The Secrecy Rate Region of the Broadcast Channel

1 The Secrec y Rate Re gion of the Broadca st Chann el Ghadamali Bagherikaram, Abolfazl S. Motahari, Amir K. Khan dani Coding a nd Sign al Transmission Laboratory , Department of Elec trical and Compu ter Enginee ring, Univ ersity of W aterloo, W aterloo, Ontario, N 2L 3G1 Emails: { gba gheri,abolfazl,khan dani } @cst.uwaterloo.ca Abstract In this paper , we co nsider a scenario where a source node wishes to broadca st two confidential message s for tw o respectiv e recei vers, wh ile a wire-t apper a lso recei ves the t ransmitted signa l. This model is motiv ated by wireless communications, where indi vidual secure messages are broadcast ove r open media and can be receive d by any illegitimate recei ver . The secrecy le vel is measured by equi voca tion rate at t he eavesdro pper . W e first study the general (non-degrad ed) broadcast channel with confidential messages. W e present an i nner bound on the secrecy capacity region for this model. The inner bound coding scheme i s based on a combination of random binning and t he Gelfand-Pinsk er bining. T his scheme matches the Marton’ s inner bound on the broadcast channel without confidentiality constraint. W e f urther study the situation where the channels are degrade d. For the degrad ed broadcast channel with confidential messages, we present t he secrecy capacity region. Our achiev able coding scheme is based on Cover’ s superposition scheme and random binning. W e refer to this scheme as Secret Superposition Scheme. In this scheme, we show that rand omization in the first l ayer increases the secrecy rate of the second layer . This capacity region matches the capacity region of the degraded broadcast channel wi thout security constraint. It also matches the secrecy capacity for the con ventional wire-tap channel. Our conv erse proof is based on a combination of the con verse proof of the con ventional degraded broadcast channel and C siszar l emma. Finally , we assume that the channels are Additiv e White Gaussian N oise (A WGN) and sho w that secret superposition scheme with Gaussian codebo ok is optimal. The con verse proof is based on the generalized entropy po wer inequality . I . I N T R O D U C T I O N The notion of information theoretic secrecy in communica tion systems was first introduced by Shannon in [1]. The information theoretic secrecy requ ires that the received signal of the eav esdropp er does no t provide even a sing le bit informa tion abou t the tran smitted messages. Shan non considere d a pessimistic situation wher e both the intende d rec eiv er an d the e av esdropp er have dir ect access to the transmitted sig nal (which is called cipher text). Un der th ese c ircumstances, he proved a negativ e re sult showing th at per fect secr ecy c an b e achieved only when th e entr opy of the secr et key is g reater than or equ al to the e ntropy of the message . In moder n cryp tograph y , all prac tical cry ptosystems are based on Sh annno n’ s pessimistic assumptio n. Due to practical co nstraints, secret keys are much shorter than messages. Ther efore, the se p ractical cry ptosystems a re theoretically susceptible of bre aking by attackers. Howev er , the go al of designin g such p ractical c iphers is to guara ntee that th ere exists n o efficient algo rithm for br eaking them. W yner in [2] showed that the a bove negativ e r esult is a co nsequence of Shan non’ s r estrictiv e assumption that the ad versary has access to precisely the same infor mation as th e legitimate receiver . W yner con sidered a scenario in which a wire-tapper receives the transmitted signal over a degrad ed ch annel with r espect to th e legitimate receiver’ s channel. He further assumed that the wire-tapper has no co mputation al limitations an d kn ows the cod ebook used by th e tr ansmitter . He measure d the lev el of igno rance at the eavesdropper by its equ i vocation an d ch aracterized the capacity- equiv ocation region . In terestingly , a non-n egati ve p erfect secrecy capacity is always a chiev able fo r this scen ario. The secrecy c apacity for th e Gaussian wire-tap chan nel is characterized by L eung-Y an-Cheong in [3]. W yner’ s work then is extended to the gener al (n on-degrad ed) broad cast channe l with confide ntial messages (BCC) by Csiszar and K orner [4]. They considered transmitting con fidential info rmation to th e legitimate rece iv er while tran smitting common inf ormation to both the legitimate receiver and the wire -tapper . They establishe d a ca pacity-equ iv o cation region o f this ch annel. The BCC is f urther studied rec ently in [5] –[7], where the sour ce node transmits a c ommon message fo r both receivers, along