Secrecy-Optimized Resource Allocation for Device-to-Device Communication Underlaying Heterogeneous Networks

Transcription

1 Secrecy-Optimized Resource Allocatio for Device-to-Device Commuicatio Uderlayig Heterogeeous Networks Kecheg Zhag, Muge Peg, Seior Member, I, Pig Zhag, Seior Member, I, ad Xuelog Li, Fellow, I arxiv: v [cs.it] 5 Sep 207 Abstract Device-to-device D2D commuicatios recetly have attracted much attetio for its potetial capability to improve spectral efficiecy uderlayig the existig heterogeeous etworks HetNets. Due to o sophisticated cotrol, D2D user equipmets DUs themselves caot resist eavesdroppig or security attacks. It is urget to maximize the secure capacity for both cellular users ad DUs. This paper formulates the radio resource allocatio problem to maximize the secure capacity of DUs for the D2D commuicatio uderlayig HetNets which cosist of high power odes ad low power odes. The optimizatio obective fuctio with trasmit bit rate ad power costraits, which is o-covex ad hard to be directly derived, is firstly trasformed ito matrix form. The the equivalet covex form of the optimizatio problem is derived accordig to the Perro-Frobeius theory. A heuristic iterative algorithm based o the proximal theory is proposed to solve this equivalet covex problem through evaluatig the proximal operator of Lagrage fuctio. Numerical results show that the proposed radio resource allocatio solutio sigificatly improves the secure capacity with a fast covergece speed. Idex Terms Device-to-device commuicatio, heterogeeous etwork, secure capacity, resource allocatio I. INTRODUCTION Heterogeeous etworks HetNets, which are expected to sigificatly improve capacity ad exted coverage, have bee widely applied to affordig the explosive growth of data traffic []. A HetNet uses a mixture of high power odes HPNs e.g., macro or micro base statios BSs ad low power odes LPNs e.g., picocell BSs, femtocell BSs, small cell BSs, wireless relay, or distributed ateas i the same coverage. HPNs are deployed to provide seamless coectivity ad guaratee the basic quality of service QoS requiremets of user equipmets Us. Sice the trasmit power of HPNs is sufficietly high to achieve seamless coverage, all Us ofte access the HPN to obtai the cotrol sigallig of the whole HetNet, which fulfills the decouple of cotrol ad user plaes [2]. I some hot spots with huge value of packet traffic, a U Copyright c 205 I. Persoal use of this material is permitted. However, permissio to use this material for ay other purposes must be obtaied from the I by sedig a request to pubs-permissios@ieee.org. Kecheg Zhag buptzkc@63.com, Muge Peg pmg@bupt.edu.c, ad Pig Zhag pzhag@bupt.edu.c are with the Key Laboratory of Uiversal Wireless Commuicatios for Miistry of ducatio, Beiig Uiversity of Posts ad Telecommuicatios BUPT, Chia. Xuelog Li xuelog_li@opt.ac.c is with the Ceter for OPTical IMagery Aalysis ad Learig OPTIMAL, State Key Laboratory of Trasiet Optics ad Photoics, Xi a Istitute of Optics ad Precisio Mechaics, Chiese Academy of Scieces, Xi a 709, Chia. prefers to access LPNs rather tha HPNs because LPNs ca provide much higher capacity tha the HPN does. Therefore, the dese deploymet of LPNs icreases spectral efficiecy S ad is oe of the key factors for boostig the capacity of the future etwork [3]. However, although the HetNet is a good alterative to improve S with seamless coverage, the iter-tier iterferece betwee HPNs ad LPNs is severe, ad the backhaul is ofte costraied because Us have to coect with the core etwork with a high capacity requiremet [4]. I order to alleviate the heavy load o backhaul, several alterative approaches have bee preseted. I particular, a hierarchical cloud computig architecture is proposed i [5] through addig a mobile dyamic cloud. To alleviate the heavy burde o the backhaul ad improve S i HetNets, device-to-device D2D commuicatio which is defied as direct commuicatio betwee Us without passig through BSs has bee proposed i the 3 rd geeratio partership proect log term evolutio-advaced [6]. With more traffic beig trasmitted by D2D commuicatio, iformatio exchage betwee Us ad core etwork is abated [7]. Recetly, D2D commuicatio has attracted much attetio with the ever icreasig demad for the local cotet exchage betwee two earby Us. I D2D uderlayig HetNets, radio resource is typically reused betwee the direct D2D lik ad the cellular air access lik. As a result, the iter-tier iterferece is critical to make D2D commuicatio rollout. May works have bee doe to improve S for the D2D commuicatio uderlayig HetNets. I [8], the authors show that D2D commuicatios ca effectively improve the total throughput without geeratig harmful iterferece to HetNets with proper maagmet. I [9], the harmful iter-system iterferece problems for D2D uderlayig cellular etworks are addressed, i which two iterferece avoidace mechaisms are applied to suppress the iter-tier iterferece betwee cellular ad D2D liks. Other recet researches, e.g., [0] [2], also have verified that the D2D commuicatio uderlayig HetNets have several advatages, icludig the high spectral utilizatio, the low eergy cosumptio, ad icurrig ew packet services. To improve S, the resource allocatio should be optimized for D2D commuicatio uderlayig HetNets, which is a o-covex optimizatio problem. The o-covex optimizatio problem has bee extesively researched, i which the Lagragia dual method has bee widely used [3], [4]. However, it is assumed that the iterferece from cellular users to D2D users is costat i [3] ad the iter-tier

