Privacy-Aware Laser Wireless Power Transfer for Aerial Multi-Access Edge Computing: A Colonel Blotto Game Approach

This article studies the integration of laser-beamed wireless power transfer (WPT) into high-altitude platform (HAP)-aided multiaccess edge computing (MEC) systems for the HAP-connected aerial user equipments (AUEs). By discretizing the 3-D coverage space of the HAP, we present a multitier tile grid-based spatial structure to provide the aerial locations in the form of tile grids to AUEs for laser charging. We identify a new privacy vulnerability caused by the openness during the WPT signaling transfer in the presence of a terrestrial adversary, which is able to launch the attacks by distributing the false tile grids to the AUEs. To address this vulnerability and enhance the location privacy of AUEs, we then propose a Colonel Blotto (CB) game framework to formulate the competitive tile grid allocation problem for the HAP and the adversary. The attack-defense interaction between the adversary and the HAP as a defender in their tile grid allocations to the AUEs is formulated as a CB game, which models the competition of two players for limited resources over multiple battlefields. Moreover, we derive the mixed-strategy Nash equilibria of the game for both symmetric and asymmetric tile grids between two players. Simulation results show that the proposed framework significantly outperforms the design baselines with a given privacy protection level in terms of system-wide expected total utilities.


I. INTRODUCTION
W ITH the recent advancements in the fifth generation (5G) enabling technologies and the Internet of Things (IoT), the phenomenal growth of smart user equipment (UE) has been driving the rapid development of numerous attractive mobile applications and multimedia services, such as extended reality, holographic telepresence, autonomous driving, etc. In addition to ground UEs, unmanned aerial vehicles (UAVs) can act as a new type of aerial UEs (AUEs), which are also referred to as the cellular-connected drone users [1], [2]. The difference of AUEs with respect to ground UEs is their ability to intelligently fly in the 3-D space and flexibly optimize their trajectories to complete their missions.
Owing to their inherent properties, the use of AUEs has raised a great upsurge of interest for both civilian and commercial applications, e.g., aerial surveillance, flying cars, disaster response, etc. [3]- [5]. Among them, most of the applications are not only computation intensive but also latency sensitive. For instance, a huge amount of data collected by the AUEs in the missions of disaster sensing generally consumes extensive computation resources [6]. However, the constrained computing capabilities of AUEs are usually difficult to fulfill the computation requirements of these applications. To tackle this issue, multi-access edge computing (MEC) is envisioned as a promising approach to provide the cloud-computing capability in close proximity to the users, by enabling them to offload the computation tasks to the network edges for execution [7].
To provide the edge computing for AUEs, a general solution is to integrate MEC servers at the edge of terrestrial networks, e.g., gNBs in 5G systems. However, it may be impractical and uneconomical to deploy such an infrastructure mounted with MEC servers for the AUEs when they are distributed above the challenging environments, e.g., disaster scenarios and ocean areas. An alternative is to extend the computation resources to the air for providing flexible computing services for AUEs. Compared to the UAVs in lower height above the Earth, high-altitude platforms (HAPs) in the stratosphere are more suitable to offer elastic computing services due to larger area coverage, bigger payload capacity, and longer endurance [8]. To this end, the integration of the HAPs with MEC servers motivates the prospect of implementing the aerial computation offloading for the HAP-connected AUEs.
Despite the appealing potential of HAP-aided MEC, computation offloading may be still interrupted due to the constrained energy storage (e.g., onboard lithium-ion polymer battery) of AUEs [9]. A promising solution is to apply the wireless power transfer (WPT) to provide a convenient and cost-effective energy supply to AUEs for power replenishment. Compared with near-field WPT via inductive coupling or magnetic resonance, far-field WPT has emerged as a prospective solution to the sustainability of future generation networks, due to the evident benefits, such as long charging ranges, highly mobility support, small form factors of the receiver, etc. [10]. Currently, laser-beamed far-field WPT is becoming a viable solution to provide a controllable, sustainable, and longer range power supply for extending the lifetime of UAVs [11], [12]. Unlike the far-field WPT enabled by radio frequency (RF), laser WPT delivers more energy to the users with higher energy concentration over long distance [11]. The JAXA has demonstrated that an orbiting satellite with a solar condenser quickly distributes kW-class laser energy to ground users [13]. Therefore, we, in this article, employ the laser WPT for HAP-aided MEC to develop a novel paradigm of laser-powered aerial MEC system, wherein a laser transmitter and an MEC server are both mounted at the HAP. The HAP transmits laser energy to charge the AUEs from the sky, and the AUEs utilize the harvested energy to support the flights and complete the computation offloading.
To obtain an effective aerial location for laser charging, the AUE is required to delivery a WPT signaling message to the HAP on a control channel via uplink. However, the WPT signaling transfer in an open radio environment makes the wireless communications vulnerable to security threats from adversaries [14]. Being a smart and programmable radio device, an external adversary is able to capture the wireless communications, and analyze the intercepted signaling messages from the AUEs to track their locations and learn about the contents of messages. With these actions, the adversary aims to actively launch the attacks of charging location data falsification (CLDF) by sending the false charging location data (e.g., 3-D location of charging points) to AUEs. Essentially, the privacy of the AUEs' charging locations is compromised by observing the signaling transfer at the adversary. Therefore, enhanced security and privacy mechanisms are of paramount importance to the WPT since a minor compromise may result in a major security problem, e.g., the leakage of aerial charging location information of the AUEs. This, however, may allow the adversary to stealthily infer and track the aerial charging location of each AUE to launch the CLDF attacks for laser WPT. Such attacks may further lead to the unsuccessful laser WPT and the failure of aerial computation offloading.
From the perspective of location privacy enhancement, while the adversary tries to allocate its attack resources regarding false charging location data over the AUEs, the HAP will act as a defender that seeks to optimally allocate its defense resources in terms of legitimate charging location data to the AUEs. The objective of the HAP is to prevent the adversary from causing the unsuccessful laser WPT and the failure of the aerial MEC. Such an attack-defense resource allocation motivates us to gain more insights into the competitive interaction between the adversary and the defender in their simultaneous distributions for the finite amount of charging location data across the AUEs. Aiming to address the above privacy concerns, there are two key bottlenecks that must be overcome: 1) how to characterize the 3-D location of charging point in 3-D coverage space of the HAP as a kind of resource assigned to the AUEs and 2) how to formulate the attackdefense resource allocation problem between the adversary and the defender in their allocations of 3-D location of charging points across the AUEs. Admittedly, these bottlenecks and challenges motivate the need for a better understanding of the interplay between the spatial structure design and the attackdefense resource allocation in the laser-powered aerial MEC systems.
Motivated by the above observations, we can find that the exploration of privacy-aware laser WPT for HAP-aided aerial MEC systems has become highly valuable. Our objective in this article is to achieve enhanced location privacy for AUEs by identifying such an interplay in the laser-powered aerial MEC. For bridging the research gap, in this article, we investigate a privacy-aware laser WPT problem via the Colonel Blotto (CB) game framework for laser-powered aerial MEC systems. The main contributions of this article can be summarized as follows.
1) We propose a multitier tile grid-based spatial structure for charging the AUEs to discretize the 3-D coverage space of the HAP in the laser-powered aerial MEC system. This is a new approach toward the 3-D spatial view of charging location design for AUEs by exploiting the equally sized square tile grids within a finite number of tiers. 2) We identify a new privacy vulnerability due to the openness during the WPT signaling transfer in the presence of a ground adversary, who can launch the CLDF attacks by allocating the false tile grids to the AUEs. To defend against the CLDF attacks, we propose a CB game framework to formulate the competitive tile grid allocation problem for the HAP as a defender and the adversary. The competitive interaction between the adversary and the defender in their tile grid allocations is modeled as the competition of two players in the game for limited resources over a finite set of battlefields. 3) We design the utility function that each player receives on an individual battlefield by balancing the local computation capacity, energy consumption, and execution delay of the AUE. The mixed-strategy Nash equilibrium (NE) solutions of the game are analytically derived for both symmetric and asymmetric tile grids between the defender and the adversary. In addition, we also obtain the mixed-strategy NE solution to the game under the privacy protection level constraint. The mixed-strategy NE points are shown to evaluate how the system performance tradeoff for the laser-powered aerial MEC systems impacts the expected total utilities of the defender. The remainder of this article is organized as follows. In Section II, we provide a review of related works. Section III describes the system model. In Section IV, we formulate the CB game between the defender and the adversary, and also design the utility function. Section V discusses the solution of the game. Simulation results are presented in Section VI. Finally, Section VII concludes this article.

