Multi-receiver Wiretap Channel with Public and Confidential Messages

Multi-receiv er Wiretap Channel with Public a n d Conﬁden tial Messages ∗ Ersen Ekrem Senn ur Ulukus Departmen t of Elec trical and Computer Engineering Univ ersit y o f Maryland, College P ark, MD 20742 ersen@umd.e du ulukus@umd.e du No vem b er 15, 2018 Abstract W e study the m ulti-receiv er wir etap c hannel with public and conﬁden tial messages. In this c hannel, there is a transmitter that wishes to comm un icate with tw o legitimate users in the presence of an external eav esdropp er. The trans m itter sends a pair of public and conﬁdent ial messages to eac h legitimate user. While there are no s ecrecy constrain ts on the p ublic messages, conﬁdentia l messages need to b e transmitted in p erfect secrecy . W e study the discrete memoryless multi-recei ver wiretap c hannel as w ell as its Gaussian m ulti-input m ulti-output (MIMO) ins tance. First, we consider the degraded discrete memoryless c hannel, and obtain an inner b ound for the capacit y re- gion by u sing an ac hiev a ble sc heme that uses su p erp osition co ding and b inning. Next, w e obtain an outer b ound, and s ho w that this outer b ound partially matc hes the inner b ound, pro viding a partial c haracterization for the capacit y r egio n of the degraded c hannel mo del. Second, we obtain an inner b oun d for the general, not n ecessarily degraded, d iscrete memoryless c hannel b y using Marton’s inn er b ound, sup erp osition co ding, r ate-splitt ing and binn in g. Third, we consider the degraded Gaussian MIMO c hannel, and show that, to ev aluate b oth the in n er and outer b ounds, considering only join tly Gaussian auxiliary rand om v ariables and c hann el inpu t is suﬃcien t. Since the inn er and outer b ound s partially matc h, these suﬃciency results pr o vide a partial c haracterizatio n of the capacit y region of the degraded Gaussian MIMO c hann el. Fi- nally , w e p r o vide an inn er b ound for th e capacit y region of the general, n ot necessarily degraded, Gaussian MIMO multi-rec eiv er wiretap c hannel. ∗ This work was suppo rted by NSF Gr an ts CCF 07 -29127 , CNS 09-6 4632, C C F 09-64 645 and CCF 10- 18185 , a nd presented in part a t the Allerton Confer e nce o n Communications, Control and Computing, Monticello, IL, September 20 10. 1 1 In tro d u ction Information t heoretic secrecy is initiated by Wyner in [1], where he introduces the wiretap c hannel and obtains the capacit y-equiv o cation region o f a sp ecial class of wiretap c hannels. Wyner considers t he degraded wiretap c hannel, where the eav e sdropp er’s o bserv atio n is a degraded v ersion of the legitimate user’s observ ation. His result is g eneralized to arbitrary , not ne c essa rily de gr ade d , wiretap channels in [2]. Recen tly , the wiretap c hannel gathered a renew ed intere st, and man y multi-user extensions of the wiretap c hannel hav e b een consid- ered. One par ticular multi-user extension of the wiretap channe l, that is relev a n t to o ur w ork here, is the multi-rece ive r wiretap c hannel considered in [3 – 5]. A recen t surv ey on the secure broadcasting problem (including the multi-receiv er wiretap c hannel) can b e found in [6]. In the m ulti- receiv er wiretap c hannel (see Figure 1) diﬀeren t from the basic wiretap c hannel in [1, 2], there are m ultiple legitimate users to whic h the transmitter sends conﬁden tial messages in t he presence of an external ea v esdropp er. These m ultiple conﬁden tial messages need to b e kept secret from the eav esdropp er. References [3 – 5] consider the degraded m ulti- receiv er wiretap c hannel (see Figure 2) , where the observ ations of the legit imat e users and the ea v esdropp er are arranged according to a degra dednes s order. References [3 – 5] study the scenario where the tra nsmitter sends a conﬁden tial message to eac h legitimate user where these conﬁden tial messages need to b e k ept p erfectly secret from t he eav esdropp er. The capacit y region o f t he degraded multi-receiv er wiretap channel for this scenario is obtained in [3] f o r t w o legitimate users, and in [4, 5] for an arbitrar y n umber o f legitimate users. W e note that to ensure the conﬁden tialit y o f the messages in a multi-receiv er wiretap c hannel, eac h conﬁdential message needs to b e randomized b y man y dumm y messages [4, 5]. These dumm y messages pro tect the conﬁden tial messages from the ea ve sdropp er, and are deco ded b y the legitimate users in addition t o the conﬁden tial messages they receiv e. Hence, indeed, the actual tra nsmiss ion ra tes ar e greater than the conﬁden tial message rates. This also implies that the use of dumm y messages can b e view ed as a waste of resources since some of the achiev able rate is sp en t on transmitting them. T o ov ercome this waste of resources, these dumm y messages can b e replaced with some public messag es on whic h there are no secrecy constraints , as w e do in this pap er. In this pap er, b oth to ov ercome this w aste of resources and also to understand the actual p ossible transmission rates in a m ulti-receiv er wiretap channel, w e consider the scenario where t he transmitter sends a pair of public and conﬁden tial messages to eac h legitimate user. While there are no secrecy concerns on the public messages, conﬁdential messages need to b e tra nsmitted in p erfect secrecy . W e call the c hannel mo del arising from this scenario the multi-r e c eiver wir etap channel with public and c onﬁdential messages . The capacity r egion of the multi-receiv er wiretap c hannel with public and conﬁden tial messages is closely related to the capacit y-equiv o cation regio n of the multi-rece ive r wiretap c hannel. In the mu lti- r eceiv er wiretap c hannel studied in [3 – 5] (see also [7] for its G aussian MIMO instance) eac h legitimate user receiv es only a single message whic h needs to b e k ept 2 p ( y 1 | x ) Ea vesdropper T ransmitter p ( y 2 | x ) p ( z | x ) X Z Y 1 Y 2 User 1 User 2 Figure 1: Multi-receiv er wiretap c hannel. p erfectly secret from the ea v esdropp er. That is, [3 – 5, 7] consider p erfect secrecy for the mes- sages. As in the single-user wiretap c hannel [1, 2], one can b e in terested in the rates and equiv o cations of b oth messages simultaneous ly , i.e., in the sim ultaneously achiev able quadru- ples ( R 1 , R 1 e , R 2 , R 2 e ). This w ould b e a four dimensional region, as in the mo del discussed in this pap er, where w e ha ve the rates R p 1 , R s 1 , R p 2 , R s 2 for the rat es of the conﬁden tial and public messages of the tw o users. Unfortunately , unlik e the single-user wiretap c hannel [1, 2], where ( R p , R s ) and ( R , R e ) ha v e a one-to- one relationship [8 – 12], w e do not ha v e a one-to-one relationship b etw een ( R p 1 , R s 1 , R p 2 , R s 2 ) and ( R 1 , R 1 e , R 2 , R 2 e ). In this pap er, w e fo cus on the public and conﬁden tial messages framew ork, a nd fo cus on the region ( R p 1 , R s 1 , R p 2 , R s 2 ). Please see Section 2.1 for a more detailed t r eat ment of the relationship b et w een this region and the capacit y-equiv o cation region of the