I Introduction
Rapid deployment of wireless communication systems has always been engraved with the issue of security of the transmitted data. The basic reason is the broadcast nature of transmission which makes the signals vulnerable to tapping by malicious users [1]. Security of the transmitted signal is generally considered as a responsibility of the higher layers which employ cryptographic techniques. This strategy relies on the basic assumption that the enciphering system is unbreakable by the malicious users [2]. With growing computing capabilities, such measures prove to be insufficient and it has motivated the research community to explore security at the physical layer. Physical layer security finds its basis from independence of the wireless communication channels and has a low implementation complexity [3]. This added layer of security is considered as the strictest kind of security, not requiring even any kind of key exchange [1, 2].
Orthogonal frequency division multiple access (OFDMA) is a potential physical layer technology for the next generation access networks such as WiMAX, LTE, and beyond. Hence, the study of physical layer security in OFDMA has gained considerable attention in recent years [4, 5, 6, 7, 8, 9, 10, 11, 12]. The subcarrier allocation and power optimization in multinode secure OFDMA system has been studied in two cases: in one scenario transmitter assumes all users to be trusted and there exists an external eavesdropper which tries to decode the data of trusted users [4, 5, 6, 7, 8], while in another scenario there is a mutual distrust among the users and a particular user may demand the source to transmit its data considering all other users as the potential eavesdroppers [11, 10, 12].
In trusted users case, the authors in [5] considered maximizing the minimum of weighted sum secure rate of all the users. They proposed a complex MILP based solution and less complex suboptimal schemes. However, the issue of resource scarcity of a user that is very near to the eavesdropper was not highlighted. Maximization of energy efficiency with multiantenna source, eavesdropper, and single antenna users was studied in [6]
with bounds on peruser tolerable secrecy outage probability. The average secrecy outage capacity maximization in the same setup with a multiantenna Decode and Forward relay was studied in
[7]. The authors in [13] studied secure rate maximization under fixed quality of service (QoS) constraints for secure communication among multiple sourcedestination pairs in the presence of multiantenna external eavesdropper, with the help of multiple single antenna Amplify and Forward (AF) relays. An extension of the above proposed schemes to multiple eavesdroppers case may be straight forward, but with untrusted users the problems present an interesting challenge with multiple unexplored facets. Looking at another domain of untrust, subcarrier assignment and power allocation problem was studied in [9] in an AF untrusted relay aided secure communication among multiple sourcedestination pairs.In an effort to tackle the untrusted users, the authors in [10] proposed to allocate a subcarrier to its best gain user in a twouser OFDMA system and presented the optimal power allocation. The scheme can be trivially extended to more than two users for sum rate maximization. In a scenario having heterogeneous demand of resources by users, the authors in [11] proposed a joint subcarrier and power allocation policy for two classes of users: secure users demanding a fixed secure rate and normal users which are served with best effort traffic. In the cognitive radio domain, secure communication over a single carrier, between multiantenna secondary transmitter and a fixed user considering other secondary and primary users as eavesdroppers was studied in [14]. Similarly, precoder design to maximize the sum secrecy rate to achieve confidential broadcast to users with multiple antennas was studied in [15, 16]. The authors in [12] solved the resource allocation problem among multiple sourcedestination pairs in a relayaided scenario with untrusted users. However, in secure OFDMA fair resource distribution among untrusted users poses new challenges which have not yet been investigated.
Ia Motivation
The study in [11] raised the feasibility issue of resource allocation problem because of the channel conditions, i.e., some users may not achieve the required secure rate due to tapping by untrusted users. This motivates us to investigate if the secure rate over a subcarrier could be any how improved. The strategy proposed in [8] utilizes interference to improve the secure rate of a user. But this scheme cannot be used in multiuser untrusted scenario because it relies on a strong assumption that the jammer affects the eavesdropper only. It was shown in [17] that, for single user multiantenna system, secrecy rate can be improved by a multiantenna source by jointly transmitting message in the range space and interference in the null space of the main channel. Since this solution is difficult to apply for single antenna systems, the authors in [18] showed the possibility of achieving a positive rate even when eavesdropper’s channel is stronger compared to main channel by using jamming power control only. The studies in [8, 17, 18] are based on single user system. The implications of utilizing interference in multiuser scenario are yet to be investigated. With regard to fair resource allocation without the help of interference, the strategy proposed in [5], which balances the secure rate among users, can limit the maximum achievable rate of the system because of a poor user facing strong eavesdropping by other users. This leads to system resource wastage. Effectively utilizing jammer power control in multiple untrusted user scenario for sum rate maximization or fair resource allocation raises challenges which to the best of our knowledge has not yet been investigated in the literature.
In this paper, we intend to explore the role of the jammer power in alleviating the following issues which plague single antenna secure OFDMA system with untrusted users:

Secure rate achievable by the users can be very low as compared to the single external eavesdropper case, because now there are wiretappers instead of one;

A large number of users can starve for subcarriers, because a group of users with very good channel gains may prohibit secure communication to the other users.
Assuming the jammer to be a node affecting all the users, we intend to use the jammer power for individual and independent jamming over the subcarriers. The resource allocation problem in presence of jammer is a complex Mixed Integer Non Linear Programming (MINLP) problem belonging to the class of NPhard [19, 20]. Due to the requirement of decision on subcarrier allocation at the source as well as to use jammer power over a subcarrier, the problem exhibits combinatorial nature having exponential complexity with number of subcarriers [21]. Hence, instead of attempting a global optimal solution, we study the resource allocation problem to improve either the sum secure rate or the overall system fairness. While extending our preliminary work in [22]
, where we presented two suboptimal solutions based on sequential source and jammer power allocation, we conduct a deeper study on the behavior of secure rate with jammer power and solve the joint source and jammer power optimization problem. For maxmin fair resource allocation, two novel methodologies of jammer power utilization are introduced. We also present the asymptotic analysis of the algorithms with
and , and discuss the computational complexity of all the proposed schemes under both the usages of jammer.IB Contribution
Our contributions can be summarized as answers to the following two questions:
(1) How to improve sum secure rate in OFDMA systems with untrusted users? To address this question, we consider the possibility of using jammer power over a subcarrier to improve the secure rate beyond what can be achieved by source power only. The features of the proposed solution are as follows:

We obtain the constraints of secure rate improvement over a subcarrier and show that the secure rate in this constrained domain is a quasiconcave function of jammer power, thereby offering a unique maxima.

The jammer aided approach introduces a new challenge  referred as SNR reordering in the rest of the paper  which makes the problem combinatorial even after subcarrier allocation and jammer utilization decision. We introduce the concept of constrainedjamming by developing jammer power bounds to handle this challenge.

We analyze the complexity of the proposed algorithm and show its convergence in finite steps.

We also propose less complex solution which allows a tradeoff between performance and complexity.

Asymptotically optimal solution is derived to assess optimality of the proposed scheme.
(2) How to remove subcarrier scarcity and have fair distribution? In secure OFDMA with untrusted users and in absence of jammer, the best gain user over a subcarrier is the strongest eavesdropper for all other users. Thus, a subcarrier should normally be given to its best gain user, which may cause some users to starve for channel resources. To attain a better fairness, we propose to take away a subcarrier from its best gain user and allocate it to another user with poor channel gain with the help of jammer. The features of the proposed solution are:

Secure rate over the snatched subcarrier is positive when the jammer power is above a certain threshold and attains a maximum value at an optimal jammer power.

The conventional maxmin fairness algorithm cannot be employed in secure OFDMA because a user with poor channel gains over all the subcarriers may force the algorithm into a deadlock. The proposed maxmin fair scheme offers a graceful exit with such users.

Two variants of maxmin fairness algorithms are proposed where the jammer power can be utilized either by preserving it for possible snatching over each subcarrier, or by allocating it based on the demand of snatching user.

The power allocation is done by PD and AO process while the issue of signal to noise ratio (SNR) reordering is handled similarly as in sum rate maximization case.

Low complexity solutions have been presented for both the variants of maxmin fairness algorithms.