A. Far-Field WPT in MEC Systems
Recently, the integration of far-field WPT into MEC systems has received considerable attention, due to the enhanced system performance by jointly optimizing computation offloading and resource allocation. Feng et al. [15] investigated the reward maximization problem to simultaneously maximize the data utility and minimize the energy consumption from the operator's perspective, by jointly considering the power allocation at base station (BS) as well as the offloaded data size and power allocation at UEs. Mao et al. [16] explored the stochastic optimization problem focusing on the tradeoff between time-average energy efficiency (EE) and delay for the multiple access-based wireless powered MEC systems, by optimizing the network EE under the energy causality and network stability constraints. Liu et al. [17] developed the optimization framework to minimize the total required energy of the MEC server-mounted UAV, via jointly optimizing the computation bits, CPU frequencies, UE's transmit power, and UAV's trajectory. However, these works mainly focused on the use of RF-based WPT at the terrestrial BS [15], [16] and the aerial infrastructure [17] to prolong the lifetime of energy-limited ground UEs.

B. Laser WPT in Network Scenarios
Several recent works have been devoted to applying the laser WPT in network scenarios for energy-demanding devices, e.g., the UAVs [9], [11], [12], [18], [19] and the IoT devices [20]. Based on the downlink communication scenarios for the laser-charged quadrotor UAV, Jaafar and Yanikomeroglu [9] identified the relationship between the motion regimes, energy consumption/harvesting, and battery dynamics of the UAV. The sum downlink throughput maximization problem was presented in [11] by jointly optimizing the UAV's trajectory, velocity, acceleration, and transmit power in laser-powered UAV networks. Lahmeri et al. [18] analyzed the performance of simultaneous information and power transfer from the laser transmitter to the laser-powered UAV via power splitting and also derived the joint energy and signal-to-noise ratio coverage probability via stochastic geometry. Zhao et al. [12] developed the general optimization framework for joint power allocation and trajectory design in the UAV-enabled mobile relaying system with laser charging. Hassan et al. [19] employed the laser-driven adaptive WPT for UAV-mounted relay-assisted IoT networks and formulated the optimization problem to maximize the number of connected IoT devices for uplink data transmission. Compared with the studies applying laser WPT for the UAVs, Zhang et al. [20] proposed to employ the distributed laser charging to provide the IoT devices with safe WPT capability with the aid of the photovoltaic cells instead of the collecting lens at the receiver. However, the above studies exploited the laser WPT only to the scenarios of UAV communications and IoT systems and did not capture the effect of laser WPT on our considered HAP-aided MEC systems.

