# Energy-Efficient Computational Offloading for Secure NOMA-Enabled Mobile Edge Computing Networks

Authors: Haiping Wang
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

Computational offloading and nonorthogonal multiple access (NOMA) are two promising technologies for alleviating the problems of limited battery capacity, insufficient computational capability, and massive deployment of terminal equipment in the Internet of Things (IoT) era. However, offloading data may be threatened by malicious eavesdroppers, which leads to more energy consumptions to avoid being eavesdropped. In this work, we study the energy-efficient way of computational offloading under the condition of certain security requirement in a secure NOMA-enabled mobile-edge computing (MEC) networks, whereK end users are intended to offload their data to the N-antenna access point (AP) through the same resource block under the threat of an eavesdropper. We first address energy-efficient local resource allocation by minimizing sum-energy consumption of end users, subject to CPU frequencies, offloading bits, secrecy offloading rate, and transmit power. We then optimize the local resources to obtain the minimum computation latency of task for each end user, with the constraint of certain energy budget. The solutions to the above two optimization problems are given and demonstrated numerically by a 3-user scenario.


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## Body

## 1. Introduction

Research shows that the contradiction between the limited resources of mobile terminal equipment and its explosive development of business and application requirements has become the most challenging tasks in the field of mobile Internet and Internet of Things (IoT) [1]. The resources of the terminal equipment are mainly reflected in the computational capability of the processor and the battery capacity. According to the theory of integrated circuits, the power consumption of the CPU is proportional to the cube of its frequency [2]. The stronger computing power, the shorter battery life.The newly developed mobile-edge computing (MEC) [3] technology helps to resolve the above contradiction. It allows terminal devices to migrate data to the MEC server for computing-computational offloading [4]. Although MEC effectively solves the contradictions faced by the terminal and improves the service experience, there are still problems of security and high energy consumption [5].In recent years, most of the research on MEC computational offloading focuses on the optimization of energy consumption or/and time delay without considering the security threat. Some [6–9] investigated how to save energy, some [10–14] studied from the perspective of improving real-time performance, and [15] took into account both energy consumption and computation latency. Although many achievements have been received in the optimization of computational offloading performance under the condition of no security threat, in practical applications, data offloading often needs to consider information security issues. Whether it is IoT data, Internet of Vehicles (IoV) data, video streaming analysis data, or other data with high computational intensity [16], data leakage needs to be avoided. After security processing, such as encrypting and decrypting data in the terminal and MEC server, the computational offloading process inevitably increases energy consumption and computation latency, resulting in distortion of the existing performance optimization research results. To this end, security should be considered when studying the performance of computational offloading in practical applications.The research on the security of the computational offloading process can mainly be divided into two categories: one is based on the traditional cryptography; the other is based on the physical layer security. [17, 18] studied the energy efficiency of computational offloading based on cryptographic encryption, taking into account the delay and energy consumption caused by encryption, making energy efficiency evaluation closer to practical applications. [17] compared the energy consumption results of computational offloading with encryption and without encryption. As expected the energy consumption with encryption is greater than that without encryption. In turn, the authors further considered compressing the data to reduce the total amount of data, shorten the transmission delay and energy consumption, and achieve desired results.Compared with traditional cryptography, computational offloading based on physical layer security is more attractive to researchers. The physical layer security is keyless, which avoids the network vulnerability problem caused by key distribution and management [19]; on the other hand, the physical layer security has been proved to be reliable from the perspective of information theory [20]. In addition, the physical layer security and traditional cryptography technology belong to different layers in the network system, and there is no replacement relationship between them, and the existing traditional security system can still be retained. Presently, the research combining physical layer security with MEC computational offloading energy efficiency is still in its infancy, and there are not many related studies. Most of them focus on secrecy offloading rate improvement and secrecy offloading resource allocation.The most effective way to improve the energy efficiency of the terminal side is to increase the secrecy offloading rate without increasing the transmit power, shorten the data offloading time, and then reduce the offloading energy consumption of the terminal. Therefore, the related secrecy capacity enhancement techniques in the physical layer security can be applied, such as improving the main channel through relays; or deteriorating the eavesdropper’s channel through artificial noise; or enhancing the main capacity through nonorthogonal multiple access technology. [21] studies the use of relay to improve the offloading rate. Since the relay is untrustworthy, the MEC server needs to interfere with it accordingly. [22] uses full-duplex artificial noise to deteriorate the eavesdropper’s channel, that is, the MEC server sends out interference signals, which does not interfere with the legitimate receiver, but can influence the eavesdropper. Cooperative jamming assisted scheme via cooperative NOMA transmission was studied in [23]. A novel jamming signal scheme was designed to multiuser multiserver MEC-enabled IoT in [15, 24] adopts nonorthogonal multiple access technology to improve the terminal’s antieavesdropping capability. [25–27] takes unmanned aerial vehicle (UAV) as the target of MEC data offloading and improves the secrecy capacity of end users by adjusting the position and power of UAV.As for secure offloading resource allocation, there are two sides: terminal side and MEC side. The terminalside resources include CPU frequency, transmit power, transmission rate, and for the data “partial offloading” mode, also include the data partition ratio. MEC side resources include server CPU frequency, channel resources, and the like. [28] optimizes energy consumption by taking the proportion of data offloading and transmit power as resource allocation objects, that is, minimizing the energy consumption of the terminal by reasonably adjusting the data ratio of local computing and offloading computing and offloading power. The scenario is extended from single user to multiuser [29, 30], and the weighted total power is minimized by rationally distributing the data offloading ratio and transmit power of each user. Since the state-of-the-art CPU architecture used by most terminal devices adopts dynamic frequency and voltage scaling (DVFS) technique, the energy consumption can be controlled by flexibly adjusting the frequency or voltage of the CPU, so the local CPU frequency can be further included in the resource optimization object. [31–33] include channel resources (such as time slots and frequencies) into optimization objects while considering local resources. In addition, [34, 35] also studied dividing the data into several parts and offloading them to different MEC servers for computation.In this work, we further investigate terminal-side resource allocation for multiterminal energy efficiency of a secure MEC system enabled with NOMA, which is of high spectral efficiency and can accommodate massive devices. Zero-forcing (ZF) combined with successive interference cancelation (ZF-SIC) is adopted on the side of AP. Different from MMSE, ZF is less complex and more energy-efficient. We consider “partial offloading” mode and take more practical secrecy performance metric of secrecy outage probability (SOP) during offloading. Although [30] has studied energy-efficient resource allocation in secure NOMA-enabled mobile MEC networks, the authors only addressed the scenario of two user NOMA and one single antenna at the AP associated with the MEC server for simplicity. In this paper, we generalize the model of secure NOMA-enabled MEC with multiple users and multiple antennas at AP and tackle the efficient problem of energy and latency via local resource allocation. The major challenges of this work are (1) modeling wiretap channel and partial offloading mode and giving the closed-form expressions; (2) formulating sum-energy consumption minimization problem and computation latency minimization problem; (3) transforming minimization problems with multiple variables into a single variable one and giving the optimal solutions.We adopt SOP as the secure QoS metric rather than secrecy capacity or ergodic secrecy capacity based on the following considerations: (1) more favorable main channel than eavesdropper channel is not always available; (2) the eavesdropper is always passive, which implies the CSI of eavesdropper channel is not available; (3) ergodic secrecy capacity needs to encode over a long period of time over all channel realizations, which incurs long delay and is suitable only for delay-tolerant applications; (4) secrecy outage is suitable for encoding confidential message over a single coherence interval or channel block.The main contributions of the paper are summarized as follows:(1)
We generalize the model of NOMA-enabled MEC networks against an external eavesdropper byK users and N antennas at AP associated with MEC server. A novel design framework is introduced by adopting zero forcing plus successive interference cancelation at the AP, jointly optimizing the number of offloading bits, local CPU frequency, and transmit power, targeting at end users’ energy and latency(2)
Aiming at energy-efficient design, we characterize the optimization problem of sum energy of end users under certain SOP and latency requirement in the secure NOMA-enabled MEC networks, subject to CPU frequency, offloading bits, secrecy offloading rate, and transmit power of each user. We transform the complex and nonconvex problem into a single-variable one and give solution by numerical-finding(3)
Aiming at latency-efficient design, we investigate the problem of minimizing the computation latency of each user’s task under certain SOP and energy budget requirement, with the constraint of CPU frequency, offloading bits, secrecy offloading rate, and transmit power. Similarly, the algorithm of the solution is given by transforming the original complex problem into a single-variable oneThe rest of the paper is organized as follows. In the following section, we model the multiuser secure NOMA-enabled mobile MEC system and derive the closed-form expression of the secrecy outage probability of an individual user. In the third section, the optimization problem of sum-energy consumption of end users under certain SOP and latency requirement is characterized, and the algorithm of the solution is given. The problem of minimizing the computation latency of each user’s task under certain SOP and energy budget requirement is addressed and solved in the fourth section. Numerical results are presented in the section of “Numerical Results” to confirm the efficient design done in the previous sections. Finally, a conclusion is drawn in the last section.