multi-rece ive r wiretap channel that consists of the quadruples ( R 1 , R 1 e , R 2 , R 2 e ). In this pap er, w e ﬁrst consider t he degraded discrete memoryless m ulti-receiv er wiretap c hannel. W e pr o p ose an inner b ound for the capacity region o f the discrete memoryless c hannel. This inner b ound is based on an ac hiev able sc heme t ha t combine s sup erp osition co ding [13] a nd binning. Binning has b een used previously for the single-receiv er and multi- receiv er wiretap channels in [1 – 5] to asso ciate eac h conﬁden tial message with man y co de- w ords, and hence to provide randomness fo r t he conﬁden tial message to protect it from the ea v esdropp er. In other w ords, by means of binning, the conﬁden tial message is em b edded in to a doubly indexed co dew or d where one index denotes the conﬁden tial message and the other index (dumm y index) denotes the necessary randomness to ensure the conﬁdentialit y of the message. This second index (dummy index) do es not carry an y information con ten t. Since in our channe l mo del there are public messages, on whic h there are no secrecy constrain ts, the protection of the conﬁden tia l messages fro m the ea v esdropp er can b e a ccom- plished b y using these public messages instead of the dumm y messages. Thus , the diﬀerence of binning used here from the binning used in [3 – 5] is that, here, the confusion messages carry information, although t here are no securit y guarantees on this information. Consequen tly , the injection of public messages in to the multi-receiv er wiretap channel can b e view ed as 3 p ( y 1 | x ) X p ( z | y 2 ) T ransmitter p ( y 2 | y 1 ) Y 1 Y 2 Z Ea vesdropper User 2 User 1 Figure 2: Degra ded m ulti-receiv er wiretap c hannel. an eﬀort to use the waste d tra nsmission rate due to no-information b earing dummy indices, since these dummy indices a r e no w replaced with informatio n-b earing public messages on whic h t here are no secrecy gua ran tees. Next, we prop ose an outer b ound for the capacit y region of the degraded discrete mem- oryless channel. W e obtain this o uter b ound b y com bining the conv erse pro of t echniq ues for the bro a dcast channel with degraded message sets [14] and the bro a dcast c ha nnel with conﬁden tial messages [2]. This outer b ound partially matc hes the inner b ound w e prop o se, and therefore, it provides a partial characterization of the capacity region of t he degra ded discrete memoryless c hannel. In particular, when w e specialize these inner and outer b ounds b y setting either the public message ra t e o f the second legitimate user or the conﬁden tial message rate of the ﬁrst legitimate user to zero, t hey match a nd prov ide the exact capacity region for these t w o scenarios. Moreov er, when w e set the rates of b oth o f the public mes- sages to zero, these inner and outer b ounds matc h, and yield the secrecy capacity region of the degraded discrete memoryless ch annel. Second, w e consider the general, not necess arily degraded, discrete memoryless m ulti- receiv er wiretap channel. W e pro p ose an inner b ound for the capacity region of the general c hannel b y using Marton’s inner b ound [15 ], sup erp osition co ding, rate-splitting and binning. This inner b ound generalizes the inner b ound w e prop osed for the degraded case by using Marton’s co ding. Third, w e consider the degraded Gaussian m ulti-input m ulti-o utput (MIMO) instance of this channel mo del. This generalizes our w ork in [7], where we consider the general, not necessarily degraded, Gaussian MIMO c hannel only with conﬁden tial messages. F or the degra ded G aussian MIMO c hannel in this pap er, w e ﬁrst sho w that it is suﬃcien t to consider join tly Gaussian auxiliary random v a r ia bles a nd channe l input for the ev aluation of the inner b ound w e prop osed for the degraded discrete memoryless c hannel. In other words, w e prov e that there is no other p o ssible selection of auxiliary random v aria bles and c hannel input whic h can pro vide a rate vec tor outside the G a ussian rate region that is obtained b y usin g join tly Gaussian auxiliary random v ariables and channel input. W e prov e the suﬃciency of Ga ussian auxiliary random v ariables and c hannel input b y using the de Bruijn 4 iden tit y [16, 17], a diﬀeren tial relationship b et w een the diﬀerential entrop y and the F isher information matrix, in conjunction with the prop erties of the Fisher inf o rmation matrix. Next, we consider the outer b ound w e prop osed for t he degraded discrete memoryless c hannel. W e show that, similar to the inner b ound, considering only j oin tly G aussian aux- iliary random v ariables a nd ch annel input is suﬃcien t to ev aluate this outer b ound fo r the degraded Gaussian MIMO channel. Indeed, this suﬃciency result is already implied b y t he suﬃciency o f join tly G a ussian auxiliary random v ariables and channel input for the inner b ound, b ecause of the partial matc h b etw een the inner and the outer b ounds. Moreo ve r, this partial match also g ives us a partial characterization of t he capacit y region of the degra ded Gaussian MIMO c hannel. T he inner and outer b ounds for the degraded Ga ussian MIMO c hannel completely matc h giving us the exact capacit y region, when either the public mes- sage rate o f the second legitimate user or the conﬁden tial message rate of the ﬁrst legitimate user is zero. Mor eov er, these inner and o uter b ounds matc h for the secrecy capacity region of the degraded Gaussian MIMO c hannel, whic h w e obta in when w e t he rates of b oth of the public messages are zero [7, 1 8]. Finally , w e consider the g eneral, not necess arily degraded, Gaussian MIMO multi-receiv er wiretap channel. W e pr o p ose an inner b ound for the capacit y region of the general Gaussian MIMO channel. W e obta in this inner b ound b y using the ac hiev able sc heme w e prop osed for the g eneral discrete memoryless c hannel. In pa rticular, w e ev aluate this achiev able sc heme b y using dirty-paper co ding [19] to obta in an inner b ound f o r the capacit y region of the general Ga ussian MIMO c hannel. 