We obtain an asymptotic upper bound of the fairness achievable by our proposed scheme.
The remainder of the paper is organized as follows: The system model is introduced in Section II. The sum secure rate maximization problem is studied in Section III, followed by fair resource allocation study in Section IV. Asymptotic analysis has been presented in Section V. Simulation results are discussed in Section VI. Section VII concludes the paper.
Ii System model
We consider the downlink of an OFDMA system with a single source (base station), a friendly jammer, untrusted mobile users (MUs) which are randomly distributed in the cell coverage area, and subcarriers. The jammer, which is considered as another node in the network (e.g., in LTEAdvanced it could be an idle relay node), is controlled by the source to help improve the overall system performance. Jammer is assumed to be capable of collecting channel state information (CSI) in the uplink and sending jamming signal which is unknown to users on the downlink. Since all users request secure communication with the source, they share their CSI with the source as well as the jammer. In the present study we assume the jammer to be located randomly in the cell area.
SourcetoMUs and jammertoMUs channels are considered to experience slow and frequencyflat fading, such that the channel parameters remain constant over a frame duration but vary randomly from one frame to another. Perfect CSI of sourcetoMU and jammertoMU channel pairs for all the MUs is assumed to be available at source [4, 11, 6, 7, 5, 8, 9, 10]. Source utilizes this information for subcarrier allocation, and source and jammer power allocation. We consider that a subcarrier is exclusively allocated to one user only, which has been proved to be optimal for sum secure rate maximization [10].
The secure OFDMA system with untrusted users is a multiple eavesdropper scenario, where for each main user there exist eavesdroppers. Out of these eavesdroppers, the strongest one is considered as the equivalent eavesdropper (hereafter referred as eavesdropper). The secure rate of the main user over a subcarrier is defined as the nonnegative capacity difference between the main user and the eavesdropper. Let be the source to th user channel coefficient and be the jammer to th user channel coefficient over subcarrier as shown in Fig. 1. Then, the secure rate of user on th subcarrier is given as [10, 11]:
(1) 
where ,
is the AWGN noise variance,
and are respectively source and jammer powers over subcarrier . is an indicator of absence or presence of jammer power on subcarrier such that , and is the eavesdropper. The nonlinearity of secure rate in source and jammer powers along with the operator complicate the optimal resource allocation which will be discussed in the Section III and IV.Iii Resource allocation for sum secure rate maximization
In this section we discuss the joint source and jammer resource optimization problem for weighted sum secure rate maximization. The problem can be stated as:
subject to  
(2) 
where is a binary allocation variable to indicate whether subcarrier is given to user or not, is the priority weight allocated by the higher layers to user , and and are source and jammer power budgets, respectively. and are budget constraints, is jammer allocation constraint, and are subcarrier allocation constraints, and denotes source and jammer power boundary constraints.
The optimization problem in (III) has total four variables per subcarrier: ,
as binary variables; and
and as continuous variables. Since the problem is a nonconvex combinatorial problem belonging to the class of NPhard, there is no polynomial time optimal solution possible [19, 20]. We tackle the problem by breaking it in parts and attempt to find a nearoptimal solution, which approaches asymptotically optimal solution (discussed in Section V) as and increases. For this, first we perform subcarrier allocation without considering the jammer. Next we distinguish those subcarriers over which jammer can improve the secure rate, and finally we complete the joint power allocation.1.1027  0.3856  0.6719  1.2101  0.7043  
0.7423  1.0735  0.6558  1.0006  0.8943  
0.7554  1.4772  0.2498  1.3572  3.5391 
3.3624  6.0713  3.4125  3.0584  0.4987  
8.1741  7.0607  4.1047  0.9860  1.6860  
0.9028  2.0636  0.5605  3.0277  4.5346 
Iiia Subcarrier allocation at source
In presence of jammer, subcarrier allocation at source, i.e., decision is difficult. In secure OFDMA with untrusted users a subcarrier can only be given to the user having maximum SNR over the subcarrier. Any change in may force the decision of subcarrier allocation to change because of the possible reordering of updated SNRs (cf. (II)). Since is in numerator, any change in does not affect the SNR ordering. This makes the problem combinatorial, as we may need to check subcarrier allocation for every update of . In order to bypass this step, we initially assume that the jammer is not present. In absence of jammer , and the secure rate definition in (II) changes to
(3) 
As observed in (IIIA), is required to have positive secure rate over subcarrier , irrespective of . Thus, the subcarrier allocation policy, allocating a subcarrier to its best gain user can be stated as:
(4) 