## 2. System Model

As depicted in Figure1, we consider a secure uplink NOMA-enabled MEC networks, where K single-antenna users are intended to offload their computation-intensive tasks to the N-antenna access point (AP) (with an MEC server integrated) by sharing the same radio resource (such as frequency and/or time) at the presence of an eavesdropper (Eve) which is equipped with a single antenna. Here, N≥K is assumed.Figure 1
A model of secure uplink NOMA-enabled MEC networks withK users, one eavesdropper, and one N-antenna AP.All the channels are experiencing quasistatic Rayleigh fading. To userkk∈K≜1,⋯,K, the symbols used hereinafter are listed in Table 1.Table 1
Symbol description.
Symbol
Description
s
k
Complex information symbol from userk
p
k
Transmit power from userk
h
a
,
k
Channel gain vector from userk to AP
h
e
,
k
Channel gain from userk to Eve
w
a
Complex Gaussian noise vector at AP
w
e
Complex Gaussian noise at Eve
d
a
,
k
Distance from userk to AP
d
e
,
k
Distance from userk to Eve
α
Path-loss exponent
g
a
,
k
g
a
,
k
~
C
N
0
,
I
N
g
e
,
k
g
e
,
k
~
C
N
0
,
1Therefore, the instantaneous composite signals received at the AP and Eve are given by(1)ya=∑k=1Kpkha,ksk+wa,(2)ye=∑k=1Kpkhe,ksk+we,where ha,k=da,k−α/2ga,k and he,k=de,k−α/2ge,k;Esk2=1;wa~CN0,NaIN and we~CN0,Ne.


### 2.1. Wiretap Channel Model

Similar to [36], successive interference cancelation (SIC) combined with zero forcing (ZF) is employed at the AP, without loss of generality, to a specific user k, its aggregate signal left before decoding can be expressed as
(3)ya,kI=pkha,ksk+∑i∈Ipiha,isi+wa,where I⊆K\k, indicating Uk is decoded just before these users in set I (index) during an SIC process.After applying zero-forcing, the SINR of userk at the AP is given by
(4)γkI=pkQkIha,k2Na,where QkI is a N−I by N matrix whose rows are the orthonormal basis of the null space of the subspace spanned by the vector set ha,i∀i∈I.According to [37], γkI obeys a chi-squared distribution with 2×N−I degrees of freedom and has nothing to do with the specific users in set I but its cardinality I. Specifically, λkγkI~χ2N−I2, where λk=Nada,kα/pk. For the ease of presentation, we denote the cardinality of I by n and rewrite γkI by γkn, whose probability density function (pdf) is given by
(5)fγknγk=λkN−nΓN−nγkN−n−1e−λkγk,γk≥0.As for the SINR of userk (denoted as ξk) at eve, we adopt the conservative assumption that all user interference can be canceled out by Eve. Therefore, ξk=he,k2pk/Ne obeys an exponential distribution with parameter μk=Nede,kα/pk, i.e., ξk~Expμk, whose cumulative distribution function (CDF) is given by
(6)Fξkξk=1−e−μkξk,ξk>0.


### 2.2. Partial Offloading Model

We assume the task for each user can be partitioned into two parts with an arbitrary ratio. LetLk and lk be the total number of bits to be processed and the number of bits to be offloaded to the AP.We consider a more practical scenario that the eavesdropper is passive and the AP only has the statistical characteristics of eavesdropper’s channels. To this end, we adopt the secrecy outage probability (SOP) for quantifying the quality-of-service (QoS) of secure transmissions.SupposeRs,k is the secrecy rate for transmit power pk that satisfies the targeted secure QoS for user k. Therefore, the energy consumption of offloading can be formulated as
(7)Ekoff=pktkoff=pklkBRs,k,where tkoff and B refer to offloading duration and the bandwidth of the channel, respectively.We present the closed-form expression of SOP in form of theorem as follows.Theorem 1.
The secrecy outage probability of userk for the required secrecy rate Rs,k at the presence of n-user interference can be expressed as
(8)Pso,kn,Rs,k=γN−n,λk2Rs,k−1ΓN−n+λkλk+μk2−Rs,kN−neμk1−2−Rs,k×ΓN−n,λk+μk2−Rs,k2Rs,k−1ΓN−n,where
(9)γv,z=∫0zuv−1e−udu,and
(10)Γv,z=∫z∞uv−1e−udu=Γv−γv,z,are the incomplete Gamma function and its complement, respectively.Proof.
See Appendix.