2 Discret e Memoryles s Mult i - Receiv er Wiretap Chan- nels In this section, w e study discrete memoryless m ulti-receiv er wiretap c hannels whic h consist of a transmitter with input alphab et X , t w o legitimate users with output alphab ets Y 1 , Y 2 , and an eav esdropp er with output a lpha b et Z . The channel is memoryless with a transition probabilit y p ( y 1 , y 2 , z | x ), where X ∈ X is the channel input, and Y 1 ∈ Y 1 , Y 2 ∈ Y 2 , Z ∈ Z denote the c hannel outputs o f the ﬁrst legitimate user, the second legitimate user, and the ea v esdropp er, resp ectiv ely . W e consider the scenario in whic h, the transmitter sends a pair of public and conﬁden tial messages to eac h legitimate user. While there are no secrec y constraints on the public messages, the conﬁden tial messages need to b e transmitted in p erfect secrec y . W e call the c hannel mo del arising from this scenario the multi-r e c eiver wir etap channel with public an d c onﬁd e n tial me ssages . An ( n, 2 nR p 1 , 2 nR s 1 , 2 nR p 2 , 2 nR s 2 ) co de for this c hannel consists of four message sets, W p 1 = { 1 , . . . , 2 nR p 1 } , W s 1 = { 1 , . . . , 2 nR s 1 } , W p 2 = { 1 , . . . , 2 nR p 2 } , W s 2 = { 1 , . . . , 2 nR s 2 } , one en- co der at the tra nsmitter f : W p 1 × W s 1 × W p 2 × W s 2 → X n , and one deco der at eac h 5 legitimate user g j : Y n j → W pj × W sj , fo r j = 1 , 2. The probabilit y of erro r is de- ﬁned as P n e = max { P n e, 1 , P n e, 2 } , whe re P n e,j = Pr[ g j ( Y n j ) 6 = ( W pj , W sj )], for j = 1 , 2 , and W p 1 , W s 1 , W p 2 , W s 2 are uniformly distributed random v ariables in W p 1 , W s 1 , W p 2 , W s 2 , re- sp ectiv ely . A rate tuple ( R p 1 , R s 1 , R p 2 , R s 2 ) is said to b e ac hiev able if t here exists an ( n, 2 nR p 1 , 2 nR s 1 , 2 nR p 2 , 2 nR s 2 ) co de whic h satisﬁes lim n →∞ P n e = 0 and lim n →∞ 1 n I ( W s 1 , W s 2 ; Z n ) = 0 (1) W e note that the p erfect secrecy requiremen t g iv en by (1) implies the follo wing individual p erfect secrecy requiremen ts lim n →∞ 1 n I ( W s 1 ; Z n ) = 0 and lim n →∞ 1 n I ( W s 2 ; Z n ) = 0 (2) The capacity r egion of the multi-receiv er wiretap c hannel with public and conﬁden tial messages is deﬁned as the con v ex closure of a ll ac hiev able ra te tuples ( R p 1 , R s 1 , R p 2 , R s 2 ), and is denoted by C . In Section 2.2, we study de gr ade d m ulti-r eceiv er wiretap c hannels whic h satisfy the fol- lo wing Mar ko v chain X → Y 1 → Y 2 → Z (3) 2.1 Capacit y-Equiv o cation Region of the Multi-Receiv er Wiretap Channel In this section, w e consider the capacit y-equiv o cation region of the m ulti-receiv er wiretap c hannel and discuss its connection to the capacit y region of the m ulti-receiv er wiretap channel with public and conﬁden tial messages. T o this end, we consider b oth rates and equiv o cations of the messages in the m ulti-receiv er wiretap c hannel. Eac h legitimate user receiv es a single message whic h needs to b e ke pt hidden as m uc h as p ossible from the ea v esdropp er. In par - ticular, each message has its own rate and a corresp onding equiv o cation. The equiv o cations are quantiﬁe d by the follow ing conditional en tropies: 1 n H ( W 1 | Z n ) and 1 n H ( W 2 | Z n ) and 1 n H ( W 1 , W 2 | Z n ) (4) An ( n, 2 nR 1 , 2 nR e 1 , 2 nR 2 , 2 nR e 2 ) co de for this c hannel consists of tw o message sets , W 1 = { 1 , . . . , 2 nR 1 } , W 2 = { 1 , . . . , 2 nR 2 } , one enco der at t he transmitter f : W 1 × W 2 → X n , and one deco der at eac h legitimate user g j : Y n j → W j , for j = 1 , 2. The probabilit y of error is deﬁned as P n e = max { P n e, 1 , P n e, 2 } , where P n e,j = Pr[ g j ( Y n j ) 6 = W j ], for j = 1 , 2 , and W 1 , W 2 are uniformly distributed random v ariables in W 1 , W 2 , resp ectiv ely . A r a te tuple ( R 1 , R e 1 , R 2 , R e 2 ) is said to b e achiev able if there exists an ( n, 2 nR 1 , 2 nR e 1 , 2 nR 2 , 2 nR e 2 ) co de 6 whic h satisﬁes lim n →∞ P n e = 0 and R e 1 ≤ lim n →∞ 1 n H ( W 1 | Z n ) (5) R e 2 ≤ lim n →∞ 1 n H ( W 2 | Z n ) (6) R e 1 + R e 2 ≤ lim n →∞ 1 n H ( W 1 , W 2 | Z n ) (7) The capacity -equiv o cation region of the m ulti-r eceiv er wiretap c hannel is deﬁned as the con v ex closure of all a c hiev able rate tuples ( R 1 , R e 1 , R 2 , R e 2 ), and is denoted b y C e . W e no w ha ve some remarks ab out the deﬁnition of the equiv o cations by (5)-(7). First, if w e consider the p erfect secrecy r a tes, i.e., R e 1 = R 1 , R e 2 = R 2 , then the last constraint in (7) is suﬃcien t, since this last constrain t implies the individual constrain ts in (5)-(6). Our second remark is ab out the sum equiv o cation constrain t in (7). I f w e consider the single-legitimate user case a s in [1, 2], w e hav e only one of the tw o constraints in (5)- (6) as equiv o cation. A direct generalization o f this deﬁnition to the multiple legitimate user case already yields the individual constrain ts in (5)-(6), ho we v er here w e put one more constrain t b y (7). The reason b ehind this additional sum equiv o cation constrain t in (7) is as follows . The individual constraints in (5 )-(6) consider the p o ssibilit y that the ea v esdropp er can try to deco de W 1 and W 2 separately , ho w ev er, these t w o individual constrain ts in ( 5)-(6) do not consider the p ossibilit y tha t the eav esdropp er might try to deco de W 1 , W 2 join tly . Th us, to reﬂect the p ossibility tha t the ea v esdropp er mig ht try to deco de W 1 , W 2 join tly o n the equiv o cations, w e include the sum equiv o cation constraint in (7). Next, we discuss the connection b etw een the capacity region o f the m ulti-receiv er wiretap c hannel with public and conﬁden tial messages and the capacit y-equiv o cation region of the m ulti-receiv er wiretap c hannel. F or the single-legitimat e user case, there is a one-to- o ne corresp ondence b etw een these tw o regions as stated in the follow ing lemma [8 – 12]. Lemma 1 ( R p 1 , R s 1 , 0 , 0) ∈ C iﬀ ( R 1 = R p 1 + R s 1 , R e 1 = R s 1 , 0 , 0) ∈ C e . The “if ” part of this lemma follow s from the deﬁnitions of the t w o regions. Ho w ev er, the “only if ” pa r t of this lemma uses the prop erties of the capacit y ac hieving co ding sc heme for the r egio n C e s 1 . Since as of now w e do not know the capacity -a c hieving sc heme f or C e , w e can provide only a partial g eneralization of Lemma 1 to t he multiple legitimate user case as stated in the following lemma. Lemma 2 If ( R p 1 , R s 1 , R p 2 , R s 2 ) ∈ C , w e have ( R 1 = R p 1 + R s 1 , R e 1 = R s 1 , R 2 = R p 2 + R s 2 , R e 2 = R s 2 ) ∈ C e . Lemma 2 can b e prov ed b y using t he deﬁnitions of C a nd C e . This lemma states that all inner b ounds w e obtain for C a lso serv e as inner b ounds for C e . Indeed, the outer b ounds w e 1 C e s denotes the capacity-equivocation region for the single le gitimate user cas e. 7 obtain for C and the capacity results we obtain for the sub-regions of C (a sub-region of C is a region that can b e obtained from C by setting one or more of the four rates inv olved in C to zero) can b e sho wn to b e v alid for C e . 