with two ad ditional confiden tial messages for two r espective receivers. T he fading BCC is investigated in [8 ], [ 9] where th e broadc ast chan nels fr om the source no de to the legitimate receiver and the eavesdropper is corru pted by mu ltiplicative fadin g gain coefficients, in ad dition to additive white Gaussian n oise terms. T he Chan nel State Infor mation (CSI) is assumed to be known a t th e transmitter . In [ 10], th e p erfect secrecy capacity is derived where the chan nels are slow fading. Mo reover , the optimal power contr ol p olicy is obtain ed for different scenar ios regarding av a ilability of CSI. In [11], the wire-tap channel is extended to the parallel broad cast channels and th e fadin g ch annels with multiple receivers. Her e, the secrecy constraint is a p erfect equiv ocation fo r each messages, even if all the other messages ar e revealed to the eavesdropper . T he secrecy sum capacity for a reverse broa dcast chann el is d erived for this r estrictiv e assump tion. The notion of the wire-tap c hannel is also extended to multiple access chann els [12]– [15], relay channe ls [16 ]–[19 ], parallel chan nels [20 ] and MIMO chan nels [21 ]–[26 ]. Some other related works on commu nication o f confid ential m essages can b e foun d in [2 7]–[3 1]. X Y 1 Y 2 Z Fig. 1. Broadc ast Channe l with Confidenti al Messages In this paper, we con sider a scena rio where a sou rce no de wishes to b roadcast two confide ntial messages fo r two respective receivers, while a wire- tapper also receives the tran smitted signa l. This model is motivated by wireless co mmun ications, where individual secure messages are broadcast over s hared med ia and can be received by any illegitimate recei ver . In fact, we simplify the restrictive constraint imposed in [11] an d assume that the eavesdropper d oes no t have access to the oth er messag es. W e first stud y the ge neral bro adcast channe l with confid ential messag es. W e present an achiev able rate r egion fo r this channel. Our achievable coding schem e is b ased on th e com bination of the rando m b inning and the Gelfand-Pinsker bining [3 2]. This scheme m atches the Marto n’ s in ner bou nd [3 3] on th e broad cast chan nel withou t c onfidentiality constraint. W e fu rther stud y the situatio n where the channels are phy sically d egraded and ch aracterize th e secrecy cap acity region. Our achievable coding scheme is based o n Cover’ s superposition coding [34 ] an d the rand om binning . W e refer to this schem e as Secret Superposition Coding. Th is cap acity region match es the capacity r egion of the degrad ed b roadcast cha nnel withou t security co nstraint. It also matches the secrecy cap acity of th e wire- tap c hannel. The r est o f the paper is organ ized as follows. In section I I we introdu ce the sy stem mod el. In Section I II, we provid e an inner bo und on the secr ecy ca pacity region when th e chan nels are not d egraded. In section I V , we specialize our ch annel to the p hysically degrad ed and establish th e secrecy capacity r egion. In Section V , we c onclud e the p aper . I I . P R E L I M I NA R I E S In this paper, a r andom variable is d enoted by a capital letter (e.g . X) and its r ealization is denoted by a cor respond ing lower case letter (e.g. x ). Th e finite alphabet of a ran dom variable is denoted by a script letter ( e.g. X ) and its prob ability d istribution is d enoted b y P ( x ) . Let X be a finite alp habet set and denote its cardinality by |X | . The memb ers of X n will b e written as x n = ( x 1 , x 2 , ..., x n ) , where subscrip ted letter s deno te the co mpon ents and sup erscripted letter s deno te the vector . The notation x i − 1 denotes the vecto r ( x 1 , x 2 , ..., x i − 1 ) . A similar no tation will be used fo r rand om variables and ra ndom vectors. Consider a Bro adcast Chan nel with Confiden tial messages as depicted in fig.1. In this c onfidential setting , the tran smitter ( X ) wants to bro adcast some secret messages to the legitimated receivers ( Y 1 , Y 2 ), and prevent th e eavesdropper ( Z ) f rom having any informa tion ab out the messages. A discrete m emory less br oadcast cha nnel with co nfidential messages is descr ibed by fin ite sets X , Y 1 , Y 2 , Z , and a conditio nal distribution P ( y 1 , y 2 , z | x ) . The input of th e channe l is x ∈ X and th e ou tputs are ( y 1 , y 2 , z ) ∈ ( Y 1 × Y 2 × Z ) for r eceiv er 1 , receiver 2 , and the eavesdropper , respectively . The transmitter wishes to send indepen dent messages ( W 1 , W 2 ) to the re spectiv e receivers in n uses of the chann el while insuring perfect secrecy . The chann el is discrete memor yless in the sense that P ( y n 1 , y n 2 , z