Control of Wireless Networks with Secrecy

1 Control of W ireless Netw orks with Secre c y C. Emre K oksal, Ozgur Ercetin, Y unus Sarikaya Abstract —W e consider the problem of cr oss-layer resource allocation in time-v arying cellular wireless networks, and in- corpora te information theoretic secrecy as a Quality of Service constraint. Speciﬁcally , each node in the network injects tw o types of trafﬁc, private and open, a t rates chosen i n order to maximize a global utility function, subj ect to n etwork stability and secrecy constraints. The secrecy constraint enfor ces an arbitrarily low mutual information leakage from the source to every node in the network, except f or th e sink node. W e ﬁrst obtain the achiev able rate r egion f or th e problem for single and mu lti-user systems ass uming that the nodes have full CSI of their neighb ors. Then, we provide a joint ﬂow control, scheduling and private encoding scheme, whi ch does not rely on the knowledge of the prior distri bution of th e gain of any channel. W e prove that our scheme achie ves a u tility , arbitrarily close to th e m aximum achiev able u tility . Numerical experiments ar e perf ormed to verify the analytical results, and to show the efﬁcacy of the dynamic control algorithm. I . I N T R O D U C T I O N In recent years, th ere have been a numb er of investigations on wireless inform ation theoretic secr ecy . These s tudies have been largely co nﬁned within the bo undarie s of the p hysical layer in the wireless scenario and they have signiﬁcantly enhanced our un derstandin g o f the fu ndamen tal limits and principles governin g the d esign and analysis of secure wire- less com munication systems. For example, [1], [2], [3] have unv eiled the opportunistic secr ecy prin ciple which allo ws for transform ing the multi-path fading variations into a secrecy advantage for the legitimate r eceiv er, even when the eaves- dropp er is en joying a high er average signal-to-n oise ratio (SNR). The fund amental role of feedback in enhan cing th e secrecy capacity of point-to-po int wireless commun ication links was established in [4], [5], [6]. More recent works h ave explored th e use of multiple antenn as to induce ambiguity at the eav esdropp er un der a variety of assumptions on the av ailab le transmitter chann el state inform ation (CSI) [7], [8], [9], [10]. Th e m ulti-user aspect of th e wir eless en vir onmen t was studied in [11], [1 2], [ 13], [14], [15], [16], [1 7], [18], [19], [2 0], [21], [ 22], [23], [ 24], [2 5] rev ealing the potential gains that can b e re aped fro m appr opriately con structed user cooper ation p olicies. Finally , th e design o f practical co des that a pproach th e p romised capa city limits was in vestigated in [26], [27]. One of the most interesting o utcomes o f this This material is based upon work supported by the National Science Founda tion un der Gra nts CNS-0831919, CCF-0916664, CAR EER-1054738, and by Marie Curie Inte rnationa l Re search Staff Exchange Sche me Fellowshi p PIRSES-GA-2010-269132 A GIL ENet within the 7th European Community Frame work Programme. Portions of this work were presented at Asilomar Conferenc e o n Signals, Systems, and Computers (Asilomar ’10), Pa ciﬁc Grove, CA. C. E. Kok sal (koksa l@ece.osu.edu) is with the Depa rtment of Electri cal and Computer Engineering at T he Ohio State Univ ersity , Columb us, OH. O. Ercetin (email: oe rcetin@sab anciuni v .edu) and Y .Sarikaya (email: sarikaya @su.sabanciuni v .edu) are with t he D epartmen t of Ele ctronic s E ngi- neering , Fac ulty of Engineeri ng and Natural Science s, Sabanc i Unive rsity , 34956 Istanb ul, T urkey . body of work is the d iscovery of th e positive imp acts on secure commun ications of som e wireless phenom ena, e.g., interferen ce, which are tra ditionally v iewed as impa irments to be o vercome. Despite the signiﬁcant p rogre ss in infor mation the oretic secrecy , most o f the work has fo cused o n ph ysical layer technique s and on a single link. The area of wireless infor- mation theo retic secrecy remains in its infancy , especially a s it relates to th e design of wireless networks an d its impact on network contro l and proto col development. Th erefore , our understan ding of the interp lay between the secrecy require - ments and the critical functio nalities of wireless networks, such as scheduling, r ou ting, and congestion co ntr ol rema ins very limited . Scheduling in wireless networks is a prominen t and ch al- lenging problem which a ttracted signiﬁcant in terest fr om the networking co mmunity . Th e challenge arises fr om th e fact that the capacity of wireless chan nel is time varying du e to multiple sup erimposed rand om effects su ch as mo bility and multipath fadin g. Optimal schedulin g in wireless networks has b een extensively studied in the literature u nder various assumptions [ 28], [ 29], [30], [31], [32], [33]. Starting with the semin al work of T assiulas and Ephrem ides [2 8] wher e throug hput optimality of backpr essure algorith m is pr oven, policies that opportu nistically explo it the time v ar ying nature of the wireless chan nel to sched ule u sers are shown to be at le ast as good as static policies [29]. In principle, th ese oppor tunistic p olicies schedule the user with th e fav orable channel cond ition to in crease the overall perfor mance of the system. Howev er, withou t impo sing in dividual pe rforman ce guaran tees for each user in the system, this typ e of sched uling results in unfair s haring of resources and may l ead to st arvation of some user s, for example, those far away from the b ase station in a cellular network. Hence, in order to address fairness issues, schedu ling prob lem was investi gated jointly with the network utility maximizatio n prob lem [34], [35], [36], and the stoch astic network optimiza tion fram ew ork [37] was developed. T o that end , in this pap er we add ress the basic wireless network co ntrol problem in ord er to develop a cro ss-layer resource a llocation solu tion th at will in corpor ate information priv acy , measur ed by equivoc ation , as a QoS metric. In partic- ular , we co nsider the single hop uplink setting, in whic h nod es collect priv ate and open informatio n, store them in sep arate queues an d tra nsmit them to th e base station . At a given point in time, only on e no de is sche duled to transmit and it may cho ose to transmit some combination of o pen and p riv ate informa tion. Our objective is to achieve pri vacy of information from the other legitimate nodes and we a ssume that th ere are no externa l malicious e av esdropp ers in th e system . Th e motiv a tion to stud y th is no tion of secrecy is the following. In some scenarios (e.g., tactical, ﬁnancial, medical), priv acy of 2 base station n 1 Q j o j p A j o A Q j p j R R j i R j 1 i j . . . Fig. 1. Uplink communicatio n with priv ate and open information. commun icated info rmation betwee n the nodes is necessary , so that data intended to ( or o riginated from) a n ode is not sha red by any oth er legitimate node. First, we evaluate the region of achiev able op en a nd priv ate data rate pairs for a single node scenario with and without joint encodin g of open and private informatio n. Then , we consider the m ulti-nod e scenario, and introduce private opportunistic scheduling . W e ﬁnd the achievable pr iv ate in formatio n rate regions associated with priv ate opportun istic sched uling and show th at it achieves the max imum sum priv ate information rate over all join t scheduling and encoding strategies. While priv ate opportun istic sched uler is based on the av ailability of full CSI on the uplin k channels, it does not rely on infor mation on the instantane ous cross-chann el (i.e., the channel between different nodes) CSI. It require s merely the long- term average rate of the cr oss-channe l rates. T o achieve pr i vacy with this lev el of CSI, priv ate op portu nistic scheduler uses an encod ing scheme th at encod es priv a te informatio n over many packets. Note that, in the seminal paper [38], it w as shown that oppor tunistic scheduling (withou t secrecy) m aximizes the sum rate. O ur r esult can be viewed as a genera lization of this resu lt to the case with secrecy . Next, we mo del th e problem as that of network utility maximization. W e provide a d ynamic joint ﬂow co ntrol, scheduling and priv ate encoding scheme, which takes into acc ount the instantaneou s direct- and cross-chan nel state inform ation but n ot a p riori channe l state distribution. In dyn amic cross-layer control scheme priv ate info rmation is divided into a sequence o f messages wher e each m essage is encoded into an individual p acket. W e prove tha t our scheme achieves a utility , arbitra rily close to the maxim um utility achiev able in this setting. W e gen eralize dynamic cross-layer control scheme to a more general case when instan taneous cross-chann el states are not known perfectly . Con sequently , we de ﬁne the no tions of privac y outage and privacy goodput . Finally , we num erically charac terize the perfor mance of the dynamic c ontrol algorithm with respect to se veral network parameters, and show tha t its perfo rmance is fairly close to that o f p riv ate opportunistic scheduler achie vable with k nown channel priors. I I . P R O B L E M M O D E L W e consider the cellu lar network illustrated in Fig. 1. T he network consists o f n n odes, each of which has bo th open and priv ate informatio n to be tr