### 2.3. Local Computation Model

LetXk be the computation intensity in CPU cycles per bit for user k. The total number of cycles for computing local part of task is Lk−lkXk. Each user adopts the advanced dynamic frequency and voltage scaling (DVFS) technique to control the energy consumption. According to [1], the energy consumption of a CPU cycle is given by κkfk2, where κk is a constant associated with the hardware architecture, and fk is the CPU frequency for user k. For the local computation task of Lk−lkXk cycles, the energy consumption can be derived:
(11)Ekloc=κkLk−lkXkfk2.The corresponding computation time for userk can be given by
(12)tkloc=Lk−lkXkfk.


## 2.1. Wiretap Channel Model

Similar to [36], successive interference cancelation (SIC) combined with zero forcing (ZF) is employed at the AP, without loss of generality, to a specific user k, its aggregate signal left before decoding can be expressed as
(3)ya,kI=pkha,ksk+∑i∈Ipiha,isi+wa,where I⊆K\k, indicating Uk is decoded just before these users in set I (index) during an SIC process.After applying zero-forcing, the SINR of userk at the AP is given by
(4)γkI=pkQkIha,k2Na,where QkI is a N−I by N matrix whose rows are the orthonormal basis of the null space of the subspace spanned by the vector set ha,i∀i∈I.According to [37], γkI obeys a chi-squared distribution with 2×N−I degrees of freedom and has nothing to do with the specific users in set I but its cardinality I. Specifically, λkγkI~χ2N−I2, where λk=Nada,kα/pk. For the ease of presentation, we denote the cardinality of I by n and rewrite γkI by γkn, whose probability density function (pdf) is given by
(5)fγknγk=λkN−nΓN−nγkN−n−1e−λkγk,γk≥0.As for the SINR of userk (denoted as ξk) at eve, we adopt the conservative assumption that all user interference can be canceled out by Eve. Therefore, ξk=he,k2pk/Ne obeys an exponential distribution with parameter μk=Nede,kα/pk, i.e., ξk~Expμk, whose cumulative distribution function (CDF) is given by
(6)Fξkξk=1−e−μkξk,ξk>0.


## 2.2. Partial Offloading Model

We assume the task for each user can be partitioned into two parts with an arbitrary ratio. LetLk and lk be the total number of bits to be processed and the number of bits to be offloaded to the AP.We consider a more practical scenario that the eavesdropper is passive and the AP only has the statistical characteristics of eavesdropper’s channels. To this end, we adopt the secrecy outage probability (SOP) for quantifying the quality-of-service (QoS) of secure transmissions.SupposeRs,k is the secrecy rate for transmit power pk that satisfies the targeted secure QoS for user k. Therefore, the energy consumption of offloading can be formulated as
(7)Ekoff=pktkoff=pklkBRs,k,where tkoff and B refer to offloading duration and the bandwidth of the channel, respectively.We present the closed-form expression of SOP in form of theorem as follows.Theorem 1.
The secrecy outage probability of userk for the required secrecy rate Rs,k at the presence of n-user interference can be expressed as
(8)Pso,kn,Rs,k=γN−n,λk2Rs,k−1ΓN−n+λkλk+μk2−Rs,kN−neμk1−2−Rs,k×ΓN−n,λk+μk2−Rs,k2Rs,k−1ΓN−n,where
(9)γv,z=∫0zuv−1e−udu,and
(10)Γv,z=∫z∞uv−1e−udu=Γv−γv,z,are the incomplete Gamma function and its complement, respectively.Proof.
See Appendix.


## 2.3. Local Computation Model

LetXk be the computation intensity in CPU cycles per bit for user k. The total number of cycles for computing local part of task is Lk−lkXk. Each user adopts the advanced dynamic frequency and voltage scaling (DVFS) technique to control the energy consumption. According to [1], the energy consumption of a CPU cycle is given by κkfk2, where κk is a constant associated with the hardware architecture, and fk is the CPU frequency for user k. For the local computation task of Lk−lkXk cycles, the energy consumption can be derived:
(11)Ekloc=κkLk−lkXkfk2.The corresponding computation time for userk can be given by
(12)tkloc=Lk−lkXkfk.


## 3. Sum-Energy Consumption Minimization

### 3.1. Problem Formulation

In this section, we focus on the problem of the sum-energy consumption minimization over transmit powerpk, offloading bits lk, secrecy offloading rate Rs,k, and CPU frequency fk, subject to the task AkLk,T,Xk for each user.By convention, we ignore the time of the data processing at the MEC as well as that of downlink transmission, due to the fact that the MEC processing speed is very fast and the processed result usually has fewer bits. Without loss of generality, the AP decodes the users’ signals in the SIC order of user 1, user 2, ..., userK. As such, the number of user interferers for user k is n=K−k.Mathematically, the minimization problem of sum-energy consumption can be formulated as(13a)P1:minf,l,p,Rs∑k=1KκkLk−lkXkfk2+pklkBRs,k,(13b)s.t.Lk−lkXkfk≤T,∀k∈K,(13c)lkBRs,k≤T,∀k∈K,(13d)Pso,kK−k,Rs,k≤ϵ,∀k∈K,(13e)0≤lk≤Lk,∀k∈K,(13f)fk≤fkmax∀k∈K,where f=f1,f2,⋯,fK, l=l1,l2,⋯,lK, p=p1,p2,⋯,pK, and Rs=Rs,1,Rs,2,⋯,Rs,K refer to the CPU frequency vector, the offloading bit vector, the transmit power vector, and secrecy offloading rate vector, respectively; T is the computation latency, and fkmax denotes the upper bound of CPU frequency for user k.