C. Security and Privacy in WPT
Despite the fact that the WPT-enabled techniques in energyconstrained wireless networks have been intensively studied in the literature, only a very few works consider the security and privacy issues related to the WPT itself. Liu et al. [14] summarized the security attacks arising from the lack of security mechanisms in WPT-enabled networks, and described the potential countermeasures to mitigate the attack effects. Jiang et al. [21] devised the two-plane-based secure WPT architecture via the blockchain, and proposed the efficient WPT mechanism by combining the contract theory and the delegated proof-of-stake-based lightweight consensus scheme, respectively. By using the directed acyclic graph and consortium blockchain, Jiang et al. [22] developed the distributed secure UAV-aided WPT framework known as the aerial-ground chain, to protect the UAVs as aerial energy transmitters against the energy attacks. Zhang et al. [23] used the chaotic energy encryption method to propose a secure inductive WPT system. The energy of the power source was encrypted based on the specific chaotic map, and the decryption was conducted via the synchronized chaotic map. The above studies are heuristic, although they only investigated the impact of security attacks on the RF-based WPT process for ground UEs via enhancing the security mechanisms. In contrast, we extensively consider the privacy issues related to the laser WPT for AUEs from the sky in the laser-powered aerial MEC systems.

D. Game Theoretical Solution for Security and Privacy
Recently, efforts have been made to deal with security and privacy problems in wireless networks using game theory. To achieve the practical location privacy in vehicular ad hoc networks, Lu et al. [24] proposed the pseudonym changing at social spots (PCS) strategy, and used the game theory to show the feasibility of the PCS strategy by maximizing the vehicle's location privacy gain. Chen et al. [25] developed a secure content-sharing system via the double-layer blockchain in vehicular named data networks, and adopted the one-tomany matching game to model the balance between the client's demands and the server's supply. Tang et al. [26] built a hierarchical structure with the eavesdropper as a leader and the legitimate user as a follower via Stackelberg game, and obtain the optimal strategy for both the legitimate transmissions and jamming attacks. Despite the research efforts devoted to security and privacy issues in wireless networks via game theory, the works in [24]- [26] cannot deal with the attack-defense resource allocation for competitive interaction between the defender and the attacker, i.e., two competitive players. Although the Stackelberg game model in [26] can be shown as a two-player strategic game, i.e., the attacker as a leader (or a follower) and the defender as a follower (or a leader), such kind of game is not referred to as a type of two-player zerosum game framework. Particularly, the leader first decides on a strategy, and then the follower makes decisions to optimize its utility based on the leader's strategy. Also, multiple entities can act as the leaders (or followers) to make decisions simultaneously. However, in this article, we only focus on two entities, i.e., the HAP as the defender and the adversary, to conduct the competitive tile grid allocation for enhancing location privacy of AUEs. From practical view points, the defender and the adversary tend to simultaneously allocate the defense resources of legitimate tile grids and the attack resources of false tile grids across the AUEs, respectively. Thereby, the simultaneous decision making for the defender and the adversary in their attack-defense resource allocation should be fully captured in the game modeling.
The CB game [27]- [30] is a zero-sum game-theoretic framework for modeling and analyzing the strategic resource allocation of two players in a competitive environment, to simultaneously distribute limited resources over a given number of battlefields. Inspired by this, we adopt the CB game instead of Stackelberg game to formulate the competitive tile grid allocation problem for the HAP and the adversary.

A. System Overview
As shown in Fig. 1, we consider a laser-powered aerial MEC system, which consists of a multiantenna HAP integrated with a laser transmitter and an MEC server, and N single-antenna AUEs located inside the 3-D coverage space of the HAP, each having a computation-intensive latency-critical task to be completed. The HAP delivers laser-beamed energy to charge the AUEs in the sky, and each AUE then uses the harvested energy to maintain the flight operation and accomplish the computation task via executing locally or offloading to the HAP. We thereby define the charging-then-MEC phase, which is undertaken over a finite-time horizon with duration T. The HAP is operating in a quasistationary position at a fixed stratospheric altitude of L H above the Earth's surface. Each AUE in a fixed-wing configuration is able to intelligently fly and optimize its trajectory in . Moreover, the cleared LoS with each AUE can be effectively established for the HAP due to its higher altitude operation and high-elevation angle.
Without loss of generality, the 3-D coverage space of interest is discretized into a multitier spatial structure consisting of equally sized square tile grids, as illustrated in Fig. 1. For the discretized spatial structure, we consider that there are Y tiers and the lowest tier relative to the ground is referred to as tier 1. We let Y = {1, 2, . . . , Y} denote the set of the tiers for such a structure, and define L 0 as the vertical distance between tier 1 and the Earth's surface. To simplify the analysis, we assume that every tier is composed of Q × Q square tile grids with equally sized areas, for Q × Q × Y N, and all the tiers are evenly spaced by an equal vertical distance, denoted by L T , for (Y − 1)L T L H . Note that the tile grid in each tier is adopted as a spatial location to charge the AUE via the laser WPT from the HAP. Here, we focus on the center of the tile grid as the 3-D location of charging point for the AUE.
For ease of exposition, a 3-D Cartesian coordinate system is adopted. The horizontal coordinate of the HAP is determined by During the charging-then-MEC phase, every AUE is assumed to be flying at the fixed 3-D location of charging point. 1 Therefore, the horizontal coordinate of AUE n in tier γ during time horizon T is specified by q In what follows, unless otherwise stated, we use the terms "AUE n in tier γ " and "AUE n" interchangeably. Accordingly, the vertical distance between AUE n in tier γ and the Earth's surface is given asd During time horizon T, the distance between the HAP and AUE n in tier γ is then calculated by For illustration convenience, we define the signaling transfer phase, the tile grid allocation phase, and the charging location determination phase, respectively. 2 As displayed in Fig. 2, we consider a time-slotted frame structure, which is composed of four successive phases, namely, the signaling transfer phase, the tile grid allocation phase, the charging location determination phase, and the charging-then-MEC phase. Here, we ignore the time of the signaling transfer as the signaling overhead is usually of small data size [31]. The time for tile grid allocation at the HAP can be also neglected due to its high capability for decision-making, control, and processing. Moreover, we ignore the time for transmitting the charging location data to the AUE during the charging location determination phase since the HAP has higher data throughput and small propagation delay [32], [33]. We adopt a time-switched "charging-then-MEC" strategy, and use the different fractions of time for laser charging, task offloading, and result downloading. The time horizon T is equally divided into K time slots with length δ = T/K. Based on the partial computation offloading model that we will discuss in the following, for the charging-then-MEC phase, each time slot is divided into three stages, i.e., the laser WPT stage, the computation offloading stage, and the computing and downloading stage, as depicted in Fig. 2. We use t W to represent the time of the laser WPT for each AUE. Similar to [34]- [36], the time consumed at the computing and downloading stage of the HAP are further neglected since the HAP has a much powerful computation capability (e.g., a highspeed multicore CPU) and a much higher transmit power than the AUEs. Therefore, at each time slot, the HAP first uses a fraction (t W /δ) of the time slot to deliver the laser-beamed energy concurrently for all the AUEs, and then each AUE completes the partial computation offloading during the remaining 1 − t W /δ fraction of time by using the harvested energy.