2.2 Degraded Ch ann els In this section, w e consider the degraded multi-receiv er wiretap c hannel. W e ﬁrst presen t an inner b ound for C in the following theorem. Theorem 1 A n achievable r ate r e gion for the multi-r e c eiver wir etap channel with public and c onﬁd e n tial me ssages is given by the union of r ate tuples ( R p 1 , R s 1 , R p 2 , R s 2 ) satisfying R s 2 ≤ I ( U ; Y 2 ) − I ( U ; Z ) (8) R s 1 + R s 2 ≤ I ( U ; Y 2 ) + I ( X ; Y 1 | U ) − I ( X ; Z ) (9) R p 2 + R s 2 ≤ I ( U ; Y 2 ) (10) R s 1 + R p 2 + R s 2 ≤ I ( U ; Y 2 ) + I ( X ; Y 1 | U ) − I ( X ; Z | U ) (11) R p 1 + R s 1 + R p 2 + R s 2 ≤ I ( U ; Y 2 ) + I ( X ; Y 1 | U ) (12) wher e ( U, X ) satisfy the fol lowing Markov c hain U → X → Y 1 → Y 2 → Z (13) The achiev able rate region give n by Theorem 1 can b e obtained fro m Theorem 3, whic h will b e in tro duced in the next section. Here, w e pro vide an outline o f the pro of o f Theorem 1. W e pro v e Theorem 1 in t w o steps. As a ﬁrst step, one can sho w that the rate tuples ( R p 1 , R s 1 , R p 2 , R s 2 ) satisfying R p 2 ≤ I ( U ; Z ) (14) R s 2 ≤ I ( U ; Y 2 ) − I ( U ; Z ) (15) R p 1 ≤ I ( X ; Z | U ) (16) R s 1 ≤ I ( X ; Y 1 | U ) − I ( X ; Z | U ) (17) are ac hiev able, where ( U, X ) satisfy (13). W e sho w the ac hiev ability of rate tuples ( R p 1 , R s 1 , R p 2 , R s 2 ) satisfying (14)- (17) b y using sup erp o sition co ding. The diﬀerences of the super- p osition co ding used here from the original sup erp osition co ding tha t attains the capacit y region of the degra ded bro adcast c hannel [1 3] are that b oth public and conﬁden tial mes- sages of a legit imat e user are transmitted by the same lay er of the co deb o ok, and the public messages provide the necessary randomness to protect the conﬁden tial messages from the ea v esdropp er. In other w ords, in a ddition to their information con ten t, the public mes- sages a lso serv e as the confusion messages that prev en t the eav esdropp er from deco ding the 8 conﬁden tial messages. W e also note that the ac hiev ability of the region given in (14)-(1 7) can b e concluded b y using the achiev able sche me in [3 – 5], whic h w as designed fo r the degraded m ulti-receiv er wire- tap c hannel only with conﬁden tial messages. This ac hiev able sc heme also uses sup erp osition co ding and binning. In this sc heme, to achiev e the conﬁden tial message rates R s 2 = I ( U ; Y 2 ) − I ( U ; Z ) ( 1 8) R s 1 = I ( X ; Y 1 | U ) − I ( X ; Z | U ) ( 1 9) eac h conﬁden tial message r ate is equipp ed with the rate of some dumm y messages, whic h pro vide the necessary protection for the conﬁden tial messages against the ea v esdropp er. In particular, the conﬁden tial message rates R s 2 and R s 1 are equipp ed with the follow ing dumm y message rates ˜ R s 2 = I ( U ; Z ) (20) ˜ R s 1 = I ( X ; Z | U ) (21) where ˜ R sj is the dummy message rate sp en t for the j th legitimate user’s conﬁdential message rate R sj . Since in our channel mo del there are public messages on whic h there are no secrecy constrain ts, these dumm y message rates can b e used as public message rates yielding the ac hiev able r a te region giv en in (14 )-(17). W e note that the presence of public messages in our c hannel mo del preve nts the w aste o f resources due to the transmission of the dumm y messages at rates given by (20)-(21) t o protect the conﬁden tial messages. As the second step to prov e Theorem 1, one can use the following f a cts • since conﬁden tial messages can b e considered as public messages as w ell, eac h legitimate user’s conﬁden tial message rate R sj can b e giv en up in fa v or of its public message rat e R pj , i.e., if ( R p 1 , R s 1 , R p 2 , R s 2 ) is achiev able, ( R p 1 + α 1 , R s 1 − α 1 , R p 2 + α 2 , R s 2 − α 2 ) is also achiev able for a n y non-negative ( α 1 , α 2 ) pairs satisfying α j ≤ R sj , • since the c hannel is degraded, the second legitimate user’s conﬁdential message rate R s 2 can b e giv en up in fa v or of the ﬁrst legitimate user’s public and conﬁden tial message rates R p 1 and R s 1 , i.e., if ( R p 1 , R s 1 , R p 2 , R s 2 ) is ac hiev able, ( R p 1 + α, R s 1 + β , R p 2 , R s 2 − α − β ) is also ac hiev able fo r a n y non-negative ( α, β ) pairs satisfying α + β ≤ R s 2 , • since the c hannel is degraded, the second legitimate user’s public message rate R p 2 can b e g iv en up in fav or o f the ﬁrst legitimate user’s public message rate R p 1 , i.e., if ( R p 1 , R s 1 , R p 2 , R s 2 ) is ac hiev able, ( R p 1 + α , R s 1 , R p 2 − α, R s 2 ) is also ac hiev able for any non-negativ e α satisfying α ≤ R p 2 , in conjunction with F ourier-Motzkin elimination, and sho w that the region giv en in (14)-(17) is equiv alen t to the one give n in (8 )-(12). 9 The reason w e state the achiev able rate region b y using the b ounds in (8)-(12) instead of the b ounds in (14)-( 1 7) is tha t the fo rmer expressions simplify the comparison of the inner b ound with t he outer b ound whic h will b e introduced in the sequel. Another reason that w e state ac hiev able region b y using the b o unds in (8 )-(12) is that they are mor e con v enien t for an explicit ev a luation for the case of degraded Gaussian MIMO channel, whic h will b e considered in Section 3 .2. No w, w e in tro duce the follow ing outer b ound for the capacity region of the degraded discrete memoryless m ulti-receiv er wiretap channe l with public and conﬁden tial messages. Theorem 2 The c ap aci ty r e gion of the de gr ade d multi-r e c eiver wir etap chann e l with pub- lic and c onﬁdential messages is c ontaine d in the union of r ate tuples ( R p 1 , R s 1 , R p 2 , R s 2 ) satisfying R s 2 ≤ I ( U ; Y 2 ) − I ( U ; Z ) (22) R s 1 + R s 2 ≤ I ( U ; Y 2 ) + I ( X ; Y 1 | U ) − I ( X ; Z ) (23) R p 2 + R s 2 ≤ I ( U ; Y 2 ) (24) R p 1 + R s 1 + R p 2 + R s 2 ≤ I ( U ; Y 2 ) + I ( X ; Y 1 | U ) (25) for some ( U, X ) such that U, X exhibit the fol lowing Markov chain U → X → Y 1 → Y 2 → Z (26) The pro of of Theorem 2 is g iv en in App endix A. This outer b ound provides a partial con v erse for the capacity region of the degraded m ulti-r eceiv er wiretap channe l b ecause the only diﬀerence b et w een the inner b ound in Theorem 1 and the outer b ound in Theorem 2 is the b ound on R s 1 + R p 2 + R s 2 giv en b y (11). In particular, in addition to the b ounds deﬁning the outer b ound for the capacity regio n, the inner b ound includes the follow ing constraint R s 1 + R p 2 + R s 2 ≤ I ( U ; Y 2 ) + I ( X ; Y 1 | U ) − I ( X ; Z | U ) (27) Besides that , the inner and outer b ounds ar e iden tical. Despite this diﬀerence, there are cases fo r whic h the exact capacity region can b e o btained. First, we note that the inner b o und in Theorem 1 and the outer b ound in Theorem 2 matc h when the conﬁden tial message ra te o f the ﬁrst legitimate user is zero, i.e., R s 1 = 0. Corollary 1 The c ap a city r e gion of the de gr ade d multi-r e c eive r wir etap channel without the ﬁrst le g i tim a te user’s c onﬁdential message is given by the union of r ate triples ( R p 1 , R p 2 , R s 2 ) 10 satisfying R s 2 ≤ I ( U ; Y 2 ) − I ( U ; Z ) (28) R s 2 + R p 2 ≤ I ( U ; Y 2 ) (29) R p 1 + R p 2 + R s 2 ≤ I ( U ; Y 2 ) + I ( X ; Y 1 | U ) (30) wher e U, X e xhibit the fo l lowin g Markov chain U → X → Y 1 → Y 2 → Z (31) Corollary 1 can b e prov ed b y se tting R s 1 = 0 in bot h Theorem 1 a nd Theorem 2 and eliminating the redundan t b ounds. Next, w e note that the inner b o und in Theorem 1 a nd t he outer b ound in Theorem 2 matc h when the public message rate of the second legitimate user is zero, i.e., R p 2 = 0. Corollary 2 The c ap a city r e gion of the de gr ade d multi-r e c eive r wir etap channel without the se c ond le gitimate user’s public message is given by the union of r ate triples ( R p 1 , R s 1 , R s 2 ) satisfying R s 2 ≤ I ( U ; Y 2 ) − I ( U ; Z ) (32) R s 1 + R s 2 ≤ I ( U ; Y 2 ) + I ( X ; Y 1 | U ) − I ( X ; Z ) (33) R p 1 + R s 1 + R s 2 ≤ I ( U ; Y 2 ) + I ( X ; Y 1 | U ) (34) wher e U, X e xhibit the fo l lowin g Markov chain U → X → Y 1 → Y 2 → Z (35) Corollary 2 can b e prov ed b y setting R p 2 = 0 in bot h Theorem 1 a nd Theorem 2 and eliminating the