2 2 iterferece betwee cellular users ad D2D users is replaced by the maximum allowed oise i [4]. These assumptios simplify the problem ad caot achieve the best performace. To tackle this problem, ew optimizatio method should be applied for D2D commuicatio uderlayig HetNets. Meawhile, to ehace the security of wireless trasmissio, physical layer security has bee developed based o iformatio theoretic cocepts [5] [7]. Physical-layer security, which exploits the physical characteristics of wireless chael to ehace the security, has sigificatly iflueced the wireless commuicatio because there exists a secured data trasmissio betwee power odes ad the eavsedropper [8]. To avoid the iformatio from power odes to Us beig leaked, the data rate of eavesdroppers should be maitaied at a low level. With a more ope cotrol plae, D2D user equipmet is more vulerable to the attacks of eavesdroppers. There have bee some studies to improve the secure capacity of D2D uderlayig systems from the viewpoit of physicallayer security. I [9], two secure capacity optimizatio problems for a multi-iput ad multi-output secrecy chael with multiple D2D commuicatios are researched ad two coservative approximatio approaches to covert the probability based costraits ito the determiistic costrait have bee addressed. Furthermore, the radio resource allocatio as a matchig problem i a weighted bipartite graph has bee formulated i [20], i which the Kuh-Mukres KM algorithm to obtai the optimal solutio is proposed. However, the D2D commuicatio oly helps to improve the secure capacity of cellular Us by cofusig the eavesdroppers i both [9] ad [20]. I other words, the D2D commuicatio acts as a kid of iterferece agaist eavesdroppig whe a eavesdropper tries to overhear the cellular commuicatio, therefore, the available rate of eavesdropper is lesseed [2]. To our best kowledge, there have bee few publicatios to optimize the secure capacity of both cellular Us ad D2D liks. I particular, few works maximize the secure capacity of the D2D commuicatio uderlayig HetNets. Note that such secure capacity optimizatio problem is o-covex ad hard to be directly solved because of the existece of iter-tier ad itra-tier iterferece. The goal of this paper is to optimize the radio resource allocatio aimig to maximize the secure capacity of D2D commuicatio uderlayig HetNets. The o-covex obective fuctio with several costraits are first trasformed ito matrix form. The, the equivalet covex form of the matrix form is derived accordig to the Perro-Frobeius theory which is based o the eigevalue ad eigevector [22] [24]. Note that the Perro-Frobeius theory has bee maily used i homogeeous etworks, there are lack of publicatios to use this theory to address the optimizatio issue i the D2D commuicatio uderlayig HetNets. To derive this Perro-Frobeius theory based equivalet covex optimizatio problem, the correspodig Lagrage fuctio has bee formulated ad the proximal operator has bee evaluated. Fially, a ovel iterative algorithm based o the proximal theory has bee proposed. The mai cotributios of this paper ca be summarized as followig: The optimizatio problem of radio resource allocatio aimig to maximize the secure capacity i D2D commuicatio uderlayig HetNets has bee addressed i this paper. To our best kowledge, previous works maily take D2D commuicatio as friedly ammer to improve the secure capacity of cellular users. Thus, oly the performace of cellular users is optimized ad the performace of D2D commuicatio is igored. Furthermore, previous works maily focus o the traditioal S ad do ot focus o the secure capacity. Besides, whe solvig the optimizatio problem, previous works maily try to simplify the iterferece betwee cellular users ad D2D users by settig the iterferece with costat, which caot take full advatage of the beefit of subcarrier reusig. This paper explores the resource allocatio scheme for both cellular users ad D2D users whe they share the same subcarrier resource to maximize the secure capacity of D2D commuicatio uderlayig HetNets. To deal with the secrecy-optimized resource allocatio problem, a o-covex obective fuctio has bee formulated. This o-covex obective fuctio is hard to be directly derived. The existig works ofte simplify the iterferece ito costat, ad the the Lagragia dual theory ca be used. However, the iterferece i the D2D uderlayig HetNets is ofte dyamical ad should ot be regarded as costat. To deal with this o-covex issue, the obective fuctio is trasformed to a equivalet covex optimizatio problem accordig to the Perro-Frobeius theory. Furthermore, to solve the equivalet covex problem, a iterative algorithm based o the proximal theory cosistig of both outer ad ier loop optimizatios has bee proposed to achieve the global optimal solutio. The S performace of the proposed resource allocatio solutio has bee umerically evaluated. Simulatio results show that the proposed iterative algorithm ca promptly coverge ad outperforms the baselie algorithms. The effects of the QoS ad power costraits with various other comparisos have bee show to evaluate the performace gais of the proposals. The remider of this paper is orgaized as follows. I Sectio II, the system model of D2D commuicatio uderlayig HetNets will be described ad the optimizatio problem will be formulated. The solutio to the optimizatio problem ad the correspodig iterative algorithm will be preseted i Sectio III. Sectio IV will provide simulatios to verify the effectiveess of the proposed algorithm ad the correspodig solutios. Fially, the paper will be cocluded i Sectio V. II. SYSTM MODL A D2D commuicatio uderlayig HetNets is show i Fig., i which L LPNs sharig N subcarriers with the HPN is cosidered. It is assumed that each subcarrier occupies B MHz badwidth. The HPN ad LPNs are all closed accessed. The closed access meas that each cell ca oly schedule the Us that belog to it, while the ope access meas that oe cell ca schedule ay U as log as the referece sigal receivig power is sufficietly high. Similar assumptio ca be