n | x n ) = n Y i =1 P ( y 1 ,i , y 2 ,i , z i | x i ) . (1) A ((2 nR 1 , 2 nR 2 ) , n ) code for a br oadcast channe l with confiden tial me ssages consists of a sto chastic encod er f : ( { 1 , 2 , ..., 2 nR 1 } × { 1 , 2 , ..., 2 nR 2 } ) → X n , (2) and two d ecoders, g 1 : Y n 1 → { 1 , 2 , ..., 2 nR 1 } (3) and g 2 : Y n 2 → { 1 , 2 , ..., 2 nR 2 } . (4) The average p robab ility of error is d efined as th e pro bability that the de coded messages ar e not equ al to the tran smitted messages; that is, P ( n ) e = P ( g 1 ( Y n 1 ) 6 = W 1 ∪ g 2 ( Y n 2 ) 6 = W 2 ) . (5) The knowledge th at the eavesdropper gets ab out W 1 and W 2 from its received signal Z n is m odeled as I ( Z n , W 1 ) = H ( W 1 ) − H ( W 1 | Z n ) , (6) I ( Z n , W 2 ) = H ( W 2 ) − H ( W 2 | Z n ) , (7) 2 and I ( Z n , ( W 1 , W 2 )) = H ( W 1 , W 2 ) − H ( W 1 , W 2 | Z n ) . (8) Perfect secr ecy revolv es aroun d th e idea th at the eavesdropper cannot g et ev en a single bit info rmation abou t the tr ansmitted messages. Perfect secrecy th us req uires that I ( Z n , W 1 ) = 0 ⇔ H ( W 1 ) = H ( W 1 | Z n ) , (9) I ( Z n , W 2 ) = 0 ⇔ H ( W 2 ) = H ( W 2 | Z n ) , and I ( Z n , ( W 1 , W 2 )) = 0 ⇔ H ( W 1 , W 2 ) = H ( W 1 , W 2 | Z n ) . (10) The secre cy levels of co nfidential messages W 1 and W 2 are mea sured at the eavesdropper in terms of eq uiv o cation rates which are defined as follows. Definition 1 The equ ivocation rates R e 1 , R e 2 and R e 12 for the Br oa dcast channel with con fidentia l messages ar e: R e 1 = 1 n H ( W 1 | Z n ) , (11) R e 2 = 1 n H ( W 2 | Z n ) , R e 12 = 1 n H ( W 1 , W 2 | Z n ) . The perfe ct secrecy rates R 1 and R 2 are the amoun t of in formatio n th at can b e sent to the legitimate receivers no t only reliab ly but also confiden tially . Definition 2 A secr ecy rate pair ( R 1 , R 2 ) is said to be achievable if for any ǫ > 0 , th er e exists a sequen ce of ((2 nR 1 , 2 nR 2 ) , n ) codes, such that fo r sufficiently lar ge n , we have : P ( n ) e ≤ ǫ, (12) R e 1 ≥ R 1 − ǫ 1 , (13) R e 2 ≥ R 2 − ǫ 2 , (14) R e 12 ≥ R 1 + R 2 − ǫ 3 . (15) In the above definition, th e first condition concerns the reliability , while the other con ditions guar antee perfect secrecy f or ea ch individual messag e and bo th messages as w ell. The capacity region is defined as follows. Definition 3 The cap acity r e g ion of the br oa dcast cha nnel with confi dential messages is the closur e of the set of a ll achievable rate pairs ( R 1 , R 2 ) . I I I . G E N E R A L B C C S In this section, we consider the general bro adcast channel with confid ential messages and p resent an a chiev able rate region. Our ach iev ab le cod ing scheme is based o n a c ombinatio n o f the ran dom bin ning and the Gelfand-Pinsker binin g schemes [32] . The following theorem illustrates the ach iev a ble rate region for this channe l. Theorem 1 Let R I denote the union of all non -negative rate pa irs ( R 1 , R 2 ) satisfying R 1 ≤ I ( V 1 ; Y 1 ) − I ( V 1 ; Z ) , (16) R 2 ≤ I ( V 2 ; Y 2 ) − I ( V 2 ; Z ) , (17) R 1 + R 2 ≤ I ( V 1 ; Y 1 ) + I ( V 2 ; Y 2 ) − I ( V 1 , V 2 ; Z ) − I ( V 1 ; V 2 ) . (18) over all joint distrib utions P ( v 1 , v 2 ) P ( x | v 1 , v 2 ) P ( y 1 , y 2 , z | x ) . Then any rate pair ( R 1 , R 2 ) ∈ R I is achievable for the br oadca st channel with con fidentia l messages. Remark 1 If we r emove th e secrecy constraints by setting Z = ∅ , then the above rate r egion reduces to Marton’s a chievable r egion for the gen eral br o adcast chann el. Remark 2 If we r emove one of the users by setting e.g., Y 2 = ∅ , th en we get the Csiszar and K o rner’ s secr ecy capa city for the o ther user . Pr oof: 3 1 2 1 2 . . . 2 nR 2 . . . · · · · · · 2 nR 1 (V n 1 , V n 2 ) ∈ A ( n ) ǫ Fig. 2. The Stochasti c Encoder 1) Codebook Generation : The structure of th e encoder is d epicted in Fig.2. Fix P ( v 1 ) , P ( v 2 ) and P ( x | v 1 , v 2 ) . The sto chastic encoder gen erates 2 n ( I ( V 1 ; Y 1 ) − ǫ ) indepen dent and identically distributed sequences v n 1 accordin g to th e distribution P ( v n 1 ) = Q n i =1 P ( v 1 ,i ) . Next, ran domly distribute the se sequences into 2 nR 1 bins such that each bin conta ins 2 n ( I ( V 1 ; Z ) − ǫ ) codewords. Similarly , it gener ates 2 n ( I ( V 2 ; Y 2 ) − ǫ ) indepen dent and identically distributed sequences v n 2 accordin g to the distribution P ( v n 2 ) = Q n i =1 P ( v 2 ,i ) . Next, ran domly distribute the se sequences into 2 nR 2 bins such that each bin conta ins 2 n ( I ( V 2 ; Z ) − ǫ ) codewords. Index each of th e above bins by w 1 ∈ { 1 , 2 , ..., 2 nR 1 } and w 2 ∈ { 1 , 2 , ..., 2 nR 2 } resp ectiv ely . 