Note that the subcarrier allocation (4) depends only on users’ gains. It is indifferent to users’ priority imposed through weights which will play their role in power allocation.
IiiB Subcarrier allocation at jammer and jammer power bounds
In the presence of jammer, even after subcarrier allocation at source is done, the problem is still combinatorial due to . Any change in may even jeopardize the earlier decision on as described in the previous section. Inorder to solve this issue, we first introduce the concept of rate improvement, which will help us decide retaining the decision on .
IiiB1 Selective jamming for secure rate improvement
Since jammer affects all the users, it appears that using jammer over a subcarrier may degrade the secure rate. But it is interesting to note that, with jammer power the secure rate can be improved beyond what is achieved without jammer. For the proof of concept, let us consider a simple OFDMA scenario having four nodes: source, jammer, and two users and . Let a subcarrier be allocated to user , and be the eavesdropper. The following proposition describes the possibility of secure rate improvement and the existence of optimal jammer power achieving maximum secure rate over subcarrier .
Proposition III.1.
The secure rate over a subcarrier having can be improved if and the source and jammer powers and are constrained as
and  (5) 
where and .
In the constrained domain of rate improvement, the rate is a quasiconcave function of having a unique maxima.
Proof.
See Appendix A. ∎
Example (continued): Observing the source gains in Table I, the best gain users of the subcarriers are , respectively. Users are their corresponding eavesdroppers. Subcarriers satisfy the condition of secure rate improvement, i.e., . Considering the unit of transmit power is in Watt, let , , and . Source power thresholds for the three subcarriers are , respectively. Assuming equal source power allocation over all subcarriers () and comparing with , subcarriers can be utilized for secure rate improvement while cannot be used. The corresponding jammer power thresholds for are respectively found as . Variation of SNRs of the users and the secure rate of user on subcarrier with jammer power are presented in Table III. As observed, with is higher compared to the value when , till . has a maxima between and , at .
0.1487  0.0317  0.0178  0.0033  0.0030  
1.1524  0.1925  0.1050  0.0189  0.0175  
2.1821  1.5304  1.1784  0.3571  0.3339  
0.6988  1.5518  1.4720  0.7239  0.6882 
Based on the results described in Proposition III.1, we create a prospective set of subcarriers, over which the jammer power can be used for secure rate improvement. The subcarrier allocation policy at jammer can be summarized as:
(6) 
While extending the result of Proposition III.1 to , there is an inherent challenge associated with the allocation of , which we refer as SNR reordering. Without loss of generality, let the channel gains from sourcetoMUs be sorted as over a subcarrier such that user 1 is assigned the subcarrier and user 2 is the eavesdropper. While optimizing source and jammer powers jointly, any update in does not disturb the resource assignments, but as is updated it raises the following two concerns:
(i) User 1 may not have the maximum SNR.
(ii) User 2 may not remain the corresponding eavesdropper.
Because of this SNR reordering challenge, for every update a new main user and the corresponding eavesdropper are to be determined. This is the problem introduced by the operator appearing in the rate definition
(II), which makes the joint source and jammer power allocation a tedious task even after and allocation.
In order to address this challenge we develop a strategy as described below.
IiiB2 Bounds on jammer power to avoid SNR reordering
In order to handle the operator, we enforce certain constraints over jammer power so as to retain the same main user and the same eavesdropper throughout the jammer power allocation. In order to retain the eavesdropper, the rate of user should be larger than all other possible eavesdroppers, i.e.,
(7) 
Similarly, to preserve the main user we need to have:
(8) 
The constraints in (7) and (8) depend on channel conditions only, which result in lower and upper bounds over such that . Since (III.1) enforces another upper bound on , the final lower and upper bounds are given as
(9) 
Example (continued): The SNR reordering issue can be explained with the help of jammer power variation over subcarrier . The respective SNRs of the users with variation are presented in Table IV. From to , and are respectively the main user and the eavesdropper. At , the eavesdropper changes from to and this reduces the secure rate from to . When increases to , even the main user changes from to , and now is the corresponding eavesdropper. Now, the secure rate of is , while that of is . Though the jammer power threshold guarantees that, if , the secure rate remains greater than that without jammer (), this assertion relies on the basic assumption that the main user and the eavesdropper do not change. The jammer power threshold, optimum jammer power, and its upper bound to avoid SNR reordering on are respectively , , and . As soon as , the SNR order changes and is also reduced. This issue did not arise in as there was no finite jammer power upper bound .
After decision of and , combinatorial aspect of the problem is resolved if remains within bounds. Next we consider the joint source and jammer power allocation.
0.4514  0.2086  0.0798  0.0662  0.0493  
0.4301  0.1602  0.0556  0.0456  0.0336  
0.0624  0.0605  0.0554  0.0539  0.0512  