redundan t b ounds. Corollary 2 also implies tha t the inner b ound in Theorem 1 and the outer b ound in Theorem 2 match o n the secrecy capacit y region of the degraded multi-receiv er wiretap c hannel. In particular, the inner b ound in Theorem 1 and the outer b ound in Theorem 2 matc h if the rates of b oth public messages are set to zero, i.e., R p 1 = R p 2 = 0. The secrecy capacit y region of the degraded m ulti-receiv er wiretap c hannel is giv en b y the followin g corollary . Corollary 3 ([3 – 5]) Th e se cr e cy c ap acity r e gion of the de gr ade d multi-r e c eiver w ir etap chan- 11 nel is g iven by the union of r ate p airs ( R s 1 , R s 2 ) satisfying 2 R s 2 ≤ I ( U ; Y 2 ) − I ( U ; Z ) (36) R s 1 + R s 2 ≤ I ( U ; Y 2 ) + I ( X ; Y 1 | U ) − I ( X ; Z ) (37) wher e U, X e xhibit the fo l lowin g Markov chain U → X → Y 1 → Y 2 → Z (38) W e note t hat in additio n to its represen tation in Corollary 3 , the secrecy capacity r egio n of the degraded m ulti-receiv er wiretap c hannel can b e stated in an alternativ e for m as the union of rate pairs ( R s 1 , R s 2 ) satisfying R s 2 ≤ I ( U ; Y 2 ) − I ( U ; Z ) (39) R s 1 ≤ I ( X ; Y 1 | U ) − I ( X ; Z | U ) (40) where U, X exhibit the Mark o v c hain in (3 8). 2.3 General Ch ann els W e now consider the general, not necessarily degraded, discrete memoryless m ulti-receiv er wiretap c hannel with public and conﬁden tial messages. W e prop ose an inner b ound f or the capacit y regio n of the general discrete memoryless mu lti- r eceiv er wiretap channel as follow s. Theorem 3 A n achievable r ate r e gio n for the discr e te memoryless multi-r e c eiver wir etap channel with publi c and c onﬁdential mes s a ges is given by the union of r ate tuples ( R p 1 , R s 1 , R p 2 , R s 2 ) satisfying R s 1 ≤ min j =1 , 2 I ( U ; Y j | Q ) + I ( V 1 ; Y 1 | U ) − I ( U, V 1 ; Z | Q ) (41) R s 2 ≤ min j =1 , 2 I ( U ; Y j | Q ) + I ( V 2 ; Y 2 | U ) − I ( U, V 2 ; Z | Q ) (42) R s 1 + R s 2 ≤ min j =1 , 2 I ( U ; Y j | Q ) + I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) − I ( U, V 1 , V 2 ; Z | Q ) (43) R s 1 + R p 1 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) (44) R s 2 + R p 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 2 ; Y 2 | U ) (45) R s 1 + R p 1 + R s 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 2 ; Z | U ) (46) 2 The se crecy ca pacit y reg ion of the deg raded multi-receiver wir etap c hannel for an ar bitrary n umber of legitimate users can b e found in [4, 5]. 12 R s 1 + R p 1 + R s 2 ≤ min j =1 , 2 I ( U ; Y j ) + 2 I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) − I ( V 1 , V 2 ; Z | U ) (47) R s 1 + R s 2 + R p 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; Z | U ) (48) R s 1 + R s 2 + R p 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) + 2 I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) − I ( V 1 , V 2 ; Z | U ) (49) R s 1 + R p 1 + R s 2 + R p 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) (50) for some Q, U, V 1 , V 2 such that p ( q , u, v 1 , v 2 , x, y 1 , y 2 , z ) = p ( q , u ) p ( v 1 , v 2 , x | u ) p ( y 1 , y 2 , z | x ) . The pro of of Theorem 3 is g iven in App endix B. W e note that if one sets Q = φ, V 2 = U, V 1 = X in Theorem 3, the achie v able rate region in Theorem 3 reduces to the one pro vided in Theorem 1 . Th us, the a chiev able sc heme in Theorem 3 can b e seen as a generalization of the a c hiev able sc heme in Theorem 1 , where w e ac hiev e this g eneralization by using Marton’s co ding and rat e- splitting in addition t o the sup erp osition co ding and binning that w ere already used for the ac hiev able sc heme in Theorem 1. Next, we provide an outline of the ac hiev able sc heme in Theorem 3. In this ac hiev able sc heme, w e ﬁrst divide each public message W pj in to three parts as W 1 pj , W 2 pj , W 3 pj , where the rates of the messages W 1 pj , W 2 pj , W 3 pj are giv en by R 1 pj , R 2 pj , R 3 pj , resp ectiv ely , and R pj = R 1 pj + R 2 pj + R 3 pj . Similarly , w e divide eac h conﬁden tial message W sj in to t w o parts as W 1 sj , W 2 sj , where the rates of the messages W 1 sj , W 2 sj are giv en b y R 1 sj , R 2 sj , respectiv ely , a nd R sj = R 1 sj + R 2 sj . The ﬁrst parts of the public messages, i.e., W 1 p 1 and W 1 p 2 , are sen t through the sequences generated by Q . The second parts of the public messages, i.e., W 2 p 1 and W 2 p 2 , and the ﬁrst parts of the conﬁdential messages, i.e., W 1 s 1 and W 1 s 2 , are sen t through the sequence s generated by U . Both legitimate receiv ers deco de these sequenc es, and hence, eac h legitimate receiv er deco des the pa r t s of the other legitimate user’s public and conﬁdential messages. The last parts of each public message and eac h conﬁden tial message, i.e., W 3 pj and W 2 sj , are enco ded by the sequences g enerated through V j . This enco ding is p erformed b y using Marto n’s co ding [15]. Each legitimate receiv er, a fter deco ding Q n and U n , deco des the sequences V n j . The details of the pro of is giv en in App endix B. 3 Gaussian MIMO Multi-Rec eiv e r W iretap Channel s Here, w e consider the Gaussian MIMO m ulti-receiv er wiretap channe l whic h is deﬁned by Y 1 = X + N 1 (51) Y 2 = X + N 2 (52) Z = X + N Z (53) 13 where the c hannel input X is sub ject to a co v ariance constrain t E  XX ⊤   S (54) where S ≻ 0 and N 1 , N 2 , N Z are zero-mean Gaussian rando m v ectors with co v ariance ma- trices Σ 1 , Σ 2 , Σ Z , resp ectiv ely . In Section 3.2 , we consider de g r ade d Ga ussian MIMO multi-receiv er wiretap c hannels for whic h t he noise co v ar ia nce matrices Σ 1 , Σ 2 , Σ Z satisfy the followin g order 0 ≺ Σ 1  Σ 2  Σ Z (55) In a multi-receiv er wiretap channel, since the capacity regio n dep ends only on the conditional marginal distributions o f the transmitter-receiv er links, but no t o n t he entire join t distribu- tion of the ch annel, the correlations a mong N 1 , N 2 , N Z do not aﬀect the capa city region. Th us, without ch anging the corresp onding capacit y region, we can adjust the correlatio n structure among these noise v ectors to ensure that they satisfy the Mark ov c hain X → Y 1 → Y 2 → Z (56) whic h is alwa ys p ossible b ecause of our assumption ab out the co v ariance matrices in (55 ). Th us, for any Gaussian MIMO m ulti-receiv er wiretap c hannel satisfying the order in (55), w e can assume tha t it also exhibits the Mark o v chain in (56) without c hanging the capacit y region of the orig ina l channel. 