ansmitted to a single b ase station over the a ssociated u plink channel. When a nod e is transmitting, every o ther no de overhears the transmission over the associated cross channe l. W e assume every ch annel to be iid blo ck fading, with a block size of N 1 channel uses. The entire session lasts for N 2 blocks, which correspo nds to a total of N = N 1 N 2 channel uses. W e denote the instantan eous achiev able rate for the uplink chan nel o f nod e j by R j ( k ) , which is the max imum mutual infor mation b etween output symbols o f no de j a nd re ceiv ed symbo ls at the base station over block k . L ike wise, we den ote the ra te of the cro ss chann el between n odes j an d i with R ji ( k ) , which is the maximu m mutual in formation between outpu t symbols o f n ode j and input sym bols o f no de i over blo ck k . Note that there is no actual data transmission between any pa ir o f n odes, but parameter R ji ( k ) will be nec essary , when we ev aluate the priv ate rates between node j and the base station. Even th ough our results ar e gener al for all channel state distributions, in numerical ev aluations, we assum e all chann els to be Gaussian and the tran smit power to be con stant, identical to P fo r all blocks k , 1 ≤ k ≤ N 2 . W e r epresent th e u plink channel for no de j and the cr oss channel betwee n n odes j and i with a power gain (m agnitude square of the chan nel gains) h j ( k ) an d h ji ( k ) respectively over block k . W e normalize the power gains such that th e (additive Gaussian) noise has un it variance. Th en, a s N 1 → ∞ , R j ( k ) = log ( 1 + Ph j ( k )) (1) R ji ( k ) = log ( 1 + Ph ji ( k )) . (2) Each node j has a private and an open message, W pri v j ∈ { 1 , . . . , 2 N R priv j } and W open j ∈ { 1 , . . . , 2 N R open j } respectively , to be transmitted to the base station over N cha nnel uses, where R pri v j and R open j denote th e (long -term) p riv ate and open inf ormation rates r espectively , for node j . Let the vector o f symbo ls received by node i be Y i . T o achieve perfect privacy , following constraint must be satisﬁed by node j : for all i 6 = j , lim N → ∞ 1 N I ( W pri v j ; Y i ) ≤ ε (3) for any given ε > 0. W e deﬁne the instantaneo us private information rate of nod e j transmitted p riv ately fr om no de i over block k as: R p ji ( k ) = [ R j ( k ) − R ji ( k )] + , (4) where [ · ] + = max ( 0 , · ) . It was shown in [ 39] that rate (4) is achiev able as N 1 → ∞ and [1] to ok it a step fur ther and showed that, as N 1 , N 2 → ∞ , a long -term p riv ate inform ation r ate of E h R p ji ( k ) i is achiev a ble. The amo unt of op en trafﬁc, A o j ( k ) , and pr iv ate trafﬁc, A p j ( k ) , injected in th e queues at node j (shown in Fig. 1) in block k are bo th selected by node j at the beginnin g o f each block. Open an d priv a te infor mation are stored in separate qu eues with sizes Q o j ( k ) and Q p j ( k ) respectively . At any g i ven block, a scheduler ch ooses which node will transmit and the amoun t of open a nd priv ate in formation to be enc oded over the blo ck. W e use th e ind icator variable I j ( k ) to re present the scheduler decision: I j ( k ) = ( 1 , priv ate inf ormation from node j 0 , otherwise . (5) When we evaluate the region of a chiev a ble open an d p riv ate data rate pa irs for the single node scenar io, in Section III-A, we assume that th e transmitting nod e has perfect causal knowledge o f its uplink c hannel and the cross-chan nel at 3 1 2 k ( ) R 12 k ( ) 1 R BS Fig. 2. Single user priv ate communicat ion scenario. ev ery blo ck k . Thus, the achievable region o f pr iv ate an d open ra tes constitutes up per boun d o n the achiev ab le rates for each n ode, wh ich we ﬁnd subsequen tly for the mu ltiuser setting with partial CSI. For pr iv ate oppor tunistic scheduler in the multiu ser setting, we assume that, each no de j has perfect c ausal k nowledge of the up link chan nel rate, R j ( k ) , and its prio r distribution. Howe ver , we assume that it o nly has the long- term averages, E [ R ji ( k )] , i 6 = j of its cross- channel rates. T o ac hieve priv a cy with this level of CSI, priv ate opportunistic scheduler uses an encoding scheme that encodes p riv ate infor mation over m any packets. Wh en we formu late ou r prob lem as that of network utility m aximization problem , we only assum e knowledge of instantaneous ch annel gains without r equiring the knowledge of prior distribution of channel g ains . Hence, priv ate encodin g is per formed ov er a single block length unlike the case with priv ate opportu nistic scheduler . Addition ally , we analyze a more rea listic scenario when the instantaneous ch annel rates are not known perfectly , but estimated with some r andom additive error . The scheduled transmitter, j , will encode at a rate ˆ R p j ( k ) = [ R j ( k ) − ρ j ( k )] + , where ρ j ( k ) is the rate margin, chosen such th at the esti- mation erro r is taken into account. Note that wh en ρ j ( k ) < max i 6 = j R ji ( k ) , then perf ect priv acy constraint (3) is v iolated over block k . In suc h a case, we say that privacy o utage h as occurre d. The p robab ility of p riv acy ou tage over block k when user j is scheduled , is represented as p out j ( ρ j ( k )) . Sin ce perfec t priv acy cann ot be en sured over ev ery blo ck, we requ ire that expected proba bility o f p riv acy o utage of each u ser j is below a gi ven th reshold γ j . I I I . A C H I E V A B L E R A T E S A N D P R I V A T E O P P O RT U N I S T I C S C H E D U L I N G In this section, we ev alu ate the region of priv ate and open rates achiev able by a sched uler for m ultiuser uplink and downlink setting. W e start w ith a sing le node transmitting, and thus, th e scheduler o nly chooses wheth er to encod e priv ate info rmation a t any given p oint in time or not. W e consider th e po ssibility of both the separate a nd the joint encodin g of p riv ate and op en data. For multiuser transmission, we introdu ce o ur scheme, private opp ortunistic scheduling , ev alua te achiev able rates and show that it maximizes the sum priv ate informatio n rate achiev able by any scheduler . Along with p riv ate oppor tunistic sch eduling, we p rovide the asso- ciated p hysical-layer private e ncoding schem e that enco des informa tion over m any blo cks. A. Sin gle User A chievable Rates Consider the single user scenario in which the primar y user (node 1 ) is tran smitting infor mation over the primary channel and a single secondar y u ser (node 2) is overhearing the transmission over the second ary channel as shown in Fig. 2. In this scenario, we assume node 2 is p assi vely listening with out transmitting info rmation and n ode 1 has perfe ct knowledge of instantaneous rates R 1 ( k ) and R 12 ( k ) for all k a s well as their sample distributions. Over each blo ck k , the primary user chooses the rate of priv ate and open information to be transmitted to the intended rece iv er . As discussed in [40] it is possible to encode op en infor mation at a rate R 1 ( k ) − R p 12 ( k ) over each b lock k , jointly with the p riv ate information at rate R p 12 ( k ) . For that, one can simply replace th e rand omization message of th e binning strategy of the achiev ab ility schem e with the open message, which is allowed to be decoded by the secondary user . In the r est of the section, we an alyze both the case in wh ich o pen in formation can and cannot be encoded along with th e priv ate information. W e ﬁnd the region of achiev able priv ate and open infor mation rates, ( R pri v 1 , R open 1 ) , over the primary channel. 1) Sepa rate en coding of private and open messag es: Fir st we assume th at each block co ntains either priv ate or op en informa tion, but jo int en coding over the same b lock is not allowed. Recall tha t I 1 ( k ) is th e indic ator variable, which takes on a value 1, if info rmation is en coded priv ately over block k and 0 otherwise. Then, one can ﬁnd R pri v 1 , associated with the point R open 1 = α by solving the following in teger progr am: max { I 1 ( k ) }∈{ 0 , 1 } E  I 1 ( k ) R p 12 ( k )  (6) subject to E [( 1 − I 1 ( k )) R 1 ( k )] ≥ α , (7) where the expectations are over the joint d istribution of the instantaneou s r ates R 1 ( k ) an d R 12 ( k ) . Note that, sinc e the channel r ates are iid, the solu tion, I ∗ 1 ( k ) = I ∗ 1 ( R 1 ( k ) , R 12 ( k )) will b e a stationar y policy . Also, a necessary condition f or the existence of a feasible so lution is E [ R 1 ( k )] ≥ α . Dropping the block index k for simplicity , the problem leads to the follo win g Lagrang ian relaxa tion: min λ > 0 max { I 1 }∈{ 0 , 1 } E  I 1 R p 12  + λ ( E [ ( 1 − I 1 ) R 1 ] − α ) = min λ > 0  max { I 1 }∈{ 0 , 1 } Z ∞ 0 Z ∞ 0  I 1 R p 12 − λ ( 1 − I 1 ) R 1  p ( R 1 , R 12 ) d R 1 d R 12 − λ α } , ( 8) where p ( R 1 , R 12 ) is th e join t pd f o f R 1 and R 12 . For any given values of the Lagr ange m ultiplier λ and ( R 1 , R 12 ) pair, the optimal policy will choo se I ∗ 1 ( R 1 , R 12 ) = 0 if the integrant is maximized for I 1 = 0, or it will choose I ∗ 1 ( R 1 , R 12 ) = 1 otherwise. If b oth I 1 = 0 an d I 1 = 1 lead to an id entical value, the policy will choose one of th em rando mly . The solution can be summ arized as fo llows: R p 12 R 1 I ∗ 1 = 1 R I ∗ 1 = 0 λ ∗ , (9) where λ ∗ is the value of λ for wh ich E [ ( 1 − I ∗ 1 ) R 1 ] = α , since λ ∗ ( E [ ( 1 − I 1 ) R 1 ] − α ) ≤ 0 . For Gau ssian uplin k and cro ss channels descr ibed in Sec- tion II, the solu tion can be obtained b y pluggin g (1,2,4) in 4 00000000000 00000000000 00000000000 00000000000 00000000000 11111111111 11111111111 11111111111 11111111111 11111111111 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1− λ ) h 1 h 12 λ=0 h 12 h 1 1 1 =1 * =0 I * I (1+ −1 = Fig. 3. Optimal decisio n regions with separate encoding of priv ate and open messages. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 R open 1 R priv 1 solid: separate encoding dashed: joint encoding E[h 1 ]=2, E[h 12 ]=1 E[h 1 ]=2, E[h 12 ]=2.5 Fig. 4. Achiev able rate regions for the single user scenario with iid Rayleigh block fading channel s. (9): ( 1 + Ph 1 ) 1 − λ ∗ I ∗ 1 = 1 R I ∗ 1 = 0 1 + P h 12 . (10) The associated solu tion I ∗ is gra phically illustrated on th e ( h 1 , h 12 ) space in Fig. 3 for P = 1. As th e value of λ varies betwe en 0 an d 1, the optimal dec ision region for I = 0 increases from the u pper half of th e ﬁrst quadr ant represented b y h 12 ≥ h 1 to the entire ﬁrst qu adrant, i.e., all h 1 , h 12 ≥ 0 . In Fig. 4, the achiev ab le pair of priv ate and open informa tion r ates, ( R pri v 1 , R open 1 ) , is illustrated for iid Ray leigh fading Gaussian chan nels, i.e., the power gains h 1 and h 12 have an expon ential distribution. W e consid ered two d ifferent scenarios in wh ich the mean power gain s, ( E [ h 1 ] , E [ h 12 ]) , are ( 2 , 1 ) and ( 2 , 2 . 5 ) , and P = 1 . The associated bou ndaries of the rate regions with separa te enco ding ar e illustrated w ith solid curves. T o p lot these boun daries, we varied λ from 0 to 1 a nd calculated the achievable rate pair for ea ch point. Note that the ﬂat p ortion on the top part of the rate region s f or separate encodin g correspo nds to the case in which Constraint ( 7) is inactive. It is also in teresting to n ote that as demon strated in Fig. 4, one can achieve non -zero pri vate information rates e ven when the mean cross channel gain between node 1 and node 2 is higher tha n the mean u plink channel g ain of n ode 1. 2) Joint e ncoding of priva te an d o pen messa ges: With the possibility of joint enco ding o f th e o pen and priv ate info r- mation ov er th e same block, the indicator v ariable I 1 ( k ) = 1 implies th at the pri vate and o pen infor mation rates are R p 12 ( k ) and R 1 ( k ) − R p 12 ( k ) resp ectiv ely over block k simultaneo usly . Otherwise, i.e., if I 1 ( k ) = 0, open en coding is used solely over the block. T o ﬁnd ach iev able R pri v 1 , associated with the point R open 1 = α , on e needs to co nsider a slightly d ifferent optimization problem this ti me: max { I 1 ( k ) }∈{ 0 , 1 } E  I 1 ( k ) R p 12 ( k )  (11) subject to E  ( 1 − I 1 ( k )) R 1 ( k ) + I 1 ( k )( R 1 ( k ) − R p 12 ( k ))  ≥ α , (12) This optimization problem can be solved in a s imilar way by employing Lagrang ian relaxation as the problem co nsidered in Section III- A1. First, we spec ify two region s of parameters for which the solution is trivial: 1) if E [ R 1 ] < α , no solution exists fo r (1 1,12), since the up link ch annel capacity is not sufﬁcient to meet the desired open rate, α ; 2) if E  R 1 − R p 12  > α , then I ∗ 1 = 1 for all block s, i.e., all open information will be encoded jointly with pr iv ate information, s ince the remaining capacity over that is necessary to support pri vate info rmation is suf ﬁcient to serve open inform ation at rate α . In this case, Constraint (12) is inacti ve and the achieved p riv ate informa tion rate is R pri v 1 = E  R p 12  . In all other cases, i.e., E  R 1 − R p 12  ≤ α ≤ E [ R 1 ] , it can be shown that the optimal solution can b e ach iev ed b y th e following pr obabilistic scheme 1 : For any gi ven b lock, I ∗ 1 = ( 1 , w .p. p p 0 , w .p. 1 − p p , (13) indepen dently o f R 1 and R 12 , where p p = E [ R 1 ] − α E [ R p 12 ] . The d etails of the deriv ation of th e descr ibed op timal sch eme is given in [41]. With this so lution, only a fractio n p p of the b locks contain jointly enco ded priv ate and open information , and the remaining 1 − p p fraction of the blocks contain solely open informa tion. Th us, f or a given α , the achieved private and open information rates can be found as R pri v 1 = p p E  R p 12  = E [ R 1 ] − α and R open 1 = p p E  R 1 − R p 12  + ( 1 − p p ) E [ R 1 ] = α respectively . Rather surp risingly , it d oes no t matter whic h blocks contain on ly open informatio n an d which ones co ntain jointly en coded priv ate and open infor mation, as lon g as the desired open inf ormation rate α is met. Consequently , a random scheme that ch ooses 1 − p p fraction of bloc ks for open informa tion only and the rest f or jointly enco ded open and priv ate inf ormation suf ﬁces to achieve the optimal solution. By the above analysis, on e can con clude th at the a chiev a ble rate region with jo int encod ing can b e summ arized b y the intersection o f two regions speciﬁed by : (i) ( R pri v 1 + R open 1 ) ≤ E [ R 1 ] and ( ii) R pri v 1 ≤ E  R p 12  . Any point on the boundar y of th e region can be ach iev ed by the simple prob abilistic scheme descr ibed above. One can realize that this region is the maximum ach iev ab le rate region, since in our system, the to tal informa tion rate (pr iv ate and open) is upper bou nded b y the capacity , E [ R 1 ] , of up link cha nnel 1 a nd the a chiev a ble private rate is upper bound ed b y th e secrecy capac ity , E  R p 12  , of the associated wiretap chan nel. Thus, there exists no other sch eme that can achie ve a larger rate re gion than the one ach iev ed by the simple probabilistic sche me. In Fig. 4, the achiev able pairs o f private and open inf or- mation rates, ( R pri v 1 , R open 1 ) with joint encod ing are illustrated for the iid Rayleigh fading Gaussian ch annels with the same parameters as the separate enco ding scenario . The b ound aries 1 Note that the solution of Problem (11 ,12) is not unique and the described probabil istic solution is just one of them. 5 base station j n i * j ( ) R ji k ( ) j R k ( ) ( i ) ji * k ( ) R . . . i Fig. 5. Multiuser priv ate communica tion system - uplink of the regions are speciﬁed with dashed cur ves, which are plotted by varying the v alu e of p p from 0 to 1 and ev alu ating ( E  R 1 − R p 12  , E  R p 12  ) pa ir for e ach value. Sim ilar to the separate encoding scenario , the ﬂat p ortion o n the top part o f the regio ns correspo nds to the ca se in which Constraint (12) is inacti ve. B. Private Opportun istic Scheduling and Multiuser Achiev- able Rates In this s ection, we consider the mu ltiuser setting descr ibed in Fig. 1. W e introd uce private opp ortunistic schedu ling (POS) for the uplink scen ario and pr ove that it achieves the m aximum ach iev able sum p riv ate inform ation rate over the set of all schedu lers. POS sch edules the n ode that h as the largest instantaneous private infor mation rate, with resp ect to the “best eav esdrop per” node, which ha s the largest mean cross-chann el r ate. E ach n ode en sures p erfect p riv acy f rom its b est eavesdropper node by using a b inning strategy , which requires only th e average c ross-chann el rates to enco de the messages over many blocks. W e c onsider the mu ltiuser uplin k scenario given in Fig. 5. W e a ssume e very node j has p erfect causal knowledge of its uplink channel rate, R j ( k ) for all blo cks k and the average cross-chann el ra tes, E [ R ji ( k )] , for all i 6 = j . 1) P rivate Opp ortunistic Sched uling for uplink: W e deﬁne the b est eavesdropper of n ode j as i ∗ ( j ) , argmax i 6 = j E [ R ji ( k )] and denote its average cross-channel rate with ¯ R m j , E  R ji ∗ ( j ) ( k )  . Note that i ∗ ( j ) d oes no t cha nge fro m o ne block to anoth er . In POS, only o ne of the n odes is scheduled for data tr ansmission in any given bloc k. I n p articular, in block k , we opportun istically schedu le no de j M ( k ) , argmax j ∈{ 1 ,..., n }  R j ( k ) − ¯ R m j  if max j ∈{ 1 ,..., n } h R j ( k ) − ¯ R m j i > 0 and no node is scheduled fo r priv ate info rmation transmission otherwise, i.e., j M ( k ) = / 0. In case of multiple nod es achieving the same max imum priv acy rate, the tie can be broken at random. Indicator variable I POS j ( k ) takes on a value 1, if n ode j is sched uled over b lock k and 0 other wise. W e den ote the prob ability that n ode j b e sch eduled with p M j , P  j M ( k ) = j  and the associated uplin k channel rate when node j is s chedu led with ¯ R M j , E  R j ( k ) | j = j M ( k )  , where the expectations are ov er the condition al joint distribution of the instantaneo us rates o f all uplink channels, gi ven j = j M ( k ) . As will be shown sho rtly , private opportunistic schedulin g achieves a p riv ate inform ation rate R pri v j = p M j ( ¯ R M j − ¯ R m j ) for all j ∈ { 1 , . . . , n } . T o achiev e this set of rates, we use the following priv ate encod ing strategy based on binn ing: T o b egin, nod e j gen erates 2 N p M j ( ¯ R M j − δ ) random binary sequences. Then, it assigns ea ch ran dom binary sequence to one o f 2 N R priv j bins, so that each bin contains exactly 2 N p M j ( ¯ R m j − δ ) binary sequences. W e call the sequences associated with a bin, the randomizatio n sequences of that bin . Each bin of n ode j is one -to-one matched with a private message w ∈ { 1 , . . . , 2 N R priv j } rand omly . This selection ( along with the binary sequence s conta ined in each bin) is revealed to the b ase station and all nodes before the commun ication s tarts. Then , whenever the message to be transmitted is selected by node j , the stochastic enco der of that node ch ooses one of th e ran domization sequenc es associated with each bin at random 2 , ind ependen tly an d u niform ly ov er all rando mization sequences associated with tha t bin. This particular ran domization message is used for the transmission of the message and is not re vealed to any of the nodes no r to the base station. Priv ate op portun istic scheduler schedule s nod e j M ( k ) in each blo ck k and the transmitter tr ansmits N 1 R j M ( k ) ( k ) bits of the binary sequ ence a ssociated with the message of no de j M ( k ) f or all k ∈ { 1 , . . . , N 2 } . Thus, asymptotically , th e rate of data transmitted by n ode j over N 2 blocks is identical to: lim N 1 , N 2 → ∞ 1 N N 2 ∑ k = 1 N 1 I POS j ( k ) R j ( k ) = lim N 1 , N 2 → ∞ 1 N 2 N 2 ∑ k = 1 I POS j ( k ) R j ( k ) ≥ p M j ( ¯ R M j − δ ) w .p. 1 (14) for any g iv en δ > 0 from stron g law of large num bers. Hence, all of N ( p M j ( ¯ R M j − δ )) bits, gen erated b y each nod e j is transmitted with probability 1 . 