B. Laser WPT Model
With the time-switched "charging-then-MEC" strategy, the HAP wirelessly transfers the laser-beamed energy concurrently to all the AUEs at each time slot through laser beaming. We assume that the HAP adopts a fixed optical laser power β of the associated laser beam at each time slot. With the LoS link established by the HAP, the received optical laser power for AUE n in tier γ at time slot k can be given by [11] and [37] where A is the area of the receiver telescope or collection lens, D is the size of the initial laser beam, θ is the angular spread, χ is the combined transmission receiver optical efficiency, and ζ is the attenuation coefficient of the medium (in m −1 ). With the linear energy harvesting model, the actual harvested laser energy of AUE n in tier γ at time slot k can be computed as a linear function of the input laser-beamed energy, i.e., where τ ∈ (0, 1) is the laser energy conversion efficiency. Accordingly, the actual harvested laser energy of AUE n in tier γ during time horizon T can be obtained as (4)

C. Communication Model
To offload the computation task to the HAP during the charging-then-MEC phase, it is assumed that each AUE is assigned to an orthogonal subchannel over a resource block (RB). 3 Each subchannel has an equally sized bandwidth of W. Thus, the co-channel interference is completely avoided for all the AUEs during the uplink transmission. We suppose that the HAP operates in Ka-band sharing the same RB based on the ITU-R spectrum regulation [8]. Due to the dominated LoS link between the AUE and the HAP, the channel fading thus depends on both the free space path loss and miscellaneous atmospheric loss [32]. As such, the path loss between AUE n in tier γ and the HAP at time slot k is expressed by where φ n is the carrier frequency of subchannel for AUE n, c is the speed of light, α ≥ 2 is the path-loss exponent, and l A is the atmospheric loss depending on the effect of oxygen and water vapor on the transmission link. Note that the path loss can be referred to as the fixed value for the AUE at each time slot due to the fixed distance between the HAP and the AUE during time horizon T. Then, the channel gain between AUE n in tier γ and the HAP at time slot k is given by g n As a result, the achievable uplink rate of AUE n in tier γ to the HAP at time slot k can be formulated as where p n [k] is the transmit power of AUE n to the HAP at time slot k, and σ 2 is the variance of the AWGN at the HAP.

D. Computing Model
We adopt the partial computation offloading model during the charging-then-MEC phase. The computation task of AUE n in tier γ at time slot k is defined by a two-tuple is the total data size of computation task for AUE n, and ξ n [k] (in bits) is the data size of computation task offloaded from AUE n to the HAP. 4 1) Local Computing: The data size of computation task executed by AUE n at time slot k is denoted as B n [k] − ξ n [k] for task w n [k]. We assume that the local computing capability chosen by each AUE is fixed when computing during time horizon T, but may continuously vary over the different AUEs. Let f n be the local computing capability (in CPU cycles/s) of AUE n. Given each time slot, we denote the number of CPU cycles required to process one bit of raw data at AUE n by n , which is determined by the nature of resource-hungry applications and is further assumed to be equal for each AUE. Thereby, the computing rate (in bps) for local computing at AUE n at time slot k is calculated by Note that r L n [k] is also equivalent to the local computing rate of AUE n during time horizon T, i.e., r L n = r L n [k], ∀k. The computation delay for processing task w n [k] locally at AUE n at time slot k is then calculated by t L . Thereby, the total computation delay for local computing at AUE n during time horizon T can be expressed as Based on the configuration of CPU architecture, the power consumption for local computing of AUE n at time slot k can be given by q L n [k] = κf 3 n , where κ is an effective switched capacitance depending on the CPU architecture of AUE [16]. Thus, the energy consumption for local computing of AUE n at time slot k is modeled as . Correspondingly, the total energy consumption for local computing of AUE n during time horizon T is then formulated by 2) Computation Offloading: By combining the transmission overhead (e.g., encryption and packet header), denoted by ϑ n , the actual data size of computation task offloaded from AUE n to the HAP at time slot k is defined as ϑ n ξ n [k]. Then, the uplink transmission delay of AUE n for offloading the task to the HAP at time slot k can be obtained by . As a result, the total uplink transmission delay for computation offloading at AUE n during time horizon T is expressed as By considering the transmit power for offloading, the energy consumption for uplink transmission of AUE n at time slot k is written by . Therefore, the total energy consumption for computation offloading at AUE n during time horizon T is given by

E. Flight Energy Consumption Model
The flight energy consumption of fixed-wing AUE mainly depends on its propulsion energy consumption to maintain the airborne and to support the velocity and acceleration. We assume that the AUE has the same flight energy consumption during the charging-then-MEC phase. Utilizing the results of [38], the flight energy consumed by AUE n during time horizon T can be analytically obtained by (12), shown at the bottom of the page, where 1 and 2 are parameters related to the weight, wing area, air density of AUE n, m is the mass of AUE n including all the payloads, g is the acceleration of gravity, the superscript a T n is the transpose of vector a n , and v n [k] and a n [k] are the velocity and acceleration vectors of AUE n at time slot k, respectively.