3.1 Commen ts on the Channel Mo del and the Co v ariance Con- strain t W e provide some commen ts a b o ut the w ay w e deﬁne the Gaussian MIMO m ulti-receiv er wiretap ch annel. The ﬁrst one is ab out the co v ariance constraint in (5 4). Though it is more common to deﬁne capacity regions under a total p o w er constraint, i.e., tr  E  XX ⊤  ≤ P , the cov ariance constrain t in (54) is more g eneral and it subsumes the to tal p o w er constrain t as a sp ecial case [20]. In particular, if we denote the capacit y region under the constrain t in (54) by C ( S ), then the capacity region under the trace constrain t, tr  E  XX ⊤  ≤ P , can b e written a s [20] C trace ( P ) = [ S :tr( S ) ≤ P C ( S ) (57) Similarly , the inner a nd outer b ounds obtained f or the cov a r iance constraint in (54 ) can b e extended to pro vide the corresp onding inner and outer b ounds for the total p ow er constrain t. The second commen t is ab out our assumption t ha t S is strictly p ositiv e deﬁnite. This 14 assumption do es not lead to any loss o f generalit y b ecause for an y G aussian MIMO m ulti- receiv er wiretap c hannel with a p ositiv e semi-deﬁnite cov ariance constrain t, i.e., S  0 and | S | = 0, w e can alw ays construct an equiv a len t c hannel with the constraint E  XX ⊤   S ′ where S ′ ≻ 0 (see Lemma 2 of [2 0]), whic h has the same capacity region. The last commen t is ab out the assumption that the transmitter and all receiv ers ha v e the same n umber of antennas . This assumption is implicit in the c hannel deﬁnition, s ee (51)-(53), and also in the deﬁnition of degradedness, see (55). Ho wev er, w e can extend the deﬁnition of the Gaussian MIMO m ulti-receiv er wiretap channe l to include the cases where the num b er of transmit antennas and the n um b er o f receiv e ante nnas at eac h r eceiv er are not necessarily the same b y intro ducing the follow ing c hannel mo del Y 1 = H 1 X + N 1 (58) Y 2 = H 2 X + N 2 (59) Z = H Z X + N Z (60) where H 1 , H 2 , H Z are the c hannel mat r ices of sizes r 1 × t, r 2 × t, r Z × t , r esp ectiv ely , and X is of size t × 1. The channel outputs Y 1 , Y 2 , Z are of sizes r 1 × 1 , r 2 × 1 , r Z × 1, resp ective ly . The Ga ussian noise v ectors N 1 , N 2 , N Z are assumed to ha v e identit y co v ariance matrices. Next, w e in tro duce the deﬁnition of the degradedness for the c hannel mo del giv en in (58)-(60) [21]. The Gaussian MIMO m ulti-receiv er wiretap c hannel in (5 8)-( 6 0) is said to b e degraded (according t o the Mark o v ch ain X → Y 1 → Y 2 → Z ) if the following t w o conditions ho ld: • There is a matrix D 21 whic h satisﬁes H 2 = D 21 H 1 and D 21 D ⊤ 21  I , • There is a matrix D Z 2 whic h satisﬁes H Z = D Z 2 H 2 and D Z 2 D ⊤ Z 2  I . In the rest of the pap er, w e consider the c hannel mo del giv en in (51) - (53) instead of the c hannel mo del giv en in (58)- ( 60), whic h is mor e general. How ev er, the inner b ounds, the outer b ounds, and the capacity regions w e obtain fo r the Gaussian MIMO multi-receiv er wiretap channel deﬁned by (51)-(53) can b e extended to pro vide the inner b ounds, the outer b ounds, the capacity regions, resp ectiv ely , for the Gaussian MIMO m ulti-receiv er wiretap c hannel deﬁned by (58)- (60) using the analysis carried out in Section V of [21] and Section 7.1 o f [7]. Thus , fo cusing on the c hannel mo del in (51)-(53) do es not result in an y loss of generalit y . 3.2 Degraded Ch ann els W e ﬁrst provide an inner b ound for the capacit y region o f the degraded Gaussian MIMO m ulti-receiv er wiretap c hannel with public and conﬁden tial messages by using Theorem 1. The corresp onding ac hiev able rate region is stated in the following theorem. 15 Theorem 4 A n a chievable r ate r e gion for the de gr ade d Gaussian MIMO m ulti-r e c eiver wir e- tap chann e l with public and c onﬁdential messages is give n by the unio n of r ate tuples ( R p 1 , R s 1 , R p 2 , R s 2 ) satisfying R s 2 ≤ 1 2 log | S + Σ 2 | | K + Σ 2 | − 1 2 log | S + Σ Z | | K + Σ Z | (61) R s 1 + R s 2 ≤ 1 2 log | S + Σ 2 | | K + Σ 2 | + 1 2 log | K + Σ 1 | | Σ 1 | − 1 2 log | S + Σ Z | | Σ Z | (62) R s 2 + R p 2 ≤ 1 2 log | S + Σ 2 | | K + Σ 2 | (63) R s 1 + R s 2 + R p 2 ≤ 1 2 log | S + Σ 2 | | K + Σ 2 | + 1 2 log | K + Σ 1 | | Σ 1 | − 1 2 log | K + Σ Z | | Σ Z | (64) R s 1 + R s 2 + R p 1 + R p 2 ≤ 1 2 log | S + Σ 2 | | K + Σ 2 | + 1 2 log | K + Σ 1 | | Σ 1 | (65) wher e K is a p ositive sem i-deﬁnite matrix satisfying K  S . This achie v able ra te region giv en in Theorem 4 can b e obtained by ev aluating the ac hiev- able r a te region in T heorem 1 for the degraded G aussian MIMO m ulti-receiv er wiretap c hannel b y using the follow ing selection for U, X : i) U is a zero-mean G aussian random ve c- tor with co v ariance matrix S − K , ii) X = U + U ′ where U ′ is a zero-mean Gaussian random v ector with cov a r iance matrix K , and is indep enden t of U . W e note t ha t b esides this j o in tly Gaussian ( U, X ) selection, there might b e other p o ssible ( U, X ) selections whic h ma y yield a larg er region than the one obtained b y using jo intly Gaussian ( U, X ). How ev er, w e sho w that join tly Gaussian ( U, X ) selec tion is suﬃcien t to ev aluate the ac hiev able ra te region in Theorem 1 for the degra ded Gaussian MIMO multi-receiv er wiretap c hannel. In o t her w ords, join tly G aussian ( U, X ) selection exhausts the achie v able ra t e r egio n in Theorem 1 for the degraded Gaussian MIMO m ulti-receiv er wiretap channel. This suﬃciency result is stated in the follow ing theorem. Theorem 5 F o r the de gr ade d Gaussian MIMO multi-r e c eiv e r wir etap channel , the achiev- able r ate r e gion in The or em 1 is exhauste d by jointly Gaussia n ( U, X ) . In p articular, for an y non-Gaussian ( U, X ) , ther e exists a Gaussian ( U G , X G ) wh ich yields a lar ger r e gion than the one obtaine d by using the non-Gaussian ( U, X ) . Next, w e prov ide an outer b ound for the capacity region of the degraded G aussian MIMO m ulti-receiv er wiretap c hannel. This outer b ound can b e obtained by ev aluating the outer b ound give n in Theorem 2 for the degraded Gaussian MIMO m ulti-receiv er wiretap c hannel. This ev alua tion is ta n tamoun t to ﬁnding the optimal ( U, X ) whic h exhausts the outer b ound in Theorem 2 for the degraded Gaussian MIMO m ulti-receiv er wiretap c hannel. W e sho w that join tly Gaussian ( U, X ) is suﬃcien t to exhaust the outer b ound in Theorem 2 for the degraded G aussian MIMO channel. The correspo nding outer b ound is stated in the following 16 theorem. Theorem 6 The c ap acity r e gion of the de gr ad e d Gau ssia n MIMO multi-r e c eiver wir e tap channel is c ontaine d in the union of r ate tuples ( R p 1 , R s 1 , R p 2 , R s 2 ) satisfying R s 2 ≤ 1 2 log | S + Σ 2 | | K + Σ 2 | − 1 2 log | S + Σ Z | | K + Σ Z | (66) R s 1 + R s 2 ≤ 1 2 log | S + Σ 2 | | K + Σ 2 | + 1 2 log | K + Σ 1 | | Σ 1 | − 1 2 log | S + Σ Z | | Σ Z | (67) R s 2 + R p 2 ≤ 1 2 log | S + Σ 2 | | K + Σ 2 | (68) R s 1 + R s 2 + R p 1 + R p 2 ≤ 1 2 log | S + Σ 2 | | K + Σ 2 | + 1 2 log | K + Σ 1 | | Σ 1 | (69) wher e K is a p ositive sem i-deﬁnite matrix satisfying K  S . The pro ofs o f Theorem 5 and Theorem 6 are giv en in App endix D. W e prov e Theorem 5 and Theorem 6 by using the de Bruijn iden tit y [16, 17], a diﬀeren tial relationship b et w een diﬀeren tial en tropy and the Fisher information matrix, in conjunction with the prop erties of the Fisher information matrix. In particular, to pro ve Theorem 5 , we consider the region in Theorem 1, and sho w that f or a n y non-Gaussian ( U, X ), there exists a Gaussian ( U G , X G ) whic h yields a larger r egio n than the one that is obtained b y ev aluating the region in The- orem 1 with the non-G aussian ( U, X ). W e no t e that this pro of of Theorem 