3 3 foud i previous works [25], [26]. Uder the closed access, Us moopolize their ow servig cells ad the privacy ad security of the commuicatio ca be guarateed, though the performace of closed access is worse tha that of ope access. Sice this paper focus o the secure capacity, the closed access is oly cosidered. Let H deote the average umber of HPN-accessed Us HUs i each HPN ad M deote the average umber of LPN-accessed Us LUs i each LPN, respectively. Besides, K D2D-worked Us DUs coexist with LUs i each LPN. It is assumed that all Us e.g., HUs, LUs or DUs have low mobility so that the chael state iformatio betwee Us ad power odes as well as the chael state iformatio betwee the trasmitter ad receiver of ay D2D user pair remai statioary [27] [29]. Therefore, subcarriers ca be assumed to be idepedet from each other, ad the chael fadig of each subcarrier ca be assumed the same withi a time slot, but may vary cross differet subcarriers. O each subcarrier, there exists a eavesdropper that leverages the iformatio of the desired Us. Sice DUs prefer to share the uplik radio resources with HUs or LUs to avoid the severe iter-tier iterferece, this paper oly focuses o the resource allocatio to optimized the secure capacity i uplik. HPN LPN HU LU DU h HH h h HL hl h HD hlk h H h h LH lm h LL lm h LD lmk h L lm h DH lk h DL lk h LD lmk h L lm p H h p L lm p D lk α H h α L lm α D lk TABL I SUMMARY OF NOTATIONS IN THIS PAPR high power ode low power ode HPN-accessed U LPN-accessed U D2D-worked U the chael gai from the h-th HU to the HPN o the -th subcarrier the chael gai from the h-th HU to the l-th LPN o the -th subcarrier the chael gai from the h-th HU to the receiver of the k-th DU pair of the l-th LPN o the -th subcarrier the chael gai from the h-th HU to the eavesdropper o the -th subcarrier the chael gai from the m-th LU of the l-th LPN to the HPN o the -th subcarrier the chael gai from the m-th LU of the l-th LPN to the -th LPN o the -th subcarrier the chael gai from the m-th LU of the l-th LPN to the receiver of the k-th DU pair of the -th LPN o the -th subcarrier the chael gai from the m-th LU of the l-th LPN to the eavesdropper o the -th subcarrier the chael gai from the trasmitter of the k-th DU pair of the l-th LPN to the HPN o the -th subcarrier the chael gai from the trasmitter of the k-th DU pair of the l-th LPN to the -th LPN o the -th subcarrier the chael gai from the trasmitter of the k-th DU pair of the l-th LPN to the receiver of the i-th DU pair of the -th LPN o the -th subcarrier the chael gai from the trasmitter of the k-th DU pair of the l-th LPN to to the eavesdropper o the -th subcarrier the trasmit power of the h-th HU o the -th subcarrier the trasmit power of the m-th LU of the l-th LPN o the -th subcarrier the trasmitter of the k-th DU pair of the l-th LPN o the -th subcarrier subcarrier allocatio idex which deotes whether the -th subcarrier is allocated to the h-th HU subcarrier allocatio idex which deotes whether the -th subcarrier is allocated to the m-th LU of the l-th LPN subcarrier allocatio idex which deotes whether the -th subcarrier is allocated to the k-th DU pair of the l-th LPN Let h HH h,hhl hl,hhd hlk, ad hh h deote the radio chael Fig.. D2D commuicatio uderlayig HetNets. gai from the h-th HU to the HPN, from the h-th HU to the l-th LPN, from the h-th HU to the receiver of the k-th DU pair of the l-th LPN, ad from the h-th HU to the eavesdropper o the -th subcarrier, respectively. Meawhile,, ad hl lm deote the radio chael gai from the m-th LU of the l-th LPN to the HPN, from the m-th LU of the l-th LPN to the -th LPN, from the m-th LU of the l-th LPN to the receiver of the k-th DU pair of the -th LPN, ad from the m-th LU of the l-th LPN to the eavesdropper o the -th subcarrier, respectively. Similarly, let h LH lm, hll lm, hld lmk, ad hd lk deote the chael gai from the trasmitter of the k-th DU pair of the l-th LPN to the HPN, to the -th LPN, to the receiver of the i-th DU pair of the -th LPN, ad to the eavesdropper o the -th subcarrier, h DH lk, hdl lk, hdd lki respectively. Deotep H h,pl lm, adpd lk by the trasmit power of the h-th HU, the trasmit power of the m-th LU of the l-th LPN, ad the trasmitter of the k-th DU pair of the l-th LPN o the -th subcarrier, respectively. I orthogoal frequecy divisio multiple access OFDMA based wireless etworks, a base statio allocates orthogoal subcarriers to differet users. Thus, i each cell e.g., a HPN or a LPN, each subcarrier ca be allocated to oly oe cellular U or a DU pair [30] [32], but oe or more subcarriers ca be allocated to the same cellular U or a DU pair. It meas that at most oe elemet of p H h,ph h2 hn, ph is positive ad the others are all zero. Let α H h,αl lm,αd lk {0,} deote the allocatio idex of subcarriers, which deotes whether the -th subcarrier is allocated to the h-th HU, the m-th LU of the l-th LPN, ad the k-th DU pair of the l-th LPN or ot, respectively. It should be oted that eavesdroppers are passive ad will ot share its CSI with the HPN or LPN. I fact, if eavesdroppers keep silet all the time, it is impossible for the etwork to estimate the CSI of them. However, whe eavesdropper become active, they ca be detected by the etwork [33]. Thus, i this paper, the CSI of eavesdropper is assumed to be kow ad similar assumptio ca be foud i previous works [34] [36]. The uplik sigal-to-iterferece-plus-oise ratio SINRfrom the h-th HU to the HPN o the -th subcarrier ρ H h ca be expressed as:

4 4 ρ H h = l=m= α H h ph h hhh h α L lm pl lm hlh lm L l= k=. α D lk pd lk hdh lk Bσ2 The uplik SINR from the h-th HU to the eavesdropper o the -th subcarrier ρ H h ca be expressed as: ρ H h = l=m= α H h ph h hh h α L lm pl lm hl l L l=k=. α D lk pd lk hd lk Bσ2 2 The uplik SINR from the m-th LU of the l-th LPN to the l-th LPN o the -th subcarrier ρ L lm ca be expressed as: h= ρ L lm = αl lm pl lm hll lml I L lm σ2, 3 where Ilm L deotes the iterferece ad is expressed as: Ilm= L α H hp H hh HL hl = k= =, l i= α L ip L ih LL il α D kp D kh DL kl. 4 The uplik SINR from the the l-th LPN to the eavesdropper o the -th subcarrier ρ L lm ca be expressed as: where I D lk I D lk = deotes the iterferece ad is expressed as: H h= L α H h ph h hh h =, l i= = m= α L m pl m hl l α L i pd i hd i. 0 Let Ch HS, CLS lm, CDS lk deote the secrecy capacity of the h-th HU, the m-th LU of the l-th LPN, ad the