2) En coding : T o send messages w 1 and w 2 , th e transmitter lo oks fo r v n 1 in bin w 1 of the first bin set and looks f or v n 2 in bin w 2 of the seco nd bin set, such that ( v n 1 , v n 2 ) ∈ A ( n ) ǫ ( P V 1 ,V 2 ) where A ( n ) ǫ ( P V 1 ,V 2 ) de notes the set of jointly typ ical sequences v n 1 and v n 2 with respect to P ( v 1 , v 2 ) . The rates are such that there exist more than one jo int typical p air , the transmitter r andom ly chooses on e of them a nd th en g enerates x n accordin g to P ( x n | v n 1 , v n 2 ) = Q n i =1 P ( x i | v 1 ,i , v 2 ,i ) . This schem e is equivalent to the scen ario in wh ich each bin is divided into subb ins and th e tran smitter rand omly chooses on e of th e subb ins of b in w 1 and one of th e subbins of b in w 2 . It then look s for a joint typ ical sequence ( v n 1 , v n 2 ) in the correspo nding subb ins and generates x n . 3) Decodin g : The re ceiv ed signa ls at the legitimate r eceiv ers, y n 1 and y n 2 , are the outpu ts of th e channels P ( y n 1 | x n ) = Q n i =1 P ( y 1 ,i | x i ) a nd P ( y n 2 | x n ) = Q n i =1 P ( y 2 ,i | x i ) , respectively . The first r eceiver looks for th e u nique sequen ce v n 1 such that ( v n 1 , y n 1 ) is jointly typical and declar es the index of th e bin con taining v n 1 as th e message received. The seco nd rece iv er uses the same m ethod to extract the message w 2 . 4) Err or Pr o bability Analysis : Since the region of (12) is a subset of Ma rton’ s region then, error pr obability analysis is the same as [33]. 5) Equivoca tion Calcula tion : The proof of secrecy requirem ent fo r each individual message (13) an d (14) is straightf orward and may therefor e be omitted. T o pr ove th e req uiremen t of (15) conside r H ( W 1 , W 2 | Z n ) , we have nR e 12 = H ( W 1 , W 2 | Z n ) ≥ H ( W 1 , W 2 , Z n ) − H ( Z n ) = H ( W 1 , W 2 , V 1 n , V n 2 , Z n ) − H ( V n 1 , V n 2 | W 1 , W 2 , Z n ) − H ( Z n ) = H ( W 1 , W 2 , V n 1 , V n 2 ) + H ( Z n | W 1 , W 2 , V n 1 , V n 2 ) − H ( V n 1 , V n 2 | W 1 , W 2 , Z n ) − H ( Z n ) ( a ) ≥ H ( W 1 , W 2 , V n 1 , V n 2 ) + H ( Z n | W 1 , W 2 , V n 1 , V n 2 ) − nǫ n − H ( Z n ) ( b ) = H ( W 1 , W 2 , V n 1 , V n 2 ) + H ( Z | V n 1 , V n 2 ) − nǫ n − H ( Z n ) ( c ) ≥ H ( V n 1 , V n 2 ) + H ( Z n | V n 1 , V n 2 ) − nǫ n − H ( Z n ) ( d ) = H ( V n 1 ) + H ( V n 2 ) − I ( V n 1 ; V n 2 ) − I ( V n 1 , V n 2 ; Z n ) − nǫ n ( e ) = I ( V n 1 ; Y n 1 ) + I ( V n 2 ; Y n 2 ) − I ( V n 1 ; V n 2 ) − I ( V n 1 , V n 2 ; Z n ) − nǫ n ≥ nR 1 + nR 2 − nǫ n , where ( a ) f ollows from Fano’ s inequality which states that for sufficiently large n we have H ( V n 1 , V n 2 | W 1 , W 2 , Z n ) ≤ h ( P ( n ) we ) + nP n we I ( V 1 , V 2 ; Z ) ≤ nǫ n . Here P n we denotes the wir etapper’ s error pr obability o f deco ding ( v n 1 , v n 2 ) in the 4 case th at the bin nu mbers w 1 and w 2 are kn own to the eavesdropper . Since the su m rate is less tha n I ( V 1 , V 2 ; Z ) , then P n we → 0 fo r sufficiently large n . ( b ) follows fr om the fo llowing Markov chain: ( W 1 , W 2 ) → ( V 1 , V 2 ) → Z . Hence, we have H ( Z n | W 1 , W 2 , V n 1 , V n 2 ) = H ( Z n | V n 1 , V n 2 ) . ( c ) follows from the fact that H ( W 1 , W 2 , V n 1 , V n 2 ) ≥ H ( V n 1 , V n 2 ) . ( d ) f ollows from that fact that H ( V n 1 ) = I ( V n 1 ; Y n 1 ) and H ( V n 2 ) = I ( V n 2 ; Y n 2 ) . I V . T H E S E C R E C Y C A PAC I T Y R E G I O N O F T H E D E G R A D E D B C C S In this section , we co nsider the degra ded bro adcast ch annel with con fidential m essages and estab lish its secrecy capacity region. Definition 4 A br oadcast channel with c onfide ntial messages is said to be phy sically degraded, if X → Y 1 → Y 2 → Z fo rms a Markov chain. In the other words, we have P ( y 1 , y 2 , z | x ) = P ( y 1 | x ) P ( y 2 | y 1 ) P ( z | y 2 ) . (19) Definition 5 A br oadcast channel with c onfide ntial messages is said to be stochastically degr aded if its condition al marginal distributions are the same as that o f a physically degraded b r oa dcast channel, i. e., if there exist two distributions P ′ ( y 2 | y 1 ) and P ′ ( z | y 2 ) such that P ( y 2 | x ) = X y 1 P ( y 1 | x ) P ′ ( y 2 | y 1 ) (20) P ( z | x ) = X y 2 P ( y 2 | x ) P ′ ( z | y 2 ) Lemma 1 The secr e cy c apacity r egion of a br oa dcast chan nel with c onfide ntial messages depen ds only on the condition al mar ginal distributions P ( y 1 | x ) , P ( y 2 | x ) an d P ( z | x ) . Pr oof: The pro of is very similar to [34 ] and m ay theref ore be omitted here. In the following the orem, we fully character ize th e capacity r egion of the p hysically degraded broad cast channel with confidential messages. Theorem 2 The capacity r e g ion for transmitting indep enden t secr et information over the d e graded br oadcast chann el is