5 implies the pro of of Theorem 6. In particular, since the region in Theorem 1 includes all the constraints in v olv ed in the outer b ound giv en in Theorem 2, the pro of of Theorem 5 r eveals that for an y non-Gaussian ( U, X ), there exists a Gaussian ( U G , X G ) whic h yields a larger regio n than the one that is obtained b y ev aluating the regio n in Theorem 2 with the non-Gaussian ( U, X ). W e no te that the o nly diﬀerence b etw een the inner and the outer b ounds f o r the degraded Gaussian MIMO m ulti-receiv er wiretap g iv en in Theorem 4 and Theorem 6 , resp ectiv ely , comes f rom the b ound in (64). In other w ords, there is o ne more constraint in the inner b ound give n by Theorem 4 than the o uter b ound g iv en b y Theorem 6. This additiona l constrain t is R s 1 + R s 2 + R p 2 ≤ 1 2 log | S + Σ 2 | | K + Σ 2 | + 1 2 log | K + Σ 1 | | Σ 1 | − 1 2 log | K + Σ Z | | Σ Z | (70) Besides this constrain t on R s 1 + R s 2 + R p 2 , the inner b ound in Theorem 4 and the outer b ound in Theorem 6 are the same. W e conclude this section b y prov iding the cases where the inner b ound in Theorem 4 and the outer b ound in Theorem 6 match . W e ﬁrst note that the inner b ound in Theorem 4 and the outer b ound in Theorem 6 matc h when the conﬁden tial message rate of the ﬁr st legitimate user is zero, i.e., R s 1 = 0. The corresp onding capacity regio n is g iven by the follo wing corollary . 17 Corollary 4 The c ap acity r e gi o n of the de g r ade d Gaussian MIMO multi-r e c eive r wir etap channel without the ﬁrst le gitimate user’s c onﬁd ential m essage is g i v en b y the union of r ate tuples ( R p 1 , R p 2 , R s 2 ) satisfying R s 2 ≤ 1 2 log | S + Σ 2 | | K + Σ 2 | − 1 2 log | S + Σ Z | | K + Σ Z | (71) R s 2 + R p 2 ≤ 1 2 log | S + Σ 2 | | K + Σ 2 | (72) R s 2 + R p 1 + R p 2 ≤ 1 2 log | S + Σ 2 | | K + Σ 2 | + 1 2 log | K + Σ 1 | | Σ 1 | (73) wher e K is a p ositive sem i-deﬁnite matrix satisfying K  S . W e note that Corolla r y 4 is the Gaussian MIMO v ersion of Corolla r y 1 whic h obtains the capacit y region of the degraded discrete memoryless multi-receiv er wiretap c hannel without the ﬁr st legitimate user’s conﬁden tial message. Corollary 4 can b e prov ed by setting R s 1 = 0 in b oth Theorem 4 a nd Theorem 6 and eliminating the redundan t b ounds. W e next note that the inner b ound in Theorem 4 and the outer b ound in Theorem 6 matc h when the public message rate of the second legitima t e user is zero, i.e., R p 2 = 0. The corresp onding capacity region is stated in the fo llowing corollary . Corollary 5 The c ap acity r e gi o n of the de g r ade d Gaussian MIMO multi-r e c eive r wir etap channel without the se c ond l e gitimate user’s public message is given by the union of r ate tuples ( R p 1 , R s 1 , R s 2 ) satisfying R s 2 ≤ 1 2 log | S + Σ 2 | | K + Σ 2 | − 1 2 log | S + Σ Z | | K + Σ Z | (74) R s 1 + R s 2 ≤ 1 2 log | S + Σ 2 | | K + Σ 2 | + 1 2 log | K + Σ 1 | | Σ 1 | − 1 2 log | S + Σ Z | | Σ Z | (75) R s 1 + R s 2 + R p 1 ≤ 1 2 log | S + Σ 2 | | K + Σ 2 | + 1 2 log | K + Σ 1 | | Σ 1 | (76) wher e K is a p ositive sem i-deﬁnite matrix satisfying K  S . W e note that Corolla r y 5 is the Gaussian MIMO v ersion of Corolla r y 2 whic h obtains the capacit y region of the degraded discrete memoryless multi-receiv er wiretap c hannel without the second legitimate user’s public message. Corollary 5 can b e prov ed b y setting R p 2 = 0 in b oth Theorem 4 a nd Theorem 6 and eliminating the redundan t b ounds. Corollary 5 also implies tha t the inner b ound in Theorem 4 and the outer b ound in Theorem 6 match fo r the secrecy capacit y region of the degraded Gaussian MIMO m ulti- receiv er wiretap c hannel. In particular, the inner b ound Theorem 4 and the outer b ound in Theorem 6 match if the rates of b oth public mess ages are set to zero, i.e., R p 1 = R p 2 = 0. The secrec y capacit y region of the degraded m ulti-receiv er wiretap c hannel is g iv en b y the 18 follo wing corollary . Corollary 6 ([7, 18]) The se cr e cy c a p acity r e gion of the de gr ade d Gaussian MIMO multi- r e c eiver w ir etap ch a nnel is given by the union of r ate p airs ( R s 1 , R s 2 ) satisfying 3 R s 2 ≤ 1 2 log | S + Σ 2 | | K + Σ 2 | − 1 2 log | S + Σ Z | | K + Σ Z | (77) R s 2 + R s 1 ≤ 1 2 log | S + Σ 2 | | K + Σ 2 | + 1 2 log | K + Σ 1 | | Σ 1 | − 1 2 log | S + Σ Z | | Σ Z | (78) wher e K is a p ositive sem i-deﬁnite matrix satisfying K  S . W e note t ha t in addition to its represen tation in Corollary 6, the secrecy capacit y region of the degraded Gaussian MIMO m ulti-receiv er wiretap c hannel can b e stated in an alternativ e form as t he union of rate pair s ( R s 1 , R s 2 ) satisfying R s 2 ≤ 1 2 log | S + Σ 2 | | K + Σ 2 | − 1 2 log | S + Σ Z | | K + Σ Z | (79) R s 1 ≤ 1 2 log | K + Σ 1 | | Σ 1 | − 1 2 log | K + Σ Z | | Σ Z | (80) where K is p ositiv e semi-deﬁnite matrix satisfying K  S . 3.3 General Ch ann els Here we consider the general, i.e., not ne c es s arily de gr ade d , Gaussian MIMO multi-receiv er wiretap c hannel with public and conﬁdential messages, and pro p ose an inner b ound for the capacit y region of the general Gaussian MIMO m ulti-receiv er wiretap c hannel as follo ws. Theorem 7 A n achievab le r ate r e gion for the gener al Gaussian MIMO multi-r e c eiver w ir e- tap chann e l with p ublic and c onﬁdential mes sages is given by con v ( R 12 ( K 0 , K 1 , K 2 ) ∪ R 21 ( K 0 , K 1 , K 2 )) (81) wher e R 21 ( K 0 , K 1 , K 2 ) is given by the union of r ate tuples ( R p 1 , R s 1 , R p 2 , R s 2 ) satisfying R s 1 ≤ min j =1 , 2 1 2 log | K 0 + K 1 + K 2 + Σ j | | K 1 + K 2 + Σ j | + 1 2 log | K 1 + Σ 1 | | Σ 1 | − 1 2 log | K 0 + K 1 + K 2 + Σ Z | | K 1 + K 2 + Σ Z | − 1 2 log | K 1 + Σ Z | | Σ Z | (82) 3 The secrecy capacity r egion o f the general, not necessarily deg raded, Gaussia n MIMO mu lti-r e ceiv er wiretap channel for an arbitrar y num b er of le g itimate users ca n be found in [7]. 19 R s 2 ≤ min j =1 , 2 1 2 log | K 0 + K 1 + K 2 + Σ j | | K 1 + K 2 + Σ j | + 1 2 log | K 1 + K 2 + Σ 2 | | K 1 + Σ 2 | − 1 2 log | K 0 + K 1 + K 2 + Σ Z | | K 1 + Σ Z | (83) R s 1 + R s 2 ≤ min j =1 , 2 1 2 log | K 0 + K 1 + K 2 + Σ j | | K 1 + K 2 + Σ j | + 1 2 log | K 1 + K 2 + Σ 2 | | K 1 + Σ 2 | + 1 2 log | K 1 + Σ 1 | | Σ 1 | − 1 2 log | K 0 + K 1 + K 2 + Σ Z | | Σ Z | (84) R s 1 + R p 1 ≤ min j =1 , 2 1 2 log | S + Σ j | | K 1 + K 2 + Σ j | + 1 2 log | K 1 + Σ 1 | | Σ 1 | (85) R s 2 + R p 2 ≤ min j =1 , 2 1 2 log | S + Σ j | | K 1 + K 2 + Σ j | + 1 2 log | K 1 + K 2 + Σ 2 | | K 1 + Σ 2 | (86) R s 1 + R p 1 + R s 2 ≤ min j =1 , 2 1 2 log | S + Σ j | | K 1 + K 2 + Σ j | + 1 2 log | K 1 + Σ 1 | | Σ 1 | + 1 2 log | K 1 + K 2 + Σ 2 | | K 1 + Σ 2 | − 1 2 log | K 1 + K 2 + Σ Z | | K 1 + Σ Z | (87) R s 1 + R s 2 + R p 2 ≤ min j =1 , 2 1 2 log | S + Σ j | | K 1 + K 2 + Σ j | + 1 2 log | K 1 + K 2 + Σ 2 | | K 1 + Σ 2 | + 1 2 log | K 1 + Σ 1 | | Σ 1 | − 1 2 log | K 1 + Σ Z | | Σ Z | (88) R s 1 + R p 1 + R s 2 + R p 2 ≤ min j =1 , 2 1 2 log | S + Σ j | | K 1 + K 2 + Σ j | + 1 2 log | K 1 + K 2 + Σ 2 | | K 1 + Σ 2 | + 1 2 log | K 1 + Σ 1 | | Σ 1 | (89) for som e p ositive sem i-deﬁnite matric es K 0 , K 1 , K 2 satisfying K 0 + K 1 + K 2  S . R 12 ( K 0 , K 1 , K 2 ) c an b e o btaine d fr om R 21 ( K 0 , K 1 , K 2 ) by swapping the subscripts 1 and 2. The pro of of Theorem 7 is giv en in App endix E. W e o btain t he achiev able rate region in Theorem 7 b y ev a luating the ac hiev able r ate region giv en in The orem 3 with jointly Gaussian ( Q, U, V 1 , V 2 , X ) having a sp eciﬁc correlatio n structure. In particular, Q, U are selected in a ccordance with sup erp osition co ding, and V 1 , V 2 are enco ded by using dir ty- pap er enco ding [19]. W e note that Theorem 4 is a sp ecial case of Theorem 7. In other w ords, the a c hiev able rate regio n in Theorem 4 can b e o btained from the a chiev able rate region in Theorem 7. T o this end, one needs to consider o nly the region R 21 ( K 0 , K 1 , K 2 ) with K 2 = φ, K 1 = K , S = K 0 + K 1 . After eliminating the r edundan t b ounds fr o m the corresp onding region, one can reco v er the achiev able rate region pro vided in Theorem 4 . 