k-th DU pair of the l-the LPN, respectively. Accordigly, Ch HS, CLS lm, ca be expressed as: lk Ch HS = Blogρ H hblogρ H h, Clm LS = BlogρL lm BlogρL lm, 2 Clk DS = Blogρ D lkblogρ D lk, 3 ρ L lm =αl lm pl lm hl lm I L lm σ2, 5 where Ilm L deotes the iterferece ad is expressed as: I L lm = h= α H hp H hh H h = k= =, l i= α L ip L ih L α D k pd k hd k. 6 The uplik SINR from the trasmitter of the k-th DU pair of the l-the LPN to the receiver of the k-th DU pair of the l-the LPN o the -th subcarrier ρ D lk ca be expressed as: ρ D lk= αd lk pd lk hdd lklk I D lk σ2, 7 where Ilk D deotes the iterferece ad is expressed as: I D lk = h= α H hp H hh HD hl =, l i= = m= α L mp L mh LD ml α D i pd i hdd ilk. 8 Similarly, the uplik SINR from the trasmitter of the k-th DU pair of the l-the LPN to the eavesdropper ρ D lk ca be expressed as: ρ D lk =αd lk pd lk hd lk I D lk σ2, 9 respectively. Therefore, the secrecy capacity of the D2D commuicatio uderlayig HetNets ca be writte as: h== C HS h L l= m= C LS lm L l= k= = lk. 4 It is assumed that the secrecy capacity of each U should be o less tha the pre-defied threshold to guaratee QoS. Meawhile, to save the eergy cosumptio, the maximum allowed trasmit power of each power ode is limited. Besides, the subcarrier allocatio idex should be restricted to prevet the same subcarrier beig allocated to more tha oe U of the same type, e.g., HUs, LUs, ad DUs. As a result, the secrecy-optimized resource allocatio problem for the D2D commuicatio uderlayig HetNets ca be formulated with the QoS ad trasmit power costraits. Problem Secrecy Capacity Optimizatio: With the costraits o the required QoS ad maximum trasmit power allowace, the secrecy-optimized resource allocatio problem for the D2D commuicatio uderlayig HetNets ca be

5 5 formulated as: max {p H h,pl lm,pd lk } s.t. h== C HS h L l= k== Ch HS CHS mi = = l= m= C LS lm lk 5, h, 6 Clm LS CLS mi, l,m, 7 lk mi, l, k, 8 p H h P H max, h, 9 p L lm PL max, l,m, 20 p D lk Pmax, l,k, D 2 α H h =,,α H h {0,}, 22 h= α L lm =, l,,α L lm {0,},23 m= α D lk =, l,,α D lk {0,}, 24 k= where the costraits 6-8 correspod to the required lower boud of the the secrecy capacity of HUs, LUs, ad DUs, respectively. The costraits 9-2 represet the maximum allowed trasmit power of HUs, LUs, ad DUs, respectively. The costraits put limitatios o the subcarrier allocatio policy so that each subcarrier ca oly be allocated to at most oe U of the same type. III. SCRCY-OPTIMIZD RSOURC ALLOCATION OPTIMIZATION Joit optimizatio of subcarriers ad trasmit power is a itractable problem [], ad the optimizatio problem 5 with costraits 6-24 listed i Sectio II is odetermiistic polyomial NP hard, which is impossible to be directly solved. But it is oted that uder ay give the subcarrier allocatio policy, problem 5 ca be trasformed to a covex form accordig to the Perro-Frobeius theory. Thus, if we solve the equivalet covex problem for every possible subcarrier allocatio policy, the global optimal solutio ca be derived. However, such exhaustive method with high complexity is hard to be practical. I this paper, we solve this trasformed covex-optimizatio problem with two steps. I the first step, we propose a suboptimal algorithm to allocate each user with proper subcarriers. I the secod step, we desig a optimal power allocatio scheme uder the subcarrier allocatio policy proposed i the first step. Similar method to tackle the NP-hard problem has bee used i previous works [37] [39]. Based o the proposed subcarrier allocatio solutio, it ca be demostrated that the optimizatio problem 5 ca be completely solved through proposig the optimal power allocatio solutio. A. Subcarrier Allocatio I this subsectio, subcarrier allocatio scheme is derived for solvig problem 5. Note that the subcarrier allocatio idex is a discrete variable ad for ay give subcarrier allocatio scheme the optimal power allocatio is idepedet from the subcarrier allocatio results. Therefore, the followig theorem ca be holde. Theorem : For ay optimizatio problem max {x,y} fx,y, where x N X is a fiite discrete variable vector ad each elemet of x is oe of S iteger values. y R Y is cotiuous variable. X, S ad Y are positive itegers. Let x i deote the i-th feasible solutio of x, where x is the optimal x, ad y i deote the optimal y uder x i ad i X S. Therefore, x ca be expressed as: x = argmax {x i} fx i,y i. 25 Proof: Please see Appedix A. Accordig to Theorem, let x deote the subcarrier allocatio scheme ad y deote the trasmit power allocatio scheme, the optimal subcarrier allocatio scheme ca be directly derived with exhaustive method. However, such exhaustive method is impractical because each subcarrier eeds to be searched to derive the optimal subcarrier allocatio, which resultig i a high complexity. I this subsectio, a suboptimal subcarrier allocatio scheme is proposed to optimize the capacity of each cell e.g., HPNs or LPNs, which ca be expressed as: max s.t. h== h= log αh h ph h hhh h Bσ 2 26 p H h hhl hl IH, 27 α H h =,αh h {0,}, 28 h= var p H h,αh h, 29 where I H is the maximum allowed iterferece caused by HUs. Obviously, it ca be solved by usig the Lagragia fuctio, i.e., Lα H h,ph h = H h== λ log αh h ph h hhh h σ 2 h= where λ is the Lagrage multiplier. p H h hhl hl IH, 30