2) Achievable up link rates with private opp ortunistic scheduling: Theor em 1: W ith priv ate opportun istic schedu ling, a priv ate informa tion rate of R pri v j = p M j ( ¯ R M j − ¯ R m j ) is ach iev able for each node j . The p roof of this theorem is based on an equ iv ocatio n analysis and it can b e found in Append ix A. Next we show that private oppo rtunistic scheduling maximize s the achievable sum pri vate info rmation rate amon g all sched ulers. Theor em 2: Among the elemen ts of the set of all schedulers, { I ( R 1 , . . . , R n ) } , private oppor tunistic schedu ler I POS ( R 1 , . . . , R n ) maximizes th e sum priv acy uplink rate, R pri v sum,up = ∑ n j = 1 R pri v j . Furthermor e, the maximum achievable sum pri vacy uplink rate is R pri v sum,up = n ∑ j = 1  p M j  ¯ R M j − ¯ R m j  . The proof of Theor em 2 can be found in Ap pendix B. There, we also show th at the in dividual p riv ate info rmation rates giv en in T heorem 1 ar e th e maximum ac hiev ab le individual rates with pri vate oppo rtunistic scheduling . He nce the converse of Theorem 1 also holds. Combining Theorems 1 and 2, one can r ealize tha t private opportu nistic scheduling achieves the maximum ach iev ab le sum pri vate information r ate. Th us, o ne cannot in crease th e individual private information rate a single node achieves with POS b y an amount ∆ > 0, without reducing another node’ s pri vate inform ation rate by more than ∆ . 2 In ca se of joint encodi ng of pri vate an d open information, the ran dom- izati on seque nce is chosen appropriat ely , correspondi ng to the desired ope n message. 6 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 R sum,up open /n R priv sum,up /n E[h ji ]=2.5 E[h ji ]=1 n=5 n=10 Fig. 6. Bounds on the achi ev able sum rate regio n for the multiuse r uplink scenari o with iid Rayleigh block fading channels. Next, we ﬁnd the b ound ary of the region o f achiev ab le sum open an d sum priv ate uplink rate pa ir with joint en - coding of private and op en information . In oppo rtunistic scheduling [38], [29 ] without any privac y con straint, th e user with the best u plink channel is scheduled for all blocks k . Hen ce, the associated achievable rate can be written as R opp sum,up = E  max j ∈{ 1 ,..., n } R j ( k )  . Since this constitutes an up- per bo und fo r th e achiev able cu mulative informatio n rate [38], the total pr iv ate and op en in formation rate in o ur system cannot exceed R opp sum,up . Combin ing this with Theo rem 2 , we can characterize an ou ter bound for the achievable r ate region for the sum r ates as f ollows: (i) R pri v sum,up + R open sum,up ≤ R opp sum,up ; (ii) R pri v sum,up ≤ ∑ n j = 1 h p M j  ¯ R M j − ¯ R m j i . Next we illustrate this region and discuss how the entire region can be ach iev ed by POS along with joint encoding of pr iv ate and open m essages. The boundaries of this re gion is illustrated in Fig. 6 for a 5 - node and a 10 -nod e system. W e assume all channels to be iid Rayleigh fading with mean u plink chann el power gain E [ h j ] = 2 an d m ean cro ss chann el power gain E [ h ji ] = 1 o r 2 . 5 in two separate scenarios for all ( i , j ) . No ise is additive Gau ssian with unit v arian ce and transmit po wer P = 1. In th ese graphs, sum rates are no rmalized with respect to the numbe r of n odes. One can obser ve th at, the achievable sum rate per nod e decreases from 0 . 31 to 0 . 19 bits/cha nnel use/n ode fo r E [ h ji ] = 1 an d from 0 . 2 to 0 . 13 bits/chann el use/node for E [ h ji ] = 2 . 5 as the number of no des increases from 5 to 10. Also, the o pen rate per no de drops from 0 . 4 7 to 0 . 27 bits/chann el use/node with the same increase in the number of nodes. Note that, any po int on the pa rt of the bo undar y speciﬁed b y (i) above (ﬂat portion on the top part) is achiev able by POS and jointly encoding th e priv ate informatio n with the ap propr iate amount of op en info rmation used as a rando mization message. For in stance, the corn er p oint of two boundaries ( intersection of (i) an d (ii)) is ac hieved when open inform ation is u sed completely in place of rand omization messages by all nodes. All points on the pa rt of the bounda ry speciﬁed by (ii) can be achieved by time- sharing b etween the corner point, and point  R opp sum,up , 0  , which correspo nds to opp ortun istic schedulin g (without pri vacy). I V . D Y N A M I C C O N T RO L O F P R I V AT E C O M M U N I C A T I O N S In Section III, we d etermined the achiev able p riv ate in- formation rate regions associated with p riv ate opp ortunistic scheduling which en codes message s over ma ny blocks . Henc e, the delay of d ecoding private infor mation m ay be extrem ely long. Also, the private opportu nistic scheduler was based on the av ailab ility of full CSI on the uplin ks, an d lo ng-term av erage of cross-ch annel rates. In this section, we investigate a dynamic contro l algo rithm which does n ot rely on any a prio ri knowledge of distributions of dire ct- or cross-chann el rates, and the pr iv ate informa tion is enc oded over a single block . Hence, a priv ate message can be decod ed with a maximum delay of on ly a sin gle block du ration. Note tha t ev en though by encodin g over many blo cks o ne m ay a chieve higher pr iv ate informa tion rates, de coding delay may be a mo re important concern in many practical scenarios. In particu lar , each message W pri v j and W open j are bro ken into a sequence of messages, W pri v j ( k ) and W open j ( k ) respec ti vely and each element of the sequence is encod ed into an i ndividual packet, encoded over blo ck k . The delay-limited d ynamic cross-layer con trol algorithm opp ortunistically schedules the nodes with th e objectiv e of maxim izing the total expected utility gained from each packet transmission while maintaining the stab ility of pri vate and open trafﬁc queues. The algorithm takes as input the queue lengths and instantan eous direct- and cross-chann el rates, and gives a s outpu t the scheduled nod e and its pri vacy encoding rate. In the sequel, we only co nsider joint encoding o f p riv ate and open information as descr ibed in Section III-A2. Let g p j ( k ) a nd g o j ( k ) b e the utilities ob tained b y nod e j f rom priv ate and open transmissions over blo ck k respectively . Let us deﬁne the instantan eous priv ate inf ormation rate of no de j as R p j ( k ) , min i 6 = j R p ji ( k ) , where R p ji ( k ) was deﬁn ed in (4 ). Also, the in stantaneou s ope n rate, R o j ( k ) , is the amount of open inform ation nod e j transm its over block k . The utility over block k de pends on rates R p j ( k ) , and R o j ( k ) . In gen eral, this depen dence can be described as g p j ( k ) = U p j ( R p j ( k )) and g o j ( k ) = U o j ( R o j ( k )) . Assume that U p j ( 0 ) = 0 , U o j ( 0 ) = 0 , and U p j ( · ) , U 0 j ( · ) are concave no n-decr easing functio ns. W e also assume that the utility of a p riv ate tran smission is higher than the utility of open transmission at the same rate. Th e amount of open traf ﬁc A o j ( k ) , and priv ate tr afﬁc A p j ( k ) injected in the queues at node j have long term arrival r ates λ o j and λ p j respectively . Our o bjective is to su pport a fraction of the trafﬁc demand to achieve a long term priv ate an d op en thr oughp ut that maximizes the sum of utilities of the no des. A. P erfect Knowledge of Instantan eous CSI W e ﬁrst consider the case when every node j has perfect causal knowledge o f its uplink chann el rate, R j ( k ) , and cro ss- channel rates to all other nod es in the network R ji ( k ) , ∀ j 6 = i , for all b locks k . The dyn amic con trol alg orithm d eveloped for this case will th en provide a basis f or the algor ithm that we are going to develop for a more realistic case when cross-chan nel rates are not known pe rfectly . W e aim to ﬁnd the solution of the follo wing optimization problem: max n ∑ j = 1  E h g p j ( k ) i + E  g o j ( k )   (15) subject to ( λ o j , λ p j ) ∈ Λ (16) The ob jectiv e f unction in (15) ca lculates the tota l expected utility of ope n an d private co mmunicatio ns where expectation is taken over the ran dom a chiev a ble r ates (r andom channe l 7 condition s), and possibly over the ran domized po licy . Th e constraint ( 16) ensur es that priv ate and open injection rates are with in the achiev able rate region suppo rted by the network denoted by Λ . In the afo remention ed o