0.0328  0.1020  0.0616  0.0315  0.0000 
IiiC Joint optimization of source and jammer power
After the decision on utilization of jammer power based on Proposition III.1, all subcarriers can be categorized in two sets:  the ones that do not use jammer power (), and  the others that use jammer power (). Let and respectively denote the SNRs of the user , without jammer power and with jammer power over subcarrier , i.e.,
(10) 
The joint power allocation problem can be stated as
subject to  
(11) 
where is the weight of the main user over the th subcarrier. and are the power budget constraints. Constraints and are imposed to tackle the SNR reordering challenge (cf. Section IIIB2), while the boundary constraints for source power are captured in .
The source power has to be shared among the subcarriers’ set and , while the jammer power has to be allocated on the subcarriers of set only.
We observe that source power is a coupling variable between the power allocation problem over the complementary sets and , and is the corresponding complicating constraint.
Since the secure rate over a subcarrier is an increasing function of source power, the problem has to be solved at full source power budget .
The optimization problem (IIIC) can be solved using
primal decomposition (PD) procedure by dividing it into one master problem (outer loop) and two subproblems (inner loop) [23].
The first subproblem is source power allocation over set , and second is joint source and jammer power allocation over set .
The subproblem1 can be stated as:
Subproblem1
subject to  
(12) 
where , representing the coupling variable, is the source power budget allotted to subproblem1.
The source power budget to subproblem2 is , as described in the following:
Subproblem2
subject to  
(13) 
The two subproblems are solved independently for a fixed . Once the solutions of both the subproblems are obtained, the master problem is solved using subgradient method by updating the coupling variable as , where is an appropriate step size [25] and , are the Lagrange multipliers [23] corresponding to the source power constraints in subproblems (IIIC) and (IIIC), respectively. This procedure of updating is repeated until either the sum secure rate saturates or the iteration count exceeds a threshold.
IiiC1 Solution of subproblem1
Since each subcarrier is allocated to the best gain user (cf. Section IIIA), . Thus, the objective function in (IIIC) is a concave function of . The optimal obtained after solving the KarushKuhnTucker (KKT) conditions is given as
(14) 
where is defined as follows:
(15) 
, associated with , is found such that .
IiiC2 Solution of subproblem2
In order to solve the subproblem2, we observe that the secure rate over the subcarriers of set is a concave function of source power for a fixed jammer power , and also a quasiconcave function of for a fixed as shown in Proposition III.1. This motivates us to use the method of alternating optimization (AO) [24] for joint power optimization, which alternates between source and jammer power optimizations. The AO method starts with , i.e., equal power over all the subcarriers of set , and for the known optimal allocation is done. For a fixed source power, the jammer power allocation problem can be stated as:
subject to  
(16) 
Since the secure rate over a subcarrier achieves a maxima at a unique jammer power (cf. Proposition III.1). So, first we evaluate optimal for achieving maximum secure rate, respecting the jammer power bounds as follows:
(17) 
where is a small positive number. Note that, in secure rate improvement and , so first case does not arise. If the condition is satisfied, then optimum jammer power is allocated over all subcarriers, i.e., . Otherwise the problem (IIIC2) is solved under jammer power budget constraint. The partial Lagrangian of the problem (IIIC2) is given in (18).
(18) 
After setting firstorder derivative of the Lagrangian in (18) equal to zero, we obtain a fourthorder nonlinear equation in having following form
(19) 
The coefficients of the equation (IIIC2) are tabulated in Table V. In the domain , the secure rate is a concave increasing function (cf. Proposition III.1) and since , depending on , there exists a single positive real root of (IIIC2). After obtaining for a fixed value of , is updated using the subgradient method [25]. For constrained jammer power budget, optimal jammer power is obtained after constraining between jammer power bounds as in (17). Thus, allocation for fixed can be written as:
(20) 
For a known , the AO method now obtains based on (14) with the source power’s coefficient being , instead of . This completes one iteration of the AO method. This procedure of optimizing source and jammer powers alternatively continues until either the secure rate saturates or the iteration count exceeds a threshold. A PD and AO based Joint Power Allocation (JPA) scheme for weighted sum secure rate maximization is presented in Algorithm 1.

During power optimization in AO, if the jammer power over some of the subcarriers in is zero, then these subcarriers are taken out of and added back into for source power allocation in the next iteration of PD procedure.
= ; = ; 
= 
; 
= ; 
= ; = ; 
= ; = ; 
IiiC3 Convergence of joint power allocation
The source power allocation is done through PD with as the subgradient. The secure rate is a concave increasing function of source power in subproblem1 and for a fixed jammer power, in subproblem2. Thus, and are positive and bounded. Hence, is bounded and the method converges to an optimal value in finite number of steps [25].
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