F. Adversary Model
Due to the openness of wireless transmissions for the WPT signaling messages on the control channels, we thus take into account an external adversary who misbehaves or deviates from the legitimate laser charging operations required by the AUEs. The adversary is assumed to be deployed on the ground with a fixed location, as shown in Fig. 1. The goal of the adversary is to capture the wireless communications during the signaling transfer phase, and analyze the intercepted messages from the AUEs to track their locations and learn about the contents of messages, e.g., the preferences of charging locations for AUEs. Based on the observation of WPT signaling transfer, the adversary aims to actively launch the CLDF attacks by sending the false charging location data to the AUEs during the tile grid allocation phase. Overall, the CLDF attacks require the adversary to eavesdrop on the signaling message first, and then to falsify the charging location data and send it to each AUE. Particularly, we thus consider the above-mentioned two types of attacks that can be mounted by the adversary, i.e., 5 as follows. 1) Eavesdropping: Being a smart and programmable radio device as mentioned in [39], the adversary listens on the control channels during the signaling transfer phase to eavesdrop on the signaling messages from the AUEs and read the contents of the signaling messages. We assume that the adversary has the capability to perfectly detect when and where a signaling message arrived from that it has the same communication range as the AUEs in the system, and also has a large amount of memory to keep track of the signaling messages that have been overheard. 5 Note that here we do not consider the adversary model that can modify or falsify the offloaded computation task, because our proposed CB game framework we will describe in Section IV is based on the assumption that all the offloaded data are correct during the charging-then-MEC phase.
2) Falsification: With the captured signaling message, the adversary falsifies the charging location data and sends a false location data in terms of tile grid to the AUE during the tile grid allocation phase, causing the AUE to receive the wrong 3-D location of charging point, which seriously violates the location privacy of AUE. The CLDF attacks apparently suggest a compelling need for protecting the location privacy of laser WPT for HAP-aided MEC. In this article, we only focus on the attack of falsification in the CLDF attacks mounted by the adversary by distributing the false tile grids over the AUEs to compromise the location privacy for laser charging. While the adversary tries to allocate its attack resources regarding false tile grids over the AUEs, the HAP will act as a defender that seeks to optimally allocate its defense resources in terms of legitimate tile grids to prevent the adversary from causing the unsuccessful laser WPT and the failure of the aerial MEC. Such an attack-defense resource allocation during the tile grid allocation phase allows us to gain more insights into the competitive interaction between the adversary and the defender in their simultaneous distributions for the fixed number of tile grids across the AUEs. To formulate the competitive resource allocation problem under such an attack-defense scenario, in Section IV, we will employ the CB game framework [27]- [29] for strategically distributing tile grids over the AUEs for both the defender and the adversary.

IV. CB GAME FORMULATION
In this section, we first propose the CB game framework in the case of pure strategy to formulate the competitive resource allocation problem, and particularly design the utility function of the player in this game. To effectively solve this game, we alternatively formulate the CB game with the mixed strategy.

A. Formulation of CB Game With Pure Strategy
The interaction between the defender and the adversary regarding their competitive allocations of tile grids among N AUEs is formulated as a CB game with N battlefields. In this game, the defender chooses the defense tile grid distribution vector over N battlefields during the tile grid allocation phase, while the adversary chooses the attack tile grid distribution vector. The player that allocates a higher amount of tile grids to the individual battlefield wins in that battlefield. Formally, the CB game framework is defined in the following.
Definition 1 (Pure Strategy): A CB game with pure strategy for tile grid allocation of N AUEs during the tile grid allocation phase is formulated as a 4-tuple

4) Utility Function {u [T]
X ,n } n∈N : The utility obtained by each player from battlefield n is denoted as u [T] X ,n . The total utilities of two players are given by u n on battlefield n, respectively. Typically, each battlefield is won by the player that allocates a larger number of tile grids on that battlefield. To this end, the main idea of the proposed privacyaware laser WPT is to compare the number of distributed tile grids for each battlefield between two players. Then, we can infer that when H [T] n > S [T] n , the defender wins battlefield n.

Otherwise, when H [T]
n ≤ S [T] n , the adversary wins battlefield n. From the above observations, we note that the player who wins more battlefields than its opponent finally wins this game. Hence, our objective in this game is to maximize the number of times that the defender wins across battlefield set N .