ptimization pr oblem, it is implicitly requ ired that perfect secrecy condition g iv en in (3) is s atisﬁed in each block as N 1 → ∞ . The pro posed cross-layer dyn amic control algorith m is based on the stoc hastic network o ptimization framework de- veloped in [3 7]. Th is f ramework allows the solution of a long-ter m stochastic optimization prob lem without requ iring explicit characterization of the achiev a ble r ate re gion, Λ . W e assume that there is an inﬁnite b acklog of data at the transport la yer of each nod e. Our pr oposed dyn amic ﬂow control algorithm deter mines the amount of open an d priv ate trafﬁc injected into the queu es at the n etwork layer . The dynamics of priv ate and open trafﬁc queues is given as follows: Q p j ( k + 1 ) = h Q p j ( k ) − R p j ( k ) i + + A p j ( k ) , (17) Q o j ( k + 1 ) =  Q o j ( k ) − R o j ( k )  + + A o j ( k ) , (18) where [ x ] + = max { 0 , x } , and the ser vice rates of private and open queues are given as , R p j ( k ) = I p j ( k )  R j ( k ) − max j 6 = i R ji ( k )  , and R o j ( k ) = I o j ( k ) R j ( k ) + I p j ( k )( R j ( k ) − R p j ( k )) . where I p j ( k ) and I o j ( k ) are indicator functions taking value I p j ( k ) = 1 when transmitting jointly encoded pri vate and open trafﬁc, or I o j ( k ) = 1 when transmitting on ly open traf ﬁc o ver block k respectively . Also n ote that at any block k , ∑ j I p j ( k ) + I o j ( k ) ≤ 1. Control A lgorithm: The algorith m is a simple index p olicy and it e xecutes the following steps in each block k : (1) Flow control: For some V > 0, eac h nod e j in jects A p j ( k ) priv ate an d A o j ( k ) open bits, where  A p j ( k ) , A o j ( k )  = argmax A p , A o n V h U p j ( A p ) + U o j ( A o ) i −  Q p j ( k ) A p + Q o j ( k ) A o o (2) Scheduling: Sch edule node j and transmit jointly en coded private and open trafﬁc ( I p j = 1 ), o r o nly open ( I o j = 1 ) trafﬁc, wher e ( I p j ( k ) , I o j ( k )) = argmax I p , I o n Q p j ( k ) R p j ( k ) + Q o j ( k ) R o j ( k ) o , and fo r each n ode j , encode priv a te data over e ach bloc k k at rate R p j ( k ) = I p j ( k )  R j ( k ) − max i 6 = j R ji ( k )  , and transmit open data at rate R o j ( k ) = I o j ( k ) R j ( k ) + I p j ( k )( R j ( k ) − R p j ( k )) Optimality o f Contr ol Algo rithm: The optima lity of the algorithm ca n be shown using the L yapun ov optimizatio n theorem [ 37]. Befo re restatin g this theorem, we deﬁn e the following p arameters. Let W ( k ) = ( W 1 ( k ) , . . . , W n ( k )) b e the queue b acklog proc ess, and let our ob jectiv e be the maxi- mization of time average o f a scalar v alued function f ( · ) of another pro cess R ( k ) while keeping W ( k ) ﬁnite. Also d eﬁne ∆ ( W ( k )) = E [ L ( W ( k + 1 )) − L ( W ( k )) | W ( k )] as th e dr ift of some appropriate L yap unov fu nction L ( · ) . Theor em 3: (L yap unov Optimization ) [3 7] For the scalar valued f unction f ( · ) , if there exists po siti ve con stants V , ε , B , such th at fo r all b locks k and all unﬁnished work vector W ( k ) the L yapunov drif t s atisﬁes: ∆ ( W ( k )) − V E [ f ( R ( k )) | W ( k ) ] ≤ B − V f ∗ − ε n ∑ j = 1 W j ( k ) , (19 ) then the tim e a verage utility a nd queue backlog satisfy: lim inf N 2 → ∞ 1 N 2 N 2 − 1 ∑ k = 0 E [ f ( R ( k )) ] ≥ f ∗ − B V (20) lim sup N 2 → ∞ 1 N 2 N 2 − 1 ∑ k = 0 n ∑ j = 1 E [ W j ( k )] ≤ B + V ( ¯ f − f ∗ ) ε , (21) where f ∗ is the maxim al value o f E [ f ( · )] and ¯ f = lim sup N 2 → ∞ 1 N 2 ∑ N 2 − 1 k = 0 E [ f ( R ( k )) ] . For ou r purpo ses, we consider p riv ate and op en unﬁn- ished work vectors as Q p ( k ) = ( Q p 1 ( k ) , Q p 2 ( k ) , . . . , Q p n ( k )) , and Q o ( k ) = ( Q o 1 ( k ) , Q o 2 ( k ) , . . . , Q o n ( k )) . Let L ( Q p , Q o ) be quadra tic L yapunov function of pri vate and op en queue b ack- logs deﬁned as: L ( Q p ( k ) , Q o ( k )) = 1 2 ∑ j h ( Q p j ( k )) 2 + ( Q o j ( k )) 2 i . (22) Also consider the o ne-step e xpected L yapunov d rift, ∆ ( k ) for the L yapunov fun ction (22) as: ∆ ( k ) = E [ L ( Q p ( k + 1 ) , Q o ( k + 1 )) − L ( Q p ( k ) , Q o ( k ))   Q p ( k ) , Q o ( k )  . (23) The following lemma provides a n upper b ound on ∆ ( k ) . Lemma 1: ∆ ( k ) ≤ B − ∑ j E h Q p j ( k )( R p j ( k ) − A p j ( k ))   Q p j ( k ) i − ∑ j E  Q o j ( k )( R o j ( k ) − A o j ( k ))   Q o j ( k )  , (24) where B > 0 is a c onstant. The proo f of Lem ma 1 is given in Appendix C. Now , we present our main result showing that our propo sed dy namic control algorithm can achieve a per forman ce arbitrar ily close to the optimal solution while keeping the queue backlo gs bound ed. Theor em 4: If R j ( k ) < ∞ for all j , k , th en dynamic control algorithm satisﬁes: lim inf N 2 → ∞ 1 N 2 N 2 − 1 ∑ k = 0 n ∑ j = 1 E h g p j ( k ) + g o j ( k ) i > g ∗ − B V lim sup N 2 → ∞ 1 N 2 N 2 − 1 ∑ k = 0 n ∑ j = 1 E h Q p j ( k ) i 6 B + V ( ¯ g − g ∗ ) ε 1 lim sup N 2 → ∞ 1 N 2 N 2 − 1 ∑ k = 0 n ∑ j = 1 E  Q o j ( k )  6 B + V ( ¯ g − g ∗ ) ε 2 , where B , ε 1 , ε 2 > 0 ar e constants, g ∗ is the optima l so lution of (15)-(16) and ¯ g is the maximum p ossible aggregate utility . The proof of Th eorem 4 is given in Appen dix D. 8 B. Imp erfect Knowledge of Instanta neous CSI In the previous section, we perform ed our analysis assuming that at every blo ck exact instantane ous cross-chann el rates are available. Howe ver , unlike the uplink d irect chann el rate which can be deter mined by the base station prior to the da ta transmission (e.g., via pilot signal transmission), cro ss-channel rates are har der to be estimated. I ndeed, in a non-coo perative network in wh ich nodes do not exch ange the ir CSI, the cross- channel rates { R ji , j 6 = i } can only be infer red by n ode j from the recei ved signals over th e r ev erse ch annel as nodes j 6 = i are tra nsmitting to the base station. H ence, at a given block, nodes only hav e a posteriori cha nnel distribution. Based on this a p osteriori ch annel distribution, n odes may estimate CSI of their cross-channels. Let us d enote the estimated rate o f the cro ss-channel ( j , i ) with ˆ R ji ( k ) . W e also d eﬁne cr oss-chan nel rate ma r g in ρ j ( k ) as the cross-ch annel rate a node uses when it enco des p riv ate informa tion. More speciﬁcally , nod e j enco des its pri vate informa tion at rate: R p j ( k ) = R j ( k ) − ρ j ( k ) , (25) i.e., ρ j ( k ) is the rate o f th e r andom ization message n ode j uses in the r andom bin ning scheme for p riv acy . Note that, if ρ j ( k ) < ma x i 6 = j R ji ( k ) , th en node j will not me et the perf ect secrecy constraint at block k , leadin g to a privacy outage . In the event of a priv acy outage, the pri vately en coded message is considered as an open message. The pr obability of p riv acy outage over block k fo r the sched uled nod e j , given the estimates of the c ross channel rates is: p out j ( ρ j ( k )) = P  max i 6 = j R ji ( k ) > ρ j ( k )    { ˆ R ji ( k ) , i 6 = j }  . (26) Compare th e afo rementio ned deﬁnition of privac y ou tage with the channel o utage [42] experienced in fast varying wire less channels. In time-varying wireless ch annels, ch annel outag e occurs wh en received signal an d interfe rence/no ise ratio dro ps below a thresh old necessary f or corr ect decodin g of the transmitted signal. Hence, the largest rate o f reliab le co mmu- nications at a given outage probab ility is an importan t measure of channel qua lity . In the f ollowing, we aim to d etermine utility max imizing achiev able privac y and open transmission rates fo r giv en privacy ou tage probabilities. In particular, we consider the solution of the following op timization problem: max n ∑ j = 1  E h g p j ( k ) i + E  g o j ( k )   (27) subject to ( λ o j , λ p j ) ∈ Λ , (28) and p out j ( ρ j ( k )) = γ j , (29) where γ j is the tolerable priv acy outage proba bility . Aforemen- tioned optimization problem is the same as the one gi ven for perfect CSI except for the last constrain t. The ad ditional co n- straint (29) requires that only a certain prescribed proportion of priv ate tran smissions are allowed to violate the perfect priv acy constraint. Due to priv acy outages w e deﬁne private goodpu t of user j as E h R p j ( k )  1 − p out j ( ρ j ( k )) i . Note th at priv ate goodp ut o nly includ es p riv a te messages for wh ich pe rfect priv acy constraint is satisﬁed. All pri vate me ssages for which (3) is violated ar e counted a s successful o pen transmissions. Similar to the per fect CSI case, we argue that a dynamic policy can be used to achiev e asy mptotically o ptimal solutio n. Unlike the algo rithm given in the perfec t CSI case, th e algorithm for im perfect CSI ﬁrst determines th e private data encodin g r ate so that the p riv acy ou tage constra int (29) is satisﬁed in cur rent block. Hence, the pr iv ate encod ing rate at a particular bloc k is de termined by the estimated ch annel rates and the pr iv acy outage con straint. Control Alg orithm: Similar to the perfect CSI case, ou r algorithm in volves two steps in each blo ck k : (1) Flow Cont rol: F o r some V > 0, e ach nod e injects A p j ( k ) priv ate an d A o j ( k ) open bits, where  A p j ( k ) , A o j ( k )  = argmax A p j , A o j V h U p j ( A p j )( 1 − γ j ) + U o j ( A o j )( 1 − γ j ) + U o j ( A o j + A p j ) γ j i − Q p j ( k ) A p j − Q o j ( k ) A o j . (30) (2) Scheduling: Sch edule no de j and tr ansmit jo intly encoded private a nd ope n trafﬁc ( I p j = 1) o r on ly op en ( I o j = 1) trafﬁc, wher e  I p j ( k ) , I