6 6 Accordig to the Karush-Kuh-Tucker KKT coditio, the optimal subcarrier allocatio scheme to maximize the data rate of HPNs ca be expressed as: { α H h = = argmax log λσ λ [ 2 λ σ2]. 3 0 others Iitialize the Lagrage multiplier ad the use the subgradiet-based method [40] to search for the subcarrier allocatio scheme. The proposed algorithm is executed i each cell e.g., HPNs or LPNs to a achieve good performace. I the whole etwork comprised of multiple HPNs, this proposal ca work efficietly i each HPN, thus the overall performace ca be optimized. B. Power Allocatio Scheme I this subsectio, based o the aforemetioed subcarrier allocatio, the power allocatio to solve problem 5 is researched. Problem 5 ca be rewritte as: h== C HS h H = h= l= m= C HS h L C LS lm l= m= l= k== C LS lm L l= k= lk = lk. 32 I OFDMA based wireless etworks, the iterferece betwee ay two differet subcarriers ca be igored due to the orthogoal characteristics. Therefore, the primal obective fuctio ca be divided ito N idepedet subproblems, which idicates that the primal problem ca be derived by solvig the N subproblems. The -th subproblem optimizatio ca be give as: max {p H h,pl lm,pd lk } s.t. h= C HS h L l= m= C LS lm L l= k= lk 33 Ch HS Cmi, HS h, 34 Clm LS mi, l,m, 35 Clk DS Cmi, DS l, k, 36 p H h PH max, h,, 37 p L lm Pmax, l,m,, L 38 p D lk PD max, l,k,. 39 Note that 33 is still o-covex due to the iter-tier iterferece. To trasform it ito a covex form, the uplik chael state iformatio CSI vector of the h-th HU o the -th subcarrier, which represets the CSI from the h-th HU to the HPN, from the h-th HU to LPNs, ad from the h-th HU to the receiver of other DUs that share the -th subcarrier, is give by G H h = [ h HH h,hhl h, hhl hl,hhd h, hhd hl ] T, 40 where h,m,l { h,l h,m,lα H h,αl lm,αd lk 0}. Accurately, uder give subcarrier allocatio scheme, at most oe LU ca be allocated to the -th subcarrier of the l-th LPN. Similarly, the uplik CSI vector for the LU o the -th subcarrier of the l-th LPN ca be give as: [ T, G L l = h LH lm,h LL lm, h LL lml,h LD lm, hlml ] LD 4 where h,m,l { h,l h,m,lα H h,αl lm,αd lk 0}. Similarly, the uplik CSI vector for the DU o the -th subcarrier of the l-th LPN ca be give as: G D l = [ h DH lk,hdl lk, hdl lkl,hdd lk,hdd lkl ] T, 42 where h,k,l { h,l h,k,lα H h,αl lm,αd lk 0}. Combie these three CSIs G H h, GL l, GD l together, it ca be derived that o the -th subcarrier, the CSI matrix amog all users o the -th subcarrier ca be expressed as: G = G H h,g L,G L 2, G L l, G D,G D 2 G D l G. }{{}}{{} LPN D2Duserpair 43 Similarly, the CSI betwee users ad the eavesdropper o the -th subcarrier is give by G = h H h,h L m,h L 2m, h L }{{ lm, } LPN h D k,h D 2k, h D lk }{{} D2D. 44 Note that for a give subcarrier allocatio scheme, G ad G are fixed, which meas that the umber of users HU, LU ad DU o the -th subcarrier is determied ad the CSI matrix ca be kow. Without loss of geerality, let G be a J J matrix which deotes that the umber of users o the -th chael is J, ad let G be a J vector. The trasmit power matrix of the J users is give by p. A oegative matrix is defied F with etries F i = { 0 ifi = G i G ifi, 45 T. σ ad the vector v = 2 σ G, 2 σ G, 2 22 G JJ Now, the SINR of the -th user i the uplik is S = p F p v, ad its correspodig capacity ca be expressed as: Similarly, we ca have C = log S. 46 F i = ad the vector v = { 0 ifi = σ 2 G G i G ifi T. σ, 2 σ G, 2 2 G J, 47

7 7 Now the SINR of the -th user i the wiretap chael ca be expressed as S = p F pv, ad the data rate is C = log S. Rewrite 33 as: max {p } s.t. C = C = 48 C C C S mi,, 49 0 p p max,, 50 where C S mi ad p max are the miimum secrecy data rate ad maximum trasmit power costraits, respectively. Due to the iter-tier iterferece, the optimizatio problem i 48 is still hard to be derived. To solve this problem, we ca defie the vector α { 0 i α i = 5 i = ad the costrait 50 ca be expressed as: α T p p max. 52 To trasfer the costrait 50, we ca have the followig theorem: Theorem 2: Let B = F the costrait 50 is equal to: p max v α T, 53 B = F v p max αt, 54 q = F p v, 55 q = F p v, 56 B diage C q IB q, 57 B diage C q IB q. 58 Proof: Please see Appedix B. Based o Theorem 2, the optimizatio problem i 48 ca be trasformed to max C C 59 {C,C,q,q } = = s.t. C C C S mi,, 60 B diage C q IB q,, 6 B diage C q IB q,. 62 Obviously, the obective fuctio 59 has bee trasferred to be liear. However, this optimizatio problem is still ocovex because of the o-covex costraits 6 ad 62. These two costraits ca be trasformed ito be covex by usig the followig theorem. Theorem 3: If the o-egative matrix B = IB B ad B = IB B holds, the optimizatio problem i 59 ca be trasformed ito a covex form as: max {C,C } s.t. C = C = 63 C C C S mi,, 64 log ρ B diag e C 0,, 65 log ρ B diag e C 0,, 66 where ρ is the the Perro-Frobeius eigevalue of a oegative matrix. Proof: Please see Appedix C. Due to the log-covexity property of the Perro-Frobeius eigevalue, the costraits 65 ad 66 are covex. Now the cocered optimizatio problem i 59 is a covex optimizatio problem ad ca be solved i polyomial time. However, such problem is very hard to solve by the traditioal sub-gradiet covex optimizatio method because it is difficult to get the closed-form expressios of C ad C, which is essetial to achieve the solutio. To deal with this challege, a ovel method which is effective for this problem is proposed i the followig cotet. The Lagragia fuctio of problem 63 ca be expressed as: LC,C,λ,β,µ = = C = = C = λ C C C mi S β log ρ B diag e C = µ log ρ B diag e C, 67 where λ, β, µ are the Lagrage multipliers correspodig to the costraits. The, the Lagragia dual fuctio ca be expressed as: gλ,β,µ = max {C,C } LC,C,λ,β,µ. 68 Therefore, the dual optimizatio problem ca be formulated as: mi {λ,β,µ} gλ,β,µ s.t. λ 0,β 0,µ Sice the optimizatio problem i 63 is covex, the duality gap betwee the primal problem ad its dual problem is zero, which illustrates that the primal problem ca be solved by solvig its dual problem69. The sub-gradiet-based method ca be utilized to solve 69 ad the sub-gradiet of the