### 3.2. Solution to Problem (P1)

The problem (P1) is complicated and nonconvex due to the nonconvex nature of constraints (13d). Although the closed-form expression of the optimal solution is not available, we can obtain a suboptimal solution numerically by (1) simplifying constraints, (2) relaxing and transforming the multivariable problem into a single variable problem, and (3) giving the solution numerically.We first simplify the constraints by the following lemma.Lemma 2.
The constraint (13b) and (13d) are strictly binding for the optimal solution of problem (P1), i.e.,
(14)fk=Lk−lkXkT,∀k∈K,(15)Pso,kK−k,Rs,k=ϵ,∀k∈K.Proof.
Observing the first term in (13a), the local computation energy consumption is an increasing function of fk. Obviously, the lowest CPU frequency that satisfies the condition achieves the minimum energy consumption.
According to the property of the SOP,Pso,kn,Rs,k is an increasing function of Rs,k. The maximum Rs,k that satisfies (13d) is the ϵ-outage secrecy capacity Cs,knϵ which makes Pso,kn,Cs,knϵ=ϵ hold. This completes the proof.The problem (P1) in (13a) is still complex and nonconvex. We continue to transform the problem by relaxing offloading duration with T, i.e.,
(16a)P1.1:minl,p,Rs∑k=1KκkXk3Lk−lk3T2+pkT,(16b)s.t.Rs,k≥lkBT,∀k∈K,(16c)Pso,kK−k,Rs,k=ϵ,∀k∈K,(16d)Lk−TfkmaxXk+≤lk≤Lk,∀k∈K,where ·+=max·,0.Since the transmit powerpk increases with Rs,k, Rs,k=lk/BT achieves the minimum energy consumption for fixing other parameters. Moreover, pk can be expressed as a function of Rs,k from (16c). As such, we can further transform the problem into the one only having a single variable.
(17a)P1.2:minl∑k=1KκkXk3Lk−lk3T2+PkK−k,lkBTT,(17b)s.t.Lk−TfkmaxXk+≤lk≤Lk,∀k∈K,where Pkn,Rs,k is the expression of the function for pk derived from Pso,kn,Rs,k=ϵ.Although we cannot get a closed-form solution to this problem, the optimal solution can be found numerically. The numerical solution to problemP1.2 is shown in Algorithm 1. Obviously, the solution to problem P1.2 is equivalent to minimize the energy consumption of each user over offloading bit lk independently. We will demonstrate it numerically in detail in Section.Algorithm 1: Optimal solution to problem (P1).
1: Setting: T, B, ϵ, N, K, α, Na, Ne;
2: Repeat
3:Setting: fkmax, κk, Xk, Lk, da,k, de,k;
4:Initialization: lk;
5:Repeat
6:Repeat
7:   Searchpk with certain lk via constraint of (13d);
8:Until SOP converges to ϵ within a prescribed accuracy;
9:  CalculateEkloc+Ekoff;
10:Untillk=Lk
11: Searchlk∗ to make Ekloc+Ekoff the least;
12: Calculatefk∗, Rs,k∗, pk∗;
13:Output: lk∗, fk∗, Rs,k∗, pk∗;
14:Untilk=K


### 3.3. Discussion

#### 3.3.1. Local Computation Only Mode

The energy consumption of local computation only mode for userk is κkXk3Lk3/T2 by setting lk=0. However, the local computation only mode is not always available unless fkmax≥LkXk/T is satisfied.


#### 3.3.2. Full Offloading Mode

Similarly, the full offloading mode is not always achievable as given the SOP requirement each userk has its asymptotic secrecy rate, which is explained by the following theorem.Theorem 3.
Given the SOP ofϵ, the limited value of secrecy rate Rs,k for user k at the presence of n-user interference is logμk/λk+1/N−nlogϵ−log1−ϵN−n.Proof.
p
x
⟶
∞ makes μk and λ approach to 0 while their ratio keep as a limited value. Then, the asymptotic secrecy outage probability for (8) can be formulated as
(18)Pso,kn,Rs,k⟶pxa.s.11+μk/λk2−Rs,kN−n.
In turn, the asymptotic -outage secrecy capacity is(19)Cs,knϵ⟶pxa.s.logμkλk+1N−nlogϵ−log1−ϵN−n.
This completes the proof.IfNa=Ne, the asymptotic-outage secrecy capacity can be further expressed as
(20)Cs,knϵ⟶pxa.s.αlogde,kda,k+1N−nlogϵ−log1−ϵN−n.To achieve full offloading mode, the following inequality must be satisfied.(21)Lk<BTαlogde,kda,k+1N−K+klogϵ1−ϵN−K+k.


#### 3.3.3. Computational Complexity

Observing Algorithm1, there roughly exist five nesting steps in the process of numerical-finding: (1) calculate SOP with variable Rs,k and find the ϵ-outage secrecy capacity; (2) calculate ϵ-outage secrecy capacity with variable transmit power and search optimal transmit power with certain lk; (3) calculate total energy consumption and search for the optimal lk to achieve minimum total energy consumption; (4) loop from user 1 to user K. Therefore, the computational complexity is OKm3 (here, m refers to the computational complexity of SOP.).


## 3.1. Problem Formulation

In this section, we focus on the problem of the sum-energy consumption minimization over transmit powerpk, offloading bits lk, secrecy offloading rate Rs,k, and CPU frequency fk, subject to the task AkLk,T,Xk for each user.By convention, we ignore the time of the data processing at the MEC as well as that of downlink transmission, due to the fact that the MEC processing speed is very fast and the processed result usually has fewer bits. Without loss of generality, the AP decodes the users’ signals in the SIC order of user 1, user 2, ..., userK. As such, the number of user interferers for user k is n=K−k.Mathematically, the minimization problem of sum-energy consumption can be formulated as(13a)P1:minf,l,p,Rs∑k=1KκkLk−lkXkfk2+pklkBRs,k,(13b)s.t.Lk−lkXkfk≤T,∀k∈K,(13c)lkBRs,k≤T,∀k∈K,(13d)Pso,kK−k,Rs,k≤ϵ,∀k∈K,(13e)0≤lk≤Lk,∀k∈K,(13f)fk≤fkmax∀k∈K,where f=f1,f2,⋯,fK, l=l1,l2,⋯,lK, p=p1,p2,⋯,pK, and Rs=Rs,1,Rs,2,⋯,Rs,K refer to the CPU frequency vector, the offloading bit vector, the transmit power vector, and secrecy offloading rate vector, respectively; T is the computation latency, and fkmax denotes the upper bound of CPU frequency for user k.