o j ( k )  = argmax I p , I o n Q p j ( k ) R p j ( k ) + Q o j ( k ) R o j ( k ) o . For each node j , en code priv ate d ata over each block k at rate R p j ( k ) = I p j ( k ) [ R j ( k ) − ρ j ( k )] , ρ j ( k ) = p out − 1 j ( γ j ) , and transmit open data at rate R o j ( k ) = I o j ( k ) R j ( k ) + I p j ( k )( R j ( k ) − R p j ( k )) . Optimality o f Contr ol Algorithm: The o ptimality of the control algorithm with imperfect CSI can be shown in a similar fashion as for the control algorithm with perfect CSI. W e use the same L yapun ov functio n deﬁn ed in (22) which results in the same on e-step L yap unov dr ift function (23). Hence, Lemma 1 also h olds f or th e case of im perfect CSI, but with a d ifferent con stant B ′ due to the fact that hig her maxim um priv ate information rates can be achieved by allowing priv acy outages. L yapun ov Optimization Theor em sug gests that a g ood con- trol strategy is the one that m inimizes the follo wing: ∆ U ( k ) = ∆ ( k ) − V E " ∑ j ( g p j ( k ) + g o j ( k ))    Q p ( k ) , Q o ( k ) # (31) In (3 1), expectation is over all p ossible chan nel states. The expected utility for priv ate and op en transmission s a re re spec- ti vely gi ven as: E h g p j ( k ) i = E h g p j ( k ) | I p j ( k ) , ρ j ( k ) i = ( 1 − γ j ) E h U p j  A p j ( k ) i , (32) E  g o j ( k )  = E h g o j ( k ) | I p j ( k ) , I o j ( k ) , ρ j ( k ) i = γ j E h U o j ( A p j ( k ) + A o j ( k )) i + ( 1 − γ j ) E  U o j ( A o j ( k ))  . (33) Note that (32)-(33) are ob tained d ue to Constraint (29). By combinin g Lemma 1 with (32)-(33) we m ay obtain an upper bound for (31), as follows: 9 ∆ U ( k ) < B ′ − ∑ j E h Q p j ( k )[ R p j ( k ) − A p j ( k )] i − ∑ j E  Q o j ( k )[ R o j ( k ) − A o j ( k )]  − V E " ∑ j ( 1 − γ j ) U p j  A p j ( k )  + ∑ j γ j U o j ( A p j ( k ) + A o j ( k )) + ( 1 − γ j ) U o j ( A o j ( k )) # . (3 4) Now , it is clear that the pro posed dyna mic control algorith m minimizes th e r ight h and side of (34). Th e steps of p roving the optim ality of the dy namic contro l algorithm are exactly the same as those given in Theorem 4, and hence, we skip the details. Theor em 5: If R j ( k ) < ∞ , for all j , k th en dynamic control algorithm satisﬁes: lim inf N 2 → ∞ 1 N 2 N 2 − 1 ∑ k = 0 n ∑ j = 1 E [ g p j ( k ) + g o j ( k )] ≥ g ′ ∗ − B ′ V (35) lim sup N 2 → ∞ 1 N 2 N 2 − 1 ∑ k = 0 n ∑ j = 1 E [ Q p j ( k )] ≤ B ′ + V ( ¯ g ′ − g ′ ∗ ) ε ′ 2 lim sup N 2 → ∞ 1 N 2 N 2 − 1 ∑ k = 0 n ∑ j = 1 E [ Q o j ( k )] ≤ B ′ + V ( ¯ g ′ − g ′ ∗ ) ε ′ 1 , where B ′ , ε ′ 1 , ε ′ 2 > 0 ar e constants, g ′ ∗ is the optimal solution of (27)-(29) and ¯ g ′ is the maxim um p ossible aggregate utility . V . N U M E R I C A L R E S U LT S In our numerica l expe riments, we consider ed a network consisting of ten n odes and a sing le base station. The direc t channel between a no de and the base statio n, and the cr oss- channels between pair s o f nodes are modeled as iid Rayleigh fading Gau ssian ch annels. Thus, d irect-chan nel an d cross- channel power gains are expo nentially distributed with means chosen uniformly ra ndomly in the intervals [ 2 , 8 ] , and [ 0 , 1 ] , respectively . The noise nor malized power is P = 1. In our simulations, we co nsider both o f the cases wh en perfect in- stantaneous CSI is av ailable, and when instantaneo us CSI can only be estimated with some erro r . Unless oth erwise indicated, in th e case of imperfect CSI, we take the tolerable pr i vacy outage probab ility as 0 . 1. W e ass umed the use o f an unbia sed estimator for the cro ss-channel po wer gains and modeled the associated estimation error wit h a Gaussian random variable: ˆ h ji ( k ) = h ji ( k ) + e ji ( k ) , where e ji ( k ) ∼ N ( 0 , σ 2 ) for all k . Gaussian estimatio n error can be justiﬁed as discussed in [43] or by the use of a recursive ML estimator as in [44]. Un less otherwise stated , we take σ = 0 . 5, i.e., the estimatio n error is rather signiﬁcan t relati ve to the mea n cr oss-channe l g ain. Note that, in th is sectio n, w e choose the margin ρ j ( k ) such that P  ρ j ( k ) < max i 6 = j [ log ( 1 + Ph ji )]     { ˆ h ji , i 6 = j }  ≤ γ j . W e consider logarith mic pr iv ate and op en utility functio ns where the p riv a te utility is κ times mo re than o pen utility at the same rate. More sp eciﬁcally , we take fo r a sch eduled nod e j , U p j ( k ) = κ · lo g ( 1 + R p j ( k )) , and U o j ( k ) = log ( 1 + R o j ( k )) . W e take κ = 5 in all the experim ents except for the on e inspecting the effect of κ . Th e rates depicted in the graphs are per node arriv al or service rate s calculated as the total arriv al or service rates achieved by the network d ivided by the num ber of nodes, i.e., th e unit of the plotted rates is bits/chan nel u se/node. Finally , f or perfe ct CSI, we o nly plot the service rates since arriv al and service rates are identical. In Fig. 7 a-7b, we investigate the effect of system parameter V in our d ynamic control algorithm. Fig. 7a shows that for V > 4, lon g-term u tilities con verge to their optimal values fairly clo sely . It is also observed tha t CSI e stimation erro r results in a r eduction of appro ximately 25% in aggr egate utility . Fig. 7b dep icts the well-k nown relationship b etween V and queu e backlo gs, where q ueue ba cklogs incr ease when V is increased. In Fig. 8a-8b, th e ef fect of increasing number o f nodes on the ac hiev ab le p riv ate and open r ates obtained with the propo sed dyn amic contr ol alg orithm is shown. In bo th ﬁg ures, the private inf ormation rate ach iev ed by POS algorithm given in Section I II is also d epicted. From Fig. 8a , we ﬁr st notice that by u sing the dynam ic con trol algo rithm which is based only on th e instantaneou s CSI, the private service rate is reduced by mo re tha n 25% as com pared to the maximum priv ate inf ormation rate achieved by POS which uses a priori CSI to e ncode over m any b locks. Th is difference increases with inc reasing numb er of nod es. Howe ver, for bo th POS and dynamic control algorithms, the achiev able r ates decrease with increasing n umber o f nodes since more n odes overhear ongoin g transmissions. M eanwhile, open service rate also decreases due to th e fact that ther e is a smaller nu mber o f transmission op portun ities per node with increasin g number of nodes. Fig. 8b dep icts that pr iv ate service r ate has decreased by app roximately 50% du e to CSI estimation e rrors. It is also interesting to note that pr iv ate arrival rate is h igher than the priv ate service rate, since all private messages f or w hich perfect privac y constraint can not be satisﬁed are co nsidered as succe ssful open messages. Hence, open service rate is observed to be higher than the o pen arriv al rate. W e next an alyze the effect of κ , which ca n also be inter- preted as the r atio of u tility of p riv ate and op en transmissions taking p lace at the same rate. W e call this ratio private utility gain . Fig. 9a shows that wh en priv ate utility gain is grea ter than 5, then the private and open service rates converge to their respective limits. Th ese limits dep end on the ch annel characteristics, and their sum is approx imately equal to the maximum a chiev a ble rate of th e c hannel. Howe ver, whe n there is CSI estimatio n er ror, Fig. 9b shows that although an id entical q ualitative relationship between a rriv al r ates and priv ate utility gain is still observed, p riv ate service rate is lo we r than the p riv a te arriv al rate by a f raction of γ almost u niform ly in the range of κ . In Fig. 10 a, we in vestigate the effect of the tolerable priv acy o utage probab ility . It is interesting to note that pr iv ate service rate increases initially with incr easing tolerable o utage probab ility . This is b ecause f or low γ values, in order to satisfy the tight priv acy outage constraint, a low instantaneo us priv ate informa tion rate is chosen. However , wh en γ is hig h more priv acy o utages a re experience d at th e expense of hig her 10 0 1 2 3 4 5 6 0 0.5 1 1.5 2 2.5 V Long−Term Utility Private Utility ( γ =0.1, σ =0.5) Open Utility ( γ =0.1, σ =0.5) Aggregate Utility ( γ =0.1, σ =0.5) Secure Utility (Perfect CSI) Open Utility (Perfect CSI) Aggregate Utility (Perfect CSI) (a) Long-term Utility 0 1 2 3 4 5 6 0 20 40 60 80 100 120 V Queue Size Private Queue Size ( γ =0.1, σ =0.5) Open Queue Size ( γ =0.1, σ =0.5) Private Queue Size (Perfect CSI) Open Queue Size (Perfect CSI) (b) A verage queue length Fig. 7. Numerical results with respect to optimiz ation paramete r V . 2 3 4 5 6 7 8 9 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Number of Nodes Rate(bits/channel use) Private Service Rate (Perfect CSI) Open Service Rate (Perfect CSI) POS Privacy Rate (a) Perfect CSI 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Number of Nodes Rate(bits/channel use) Private Service Rate ( γ = 0.1, σ = 0.5) Open Service Rate ( γ = 0.1, σ = 0.5) Private Arrival Rate ( γ = 0.1, σ = 0.5) Open Arrival Rate ( γ = 0.1, σ = 0.5) POS Privacy Rate (b) Imperfect CSI Fig. 8. Pri va te and open rates with respec t to number of nodes 1 2 3 4 5 6 7 8 9 10 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Private Utility Gain Rate(bits/channel use) Private Service Rate (PerfectCSI) Open Service Rate (Perfect CSI) (a) Perfect CSI 1 2 3 4 5 6 7 8 9 10 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.25 0.26 Private Utility Gain Rate(bits/channel use) Private Service Rate ( γ = 0.1, σ =0.5) Open Service Rate ( γ = 0.1, σ =0.5) Private Arrival Rate ( γ = 0.1, σ =0.5) Open Arrival Rate ( γ = 0.1, σ =0.5) (b) Imperfect CSI Fig. 9. Pri va te and open rates with respec t to increa sing amount of priv ate util ity gain. instantaneou s priv ate inf ormation rates. This is also the rea- son why we o bserve that the difference betwee n the pr i vate service and arr iv al rates is increasing. W e n ote that wh en CSI estimation error is presen t, the high est priv a te service rate is obtained wh en γ is approximate ly eq ual to 0 . 