8 8 Lagrage multipliers i the dual fuctio i the i-th iteratio ca be writte as: λ i = C i C i C mi S, J, 70 β i = logρ B diage Ci, J, 7 µ i = logρ B diageci, J 72 where C i ad C i are the capacity ad leakage capacity of the-th U o the-th subcarrier i thei-th iteratio, respectively. λ i, β i ad µ i deote the -th sub-gradiet correspodig to λ, β ad µ utilized i the i -th iteratio. Therefore, the update equatios for the dual variables i the i-th iteratio ca be expressed as: λ i = [ λ i ξ λ i β i = µ i = C i C i C mi S ] 73 [β i ξ β ilogρ B diage Ci ] 74 ] [µ i ξ µ ilogρ B diageci 75 where ξ λ i, ξ β i ad ξ µ i are positive step sizes. Accordig to the above derivatios, a sub-gradiet method based iteratio algorithm is preseted to solve 69 as show i Algorithm. To solve 76, rewrite 67 with give Lagrage multiplier λ i,β i,µ i as: LC,C = C = = = = C = λ i C C C mi S β i log ρ B diag e C µ i ρ B log diag e C 77 = f C,C g C,C, 78 Algorithm Sub-gradiet method based iteratio algorithm for the outer loop optimziatio : Set the iteratio idex i =. Iitialize the Lagrage multiplier λ i,β i,µ i, the step size ξ λ,ξ β,ξ µ, the maximum umber of iteratios I max ad the iteratio threshold δ. 2: for i I max 3: Calculate C i,c i which ca be expressed as 4: C i,c i = argmi LC i,c i,λ i,β i,µ i 76 {C i,c i } λ i = [ λ i ξ λ i C i C i ] C mi S ; [ ] ; 5: β i = β i ξ β ilogρ B diage Ci [ ; 6: µ i = µ i ξ µ ilogρ B diageci ] 7: if λ i λ i β i β i µ i µ i δ 8: break out; 9: ed if 0: i = i; : ed for 2: retur C i C i. where f C,C = β i log ρ B diag e C = = g C,C = C = = µ i ρ B log diag e C C = 79 λ i C C C mi S. 80 It is obvious that g C,C is closed proper covex fuctio ad f C,C is differetiable. So the proximal gradiet method ca be used to solve 76 accordig to the proximal theory. The proximal operator of f C,C is calculated as: prox g C,C = λ i C C I. 8 The partial derivatives of f C,C with respect to C,C are give by

9 9 f dc = f dc = = = β i x B diage C y B diage C, µ i x B diage C y B diage C., i.e., OJ. Therefore, the complexity of the power allocatio scheme is OJ 2. Therefore, the total complexity of the proposed algorithm is ONJ 2 HLM LK. The algorithm proposed i [3] has a complexity of ONKN M. Compared with the algorithm preseted i [3], the algorithm proposed i this paper cosiders both subcarrier ad power allocatio scheme at the expese of the complexity. Due to icludig subcarrier ad power allocatio executios, the proposed algorithm i this paper ca be solved i polyomial time ad is acceptable i practice. where x ad y represet the right ad left Perro- Frobeius eigevalue, respectively. It obvious that problem 76 is covex because it is a dual problem [40]. Due to the proximal gradiet method, the proximal operator of 76 has a fixed poit, which is the optimal value of 76, too. Therefore, a iterative algorithm is proposed to fid the fixed poit as show by the Algorithm 2. I fact, Algorithm ad Algorithm 2 are the outer loop ad ier loop, respectively. O each subcarrier, the Algorithm is executed to solve the equivalet covex optimizatio problem of D2D uderlayig HetNets, while the Algorithm 2 is utilized to solve the dual problem proposed i Algorithm. The complexity of the proposed algorithm is aalyzed ad compared with the algorithm preseted i [3] as follows. The complexity of the subcarrier allocatio scheme is liear with ONH LM LK. The complexity of the power allocatio algorithm is determied by Algorithm ad Algorithm 2. The complexity of Algorithm is liear with the umber of dual factors, i.e., OJ. The complexity of Algorithm 2 is liear with the umber of the elemets of C i,c i IV. SIMULATION RSULTS AND DISCUSSIONS I this sectio, the secrecy capacity performace of the D2D commuicatio uderlayig HetNets ad the correspodig optimizatio results are umerically evaluated with simulatios. I our simulatios, oe HPN is cocered, ad all LPNs are uiformly located i the circle with the distace of 000m whose ceter is the HPN. The cell radius of HPN ad LPN are 800m ad 200m, respectively. The miimum allowed distace betwee HPN ad users is 50 m ad the miimum allowed distace betwee LPN ad users is 20 m to protect users from radiatio. It is assumed that H = 2, i.e., there are 2 HUs who access to the HPN. Deote M = 5 ad K = 5, which suggests that 5 LUs ad 5 D2D liks radomly locate i the coverage of each LPN. All of LPNs share N = 8 subcarriers Algorithm 2 The dual problem solutio for the ier loop optimizatio : Set the iteratio idex s =. Iitialize the Lagrage multiplier λ i,β i,µ i of the i-th iteratio of Algorithm, the maximum umber of iteratios S max ad the iteratio threshold η. 2: for s S max 3: Calculate C s C s 4: if C s 5: break out; 6: ed if 7: s = s C s = prox g C s f dc s = prox g C s,c s 8: ed for 9: retur C s ad C s C s C s,c s, 84 f dc s η. 85 ad each subcarrier occupies 200 KHz. O each subcarrier, a eavesdropper exists i the cocered sceario ad it locates radomly. The total trasmit power of HPN is 43 dbm ad equally allocated o all subcarriers. The trasmit power of the trasmitter of DU is 5 dbm ad the distace betwee the trasmitter ad the receiver is 0 m. It is assumed that the path loss model is expressed as log 0 d for the D2D lik ad the LPN-to-LU, LPN-D2D ad D2D-LU lik with short trasmit distace, while log 0 d is used for other log liks, where d deotes the distace betwee the trasmitter ad the receiver i meters. The umber of simulatio sapshots is set at 000. I all sapshots, the fast-fadig coefficiets are all geerated as idepedetly ad idetically distributed i.i.d. Rayleigh radom variables with uit variaces. A. Covergece of the Proposed Algorithm To evaluate the covergece performace of the algorithm uder more users, we set H = 0, N = 20, M = 5 ad K = 5, respectively. The covergece of the proposed algorithm with differet QoS requiremets is show i Fig. 2. It ca be obviously illustrated that the proposed algorithm coverges withi 5 iteratio umbers for differet QoS requiremets, which suggests that the proposed algorithm ca work efficietly. Besides, the differet levels of QoS requiremets have sigificat impacts o the performace of the secrecy capacity. To evaluate the ifluece of the QoS requiremets, three QoS levels are set ad the maximum allowed trasmit power of LPNs is set at 24 dbm. Fig. 2 shows that the secrecy capacity of the system decrease with the QoS requiremet icreasig. I the case of high QoS requiremet, more trasmit power should be allocated to the user who is uable to achieve the QoS requiremet, thus the system performace has a decrease. So there exists a balace betwee the system performace ad the user performace.