## 3.2. Solution to Problem (P1)

The problem (P1) is complicated and nonconvex due to the nonconvex nature of constraints (13d). Although the closed-form expression of the optimal solution is not available, we can obtain a suboptimal solution numerically by (1) simplifying constraints, (2) relaxing and transforming the multivariable problem into a single variable problem, and (3) giving the solution numerically.We first simplify the constraints by the following lemma.Lemma 2.
The constraint (13b) and (13d) are strictly binding for the optimal solution of problem (P1), i.e.,
(14)fk=Lk−lkXkT,∀k∈K,(15)Pso,kK−k,Rs,k=ϵ,∀k∈K.Proof.
Observing the first term in (13a), the local computation energy consumption is an increasing function of fk. Obviously, the lowest CPU frequency that satisfies the condition achieves the minimum energy consumption.
According to the property of the SOP,Pso,kn,Rs,k is an increasing function of Rs,k. The maximum Rs,k that satisfies (13d) is the ϵ-outage secrecy capacity Cs,knϵ which makes Pso,kn,Cs,knϵ=ϵ hold. This completes the proof.The problem (P1) in (13a) is still complex and nonconvex. We continue to transform the problem by relaxing offloading duration with T, i.e.,
(16a)P1.1:minl,p,Rs∑k=1KκkXk3Lk−lk3T2+pkT,(16b)s.t.Rs,k≥lkBT,∀k∈K,(16c)Pso,kK−k,Rs,k=ϵ,∀k∈K,(16d)Lk−TfkmaxXk+≤lk≤Lk,∀k∈K,where ·+=max·,0.Since the transmit powerpk increases with Rs,k, Rs,k=lk/BT achieves the minimum energy consumption for fixing other parameters. Moreover, pk can be expressed as a function of Rs,k from (16c). As such, we can further transform the problem into the one only having a single variable.
(17a)P1.2:minl∑k=1KκkXk3Lk−lk3T2+PkK−k,lkBTT,(17b)s.t.Lk−TfkmaxXk+≤lk≤Lk,∀k∈K,where Pkn,Rs,k is the expression of the function for pk derived from Pso,kn,Rs,k=ϵ.Although we cannot get a closed-form solution to this problem, the optimal solution can be found numerically. The numerical solution to problemP1.2 is shown in Algorithm 1. Obviously, the solution to problem P1.2 is equivalent to minimize the energy consumption of each user over offloading bit lk independently. We will demonstrate it numerically in detail in Section.Algorithm 1: Optimal solution to problem (P1).
1: Setting: T, B, ϵ, N, K, α, Na, Ne;
2: Repeat
3:Setting: fkmax, κk, Xk, Lk, da,k, de,k;
4:Initialization: lk;
5:Repeat
6:Repeat
7:   Searchpk with certain lk via constraint of (13d);
8:Until SOP converges to ϵ within a prescribed accuracy;
9:  CalculateEkloc+Ekoff;
10:Untillk=Lk
11: Searchlk∗ to make Ekloc+Ekoff the least;
12: Calculatefk∗, Rs,k∗, pk∗;
13:Output: lk∗, fk∗, Rs,k∗, pk∗;
14:Untilk=K


## 3.3. Discussion

### 3.3.1. Local Computation Only Mode

The energy consumption of local computation only mode for userk is κkXk3Lk3/T2 by setting lk=0. However, the local computation only mode is not always available unless fkmax≥LkXk/T is satisfied.


### 3.3.2. Full Offloading Mode

Similarly, the full offloading mode is not always achievable as given the SOP requirement each userk has its asymptotic secrecy rate, which is explained by the following theorem.Theorem 3.
Given the SOP ofϵ, the limited value of secrecy rate Rs,k for user k at the presence of n-user interference is logμk/λk+1/N−nlogϵ−log1−ϵN−n.Proof.
p
x
⟶
∞ makes μk and λ approach to 0 while their ratio keep as a limited value. Then, the asymptotic secrecy outage probability for (8) can be formulated as
(18)Pso,kn,Rs,k⟶pxa.s.11+μk/λk2−Rs,kN−n.
In turn, the asymptotic -outage secrecy capacity is(19)Cs,knϵ⟶pxa.s.logμkλk+1N−nlogϵ−log1−ϵN−n.
This completes the proof.IfNa=Ne, the asymptotic-outage secrecy capacity can be further expressed as
(20)Cs,knϵ⟶pxa.s.αlogde,kda,k+1N−nlogϵ−log1−ϵN−n.To achieve full offloading mode, the following inequality must be satisfied.(21)Lk<BTαlogde,kda,k+1N−K+klogϵ1−ϵN−K+k.


### 3.3.3. Computational Complexity

Observing Algorithm1, there roughly exist five nesting steps in the process of numerical-finding: (1) calculate SOP with variable Rs,k and find the ϵ-outage secrecy capacity; (2) calculate ϵ-outage secrecy capacity with variable transmit power and search optimal transmit power with certain lk; (3) calculate total energy consumption and search for the optimal lk to achieve minimum total energy consumption; (4) loop from user 1 to user K. Therefore, the computational complexity is OKm3 (here, m refers to the computational complexity of SOP.).


## 3.3.1. Local Computation Only Mode

The energy consumption of local computation only mode for userk is κkXk3Lk3/T2 by setting lk=0. However, the local computation only mode is not always available unless fkmax≥LkXk/T is satisfied.


## 3.3.2. Full Offloading Mode

Similarly, the full offloading mode is not always achievable as given the SOP requirement each userk has its asymptotic secrecy rate, which is explained by the following theorem.Theorem 3.
Given the SOP ofϵ, the limited value of secrecy rate Rs,k for user k at the presence of n-user interference is logμk/λk+1/N−nlogϵ−log1−ϵN−n.Proof.
p
x
⟶
∞ makes μk and λ approach to 0 while their ratio keep as a limited value. Then, the asymptotic secrecy outage probability for (8) can be formulated as
(18)Pso,kn,Rs,k⟶pxa.s.11+μk/λk2−Rs,kN−n.
In turn, the asymptotic -outage secrecy capacity is(19)Cs,knϵ⟶pxa.s.logμkλk+1N−nlogϵ−log1−ϵN−n.
This completes the proof.IfNa=Ne, the asymptotic-outage secrecy capacity can be further expressed as
(20)Cs,knϵ⟶pxa.s.αlogde,kda,k+1N−nlogϵ−log1−ϵN−n.To achieve full offloading mode, the following inequality must be satisfied.(21)Lk<BTαlogde,kda,k+1N−K+klogϵ1−ϵN−K+k.


## 3.3.3. Computational Complexity

Observing Algorithm1, there roughly exist five nesting steps in the process of numerical-finding: (1) calculate SOP with variable Rs,k and find the ϵ-outage secrecy capacity; (2) calculate ϵ-outage secrecy capacity with variable transmit power and search optimal transmit power with certain lk; (3) calculate total energy consumption and search for the optimal lk to achieve minimum total energy consumption; (4) loop from user 1 to user K. Therefore, the computational complexity is OKm3 (here, m refers to the computational complexity of SOP.).