the conve x h ull of the closur e of a ll ( R 1 , R 2 ) satisfying R 1 ≤ I ( X ; Y 1 | U ) + I ( U ; Z ) − I ( X ; Z ) , (2 1) R 2 ≤ I ( U ; Y 2 ) − I ( U ; Z ) . (22) for some joint distribution P ( u ) P ( x | u ) P ( y 1 , y 2 , z | x ) . Remark 3 If we r emove th e secr e cy con straints by setting Z = ∅ , then the ab ove theorem r ed uces to the cap acity re gion of the d e graded br o adcast channel. Pr oof: Achievability : The co ding scheme is based on Cover’ s super position coding and the ran dom bining. W e refer to this sche me as Secure Su perpo sition Coding scheme . T he av a ilable resources at the encoder are used f or two purp oses: to co nfuse the eav esdropp er so that perfect secrecy can be ach iev ed for both layer s, and to tran smit the message s in the main channels. T o satisfy c onfidentiality , the rand omization used in the first layer is again used in the second layer . This m akes a shift o f I ( U ; Z ) in the b ound of R 1 . The formal p roof of th e achiev ability is as follows: 1) Codeboo k Generation : The structu re of the encod er is depicted in Fig .3. Let us fix P ( u ) and P ( x | u ) . The stocha stic encoder generates 2 n ( I ( U ; Y 2 ) − ǫ ) indepen dent and id entically distributed sequen ces u n accordin g to th e distribution P ( u n ) = Q n i =1 P ( u i ) . Next, we random ly d istribute these sequences into 2 nR 2 bins such that each bin contain s 2 n ( I ( U ; Z ) − ǫ ) codewords. W e index eac h of the above bins by w 2 ∈ { 1 , 2 , ..., 2 nR 2 } . For each codeword of u n , it also genera tes 2 n ( I ( X ; Y 1 | U ) − ǫ ) indepen dent and identically distributed sequen ces x n accordin g to the distribution P ( x n | u n ) = Q n i =1 P ( x i | u i ) . W e rand omly distribute these sequ ences into 2 nR 1 bins such that each bin contain s 2 n ( I ( X ; Z ) − I ( U ; Z ) − ǫ ) codewords. W e in dex each o f the above bins b y w 1 ∈ { 1 , 2 , ..., 2 nR 1 } . 2) Encodin g : T o send messages w 1 and w 2 , th e transmitter r andomly cho oses on e o f the cod ew ords in bin w 2 , say u n . T hen giv en u n , the tr ansmitter ran domly c hooses on e of x n in bin w 1 of the secon d layer a nd sends it. 3) Deco ding : The re ceiv ed signal a t th e legitimate receivers, y n 1 and y n 2 , a re the ou tputs of th e c hannels P ( y n 1 | x n ) = Q n i =1 P ( y 1 ,i | x i ) and P ( y n 2 | x n ) = Q n i =1 P ( y 2 ,i | x i ) , respectively . Receiver 2 determin es the u nique u n such that ( u n , y n 2 ) are jointly typical an d de clares the ind ex of the bin containing u n as the message received. If there is none of su ch or more than o f 5 1 2 1 2 . . . 2 nR 2 2 nR 1 . . . x n u n Fig. 3. Secret Superposi tion structure one such , an error is declare d. Receiv er 1 looks fo r the uniqu e ( u n , x n ) su ch that ( u n , x n , y n 1 ) a re jo intly typica l and de clares the indexes of the b ins containin g u n and x n as th e messages received. I f th ere is none of such or more than of on e such , an error is declared. 4) Err or Pr ob ability Analysis : Since each rate pair of (21) is in the capa city region o f th e d egraded broad cast chann el without confidentiality con straint, then it can b e readily shown that the error p robab ility is arbitrarily small, c.f . [3 4]. 5) Equ ivocation Calculation : T o prove th e secrecy requ irement o f (13), we have nR e 1 = H ( W 1 | Z n ) ≥ H ( W 1 | Z n , U n ) = H ( W 1 , Z n | U n ) − H ( Z n | U n ) = H ( W 1 , X n , Z n | U n ) − H ( Z n | U n ) − H ( X n | W 1 , Z n , U n ) ( a ) = H ( W 1 , X n | U n ) + H ( Z n | W 1 , U n , X n ) − H ( Z n | U n ) − nǫ n ( b ) ≥ H ( X n | U n ) + H ( Z n | X n ) − H ( Z n | U n ) − nǫ n ( c ) = H ( X n ; Y n 1 | U n ) + I ( U n ; Z n ) − I ( X n ; Z n ) − nǫ n ≥ nR 1 − nǫ n , where ( a ) follows from Fano’ s ine quality which states that H ( X n | W 1 , Z n , U n ) ≤ h ( P ( n ) we ) + nP n we I ( X ; Z ) ≤ nǫ n for sufficiently large n . Here P n we denotes the wir etapper’ s error p robability of deco ding x n giv en the bin numb er and the co dew ord u n are known to th e eavesdropper . Since the rate is less th an I ( X ; Z ) , th en P n we → 0 for sufficiently large n . ( b ) follows from the fact tha t ( W 1 , U ) → X → Z for ms a M arkov ch ain. Thu s we have I ( W 1 , U n ; Z n | X n ) = 0 , wher e it is imp lied that H ( Z n | W 1 , U n , X n ) = H ( Z n | X n ) . ( c ) follows fr om two identities: H ( X n | U n ) = I ( X n ; Y n 1 | U n ) and H ( Z n | X n ) − H ( Z n | U n ) = I ( U n ; Z n ) − I ( X n ; Z n ) . Since the proof of the requirem ent (14) is straightf orward, we need to prove the requirem ent of ( 15). nR e 12 = H ( W 1 , W 2 | Z n ) ≥ H ( W 1 , W 2 , Z n ) − H ( Z n ) = H ( W 1 , W 2 , U n , X