10 0.5.5 Secure capacitybps/hz 0.5 QoS requiremet=0 bps/hz QoS requiremet=0. bps/hz QoS requiremet=0.2 bps/hz QoS requiremet=0.3 bps/hz QoS requiremet=0.4 bps/hz Secure capacity bps/hz 0.5 Maximum trasmit power allowace of LPN=24dBm Maximum trasmit power allowace of LPN=2dBm Maximum trasmit power allowace of LPN=8dBm Maximum trasmit power allowace of LPN=5dBm Iteratios QoS requiremet bps/hz Fig. 2. Covergece evolutio uder differet QoS requiremets B. Secure Capacity Performaces of the Proposed Solutios I this part, key factors impactig o the secrecy capacity performaces of the proposed algorithm are evaluated. Sice the trasmit power costrait ad the QoS requiremet are two mai costraits i the proposed optimizatio problem, the impact of these two factors are evaluated i Fig. 3 ad Fig. 4, respectively. Besides, the average secrecy capacity per U is more sigificat tha the total secrecy capacity to evaluate the performace of the D2D uderlayig HetNets whe the umber of Us chages frequetly. Therefore, to evaluate the impact of the umber of Us to the D2D uderlayig HetNets, the relatioship betwee the umber of Us ad the average secrecy capacity per U should be evaluated. Sice LUs usually cotribute more capacity to the etwork tha HUs ad DUs, without loss of geerality, oly the relatioship betwee the umber of LUs per LPN ad the average secrecy capacity per LU is evaluated i this part ad the correspodig simulatio result is show i Fig. 5. It should be oted that i HetNets, HPNs are maily deployed to provide seamless coverage ad LPNs which are closer to users are deployed i hot spots to provide high capacity. Therefore, the capacity of the etwork is maily determied by LPNs ad oly the maximum trasmit power allowace of LPN is simulated i this paper. I Fig. 3, the secrecy capacity performaces uder varied QoS requiremets with a step size of 0.02 are compared amog differet maximum allowed trasmit power. Whe the QoS requiremet is ot sufficietly large, the secrecy performace decreases slowly with the icreasig QoS requiremet because most users ca set a sufficietly high trasmit power to satisfy its QoS requiremet. However, whe the QoS requiremet becomes large eough, more users ca afford the QoS requiremet ad the more trasmit power has to be allocated to these users to meet the QoS requiremet. Furthermore, it ca be illustrated from Fig. 3 that with the larger maximum allowed trasmit power of LPN, the secrecy capacity performace icreases, which has a similar tred with Fig. 4. Fig. 3. Secure capacity performace comparisos uder differet QoS requiremets To further evaluate the impact of the maximum allowed trasmit power of LPNs o the secure capacity performaces, four baselies are preseted. The first baselie is the upper boud of this problem ad it is calculated by the exhaustive method. We do the power allocatio algorithm for each possible subcarrier allocatio scheme ad get the optimal subcarrier ad power allocatio for the problem. The secod baselie is proposed by [3]. I [3], users that cause severe iterferece are scheduled to the other subcarriers to alleviate iterferece. For example, if user A iterferes other users seriously o the i-th subcarrier, it ca be scheduled to the -th subcarrier, o which the iterferece ca be almost avoided. The third baselie is based o the classic orthogoal subcarrier allocatio which has bee widely used i HetNets. The fourth baselie is based o the fixed power scheme, i.e., the trasmit power allocated to LUs is same ad fixed. Fig. 4 compares the secure capacity performace of differet algorithms i terms of maximum allowed trasmit power of LPNs. I this simulatio case, the QoS requiremet is set at 0 ad the maximum allowed trasmit power of LPNs varies withi [4 dbm, 36 dbm] with the step size of 2 dbm. Fig. 4 shows that whe the trasmit power is ot sufficietly high, the secure capacity performace icreases with the trasmit power rises because the iterferece is still ot the bottleeck ad the icremet of secure capacity maily depeds o icreasig trasmit power. However, whe the maximum allowed trasmit power of LPNs is sufficietly high, the iterferece limits the icrease of the secure capacity. Thus, o more trasmit power is allocated to users. As show i Fig. 4, the proposed algorithm is closed to the upper boud ad outperforms the algorithm preseted i [3] because the referece [3] ust proposed some mechaisms to mitigate the ifluece of iterferece. Ad the algorithm preseted i [3] is better tha the orthogoal subcarrier allocatio due to the beefit of frequecy reuse. Furthermore, the fixed power scheme has the worst performace because it takes o measures to optimize the secure capacity performace

11 Secure capacity bps/hz Upper boud Proposed algorithm Algorithm proposed by [3] Orthogoal subcarrier allocatio Fixed powr scheme Secure rate per LU bps/hz Maximum trasmit power allowace of LPN=24dBm Maximum trasmit power allowace of LPN=2dBm Maximum trasmit power allowace of LPN=8dBm Maximum trasmit power allowace of LPN=5dBm Maximum trasmit power allowac of LPNdBm Number of LUs Fig. 4. Secure capacity performace comparisos uder differet maximum trasmit power allowaces of LPNs Fig. 5. Average secure capacity per LU comparisos uder differet umber of LUs per LPN of the whole system. Fig. 5 illustrates the relatioship betwee the umber of LUs ad the average secrecy capacity per LU to evaluate the impact of the umber of Us to the whole etwork. I this simulatio case, the QoS requiremet is set at 0. bps/hz ad the maximum allowed trasmit power of LPNs varies from 8 dbm to 22 dbm with a step size of 2 dbm. The miimum umber of LUs per LPN is set to. Sice the umber of subcarriers is set to 8, which suggests that at most 8 LUs ca simultaeously coect to the LPN i each time slot i each LPN, the maximum umber of LUs per LPN are set to 8. It ca be see i Fig. 5 that as the umber of LUs per LPN icreases, the average secrecy capacity per LU decreases because the itra-tier iterferece icreases with more LUs coectig to the same LPN. The augmet of itra-tier iterferece leads to the abatemet of average secrecy capacity per LU. However, it ca be calculated from Fig. 5 that the total secrecy capacity of the whole LUs keeps icreasig as the umber of LUs per LPN rises, because whe more LUs coect to the LPN, more subcarriers ca be allocated to these LUs util the umber of LUs is equal to the umber of the subcarriers. Besides, Fig. 5 shows that the higher the maximum trasmit power allowace of LPN is, the better the secrecy capacity of the whole etwork achieves, which has the same tred as show i Fig. 3. V. CONCLUSION I this paper, the secrecy-optimized resource allocatio for the device-to-device D2D commuicatio uderlayig heterogeeous etworks HetNets has bee researched. I the cocered system model, there desely exist high power ode ad low power ode with D2D commuicatio, therefore, the iter-tier iterferece is always severe, which leads to the secrecy-optimized maximizatio problem beig o-covex. To solve this o-covex optimizatio problem, the primal o-covex optimizatio problem has bee trasformed ito the covex issue with several steps. Firstly, the obective fuctio with several costraits are trasformed ito a matrix form. Secodly, the equivalet covex form of the maximizatio problem for the matrix form is derived accordig to the Perro-Frobeius theory. Thirdly, the proximal operator of the trasformed covex problem is evaluated. The, a ovel iterative algorithm based o the proximal theory to solve the equivalet covex problem is proposed. Simulatio results have demostrated that the secrecy capacity has a sigificat improvemet ad the proposed algorithm is effective ad coverges fast. Furthermore, comparig with four baselies, the proposal has a sigificat performace gai o the secrecy capacity of the whole etwork. I the future, the o-ideal CSIs should be cosidered to optimize the secrecy capacity. I additio, the advaced iter-tier iterferece i the physical layer ad the dyamical queue characteristics i the upper layer should be oitly cosidered with the radio resource allocatio to optimize the secrecy capacity i the D2D commuicatio uderlyig HetNets. APPNDIX A PROOF OF THORM Without loss of geerality, let x p deote the optimal x. Thus, It ca be derived as: x p = argmax {x i} fx i,y i. 86 Assume that there exists x x which is better tha x i. Therefore, the followig equatio holds. which meas that fx p,yp < fx,y, 87 x = argmax {x i} fx i,yi. 88 It is obviously that 88s coflict with 86. Thus x p is the optimal x ad theorem holds.