## 4. Computation Latency Minimization

### 4.1. Problem Formulation

We continue to study the optimization problem of task computation latency over transmit powerpk, offloading bits lk, secrecy offloading rate Rs,k, and CPU frequency fk for user k with the constraint of certain energy budget in this section.All the assumptions and notations are the same as in section. We form the problem as follows(22a)P2:minfk,lk,Rs,k,pkmaxtkloc,tkoff.(22b)s.t.κkLk−lkXkfk2+pklkBRs,k≤Ekbu,(22c)Pso,kK−k,Rs,k≤ϵ,(22d)0≤lk≤Lk,(22e)fk≤fkmax,where tkloc=Lk−lkXk/fk and tkoff=lk/BRs,k according to the definitions in section, and Ekbu refers to the energy budget of user k for its task.


### 4.2. Solution to Problem (P2)

Before solving problem (P2) numerically, we need to simplify the problem via the following lemma.Lemma 4.
The optimal solution of the variablesfk, lk, Rs,k, and pk should make tkloc=tkoff.Proof.
We prove this lemma via contradiction. Letfk∗, lk∗, Rs,k∗, and pk∗ be the jointly optimal values of fk, lk, Rs,k, and pk. We assume Lk−lk∗Xk/fk∗<lk∗/BRs,k∗. One can find by keeping pk and Rs,k fixed, reducing lk makes tkoff and Ekoff decline. Although Ekloc increases with lk decreasing (fk fixed), the increase in Ekloc can be compensated by the decrease in Ekoff. Therefore, there exists another fk∗,lk′,Rs,k∗,pk∗, where lk′=lk∗−τk and τk is a small positive value, making Lk−lk′Xk/fk∗≤lk′/BRs,k∗. Similarly, if Lk−lk∗Xk/fk∗>lk∗/BRs,k∗ is supposed, there exists another fk∗,lk",Rs,k∗,pk∗ that makes Lk−lk"Xk/fk∗≥lk"/BRs,k∗ hold, where lk"=lk∗+τk.
We complete the proof.Combined with Lemma2 and Lemma 4, we transform the problem (P2) into the following form.
(23a)P2.1:minlkTk,(23b)s.t.κkXk3Lk−lk3Tk2+PkK−k,lkBTkTk≤Ekbu,(23c)Lk−TkfkmaxXk+≤lk≤Lk,The problem is changed into the form with a single variable, and we can find the optimal solution numerically, the algorithm of which is shown in Algorithm2.Different from Algorithm1, Algorithm 2 has one more nesting step in the process of numerical-finding: Search for the minimum Tk for the fixed lk, which makes its computational complexity be OKm4.Algorithm 2: Optimal solution to problem (P2).
1: Setting: B, ϵ, N, K, α, Na, Ne;
2: Repeat
3:Setting: fkmax, κk, Xk, Lk, da,k, de,k, Ekbu;
4:Initialization: lk, Tk;
5:Repeat
6:Repeat
7:Repeat
8:    Searchpk with certain lk and Tk via constraint of (22c);
9:Until SOP converges to ϵ within a prescribed accuracy;
10:   CalculateEkloc+Ekoff;
11:UntilTk satisfies the condition in (23b)
12:Untillk=Lk
13: Searchlk∗ to make Tk the least;
14: Calculatefk∗, Rs,k∗, pk∗;
15:Output: lk∗, fk∗, Rs,k∗, pk∗;
16:Untilk=K


## 4.1. Problem Formulation

We continue to study the optimization problem of task computation latency over transmit powerpk, offloading bits lk, secrecy offloading rate Rs,k, and CPU frequency fk for user k with the constraint of certain energy budget in this section.All the assumptions and notations are the same as in section. We form the problem as follows(22a)P2:minfk,lk,Rs,k,pkmaxtkloc,tkoff.(22b)s.t.κkLk−lkXkfk2+pklkBRs,k≤Ekbu,(22c)Pso,kK−k,Rs,k≤ϵ,(22d)0≤lk≤Lk,(22e)fk≤fkmax,where tkloc=Lk−lkXk/fk and tkoff=lk/BRs,k according to the definitions in section, and Ekbu refers to the energy budget of user k for its task.


## 4.2. Solution to Problem (P2)

Before solving problem (P2) numerically, we need to simplify the problem via the following lemma.Lemma 4.
The optimal solution of the variablesfk, lk, Rs,k, and pk should make tkloc=tkoff.Proof.
We prove this lemma via contradiction. Letfk∗, lk∗, Rs,k∗, and pk∗ be the jointly optimal values of fk, lk, Rs,k, and pk. We assume Lk−lk∗Xk/fk∗<lk∗/BRs,k∗. One can find by keeping pk and Rs,k fixed, reducing lk makes tkoff and Ekoff decline. Although Ekloc increases with lk decreasing (fk fixed), the increase in Ekloc can be compensated by the decrease in Ekoff. Therefore, there exists another fk∗,lk′,Rs,k∗,pk∗, where lk′=lk∗−τk and τk is a small positive value, making Lk−lk′Xk/fk∗≤lk′/BRs,k∗. Similarly, if Lk−lk∗Xk/fk∗>lk∗/BRs,k∗ is supposed, there exists another fk∗,lk",Rs,k∗,pk∗ that makes Lk−lk"Xk/fk∗≥lk"/BRs,k∗ hold, where lk"=lk∗+τk.
We complete the proof.Combined with Lemma2 and Lemma 4, we transform the problem (P2) into the following form.
(23a)P2.1:minlkTk,(23b)s.t.κkXk3Lk−lk3Tk2+PkK−k,lkBTkTk≤Ekbu,(23c)Lk−TkfkmaxXk+≤lk≤Lk,The problem is changed into the form with a single variable, and we can find the optimal solution numerically, the algorithm of which is shown in Algorithm2.Different from Algorithm1, Algorithm 2 has one more nesting step in the process of numerical-finding: Search for the minimum Tk for the fixed lk, which makes its computational complexity be OKm4.Algorithm 2: Optimal solution to problem (P2).
1: Setting: B, ϵ, N, K, α, Na, Ne;
2: Repeat
3:Setting: fkmax, κk, Xk, Lk, da,k, de,k, Ekbu;
4:Initialization: lk, Tk;
5:Repeat
6:Repeat
7:Repeat
8:    Searchpk with certain lk and Tk via constraint of (22c);
9:Until SOP converges to ϵ within a prescribed accuracy;
10:   CalculateEkloc+Ekoff;
11:UntilTk satisfies the condition in (23b)
12:Untillk=Lk
13: Searchlk∗ to make Tk the least;
14: Calculatefk∗, Rs,k∗, pk∗;
15:Output: lk∗, fk∗, Rs,k∗, pk∗;
16:Untilk=K