B. Utility Function Design
To determine the utility function of each player obtained from a battlefield, we here explore the impact of the chargingthen-MEC phase on the competitive allocation of tile grids between two players during time horizon T. Matter of fact, the total energy consumption for local computing of AUE n during time horizon T should be less than E H n −E F n −E U n . Considering the power consumption for local computing of AUE n at time slot k, the maximum time used for local computing of AUE n during time horizon T can be calculated as where q L n = q L n [k] ∀k, due to the fixed local computing capability for each AUE at each time slot. With the computing rate r L n , the maximum processing input bits of AUE n during time horizon T are given by B max n = r L n T max n . Let B n = K k=1 B n [k] be the total data size of computation task for AUE n during time horizon T. Therefore, the local computation capacity for AUE n during time horizon T can be defined by From the defender perspective, the local computation capacity for AUE n during time horizon T is rewritten as Since different AUEs may have different energy states and different requirements for the quality of MEC service, both the energy consumption and execution delay are critical to the AUEs during the charging-then-MEC phase. To this aim, the utility function of each player from a battlefield should be well developed by balancing the tradeoff among the local computation capacity, energy consumption, and execution delay of the AUE. Let λ 1 , λ 2 , and λ 3 denote the positive weighting factors used to ensure the same range for local computation capacity, energy consumption, and execution delay, respectively. Based on the above analysis, the utility that the defender receives on battlefield n during time horizon T in CB 1 can be determined by (19), shown at the bottom of the page.
To simplify the presentation, let us employ 1 n to stand for the utility that the defender receives on battlefield n when H [T] n > S [T] n . We also define 2 n as the utility that the defender receives on battlefield n when H [T] n ≤ S [T] n . For clarity of exposition, we then introduce a binary variable to represent the association relationship between H [T] n and S [T] n , i.e., By combining (19) and (20), the total utilities obtained by the defender over battlefield set N during time horizon T in CB 1 with pure strategy is thus written as

C. Formulation of CB Game With Mixed Strategy
However, it has been revealed that the pure-strategy NE for CB 1 does not always exist [27]. Given pure strategy, the adversary can always modify its tile grid allocation strategy accordingly to improve its utility and win the game.
Conceptually, the pure strategy indicates that each player chooses a unique and definite strategy in its action space. Both players are unwilling to adjust their strategies based on the other player's strategy. We note that there will be no dominant strategy of CB 1 with pure strategy. Alternatively, for solving CB 1 , a more general method is to find the mixed-strategy NE solution. In the case of mixed strategies, each player selects a probability distribution over the action space to maximize the potential expected total utilities. Definition 2 (Mixed Strategy): A CB game with mixed strategy for tile grid allocation of N AUEs during the tile grid allocation phase is formulated as a 4-tuple X (h, s)}: The expected total utilities obtained by each player over battlefield set N are described as U [T] X (h, s), which can be expressed as (23), shown at the bottom of the page. Based on CB 2 in Definition 2, we define ψ [T] i,r and ϕ [T] i,r by the ith highest probability of H [T] n,r and the ith highest probability of S [T] n,r , respectively, for i ∈ N . Then, the mixed-strategy action sets for two players can be, respectively, determined by n,r , V. SOLUTION TO THE CB GAME In this section, we will describe the mixed-strategy NE solution to the proposed CB game with symmetric players and asymmetric players, respectively. Through the solution derivation with symmetric and asymmetric structures, we will then explore the effect of privacy protection level on the mixed-strategy NE solution.

A. Mixed-Strategy NE Solution Under Symmetric Players
In this section, we begin by finding the mixed-strategy NE solution to CB 2 in Definition 2, which can be formally defined as follows.
Definition 3 (NE): A tile grid allocation profile (h * , s * ) is a mixed-strategy NE point of CB 2 if and only if no player can improve its utility by deviating unilaterally, i.e., Let us first consider CB 2 with symmetric players, in which two players have the same amount of resources to distribute across battlefield set N . This implies that the total number of tile grids distributed by two players is completely equal, i.e., S . In this case, two players uniformly deploy the tile grids over battlefield set N , respectively. Due to the nonnegative integer constraints imposed on the tile grid allocation, the mixed-strategy CB game is actually a constant-sum two-player matrix game with finite number of strategies [40]. We then denote by 1 h×s and 0 h×s the all-one and all-zero h×s matrices, respectively. More concretely, the mixed-strategy NE solution to CB 2 with symmetric resources is analytically derived as the following theorem.
where ϒ = Z [T] H / B and · is the lower floor function. Proof: See Appendix A. Proposition 1: With the mixed-strategy NE (h * , s * ) of CB 2 with symmetric structure, the expected total utilities of two players are all equal to zero, i.e., Proof: With the mixed-strategy NE solution in Theorem 1, the number of tile grids assigned to each battlefield by the defender is equal to those of the adversary, i.e., the same uniform allocation of tile grids for two players. Referring to (38) and (39) in Appendix A, it can be obtained that no player can receive better utility than the other one. Intuitively, each battlefield will obtain the same probability of tile grid allocation from either the defender or the adversary. Based on (20), (23), and (28), we are able to directly derive the expected total utilities of two players as follows: This completes the proof. Remark 1: Under the condition of same amount of resources to distribute for CB 2 , two players both choose the number of tile grids to uniformly allocate for each battlefield within the interval [0, 2Z [T] H B n / B ], and no battlefield dominates in the game, i.e., B n ≤ N m=1,m =n B m . Moreover, two players then have the same uniform allocation probability. From this result, we observe that no player wins the game, and no AUE can receive the spatial location to charge via laser WPT. Therefore, CB 2 with symmetric players yields zero utility for the defender.