n , Z n ) − H ( U n , X n | W 1 , W 2 , Z n ) − H ( Z n ) = H ( W 1 , W 2 , U n , X n ) + H ( Z n | W 1 , W 2 , U n , X n ) − H ( U n , X n | W 1 , W 2 , Z n ) − H ( Z n ) ( a ) ≥ H ( W 1 , W 2 , U n , X n ) + H ( Z n | W 1 , W 2 , U n , X n ) − nǫ n − H ( Z n ) ( b ) = H ( W 1 , W 2 , U n , X n ) + H ( Z | U n , X n ) − nǫ n − H ( Z n ) ( c ) ≥ H ( U n , X n ) + H ( Z n | U n , X n ) − nǫ n − H ( Z n ) = H ( U n ) + H ( X n | U n ) − I ( U n , X n ; Z n ) − nǫ n ( d ) = I ( U n ; Y n 2 ) + I ( X n ; Y n 1 | U n ) − I ( X n ; Z n ) − I ( U n ; Z n | X n ) − nǫ n ≥ nR 1 + nR 2 − nǫ n , 6 where ( a ) follows from Fano’ s inequ ality that H ( U n , X n | W 1 , W 2 , Z n ) ≤ h ( P ( n ) we ) + nP n we I ( U, X ; Z ) ≤ nǫ n for sufficiently large n . Here P n we denotes the wiretapper ’ s erro r p robability o f decod ing ( u n , x n ) in the case th at th e bin num bers w 1 and w 2 are kn own to the eavesdropper . T he eavesdropper first look s f or the uniq ue u n in bin w 2 of the first layer, such that it is jointly ty pical with z n . Since th e n umber of candidate codewords is less th an I ( U ; Z ) , then th e p robab ility of error is arbitrarily small for a sufficiently large n . Next, given u n , the eavesdropper loo ks for the uniq ue x n in bin w 1 which is jointly typical with z n . Sim ilarly , since the num ber of available cand idates is less than I ( X ; Z ) , then the p robab ility of error decoding is arbitrarily small. ( b ) f ollows from the fact that ( W 1 , W 2 ) → U → X → Z forms a Markov ch ain. Th erefore , we have I ( W 1 , W 2 ; Z n | U n , X n ) = 0 , where it is implied that H ( Z n | W 1 , W 2 , U n , X n ) = H ( Z n | U n , X n ) . ( c ) f ollows from the fact that H ( W 1 , W 2 , U n , X n ) ≥ H ( U n , X n ) . ( d ) follows fr om that fact tha t H ( U n ) = I ( U n ; Y n 2 ) and H ( X n | U n ) = I ( X n ; Y n 1 | U n ) . Con verse : The tran smitter sends two indepen dent secr et message s W 1 and W 2 to re ceiv er 1 and receiver 2 respectively . Let us defin e U i = ( W 2 , Y i − 1 1 ) . The f ollowing chain of ineq uality clarifies the proo f: nR 1 ( a ) ≤ n X i =1 I ( W 1 ; Y 1 ,i | W 2 , Z i , Y i − 1 1 , e Z i +1 ) + nδ 1 + nǫ 3 = n X i =1 I ( W 1 ; Y 1 ,i | U i , Z i , e Z i +1 ) + nδ 1 + nǫ 3 ( b ) ≤ n X i =1 I ( X i ; Y 1 ,i | U i , Z i , e Z i +1 ) + nδ 1 + nǫ 3 ( c ) = n X i =1 I ( X i ; Y 1 ,i , U i , Z i | e Z i +1 ) − I ( X i ; Z i | e Z i +1 ) − I ( X i ; U i | Z i , e Z i +1 ) + nδ 1 + nǫ 3 ( d ) = n X i =1 I ( X i ; Y 1 ,i | U i , e Z i +1 ) + I ( X i ; U i | e Z i +1 ) − I ( X i ; Z i | e Z i +1 ) − I ( X i ; U i | Z i , e Z i +1 ) + nδ 1 + nǫ 3 ( e ) = n X i =1 I ( X i ; Y 1 ,i | U i , e Z i +1 ) − I ( X i ; Z i | e Z i +1 ) + I ( Z i ; U i | e Z i +1 ) − I ( Z i ; U i | X i , e Z i +1 ) + nδ 1 + nǫ 3 ( f ) = n X i =1 I ( X i ; Y 1 ,i | U i , e Z i +1 ) − I ( X i ; Z i | e Z i +1 ) + I ( Z i ; U i | e Z i +1 ) + nδ 1 + nǫ 3 , ( a ) f ollows from th e following lemma ( 2). ( b ) follows from the d ata p rocessing th eorem. ( c ) f ollows from th e chain r ule. ( d ) follows from the fact tha t I ( X i ; Y 1 ,i , U i , Z i | e Z i +1 ) = I ( X i ; U i | e Z i +1 ) + I ( X i ; Y 1 ,i | U i , e Z i +1 ) + I ( X i ; Z i | Y 1 ,i , U i , e Z i +1 ) and fro m the fact that e Z i +1 U i → X i → Y 1 ,i → Y 2 ,i → Z i forms a Markov chain, wh ich me ans that I ( X i ; Z i | Y 1 ,i , U i , e Z i +1 ) = 0 . ( e ) follows from the fact that I ( X i ; U i | e Z i +1 ) − I ( X i ; U i | Z i , e Z i +1 ) = I ( Z i ; U i | e Z i +1 ) − I ( Z i ; U i | X i , e Z i +1 ) . ( f ) follows from the fact that e Z i +1 U i → X i → Z i forms a Mar kov chain. Thus I ( Z i ; U i e Z i +1 | X i ) = 0 which imp lies th at I ( Z i ; U i | X i , e Z i +1 ) = 0 . Lemma 2 : F or the br oadcast channel with confid ential messages of ( W 1 , W 2 ) → X n → Y n 1 Y n 2 Z n , the perfect secr ecy rates ar e bou nded as follows, nR 1 ≤ n X i =1 I ( W 1 ; Y 1 i | W 2 , Z i , Y i − 1 1 , e Z i +1 ) + nδ 1 + nǫ 3 , (23) nR 2 ≤ n X i =1 I ( W 2 ; Y 2 i | Z i , Y i − 1 2 , e Z i +1 ) + nδ 1 + nǫ 2 . (24) Pr oof: W e ne ed to prove th e first bo und. The seco nd boun d can similarly b e pr oven. Acc ording to the above discu ssion nR 1 is b ounde d as fo llows: nR 1 ( a ) ≤ H ( W 1 | W 2 , Z n ) + nǫ 3 ( b ) ≤ H ( W 1 | W 2 , Z n ) − H ( W 1 | Y n 1 , W 2 ) + nδ 1 + nǫ 3 = I ( W 1 ; Y n 1 | W 2 ) − I ( W 1 ; Z n | W 2 ) + nδ 1 + nǫ 3 where ( a ) follows from the secr ecy constraint tha t H ( W 1 , W 2 | Z n ) ≥ H ( W 1 , W 2 ) − nǫ 3 , the fact that H ( W 2 | Z n ) ≤ H ( W 2 ) and the fact th at two messages are ind epende nt. ( b ) follows from Fano’ s inequ ality that H ( W 1 | Y n 1 , W 2 ) ≤ nδ 1 . Next, we 7 expand I ( W 1 ; Y n 1 | W 2 ) and I ( W 1 ; Z n | W 2 ) starting with I ( W 1 ; Y 1 | W 2 ) and I ( W 1 ; e Z n | W 2 ) , respectively . I ( W 1 ; Y n 1 | W 2 ) = n X i =1 I ( W 1 ; Y 1 i | W 2 , Y i − 1 1 ) = n X i =1 I ( W 1 , e Z i +1 ; Y 1 i | W 2 , Y i − 1 1 ) − I ( e Z i +1 ; Y 1 i | W 1 , W 2 , Y i − 1 1 ) = n X i =1 I ( W 1 ; Y 1 i | W 2 , Y i − 1 1 , e Z i +1 ) + I ( e Z i +1 ; Y 1 i | W 2 , Y i − 1 1 ) − I ( e Z i +1 ; Y 1 i | W 1 , W 2 , Y i − 1 1 ) = n X i =1 I ( W 1 ; Y 1 i | W 2 , Y i − 1 1 , e Z i +1 ) + ∆ 1 − ∆ 2 , where, ∆ 1 = P n i =1 I ( e Z i +1 ; Y 1 i | W 2 , Y i − 1 1 ) an d ∆ 2 = P n i =1 I ( e Z i +1 ; Y 1 i | W 1 , W 2 , Y i − 1 1 ) . Similarly , we have, I ( W 1 ; Z n | W 2 ) = n X i =1 I ( W 1 ; Z i | W 2 , e Z i +1 ) = n X i =1 I ( W 1 , Y i − 1 1 ; Z i | W 2 , e Z i +1 ) − I ( Y i − 1 1 ; Z i | W 1 , W 2 , e Z i +1 ) = n X i =1 I ( W 1 ; Z i | W 2 , Y i − 1 1 , e Z i +1 ) + I ( Y i − 1 1 ; Z i | W 2 , e Z i +1 ) − I ( Y i − 1 1 ; Z i | W 1 , W 2 , e Z i +1 ) = n X i =1 I ( W 1 ; Z i | W 2 , Y i − 1 1 , e Z i +1 ) + ∆ ∗ 1 − ∆ ∗ 2 , where, ∆ ∗ 1 = P n i =1 I ( Y i − 1 1 ; Z i | W 2 , e Z i +1 ) an d ∆ ∗ 2 = P n i =1 I ( Y i − 1 1 ; Z i | W 1 , W 2 , e Z i +1 ) . According to lemma 7 of [4], ∆ 1 = ∆ ∗ 1 and ∆ 2 = ∆ ∗ 2 . Thu s, we h av e, nR 1 ≤ n X i =1 I ( W 1 ; Y 1 i | W 2 , Y i − 1 1 , e Z i +1 ) − I ( W 1 ; Z i | W 2 , Y i − 1 1 , e Z i +1 ) + nδ 1 + nǫ 3 = n X i =1 H ( W 1 | W 2 , Z i , Y i − 1 1 , e Z i +1 ) − H ( W 1 | W 2 , Y 1 i , Y i − 1 1 , e Z i +1 ) + nδ 1 + nǫ 3 ( a ) ≤ n X i =1 H ( W 1 | W 2 , Z i , Y i − 1 1 , e Z i +1 ) − H ( W 1 | W 2 , Y 1 i , Z i , Y i − 1 1 , e Z i +1 ) + nδ 1 + nǫ 3 = n X i =1 I ( W 1 ; Y 1 i | W 2 , Z i , Y i − 1 1 , e Z i +1 ) + nδ 1 + nǫ 3 , where ( a ) f ollows fro m the fact th at con ditioning always decr eases the en tropy . 8 For th e secon d r eceiver , w e have nR 2 ( a ) ≤ n X i =1 I ( W 2 ; Y 2 ,i | Y i − 1 2 , Z i , e Z i +1 ) + nδ 2 + nǫ 1 = n X i =1 H ( Y 2 ,i | Y i − 1 2 , Z i , e Z i +1 ) − H ( Y 2 ,i | W 2 , Y i − 1 2 , Z i , e Z i +1 ) + nδ 2 + nǫ 1 ( b ) ≤ n X i =1 H ( Y 2 ,i | Z i , e Z i +1 ) − H ( Y 2 ,i | W 2 , Y i − 1 1 , Y i − 1 2 , Z i , e Z i +1 ) + nδ 2 + nǫ 1 ( c ) = n X i =1 H ( Y 2 ,i | Z i , e Z i +1 ) − H ( Y 2 ,i | U i , Z i , e Z i +1 ) + nδ 2 + nǫ 1 = n X i =1 I ( Y 2 ,i ; U i | Z i , e Z i +1 ) + nδ 2 + nǫ 1 = n X i =1 I ( Y 2 ,i ; U i | e Z i +1 ) + I ( Y 2 ,i ; Z i | U i , e Z i +1 ) − I ( Y 2 ,i ; Z i | e Z i +1 ) + nδ 2 + nǫ 1 = n X i =1 I ( Y 2 ,i ; U i | e Z i +1 ) − I ( Z i ; U i | e Z i +1 ) + I ( Z i ; U i | Y 2 ,i , e Z i +1 ) + nδ 2 + nǫ 1 ( d ) = n X i =1 I ( Y 2 ,i ; U i | e Z i +1 ) − I ( Z i ; U i | e Z i +1 ) + nδ 2 + nǫ 1 , where ( a ) follows from the lemm a (2 ). ( b ) follows from the fact that con ditioning always d ecreases the entro py . ( c ) follows from the fact that Y i − 1 2 → W 2 e Z i +1 Y i − 1 1 → Y 2 i → Z i forms a Markov ch ain. ( d ) follows from th e fact th at e Z i +1 U i → Y 2 ,i → Z i forms a Markov chain . Thus I ( Z i ; U i e Z i +1 | Y 2 i ) = 0 which implies th at I ( Z i ; U i | Y 2 i , e Z i +1 ) = 0 . Now , f ollowing [34], let us d efine th e time sh aring random variable Q wh ich is unifo rmly distributed over { 1 , 2 , ..., n } and indepen dent o f ( W 1 , W 2 , X n , Y n 1 , Y n 2 ) . Let us define U = U Q , V = ( e Z Q +1 , Q ) , X = X Q , Y 1 = Y 1 ,Q , Y 2 = Y 2 ,Q , Z = Z Q , then we can bound R 1 and R 2 as follows R 1 ≤ I ( X ; Y 1 | U, V ) + I ( U ; Z | V ) − I ( X ; Z | V ) , (25) R 2 ≤ I ( U ; Y 2 | V ) − I ( U ; Z | V ) . (26) Since Cond itional mutua l infor mations are av erage of unco nditional ones, the max imum region is achieved whe n V is a constant. Th is proves the conv erse part. V . G AU S S I A N B C C S In this section we consider the physically degraded A WGN broadcast channel with confidential messages. W e show that secret superpo sition codin g with G aussian codeb ook is op timal. At time i the r eceived signals are Y 1 i = X i + n 1 i , Y 2 i = X i + n 2 i and Z i = X i + n 3 i , where n 1 i ’ s, n 2 i ’ s and n 3 i ’ s are e ach in depend ent id entically distributed Gau ssian ran dom variables with zero means and V ar ( n j i ) = N j , j=1,2,3 . All n oises are in depend ent of X i and N 1 ≤ N 2 ≤ N 3 . Assume th at transmitted power is limited to E [ X 2 ] ≤ P . Since the ch annels ar e degraded , at tim e i , Y 1 i = X i + n 1 i , Y 2 i = Y 1 i + n ′ 2 i and Z i = Y 2 i + n ′ 3 i , where n 1 i ’ s ar e i.i.d N (0 , N 1 ) , n ′ 2 i ’ s are i.i.d N (0 , N 1 − N 2 ) , an d n ′ 3 i ’ s are i.i.d N (0 , N 2 − N 3 ) . Fig.4 sh ows the equiv alent chann els for the p hysically degraded A WGN- BCCs. The f ollowing theo rem illustrates the secrecy ca pacity region of A WGN-BCCs. Theorem 3 The secr ecy capa city