1. The hig hest priv ate serv ice rate with CSI estimation error is a pprox imately 30% lo wer than the p riv ate service rate w ith the perfect CSI. W e ﬁnally in vestigate the ef fect of the quality of CSI estimator in Fig. 10b. For this purpose, we v ary the standard deviation of the Gaussian random variable m odeling the es- timation error . As expec ted th e highest priv ate ser vice rate is obtained when σ = 0. Howe ver , it is im portan t to note that this value is still lower th an the priv ate service rate with perfect CSI, since p riv a cy outages are still per mitted in 10% of pri vate transmissions. W e have also investi gated th e p erfor- mance of the dy namic control alg orithm when a posterio ri CSI distribution is not available. In this case, scheduling and ﬂow control decisions a re based only on th e mean cross channel gains. When only mean cross channel g ains are a vailable, the achieved priv ate serv ice rate per no de is app roximately equal to 0 . 16 bits per ch annel use, wh ich is signiﬁcantly lower than the private service rate with per fect CSI. In p articular, it is only when the stan dard deviation o f the estimatio n er ror is 0 . 7 that the pr i vate service rate with noisy chan nel estimator has the same pri vate service rate ach iev ab le utilizing only mean channel gains. V I . C O N C L U S I O N S In this paper, we studied the achievable priv ate and open informa tion rate regions of single- a nd multi- user wireless net- works with n ode schedulin g. W e introdu ce p riv ate op portun is- tic scheduling along with a pri vate encoding strategy , and show 11 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 γ Rate(bits/channel use) Private Service Rate Open Service Rate Private Arrival Rate Open Arrival Rate Privacy Rate with Perfect CSI (a) Effec t of γ . 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.15 0.2 0.25 0.3 0.35 σ Rate(bits/channel use) Private Service Rate Open Service Rate Private Arrival Rate Open Arrival Rate Privacy Rate with Perfect CSI Privacy Rate with Channel Mean (b) Effect of σ . Fig. 10. Pri va te and open rates with respect to tolera ble pri va cy outage probab ility . that it max imizes the sum p riv ate inform ation rate fo r bo th multiuser up link com municatio n when per fect CSI is av ailable for only the m ain uplin k channels. Then , we described a cro ss- layer dynamic algorithm th at works without pr ior distribution of chan nel states. W e prove that ou r algorith m, which is based on simple in dex policies, achiev es u tility arbitrarily close to achiev able op timal utility . Th e simu lation results also verify the efﬁcac y o f the algorithm. As a f uture directio n, we will investigate the co operation among nodes, e.g. , intelligent jamming fro m co operating nodes, as a means to improve the achiev able priv ate inf orma- tion rates. W e will also investigate an extension of the dynamic control policy for imperfect CSI, where th e optimal priv acy outage probability is also determ ined by the algorithm . R E F E R E N C E S [1] P . K. Gopala, L. Lai, and H. El Gamal, “On the secrec y capac ity of fadi ng chann els, ” IEE E T rans. Inform. Theory , vol. 54, pp. 4687–4 698, Oct. 2008. [2] J. Barros and M. R. D. Rodrigues, “Secrecy capacit y of wireless channe ls, ” in Proc . IE EE Int. Symposium Inform. Theory , (Seattle , W A), pp. 356–360, July 2006. [3] Y . Liang, H . V . Poor , and S. 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A P P E N D I X A P R O O F O F T H E O R E M 1 Let us further introduce the following n otation: W rand j : rando mization sequence associated with message W pri v j , X ( k ) : transmitted vector of ( N 1 ) symbols over block k , X j = { X ( k ) | I POS j ( k ) = 1 } : the transmitted signal over block k , wh enever I POS j ( k ) = 1 (i.e., node j is the active transmitter) Y i ( k ) : th e receiv ed vector of s ymbo ls at no de i ( Y b ( k ) f or the base station) ov er block k , Y j i = { Y i ( k ) | I POS j ( k ) = 1 } : the receiv ed signal at n ode i over block k , wh enever I POS j ( k ) = 1 (i.e. , node j is th e activ e transmitter). W e use Y j j for the r eceiv ed signal b y the base station. The equ iv oca tion a nalysis fo llows directly for the described priv acy scheme: For an y gi ven n ode j , we ha ve H ( W pri v j | Y j i ) ≥ H ( W pri v j | Y j i ∗ ( j ) ) (36) = I ( W pri v j ; Y j 1 , . . . , Y j n | Y j i ∗ ( j ) ) + H ( W pri v j | Y j 1 , . . . , Y j n ) ≥ I ( W pri v j ; Y j 1 , . . . , Y j n | Y j i ∗ ( j ) ) = I ( W pri v j , W rand j ; Y j 1 , . . . , Y j n | Y j i ∗ ( j ) ) − I ( W rand j ; Y j 1 , . . . , Y j n | Y j i ∗ ( j ) , W pri v j ) (37) = I ( W pri v j , W rand j ; Y j 1 , . . . , Y j n | Y j i ∗ ( j ) ) − H ( W rand j | Y j i ∗ ( j ) , W pri v j ) + H ( W rand j | Y j 1 , . . . , Y j n , W pri v j ) ≥ I ( W pri v j , W rand j ; Y j 1 , . . . , Y j n | Y j i ∗ ( j ) ) − H ( W rand j | Y j i ∗ ( j ) , W pri v j ) ≥ I ( W pri v j , W rand j ; Y j 1 , . . . , Y j n | Y j i ∗ ( j ) ) − N ε 1 (38) = I ( X j ; Y j 1 , . . . , Y j n | Y j i ∗ ( j ) ) − I ( X j ; Y j 1 , . . . , Y j n | Y j i ∗ ( j ) , W pri v j , W rand j ) − N ε 1 (39) ≥ I ( X j ; Y j 1 , . . . , Y j n | Y j i ∗ ( j ) ) − N ( ε 1 + ε 2 ) (40) = I ( X j ; Y j 1 , . . . , Y j n ) − I ( X j ; Y j i ∗ ( j ) ) − N ( ε 1 + ε 2 ) (41) ≥ I ( X j ; Y j j ) − I ( X j ; Y j i ∗ ( j ) ) − N ( ε 1 + ε 2 ) (42) = ∑ k : I POS j ( k )= 1  I ( X ( k ) ; Y j ( k )) − I ( X ( k ) ; Y i ∗ ( j ) ( k ))  − N ( ε 1 + ε 2 ) (43) ≥ N  p M j  ( ¯ R M j − δ ) − ¯ R m j  − ( ε 1 + ε 2 + ε 3 )  (44) with probab ility 1, for any positive ( ε 1 , ε 2 , ε 3 ) triplet and arbitrarily small δ , as N 1 , N 2 go to ∞ . Here, (36) follows since i ∗ ( j ) = argmax i ∈{ 1 ,..., n } I ( W pri v j ; Y j i ) ( W pri v j ↔ X j ↔ Y j i ∗ ( j ) ↔ Y j i forms a M arkov chain for all i and data p rocessing inequ al- ity), (37) is by the chain ru le, (38) follows from the app lication of Fano’ s inequality (as we choose the rate of the ran domiza- tion sequence to be N ( ¯ R m j − δ ) < I ( W rand j ; Y j i ∗ ( j ) ) , which allows for th e r andom ization message to be decode d at node i ∗ ( j ) , giv en the bin ind ex), (3 9) fo llows fr om the chain ru le and that ( W pri v j , W rand j ) ↔ X j ↔ ( Y j 1 , . . . , Y j n ) forms a Mar kov chain , (40) hold s sin ce I ( X j ; Y j 1 , . . . , Y j n | Y j i ∗ ( j ) , W pri v j , W rand j ) ≤ N ε 2 as the transmitted symb ol seq uence X j is d etermined w .p.1 given ( Y j i ∗ ( j ) , W pri v j , W rand j ) , (4 1) follows from the chain rule, (42) holds since Y j j ( k ) is an en try of vector [ Y j 1 ( k ) , . . . , Y j n ( k )] , (43) holds b ecause the fadin g pr ocesses are iid, and ﬁnally (44) follows fro m stro ng law o f large number s. Thus, with the described priv a cy sch eme, the perfect priv a cy constraint is satisﬁed for all nodes, since for any j ∈ { 1 , . . . , n } , we hav e 1 N I ( W pri v j ; Y j i ) = 1 N ( H ( W pri v j ) − H ( W pri v j | Y j i )) ≤ R pri v j − [ p M j  ( ¯ R M j − δ ) − ¯ R m j  − ( ε 1 + ε 2 + ε 3 )] ≤ ε , (45) for any gi ven ε > 0 . W e ju st showed th at, with pr iv ate oppor tunistic sch eduling, a pri vate info rmation rate of R pri v j = p M j ( ¯ R M j − ¯ R m j ) is achie vable for any given no de j . A P P E N D I X B P R O O F O F T H E O R E M 2 The pro of uses the n otation introd uced in the ﬁr st paragra ph of App endix A. T o meet th e perfec t secrecy constraint, it is necessary and sufﬁcient to guarantee lim N → ∞ 1 N I ( W pri v j ; Y j i ∗ ( j ) ) ≤ ε for all node s j ∈ { 1 , . . . , n } . Since n < ∞ , o ne can write an equiv alent co ndition on the sum 13 mutual information ov er each no de: ε ′ ≥ 1 N n ∑ j = 1 I ( W pri v j ; Y j i ∗ ( j ) ) = 1 N n ∑ j = 1 h H ( W pri v j ) − H ( W pri v j | Y j i ∗ ( j ) ) i = R pri v sum − 1 N n ∑ j = 1 H ( W pri v j | Y j i ∗ ( j ) ) (46) = R pri v sum − 1 N n ∑ j = 1 h I ( W pri v j ; Y j 1 , . . . , Y j n | Y j i ∗ ( j ) ) + H ( W pri v j | Y j 1 , . . . , Y j n ) i ≥ R pri v sum − 1 N n ∑ j = 1 h I ( W pri v j , W rand j ; Y j 1 , . . . , Y j n | Y j i ∗ ( j ) ) − I ( W rand j ; Y j 1 , . . . , Y j n | Y j i ∗ ( j ) , W pri v j ) + N ε 4 i (47) ≥ R pri v sum − 1 N n ∑ j = 1 h I ( W pri v j , W rand j ; Y j 1 , . . . , Y j n | Y j i ∗ ( j ) ) + N ε 4 i ≥ R pri v sum − 1 N n ∑ j = 1 h I ( X j ; Y j 1 , . . . , Y j n | Y j i ∗ ( j ) ) + N ε 4 i (48) = R pri v sum − 1 N n ∑ j = 1 h I ( X j ; Y j 1 , . . . , Y j n ) − I ( X j ; Y j i ∗ ( j ) ) + N ε 4 i (49) = R pri v sum − 1 N n ∑ j = 1 h I ( X j ; Y j j ) + I ( X j ; Y j 1 , . . . , Y j n | Y j j ) − I ( X j ; Y j i ∗ ( j ) ) + N ε 4 i (50) ≥ R pri v sum − 1 N n ∑ j = 1 h I ( X j ; Y j j ) + H ( X j | Y j j ) − I ( X j ; Y j i ∗ ( j ) ) + N ε 4 i ≥ R pri v sum − 1 N n ∑ j = 1 h I ( X j ; Y j j ) + H ( W pri v j | Y j j ) − I ( X j ; Y j i ∗ ( j ) ) + N ε 4 i (51) ≥ R pri v sum − 1 N n ∑ j = 1 h I ( X j ; Y j j ) − I ( X j ; Y j i ∗ ( j ) ) + N ( ε 4 + ε 5 ) i (52) = R pri v sum − 1 N n ∑ j = 1    ∑ k : I j ( k )= 1 [ I ( X ( k ) ; Y b ( k )) − I ( X ( k ) ; Y i ∗ ( j ) ( k ))  + ε 4 + ε 5  (53) ≥ R pri v sum − 1 N n ∑ j = 1 max I j ( k )    ∑ k : I j ( k )= 1 [ I ( X ( k ) ; Y b ( k )) − I ( X ( k ) ; Y i ∗ ( j ) ( k ))  + ε 4 + ε 5  = R pri v sum − 1 N n ∑ j = 1    ∑ k : I POS j ( k )= 1 [ I ( X ( k ) ; Y b ( k )) − I ( X ( k ) ; Y i ∗ ( j ) ( k ))  + ε 4 + ε 5  (54) ≥ R pri v sum − n ∑ j = 1  p M j  ¯ R M j − ¯ R m j  + ε 4 + ε 5 + ε 6  (55) with probab ility 1, for any p ositiv e ε ′ and ( ε 4 , ε 5 , ε 6 ) triplet as N 1 , N 2 go to ∞ . Here, (4 6) follows from the d eﬁnition of R pri v sum and that 1 N H ( W pri v j ) = R pri v j ; (4 7) follows from the chain rule and Fano’ s ine quality (a s H ( W pri v j | Y j 1 , . . . , Y j n ) ≤ H ( W pri v j , W rand j | Y j 1 , . . . , Y j n ) ≤ N ε 4 since the message pair ( W pri v j , W rand j ) can b e d ecoded with arb itrarily low proba- bility of error gi ven ( Y j 1 , . . . , Y j n ) ); (48) is from the data processing ineq uality as ( W pri v j , W rand j ) ↔ X j ↔ ( Y j 1 , . . . , Y j n ) forms a Markov chain; (49) an d (50) follow from the ch ain rule; (51) fo llows from the data processing inequa lity; (5 2) follows since nod e j decod es message W pri v j with arb itrarily low p robability o f erro r ε 5 ; (5 3) hold s since the fading pro- cesses are iid; (5 4) holds because private opp ortunistic sched - uler chooses I POS j ( k ) = argmax I j ( k ) [ R j ( k ) − R ji ∗ ( j ) ( k )] = argmax I j ( k )  I ( X ( k ) ; Y b ( k )) − I ( X ( k ) ; Y i ∗ ( j ) ( k ))  for all k ; and ﬁnally (55) follows by an application of the stron g law of large number s. Th e abov e derivation leads to the desired r esult: R pri v sum ≤ n ∑ j = 1  p M j  ¯ R M j − ¯ R m j  . (56) W e comp lete the p roof no ting that the ab ove sum rate is achiev able by priv ate oppo rtunistic sched uling as shown in (44). Note that, fr om th e above steps, we can also see th at the individual priv ate infor mation r ates g i ven in Theo rem 1 are the max imum achievable individual rates with priv ate oppor tunistic sch eduling. This is du e to the fact that, f or any no de j , with priv ate o pportu nistic schedulin g, th e above deriv ation lead to: 1 N H ( W pri v j | Y j i ∗ ( j ) ) ≤ p M j  ¯ R M j − ¯ R m j  + ε (57) for any ε > 0 as N → ∞ . Con sequently , with p riv ate oppor- tunistic scheduling, no node can achiev e any individual privac y rate ab ove that given in (57), h ence the converse of Theorem 1 also holds. A P P E N D I X C P R O O F O F L E M M A 1 Pr oof: Since th e maxim um tran smission power is ﬁ- nite, in any interf erence-limited system transmission r ates are bound ed. Let R p , max j and R o , max j be the max imum private an d open rates for user j , which d epends on the chann el states. Also assume that the arriv al rates are b ounde d, i.e., A p , max j and A o , max i be the maximu m n umber of priv ate and open bits that may arrive in a b lock for each user . He nce, the following inequalities can be ob tained for eac h pri vate que ue: ( Q p j ( k + 1 )) 2 − ( Q p j ( k )) 2 =  h Q p j ( k ) − R p j ( k ) i + + A p j ( k )  2 − ( Q p j ( k )) 2 ≤ ( Q p j ( k )) 2 + ( A p j ( k )) 2 + ( R p j ( k )) 2 − 2 Q p j ( k ) h R p j ( k ) − A p j ( k ) i − ( Q p j ( k )) 2 ≤ ( R p j ( k )) 2 + ( A p j ( k )) 2 − 2 Q p j ( k )[ R p j ( k ) − A p j ( k )] ≤ B 1 − 2 Q p j ( k )[ R p j ( k ) − A p j ( k )] (58) 14 where B 1 = ( R p , max j ) 2 + ( A p , max j ) 2 . The same line of der iv ation can be performed for open queues to obtain: ( Q o j ( k + 1 )) 2 − ( Q o j ( k )) 2 =   Q o j ( k ) − R o j ( k )  + + A o j ( k )  2 − ( Q o j ( k )) 2 ≤ B 2 − 2 Q o j ( k )[ R o j ( k ) − A o j ( k )] (59) where B 2 = ( R o , max j ) 2 + ( A o , max j ) 2 . Hence, by taking expectation , multiplying by 1 2 , and sum- ming (58)-(59) over all j = 1 , . . . , n , we obtain the uppe r bound on ∆ ( k ) as gi ven in the Lemma, where B = n ( B 1 + B 2 ) / 2. A P P E N D I X D P R O O F O F T H E O R E M 4 Pr oof: L yapu nov Optimization T heorem [37] sug gests that a goo d contro l strategy is the one that minimizes the following: ∆ U ( k ) = ∆ ( k ) − V E " ∑ j  g p j ( k ) + g o j ( k )     Q p ( k ) , Q o ( k ) # (60) By using (24), we may obtain an upper bound for ( 60), as follows: ∆ U ( k ) < B − ∑ j E h Q p j ( k )[ R p j ( k ) − A p j ( k )]   Q p j ( k ) i − ∑ j E  Q o j ( k )[ R o j ( k ) − A o j ( k )]   Q o j ( k )  − V E " ∑ j U p j ( A p j ( k )) + ∑ j U o j ( A o j ( k )) # (61) By rear ranging the terms in (61) it is easy to ob serve that our pro posed d ynamic network control algorithm minimize s the right hand side of (61). If the priv ate and open arriv al rates are in the feasible region, it has been shown in [45] that there must exist a stationary scheduling and rate control policy t hat chooses the users and their tran smission ra tes indep enden t of qu eue ba cklogs and only with respect to the chan nel statistics. In particular, the optimal station ary p olicy can b e fo und as the solution of a deterministic policy if a priori channel statistics a re known. Let U ∗ be th e op timal value of the ob jectiv e functio n o f th e problem (15)-(16) o btained b y th e aforemen tioned stationary policy . Also let λ p j ∗ and λ o j ∗ be o ptimal private and open trafﬁc arrival rate s f ound as the solution of th e same prob lem. In particular , the op timal input rates λ p j ∗ and λ o j ∗ could in p rinciple be achieved b y th e simple back log-ind ependen t admission control alg orithm of includ ing all new arrivals ( A p j ( k ) , A o j ( k )) for a given node j in block k indepe ndently with probability ( ζ p j , ζ o j ) = ( λ p j ∗ / λ p j , λ o j ∗ / λ o j ) . Then, the right hand side (RHS) o f (61) can be rewritten as B − ∑ j E h Q p j ( k ) i E h R p j ( k ) − A p j ( k ) i − ∑ j E  Q o j ( k )  E  R o j ( k ) − A o j ( k )  − V U ∗ . (62) Also, since ( λ p j ∗ , λ o j ∗ ) ∈ Λ , i.e., arri val rates are st rictly interior of the r ate region , there m ust exist a stationar y schedu ling and rate allocation policy that is independen t of qu eue backlogs and satisﬁes the fo llowing: E h R p j   Q p i ≥ λ p j ∗ + ε 1 (63) E  R o j   Q o  ≥ λ o j ∗ + ε 2 (64) Clearly , any stationary po licy shou ld satisfy (6 1). Recall that our pro posed policy minimizes RHS of (61), and hence, any oth er station ary policy (includin g the optimal policy) has a higher RHS value th an the one attained by our policy . In particular, the stationar y policy that satisﬁes (63)-(64), an d implements afo remention ed pr obabilistic admission control can be used to obtain an up per b ound for the RHS of our propo sed p olicy . Inserting (6 3)-(64) into (62), we obtain the following up per bound for our policy: RH S < B − ∑ j ε 1 E [ Q p j ( k )] − ∑ j ε 2 E [ Q o j ( k )] − V U ∗ . (65) This is exactly i n the form of L yapun ov Optim ization Theorem giv en in Theorem 3, and h ence, we can obtain bound s on the perfor mance of the pro posed policy an d the sizes of qu eue backlog s as giv en in Theorem 4. C. Emre Koksal C. Emre Koksal recei ved the B.S. degre e in el ectrica l engineering from the Middle East T echnica l Unive rsity , Anka ra, Tur ke y , in 1996, and the S.M. and Ph.D. degr ees from the Massachusetts Institut e of T echnol ogy (MIT), Cambridge, in 1998 and 2002, respecti vely , in electrica l enginee ring and computer science. He was a Postdoctora l Fello w in the Networks and Mobile Systems Group in the Computer Science and Artiﬁc ial Intelligenc e Lab- oratory , MIT and a Senior Researc her jointl y in the Laboratory fo r Comput er Communic ations and t he Laboratory for Informatio n Theory at EPFL, Lausanne, Switzerl and. Since 2006, he has been an As sistant Professor in the Electrical and Computer Engineeri ng Department, Ohio State Univ ersity , Columbus, Ohio. His genera l areas of interest are wireless communicati on, communica tion networks, infor- mation theory , stochastic processes, and ﬁnanci al economi cs. He is the rec ipient of t he Nationa l Scienc e Foundation CAREER A ward (2011), the OSU College of E ngineeri ng Lumley Research A ward (2011), and the co-reci pient of an HP Labs - Innov ation Re search A ward. T he paper he co-auth ored was a best student paper candidate in MOBICOM 2005. Ozgur Ercetin recei ved the BS degree in electric al and elec tronics engin eering from the Middle East T echnica l Univ ersity , Ankara, Tur key , in 1995 and the MS and PhD de grees in electric al engineerin g from t he Uni versity of Maryland, Col lege Pa rk, in 1998 and 2002, respecti vely . Since 2002, he has been with the Facult y of Engineeri ng and Natural Science s, Sabanci Unive rsity , Istanb ul. He was also a visiting researcher at HRL Labs, Malibu, CA, Docomo USA Labs, CA, and The Ohio State Uni- versi ty , OH. His research interests are in the ﬁeld of computer and communicatio n networks with emphasis on fundamental mathemati cal models, archit ectures and protocol s of wireless systems, and stochasti c optimiza tion. Y unus Sar ikaya recei ved the BS and MS de grees in tele communicati ons engineering from Sabanci Uni versit y , Istanb ul, T urke y , i n 2006 and 2008, re- specti vely . He is currently PhD student in electr ical enginee ring at Sabanci Univ ersity . His research intere sts incl ude optimal c ontrol of wireless networks, stochastic optimizatio n and infor- mation theoreti cal security .

Control of Wireless Networks with Secrecy

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