B. Mixed-Strategy NE Solution Under Asymmetric Players
In contrast with the symmetric structure, herein we consider CB 2 with asymmetric players. That is, two players have different numbers of tile grids, i.e., Z [T] H > Z [T] S , to deploy over battlefield set N . Formally, the mixed-strategy NE solution of CB 2 with asymmetric resources is specified as follows.
H ≤ 1 and N ≥ 3, a tile grid allocation profile (h * , s * ) constitutes a mixed-strategy NE to CB 2 with asymmetric resources, given by (31) and (32), shown at the bottom of the next page, respectively, where = Z [T] H /N. Proof: See Appendix B. Proposition 2: With the mixed-strategy NE (h * , s * ) to CB 2 with asymmetric structure, the expected total utilities of two players are achieved as Proof: As observed from the mixed-strategy NE to CB 2 in Theorem 2, the number of tile grids assigned to each battlefield by the defender is different from that of the adversary, i.e., H * n > S * n and H * n ≤ S * n . Moreover, the probability of such the different cases can be denoted by Pr(H * n > S * n ) and Pr(H * n ≤ S * n ), respectively. Referring to (18) and (20), it is readily seen that the defender can obtain the expected total utilities, i.e., Pr(H * n > S * n ), if and only if H * n > S * n . Based on (20), (23), (31), and (32), we thus derive the expected total utilities of two players as follows: As a result, we obtain the desired expression as (33). The proof is complete.
Complexity Analysis: The complexity of the mixed-strategy NE solution derivation under asymmetric players is analyzed in terms of the time and space complexity as follows. The proposed game framework has N battlefields, where each battlefield requires two players to allocate battlefield resources (i.e., tile grids) separately, and two players will compete across N battlefields at the same time. Then, the process of N competitive interactions between two players in their simultaneous distributions for battlefield resources can be expressed by a linear order time complexity, denoted as O(N).
We further consider that the results of competitive tile grid allocations are temporarily stored at two players. Then, each player requires at least a memory of N to keep track of these results. Thereby, the space complexity can be calculated as O(N), which is proportional to the number of battlefields.
Remark 2: Our objective in this work is to achieve the improvement of the total utilities of the defender by effectively identifying a tradeoff among the local computation capacity, energy consumption, and execution delay for each AUE. The defender has to distribute more tile grids than the adversary to protect the location privacy for laser charging of the AUEs. When S , the defender guarantees that more AUEs will obtain the laser WPT, which ensures that more computation tasks can be completed locally or offloading to the HAP. In other words, the defender wins the game and the expected total utilities increase accordingly with the growing number of tile grids allocated by the defender.

C. Mixed-Strategy NE Under Privacy Protection Level
So far, we have obtained the mixed-strategy NE solution to CB 2 with asymmetric players in Theorem 2. However, as for this solution, we focus on all the AUEs, i.e., a one hundred percent of the AUEs, applying our proposed CB game framework to enhance the location privacy in laser WPT. This solution should be a perfect case wherein all the AUEs obtain the laser WPT since the defender wins the game. In fact, not all of the AUEs employ our game framework to achieve the enhanced location privacy. In order to demonstrate the performance superiority of our game framework with asymmetric players, we thereby consider a realistic case wherein the proportion of AUEs with the enhanced location privacy for laser WPT by help of our game framework. In the following, we first define a paradigm of privacy protection level to reveal the proportion of AUEs with the enhanced location privacy, and then derive the expected total utilities of two players in this case.
Definition 4 (Privacy Protection Level): The privacy protection level in CB 2 with asymmetric resources is defined as a protection ratio Φ = N /N ∈ (0, 1) of N AUEs applying our game framework.
Based on Definitions 2 and 4, the expected total utilities of two players over battlefield set N under privacy protection level constraint can be calculated by (35), shown at the bottom of the next page. In the case of asymmetric resources Z [T] H > Z [T] S , by applying Theorem 2, we can solve for the mixedstrategy NE solution to CB 2 with asymmetric resources under privacy protection level Φ, as described in the following.
Proposition 3: With the mixed-strategy NE (h * , s * ) to CB 2 with asymmetric structure, the expected total utilities of two players depend on privacy protection level Φ, i.e., Proof: By applying (31), and (32) in Theorem 2, the expected total utilities of two players can be derived as follows: Thereby, we have the result of (36) to conclude the proof.