12 2 APPNDIX B PROOF OF THORM 2 Accordig to 46, it ca be derived that e C = p S = F p v which is the same as: APPNDIX C PROOF OF THORM 3 Sice IB exists, multiply both sides of costrait 6 by IB, ad the followig equatio ca be derived as: diage C I F p v = p. 89 Substitutig 55 ito 89, it ca be derived as: diage C q = p q. 90 Multiply both sides of 90 by B which is give by 53, we ca have = F F =F p p max v α T p max v α T diage C q p F v α T q p max 9 v α T p F v α T p max p max q. Accordig to 52, α T p p max. Thus 92 p max v α T p = v α T p p max v, 93 ad F p v α T p max p F v α T q p max F p v F v α T q 94 p max = q F v α T q 95 p max = IF v α T q, 96 p max where 95 is derived by 55. Now the formula ca be derived as: F v α T diage C q p max IF v α T q. 97 p max Substitutig 53 ito the above iequality, 57 ca be derived. Similarly, 58 ca be derived accordig to 54 ad 56. B diage C q IB q IB B diage C q q B diage C q q. 98 Accordig to the subivariace theorem [4], if a oegative matrix A, a positive umber a ad a oegative vector v satisfy Av av, the ρa a ad the equality holds if ad oly if Av = av. Let A = B diage C, a = ad v = q ad rewrite 6 as ρ B diag e C, which is the same as log ρ B diag e C 0. Similarly, it ca be derived that log ρ B diag e C 0. RFRNCS [] V. Suryaprakash, J. Moller, ad G. Fettweis, O the modelig ad aalysis of heterogeeous radio access etworks usig a poisso cluster process, I Tras. Wireless Commu., vol. 4, o. 2, pp , Feb [2] J. Youg, M. Hasa, ad A. Ghrayeb, Modelig heterogeeous cellular etworks iterferece usig poisso cluster processes, I J. Sel. Areas Commu., vol. 33, o. 0, pp , Oct [3] S. Sigh ad J. Adrews, Joit resource partitioig ad offloadig i heterogeeous cellular etworks, I Tras. Wireless Commu., vol. 3, o. 2, pp , Feb [4] M. Peg, C. Wag, J. Li, H. Xiag, ad V. Lau, Recet advaces i uderlay heterogeeous etworks: Iterferece cotrol, resource allocatio, ad self-orgaizatio, I Commu. Surveys & Tutorials, vol. 7, o. 2, pp , secod quarter, 205. [5] M. Jo, T. Maksymyuk, B. Strykhalyuk, ad C.H. Cho, Device-todevice-based heterogeeous radio access etwork architecture for mobile cloud computig, I Wireless Commu., vol. 22, o. 3, pp , Ju [6] K. Doppler et. al., Device-to-device commuicatio as a uderlay to LT-advaced etworks, I Commuicatios Magazie, vol. 47, o. 2, pp , Dec [7] H. Dog, W. Kae, S. Wha, ad G. Dog, Two-stage semi-distributed resource maagemet for Device-to-Device commuicatio i cellular etworks, I Tras. Wireless Commu., vol. 3, o. 4, pp , Apr [8] C. Yu, K. Doppler, C. B. Ribeiro, ad O. Tirkkoe, Resource allocatio optimizatio for device-to-device commuicatio uderlayig cellular etworks, I Tras. Wireless Commu., vol. 0, o. 8, pp , Aug. 20. [9] T. Peg, Q. Lu, H. Wag, S. Xu, ad W. Wag, Iterferece avoidace mechaisms i the hybrid cellular ad device-to-device systems, i Proc I PIMRC, pp [0] H. Mi, J. Lee, S. Park, ad D. Hog, Capacity ehacemet usig a iterferece limited area for device-to-device uplik uderlayig cellular etworks, I Tras. Wireless Commu., vol. 0, o. 2, pp , Dec. 20. [] X. Che, L. Che, M. Zeg, X. Zhag, ad D. Yag, Dowlik resource allocatio for device-to-device commuicatio uderlayig cellular etworks, i Proc. Pers. Idoor, Mobile Radio Commu. Cof., pp [2] F. Li, M. Lei, ad F. Gao, Device-to-device D2D commuicatio i MU-MIMO cellular etworks, i Proc. Global Commu. Cof.., pp , Dec [3] B. Peg, C. Hu, T. Peg ad W. Wag, Optimal resource allocatio for multi-d2d liks uderlyig OFDMA-based commuicatios, i Proc. Iteratioal Coferece o Wireless Commuicatios, Networkig ad Mobile Computig WiCOM, pp. 4, Shaghai, 202. [4] W. Liu, Y. Yag, T Peg, ad W. Wag, Optimal resource allocatio scheme for satisfyig the data rate requiremet i hybrid etwork of D2D-cellular, Joural of Computers, vol. 9, o. 5, may. 204.

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