraint is given by
(3) 
where is the peak optical intensity of the LED.
For a practical LED, its average optical intensity is constrained by the nominal optical intensity. In order to satisfy the illumination requirements in VLC, the average optical intensity cannot change with time. Mathematically, the average optical intensity constraint is given by
(4) 
where is the dimming target, is the nominal optical intensity of the LED.
In indoor VLC, the channel gain () can be expressed as [30]
(5) 
where is the order of the Lambertian emission; is the physical area of the PD; and are the optical filter gain and the concentrator gain of the PD. is the field of view (FOV) of the PD; , and are respectively the distance, the irradiance angle and the incidence angle from Alice to Bob () or Eve (), as shown in Fig. 2. Obviously, when the positions of Alice, Bob and Eve are fixed, the channel gains and are constants.
Iii Secrecy Capacity for VLC with Only an Average Optical Intensity Constraint
In this section, we focus on the VLC with only an average optical intensity constraint. Therefore, we only consider the constraints (2) and (4). If the main channel is worse than the eavesdropping channel (i.e., ), then the main channel is stochastically degraded with respect to the eavesdropping channel, and the secrecy capacity is essentially zero. Alternatively, if , the secrecy capacity^{1}^{1}1In this paper, the natural logarithms are employed, and thus the secrecy capacity is in nats/transmission. can be expressed as [23, 31]
(6)  
where denotes the secrecy capacity, and denotes the PDF of the input signal. In this section, the lower and upper bounds of the secrecy capacity will be derived. Moreover, the asymptotic behavior of the secrecy capacity is also analyzed.
Iiia Lower Bound on Secrecy Capacity
By using the channel capacity bounds derived in [32], a lower bound on the secrecy capacity is derived as (7) in the following theorem. Moreover, by using the EPI and the variational method, another lower bound on the secrecy capacity can be derived as (8) in the following theorem.
Theorem 1
The secrecy capacity for the channel in (1) with constraints (2) and (4) is lowerbounded by each of the following two bounds (7) and (8), i.e.,
(7)  
and
(8) 
where and in (7) are free parameters. Suboptimal but useful choices for and are given, respectively, by (9) and (10) as shown at the top of the next page.
(9) 
(10) 
Remark 1
When , it is obvious that the lower bound (8) is a monotonically increasing function with respect to the dimming target . In this case, the performance of (8) can be improved by increasing the value of . Moreover, it is challenging to analyze the monotonicity of the lower bound (7) with respect to , which will be given by using a numerical method in Section V.
IiiB Upper Bound on Secrecy Capacity
To obtain the upper bound on the secrecy capacity, the dual expression of the secrecy capacity is employed [33]. For an arbitrary conditional PDF , the following equality always holds [34]
(11) 
According to the nonnegativity property of the relative entropy, we have
(12) 
Note that selecting any will result in an upper bound of . Therefore, we have
(13) 
Note that, to achieve the secrecy capacity, there exists a unique input PDF that maximizes subject to the constraints in (6). Therefore, we have
(14)  
where denotes the optimal input, and its corresponding PDF is .
According to (13) and (14), we have
(15) 
It can be seen from (15) that selecting any will lead to an upper bound of the secrecy capacity. To obtain a good upper bound, a clever choice of should be found. By using the principle of dual expression of the secrecy capacity, an upper bound of the secrecy capacity in (6) is derived in the following theorem.
Theorem 2
See Appendix C.
Remark 2
IiiC Asymptotic Behavior Analysis
In indoor VLC environment, typical illumination requirement leads to a large transmit optical intensity, which can offer a high SNR at the receiver [35]. Therefore, we are more interested in the behavior of the VLC system in the high SNR regime. By analyzing Theorem 1 and Theorem 2, the asymptotic behavior of the secrecy capacity bounds at high SNR is derived in the following corollary.
Corollary 1
See Appendix D.
Remark 3
From Corollary 1, it can be found that the asymptotic upper and lower bounds on secrecy capacity do not coincide in the sense that their difference equals nats/transmission instead of zero. Although a performance gap between the asymptotic upper and lower bounds exists, the difference is so small and thus it can be ignored.
Iv Secrecy Capacity for VLC with Both Average and Peak Optical Intensity Constraints
As is well known, the peak optical intensity of an LED is also limited. In this section, we consider the constraints (2), (3) and (4). Similarly, if , the secrecy capacity is zero. If , the secrecy capacity can be expressed as [23, 31]
(18)  
In this section, the bounds of the secrecy capacity and their asymptotic behaviors for this case will be derived.
Iva Lower Bound on Secrecy Capacity
Let denote the average to peak optical intensity ratio (APOIR). Two lower bounds on the secrecy capacity can be derived by analyzing (18), and these lower bounds are shown in the following theorem.
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