VI. SIMULATION RESULTS
In this section, we provide the simulation results to illustrate the theoretical analyses and validate the performance of the proposed CB game framework with different parameter settings. Since no other existing work studies the same problem, we here compare the proposed framework against the design baselines constrained by privacy protection level Φ ∈ (0, 1) in CB 2 with asymmetric resources. More precisely, we consider two baseline schemes listed as follows for comparison.
1) Privacy Protection Level Φ = 0.8: A protection ratio of 80% of AUEs applying our game framework. 2) Privacy Protection Level Φ = 0.4: A protection ratio of 40% of AUEs applying our game framework. Throughout the simulations, all the results are obtained with the following default settings. We consider the laserpowered aerial MEC system consisting of an HAP deployed  The HAP coverage space is discretized into Y = 11 tiers. All the tiers are evenly spaced by an equal vertical distance L T = 200 m, and the vertical distance is set to L 0 = 1 km between tier 1 and the Earth's surface. In addition, each tier is composed of 500 × 500 square tile grids with equally sized areas, specified as 5 m 2 .
Regarding communication and computing models, unless otherwise noted, we set the bandwidth of orthogonal subchannel as W = 25 MHz. The transmit power of AUE is randomly assigned from the range of [250 mW, 400 mW] at each time slot. We set the transmission overhead to be ϑ n = 50 bits for the AUE. Furthermore, we assume that the AUE is flying at a constant speed during the charging-then-MEC phase. Particularly, the velocity vector of the AUE is selected randomly from the range of [(8, 8), (15,20)] and the acceleration vector is always set to a n [k] = (0, 0). The positive weighting factors associated with the utility function are set to be λ 1 = 0.97, λ 2 = 7 × 10 −4 , and λ 3 = 0.3, respectively. The other relevant parameters in simulations are summarized in Table I. Before validating the system performance against the baseline schemes, we first provide insight on the effect of number of battlefields and tile grids distributed by two players for the proposed framework. Fig. 3(a) shows the effect of number of battlefields N on the expected total utilities of the defender with the increase of Z [T] H when Z [T] S = 200. It can be seen that the expected total utilities of the defender gradually increase as N increases under a given Z [T] S . That is, larger number of battlefields, more obtained expected total utilities of the defender. Referring to Fig. 3(a), we can also find that the defender with higher value of Z [T] H obtains more expected total utilities by increasing N. The main reason for this phenomenon is that the defender allocates the tile grids to more battlefields, and then wins in more battlefields to gain more expected total utilities. This result indicates that the choice of number of battlefields and battlefield resources of the defender has a negligible effect on the system performance and the expected total utilities of the proposed framework.
In Fig. 3(b), the expected total utilities of the defender is presented versus the tile grids distributed by the defender Z H , the global curve becomes gradually convergent and slowly rising. Especially, along with the increasing value of Z [T] H , the expected total utilities of the defender have been heightened under a given Z [T] S . For a given Z [T] H , the more tile grids distributed by the adversary will generate the lower expected total utilities of the defender. This can be explained as follows: 1) with the increasing value of Z [T] H , the defender has more resources to counterbalance with the adversary, resulting in more expected total utilities obtained by the defender and 2) for winning more battlefields, the adversary also reduces the expected total utilities of the defender by improving the value of Z [T] S and competing with the defender. This result provides a hint to choose the appropriate amount of battlefield resources for the defender to improve the system performance.
The results in Fig. 4 show the comparison of the expected total utilities of the defender between the proposed framework and two baselines with privacy protection level Φ = 0.8 and Φ = 0.4, respectively, versus number of battlefields N, when Z [T] H = 550, Z [T] S = 200, f n = 3.5 × 10 9 cycles/s, φ n = 3 GHz, and n = 110 cycles/bit. Here, we set the optical laser power of laser beam at the HAP as β = 800 W for two baselines. For comparison, we employ β = 850 W and β = 800 W for the proposed framework, respectively. As expected, all the curves grow up as N increases. By increasing the value of β, we can also find that the expected total utilities of the defender improve significantly for the proposed framework. This is attributed to the fact that the AUEs can obtain more harvested energy E H n to accomplish the computation task, which not only enhances the local computation capacity R n but also gains the ever-growing expected total utilities for the system. As can be seen from Fig. 4, the proposed framework outperforms two baselines in terms of the expected total utilities of the defender. This is mainly because the privacy protection level Φ of the proposed framework is much bigger than that of the two baselines, incurring more battlefields won by the defender and thus, more expected total utilities obtained by the defender. H is. This can be explained as follows: 1) the AUEs can harvest more energy to complete the computation tasks due to the improved privacy protection level Φ and 2) the local computing capacity R n can be also increased to improve the expected total utilities of the defender when the value of β is incorporated into the proposed framework.
Finally, in Fig. 6, we illustrate the expected total utilities of the defender versus tile grids distributed by two players for the proposed framework and two baselines with privacy protection level Φ = 0.8 and Φ = 0.4, respectively, in the case of β = 800 W, φ n = 1 GHz, and f n = 4 × 10 9 cycles/s. From Fig. 6(a)-(c) as a whole, we easily observe that the increase of N drastically affects the expected total utilities of the defender achieved by the proposed framework and two baselines, respectively. It can be also seen that the baseline with Φ = 0.8 can achieve a near-optimal performance close to that with the proposed framework. Moreover, for given N, the lower value of n results in a significant utility improvement compared to higher value of n in the proposed framework. The intuition behind the observation is the fact that the AUEs require more CPU cycles to process one bit of raw data, resulting in fewer computation tasks locally executed during a fixed time horizon, and thereupon reducing R n for each AUE. Therefore, the expected total utilities of the defender are clearly inversely related to n . Such observations above bolster the importance of choosing an appropriate number of CPU cycles to process one bit of raw data at the AUEs for improving the expected total utilities of the system.

VII. CONCLUSION
In this article, we explored the privacy-aware laser WPT problem for HAP-aided aerial MEC systems in the presence of the terrestrial adversary. For addressing the new privacy vulnerability arising from the WPT signaling transfer, we formulated the interaction between the HAP as the defender and the adversary in their allocations of limited number of tile grids as charging locations to the AUEs as the CB game framework. This game framework modeled the competition of two players for limited resources over a finite set of battlefields. The mixed-strategy NE solutions of this game were derived with the symmetric and asymmetric tile grids between the defender and the adversary, respectively. Simulation results demonstrated the effectiveness and practicality of our proposed framework, which achieves beneficial performance gains on the system-wide expected total utilities compared with baseline schemes. This insight is expected to shed light on the application of practical laser-powered aerial MEC systems with enhanced security and privacy. As part of the future work, the optimization of bandwidth allocation needs to be well addressed for improving the system performance, and the considered problem can be extended to more diverse scenarios with multiple adversaries.

APPENDIX A PROOF OF THEOREM 1
For CB 2 with symmetric players, we observe that regardless of the purpose of two players, the price paid on each battlefield belongs to the same value. For analytical tractability, the value of battlefield n for both players is thus defined as the total data size of computation task for AUE n during time horizon T, i.e., B n . Thereby, the total value of N battlefields for two players is then given by B = N n=1 B n . We assume CB 2 has an NE (h * , s * ), where h * (or s * ) is the uniform allocation probability vector of H (or S). The number of tile grids distributed by the defender (or the adversary) to battlefield n is denoted by H * n (or S * n ). Since the total number of tile grids allocated by two players is completely equal, both players will distribute the tile grids reasonably to each battlefield for wining more battlefields. Thus, the number of tile grids allocated by both players for battlefield n are also equal at the mixed-strategy NE, i.e., H * n = S * n . According to [41,Proposition 1], H * n and S * n are uniformly distributed on [0, 2Z [T] H B n / B ]. Combining (14) and (15), we can obtain the uniform allocation vector H * (or S * ) of the defender (or the adversary) to battlefield n as follows: where U (·) denotes the uniform allocation of the elements. Due to the symmetric resources and the uniform allocation interval of both players, by combining (24), (25), and (38), the uniform allocation probability h * (or s * ) of the defender (or the adversary) to battlefield n is also completely equal when to battlefield n can be determined by