## 5. Numerical Results

In this section, numerical results are presented to further validate the previous research results of the energy-efficient computational offloading design in this secure uplink NOMA-enabled MEC networks.To demonstrate the numerical results more clearly, we take a 3-user scenario for instance, the layout of which is shown in Figure2. The distances (grid) between each user and the AP as well as eve are listed in Table 2. Each grid is supposed to be 20 meters, thus, the distance from user 1 to the AP is 80 meters and about 152.3 meters to eve.Figure 2
A 3-user scenario of secure uplink NOMA MEC system.Table 2
Distances between each user and AP as well as eve.
Distance (grid)
User 1
User 2
User 3
AP
4
13
4
Eve
58
37
10.We assume the number of antennas equipped at the AP is 10, i.e.,N=10, the path loss in dB is expressed as PL=10αlgd+43.5, where the path loss exponent α is set to 3.76 according to 3GPP urban path loss model. The bandwidth B=10 MHz and the equal AWGN power of the AP and eve are supposed, i.e., Na=Ne=N0B, where N0=−160 dBm/Hz is the AWGN power spectral density.For simplicity, we assume each end user has the same performance and tasks to process. The total bits of task for each user isL1=L2=L3=3 Mbits. The computation intensity is X1=X2=X3=50 cycles/bit. The maximum CPU frequency is 2 GHz. The effective capacitance coefficient is set as κ1=κ2=κ3=10−28. Without loss of generality, the SIC order is from user 1 to user 2 to user 3.


### 5.1. Energy Consumption Minimization

Based on the above assumptions, the curves of the energy consumption versus offloading bits (Mbits) for each end user are shown in Figure3, where the computation latency T=0.15 s and the secrecy outage probability ϵ=0.2.Figure 3
Energy consumption versus offloading bits for each user at the assumption ofT=0.15 s and ϵ=0.2We note each curve starts from the same value (0.015 Joule) due to the similar parameter assumption for each user, goes down until the minimum energy consumption, then rises up, with the increase of offloading bits.lk=0 means local computation only mode, and lk=3 Mbits specifies full offloading mode. In this case, both modes cannot obtain the minimum energy consumption. The minimum energy consumptions for user 1, user 2, and user 3 are 0.0096 Joule, 0.0071 Joule, and 0.012 Joule, respectively. The corresponding offloading bits are l1=1 Mbits, l2=1.3 Mbits, and l3=0.6 Mbits. Here, the resolution of offloading bits is 0.1 Mbits.Although user 1 and user 3 have the same distances to the AP, due to the different distances to eve and the different SIC order, which result in the different secrecy outage capacities, user 1 has less energy consumption than user 3. Specifically, user 3 needs 0.0024 Joule more energy to overcome the distance difference of eve. Yet, user 2 has the least energy consumption even if it is not SIC decoded at last.By the way, each user has an asymptoticϵ-secrecy outage capacity. They are Cs,12ϵ=0.2=5.6589 bps/Hz, Cs,21ϵ=0.2=5.1894 bps/Hz, and Cs,30ϵ=0.2=1.2429 bps/Hz. The asymptotic offloading bits for user 3 is about 1.86 Mbits, which indicates user 3 cannot conduct full offloading mode as the total bits of the task are 3 Mbits.We continue to study the impacts of the latencyT and the secrecy outage probability ϵ on the minimum energy consumptions for each user and their sum.We first illustrate the curves of the minimum energy consumption versus latency fromT=0.1 s to T=0.2 s in Figure 4. For each user, the minimum energy consumption decreases with the increase of T. When T=0.1 s, the minimum energy consumptions for user 1, user 2, user 3, and their sum are 0.015 Joule, 0.011 Joule, 0.023 Joule, and 0.049 Joule, respectively, while they are reduced to 0.007 Joule, 0.0054 Joule, 0.0078 Joule, and 0.0202 Joule when T=0.2 s. The values for T=0.15 s are the same as those achieved in Figure 3, which also confirms the reliability of the numerical results in Figure 4. As expected, user 2 has the least energy consumptions for the same T, while user 3 needs more energy consumption than user 1, just as in Figure 3.Figure 4
Minimum energy consumption versus latency for each user and their sum from 0.1 s to 0.2 s.Figure5 depicts the curves of the minimum energy consumptions versus secrecy outage probability for each user and their sum from ϵ=0.1 to ϵ=0.5. It is easy to confirm the reliability of the numerical results in Figure 5 by observing the values for ϵ=0.2. Similar to the impact of latency T, the minimum energy consumptions decline with the secrecy outage probability increasing from 0.0106 Joule, 0.0079 Joule, 0.0142 Joule, 0.0327 Joule for ϵ=0.1 to 0.008 Joule, 0.0058 Joule, 0.0087 Joule, and 0.0225 Joule for ϵ=0.5. We note the minimum energy consumption for user 3 at ϵ=0.1 (0.0142 Joule) is very close to that of local computation only mode (0.015 Joule), which points out offloading is not always necessary especially when the eavesdropper’s channel is much better than the legal channel.Figure 5
Minimum energy consumption versus secrecy outage probability for each user and their sum fromϵ=0.1 to ϵ=0.5.


### 5.2. Latency Minimization

Figure6 shows the latency of the three end users versus offloading bits. The curves are generated on the assumption of the secrecy outage probability ϵ=0.2 and the energy budget Ebu=0.03 Joule. The resolutions of the latency and offloading bits are 0.005 s and 0.05 Mbits, respectively. One can note each curve declines first and then climbs up, which indicates there exists an optimal value. In this case, the minimum values of the latency for user 1, user 2, and user 3 are 0.06 s, 0.055 s, and 0.09 s, respectively, which are achieved by setting offloading bit l1∗=1.35 Mbits, l2∗=1.30 Mbits, and l3∗=0.45 Mbits, respectively. As such, f1∗=1375 MHz, f2∗=1545.45 MHz, f3∗=1416.67 MHz; Rs,1∗=2.25 bps/Hz, Rs,2∗=2.36 bps/Hz, Rs,3∗=0.5 bps/Hz; and p1∗=234.4 mW, p2∗=158.8 mW, p3∗=40.6 mW.Figure 6
Latency versus offloading bits for each user at the assumption ofEbu=0.03 Joule and ϵ=0.2.Figure7 shows the minimum latency versus the energy budget from Ebu=0.01 Joule to Ebu=0.1 Joule. As expected, the minimum latency decreases with the energy budget increasing. However, the decrease is significant from Ebu=0.01 Joule to Ebu=0.04 Joule, where the minimum latency for user 1 is reduced from 0.145 s to 0.05 s. For user 3 whose channel condition is not good enough, the reduction is still effective when Ebu is greater than 0.04 Joule. Meanwhile, the curves of user 1 and user 2 evert such that the minimum latency of user 2 is greater than that of user 1 from Ebu=0.05.Figure 7
Minimum latency versus energy budget for each user fromEbu=0.01 Joule to Ebu=0.1 Joule.Here so far, all these numeric results and observations in this section are consistent with the expectations.