r egion of the A WGN br oadcast channel with con fidentia l messages is given by the set of rates pairs ( R 1 , R 2 ) such that R 1 ≤ C  αP N 1  + C  (1 − α ) P αP + N 3  − C  P N 3  , (27) R 2 ≤ C  (1 − α ) P αP + N 2  − C  (1 − α ) P αP + N 3  . (28) for some α ∈ [0 , 1] . Pr oof: Achievability : Le t U ∼ N (0 , (1 − α ) P ) an d X ′ ∼ N (0 , αP ) be ind ependen t and X = U + X ′ ∼ N (0 , P ) . Theref ore, the amoun t o f I ( X ; Y 1 | U ) , I ( U ; Z ) , I ( X ; Z ) , and I ( U ; Y 2 ) can be easily evaluated. Now conside r th e following secure superpo sition codin g sch eme: 9 X Z n 1 n ′ 2 n ′ 3 Y 1 Y 2 Fig. 4. equi vale nt channels for the A WGN-BCCs 1) Codebook Gene ration : Generate 2 nI ( U ; Y 2 ) i.i.d Gaussian co dew ords u n with average power (1 − α ) P an d r andom ly distribute these codewords into 2 nR 2 bins. Then ind ex each bin by w 2 ∈ { 1 , 2 , ..., 2 nR 2 } . Gener ate a n in depend ent set of 2 nI ( X ′ ; Y 1 ) i. i.d Gaussian codewords x ′ n with average power αP . Then, Randomly distribute them in to 2 nR 1 bins. Index each bin by w 1 ∈ { 1 , 2 , ..., 2 nR 1 } . 2) Encodin g : T o sen d messages w 1 and w 2 , the tran smitter ran domly chooses one of the co dewords in b in w 2 , (say u n ) and one of th e co dew ords in bin w 1 (say x ′ n ). Then, simply transm its x n = u n + x ′ n . 3) Decodin g : Th e received sign al at the legitimate receivers are y n 1 and y n 2 respectively . Receiver 2 d etermines the uniqu e u n such th at ( u n , y n 2 ) ar e jointly typical and declares the index of the bin conta ining u n as th e message r eceived. If there is none of such or more than of one such , a n error is declar ed. Recei ver 1 uses successiv e cancelation method ; first deco des u n and subtracts o ff y n 1 and then looks f or the u nique x ′ n such th at ( x ′ n , y n 1 ) ar e join tly typ ical an d de clares the in dexes of the bin conta ining x ′ n as the message received. The erro r prob ability ana lysis and equ i vocation calc ulation is straigh tforward and may ther efor be omitted. Con verse : According to th e previous sectio n, R 2 is b ounde d as fo llows: R 2 ≤ I ( Y 2 ; U | Z ) = H ( Y 2 | Z ) − H ( Y 2 | U, Z ) (29) The classical en tropy power in equality states that: 2 2 n H ( Y 2 + n ′ 3 ) ≥ 2 2 n H ( Y 2 ) + 2 2 n H ( n ′ 3 ) Therefo re, H ( Y 2 | Z ) can be written as follows: H ( Y 2 | Z ) = H ( Z | Y 2 ) + H ( Y 2 ) − H ( Z ) = n 2 log( N 3 − N 2 ) + H ( Y 2 ) − H ( Y 2 + n ′ 3 ) ≤ n 2 log( N 3 − N 2 ) + H ( Y 2 ) − n 2 log(2 2 n H ( Y 2 ) + N 3 − N 2 ) On the other ha nd, for any fixed a ∈ R , the functio n f ( t, a ) = t − n 2 log(2 2 n t + a ) is concave in t and ha s a glob al maxim um at t = t max . Thus, H ( Y 2 | Z ) is maximized when Y 2 (or equiv alently X ) has Gaussian distribution. Hence, H ( Y 2 | Z ) ≤ n 2 log( N 3 − N 2 ) + n 2 log( P + N 2 ) − n 2 log( P + N 3 ) (30) = n 2 log  ( N 3 − N 2 )( P + N 2 ) P + N 3  Now consider the term H ( Y 2 | U, Z ) . T his term is lower b ound ed with H ( Y 2 | U, X , Z ) = n 2 log( N 2 ) which is greater than n 2 log( N 2 ( N 3 − N 2 ) N 3 ) . Hence, n 2 log( N 2 ( N 3 − N 2 ) N 3 ) ≤ H ( Y 2 | U, Z ) ≤ H ( Y 2 | Z ) (31) Inequa lities (30) and (3 1) imply that there exists a α ∈ [0 , 1] su ch th at H ( Y 2 | U, Z ) = n 2 log  ( N 3 − N 2 )( αP + N 2 ) αP + N 3  (32) 10 Substituting (32) and (3 0) into (29) yields the desired bou nd R 2 ≤ H ( Y 2 | Z ) − H ( Y 2 | U, Z ) (33) ≤ n 2 log  ( P + N 2 )( αP + N 3 ) ( P + N 3 )( αP + N 2 )  = C  (1 − α ) P αP + N 2  − C  (1 − α ) P αP + N 3  T o bo und th e rate R 1 , we ne ed the following gener alized e ntropy power ineq uality which is proven in [35] . Lemma 3 [35]: Let n 1 , n 2 be two gau ssian random variab les. Let U be a random variab le indep endent of n 1 and n 2 . Consider The optimization pr o blem max P ( X | U ) H ( X + n 1 | U ) − H ( X + n 2 | U ) (34) subject to V ar ( X | U ) ≤ s (35) wher e the maximization is over all distribution of X given U inde penden t of n 1 and n 2 . A Gaussian P ( x | u ) with same varian ce for each u is an optimal solution for this optimization pr oblem. The rate R 1 is bou nded as fo llows R 1 ≤ I ( X ; Y 1 | U ) − I ( X ; Z ) + I ( U ; Z ) (36) = H ( Y 1 | U ) − H ( Y 1 | X , U ) + H ( Z | X ) − H ( Z | U ) = H ( Y 1 | U ) − H ( Z | U ) + n 2 log( N 3 N 1 ) = H ( X + n 1 | U ) − H ( X + n 3 | U ) + n 2 log( N 3 N 1 ) On th e oth er han d using (32), w hen Z = 0 and n 2 = 0 th en V ar ( X | U ) = αP . Th erefore , Accor ding to the above lemma the Gaussian distribution is op timum and R 1 is b ounde d as R 1 ≤ n 2 log  αP + N 1 αP + N 3 N 3 N 1  (37) = C  αP N 1  + C  (1 − α ) P αP + N 3  − C  P N 3  V I . 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