## 5.1. Energy Consumption Minimization

Based on the above assumptions, the curves of the energy consumption versus offloading bits (Mbits) for each end user are shown in Figure3, where the computation latency T=0.15 s and the secrecy outage probability ϵ=0.2.Figure 3
Energy consumption versus offloading bits for each user at the assumption ofT=0.15 s and ϵ=0.2We note each curve starts from the same value (0.015 Joule) due to the similar parameter assumption for each user, goes down until the minimum energy consumption, then rises up, with the increase of offloading bits.lk=0 means local computation only mode, and lk=3 Mbits specifies full offloading mode. In this case, both modes cannot obtain the minimum energy consumption. The minimum energy consumptions for user 1, user 2, and user 3 are 0.0096 Joule, 0.0071 Joule, and 0.012 Joule, respectively. The corresponding offloading bits are l1=1 Mbits, l2=1.3 Mbits, and l3=0.6 Mbits. Here, the resolution of offloading bits is 0.1 Mbits.Although user 1 and user 3 have the same distances to the AP, due to the different distances to eve and the different SIC order, which result in the different secrecy outage capacities, user 1 has less energy consumption than user 3. Specifically, user 3 needs 0.0024 Joule more energy to overcome the distance difference of eve. Yet, user 2 has the least energy consumption even if it is not SIC decoded at last.By the way, each user has an asymptoticϵ-secrecy outage capacity. They are Cs,12ϵ=0.2=5.6589 bps/Hz, Cs,21ϵ=0.2=5.1894 bps/Hz, and Cs,30ϵ=0.2=1.2429 bps/Hz. The asymptotic offloading bits for user 3 is about 1.86 Mbits, which indicates user 3 cannot conduct full offloading mode as the total bits of the task are 3 Mbits.We continue to study the impacts of the latencyT and the secrecy outage probability ϵ on the minimum energy consumptions for each user and their sum.We first illustrate the curves of the minimum energy consumption versus latency fromT=0.1 s to T=0.2 s in Figure 4. For each user, the minimum energy consumption decreases with the increase of T. When T=0.1 s, the minimum energy consumptions for user 1, user 2, user 3, and their sum are 0.015 Joule, 0.011 Joule, 0.023 Joule, and 0.049 Joule, respectively, while they are reduced to 0.007 Joule, 0.0054 Joule, 0.0078 Joule, and 0.0202 Joule when T=0.2 s. The values for T=0.15 s are the same as those achieved in Figure 3, which also confirms the reliability of the numerical results in Figure 4. As expected, user 2 has the least energy consumptions for the same T, while user 3 needs more energy consumption than user 1, just as in Figure 3.Figure 4
Minimum energy consumption versus latency for each user and their sum from 0.1 s to 0.2 s.Figure5 depicts the curves of the minimum energy consumptions versus secrecy outage probability for each user and their sum from ϵ=0.1 to ϵ=0.5. It is easy to confirm the reliability of the numerical results in Figure 5 by observing the values for ϵ=0.2. Similar to the impact of latency T, the minimum energy consumptions decline with the secrecy outage probability increasing from 0.0106 Joule, 0.0079 Joule, 0.0142 Joule, 0.0327 Joule for ϵ=0.1 to 0.008 Joule, 0.0058 Joule, 0.0087 Joule, and 0.0225 Joule for ϵ=0.5. We note the minimum energy consumption for user 3 at ϵ=0.1 (0.0142 Joule) is very close to that of local computation only mode (0.015 Joule), which points out offloading is not always necessary especially when the eavesdropper’s channel is much better than the legal channel.Figure 5
Minimum energy consumption versus secrecy outage probability for each user and their sum fromϵ=0.1 to ϵ=0.5.


## 5.2. Latency Minimization

Figure6 shows the latency of the three end users versus offloading bits. The curves are generated on the assumption of the secrecy outage probability ϵ=0.2 and the energy budget Ebu=0.03 Joule. The resolutions of the latency and offloading bits are 0.005 s and 0.05 Mbits, respectively. One can note each curve declines first and then climbs up, which indicates there exists an optimal value. In this case, the minimum values of the latency for user 1, user 2, and user 3 are 0.06 s, 0.055 s, and 0.09 s, respectively, which are achieved by setting offloading bit l1∗=1.35 Mbits, l2∗=1.30 Mbits, and l3∗=0.45 Mbits, respectively. As such, f1∗=1375 MHz, f2∗=1545.45 MHz, f3∗=1416.67 MHz; Rs,1∗=2.25 bps/Hz, Rs,2∗=2.36 bps/Hz, Rs,3∗=0.5 bps/Hz; and p1∗=234.4 mW, p2∗=158.8 mW, p3∗=40.6 mW.Figure 6
Latency versus offloading bits for each user at the assumption ofEbu=0.03 Joule and ϵ=0.2.Figure7 shows the minimum latency versus the energy budget from Ebu=0.01 Joule to Ebu=0.1 Joule. As expected, the minimum latency decreases with the energy budget increasing. However, the decrease is significant from Ebu=0.01 Joule to Ebu=0.04 Joule, where the minimum latency for user 1 is reduced from 0.145 s to 0.05 s. For user 3 whose channel condition is not good enough, the reduction is still effective when Ebu is greater than 0.04 Joule. Meanwhile, the curves of user 1 and user 2 evert such that the minimum latency of user 2 is greater than that of user 1 from Ebu=0.05.Figure 7
Minimum latency versus energy budget for each user fromEbu=0.01 Joule to Ebu=0.1 Joule.Here so far, all these numeric results and observations in this section are consistent with the expectations.


## 6. Conclusion

In this work, we investigated a secure uplink NOMA-enabled MEC network under the threat of an eavesdropper, whereK end users simultaneously offload their partial computation-intensive tasks to the MEC server in the same resource block and ZF-SIC is adopted at the multiantenna AP associated with MEC server. We first derived the closed-form expression of individual SOP for an arbitrary SIC order. We then characterize the optimization problem of sum-energy consumption subject to offloading bits, secrecy offloading rate, local CPU frequency, and transmit power and gave the solution. We further studied the problem of minimizing computation latency under condition of certain SOP requirement and energy budget through proving the equality of local computing time and offloading duration and transforming it into a single-variable problem. All the solutions are demonstrated and validated numerically by a 3-user case of a secure NOMA-enabled MEC network.



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*Source: 5230594-2022-04-27.xml*