20 4 Conclus ions W e study the m ulti-receiv er wiretap c hannel with public and conﬁdential messages. W e ﬁrst consider the degraded discrete memoryless c hannel. W e provide inner and outer b o unds for t he capacity regio n of the degraded discrete memoryless m ulti-receiv er wiretap c hannel. These inner and outer b ounds partially matc h. Th us, we pro vide a partial characterization of the capacit y region o f the degraded discrete memoryless m ulti-receiv er wiretap ch annel with public and conﬁden tial messages. Second, w e pro vide an inner b o und for the capacit y region of the general, not necessarily degraded, m ulti-receiv er wiretap c hannel. W e obtain the inner b ound for the general case b y using an ac hiev able sc heme that relies on sup erp osition co ding, rate-splitting, binning and Marto n’s co ding. Third, we consider the degraded Gaussian MIMO multi-rece ive r wiretap c hannel. W e show that, to ev aluate t he inner and outer b ounds, that w e already prop osed fo r the degraded discrete memoryless channel, for the Gaussian MIMO case, it is suﬃcien t to consider only the join tly Gaussian auxiliary random v ar ia bles and c hannel input. Consequen tly , since these inner and outer b ounds part ially match, w e obtain a part ia l c haracterization of the capacity region of the degra ded G aussian MIMO m ulti-receiv er wiretap c hannel with public and conﬁdential messages. Finally , w e consider the general, not necess arily degraded, Gaussian MIMO multi-receiv er wiretap c hannel and prop ose an inner b ound fo r the capa city region of the general Gaussian MIMO c hannel. A Pro of of Theorem 2 W e deﬁne the following auxiliary rando m v ar ia bles U i = W s 2 W p 2 Y i − 1 1 Z n i +1 , i = 1 , . . . , n (90) whic h satisfy the follow ing Mark ov chains U i → X i → Y 1 i → Y 2 i → Z i , i = 1 , . . . , n (91) due to the fact t ha t the channel is degraded and memoryless. F or any ( n, 2 nR p 1 , 2 nR s 1 , 2 nR p 2 , 2 nR s 2 ) co de achiev ing the rate t uple ( R p 1 , R s 1 , R p 2 , R s 2 ), due to F ano ’s lemma, we hav e H ( W s 2 , W p 2 | Y n 2 ) ≤ nǫ n (92) H ( W s 1 , W p 1 | W s 2 , W p 2 , Y n 1 ) ≤ nǫ n (93) where ǫ n → 0 as n → ∞ . Moreov er, due to the p erfect secrecy r equiremen t in (1), we ha v e I ( W s 1 , W s 2 ; Z n ) ≤ nγ n (94) 21 where γ n → 0 as n → ∞ . Next, w e obtain t he following H ( W s 1 , W s 2 ) ≤ H ( W s 1 , W s 2 | Z n ) + nγ n (95) ≤ H ( W s 1 , W s 2 , W p 1 , W p 2 | Z n ) + nγ n (96) where (95) comes fro m (9 4). Similarly , w e obtain the follow ing H ( W s 2 ) ≤ H ( W s 2 | Z n ) + nγ n (97) ≤ H ( W s 2 , W p 2 | Z n ) + nγ n (98) where (97) comes from ( 9 4). Next, we in tro duce the follow ing lemmas which will b e used frequen tly . Lemma 3 ([2, Lemma 7]) n X i =1 I ( T n 1 ,i +1 ; T 2 i | Q, T i − 1 2 ) = n X i =1 I ( T i − 1 2 ; T 1 i | Q, T n 1 ,i +1 ) (99) Lemma 4 I ( W ; T n 1 | Q ) − ( W ; T n 2 | Q ) = n X i =1 I ( W ; T 1 i | Q, T i − 1 1 , T n 2 ,i +1 ) − I ( W ; T 2 i | Q, T i − 1 1 , T n 2 ,i +1 ) (100) Lemma 4 can b e prov ed by using Lemma 3. 22 First, we o btain an outer b ound for the second legitimate user’s conﬁden tial message rate R s 2 as follows nR s 2 = H ( W s 2 ) (101) ≤ H ( W s 2 , W p 2 | Z n ) + nγ n (102) = H ( W s 2 , W p 2 ) − I ( W s 2 , W p 2 ; Z n ) + nγ n (103) ≤ I ( W s 2 , W p 2 ; Y n 2 ) − I ( W s 2 , W p 2 ; Z n ) + n ( ǫ n + γ n ) (104) = n X i =1 I ( W s 2 , W p 2 ; Y 2 i | Y i − 1 2 , Z n i +1 ) − I ( W s 2 , W p 2 ; Z i | Y i − 1 2 , Z n i +1 ) + n ( ǫ n + γ n ) (105) = n X i =1 I ( W s 2 , W p 2 ; Y 2 i | Y i − 1 2 , Z n i +1 , Z i ) + n ( γ n + ǫ n ) (106) ≤ n X i =1 I ( W s 2 , W p 2 , Y i − 1 1 , Y i − 1 2 , Z n i +1 ; Y 2 i | Z i ) + n ( γ n + ǫ n ) (107) = n X i =1 I ( W s 2 , W p 2 , Y i − 1 1 , Z n i +1 ; Y 2 i | Z i ) + n ( γ n + ǫ n ) (108) = n X i =1 I ( U i ; Y 2 i | Z i ) + n ( γ n + ǫ n ) (109) = n X i =1 I ( U i ; Y 2 i ) − I ( U i ; Z i ) + n ( γ n + ǫ n ) (110) where (102) comes from (98), (105) is due to Lemma 4. The equalities in (10 6), (108), and (110) come from the f ollo wing Marko v chains W s 2 , W p 2 , Y i − 1 2 , Z n i +1 → Y 2 i → Z i (111) W s 2 , W p 2 , Z n i , Y 2 i → Y i − 1 1 → Y i − 1 2 (112) U i → Y 2 i → Z i (113) whic h f ollo w from the fact that the c hannel is degraded and memoryless. 23 Next, w e obtain an outer b ound for the conﬁdential message sum rate R s 1 + R s 2 as f ollo ws n ( R s 1 + R s 2 ) = H ( W s 1 , W s 2 ) (114) ≤ H ( W s 1 , W p 1 , W s 2 , W p 2 | Z n ) + nγ n (115) = H ( W s 1 , W p 1 , W s 2 , W p 2 ) − I ( W s 1 , W p 1 , W s 2 , W p 2 ; Z n ) + nγ n (116) ≤ I ( W s 1 , W p 1 ; Y n 1 | W s 2 , W p 2 ) + I ( W s 2 , W p 2 ; Y n 2 ) − I ( W s 1 , W p 1 , W s 2 , W p 2 ; Z n ) + n ( γ n + 2 ǫ n ) (117) = I ( W s 1 , W p 1 ; Y n 1 | W s 2 , W p 2 ) − I ( W s 1 , W p 1 ; Z n | W s 2 , W p 2 ) + I ( W s 2 , W p 2 ; Y n 2 ) − I ( W s 2 , W p 2 ; Z n ) + n ( γ n + 2 ǫ n ) (118) ≤ I ( W s 1 , W p 1 ; Y n 1 | W s 2 , W p 2 ) − I ( W s 1 , W p 1 ; Z n | W s 2 , W p 2 ) + n X i =1 I ( U i ; Y 2 i ) − I ( U i ; Z i ) + n ( γ n + 2 ǫ n ) (119) = n X i =1 I ( W s 1 , W p 1 ; Y 1 i | W s 2 , W p 2 , Y i − 1 1 , Z n i +1 ) − I ( W s 1 , W p 1 ; Z i | W s 2 , W p 2 , Y i − 1 1 , Z n i +1 ) + I ( U i ; Y 2 i ) − I ( U i ; Z i ) + n ( γ n + 2 ǫ n ) (120) = n X i =1 I ( W s 1 , W p 1 ; Y 1 i | U i ) − I ( W s 1 , W p 1 ; Z i | U i ) + I ( U i ; Y 2 i ) − I ( U i ; Z i ) + n ( γ n + 2 ǫ n ) (121) = n X i =1 I ( W s 1 , W p 1 ; Y 1 i | U i , Z i ) + I ( U i ; Y 2 i ) − I ( U i ; Z i ) + n ( γ n + 2 ǫ n ) (122) ≤ n X i =1 I ( W s 1 , W p 1 , X i ; Y 1 i | U i , Z i ) + I ( U i ; Y 2 i ) − I ( U i ; Z i ) + n ( γ n + 2 ǫ n ) (123) = n X i =1 I ( X i ; Y 1 i | U i , Z i ) + I ( U i ; Y 2 i ) − I ( U i ; Z i ) + n ( γ n + 2 ǫ n ) (124) = n X i =1 I ( X i ; Y 1 i | U i ) − I ( X i ; Z i | U i ) + I ( U i ; Y 2 i ) − I ( U i ; Z i ) + n ( γ n + 2 ǫ n ) (125) = n X i =1 I ( X i ; Y 1 i | U i ) + I ( U i ; Y 2 i ) − I ( X i ; Z i ) + n ( γ n + 2 ǫ n ) (126) where (115) comes from (9 6), (119) is due to (110), (120) comes fr o m Lemma 4, (122 ), (124), (125) and ( 1 26) are due to the follow ing Mark ov c hain W s 1 , W p 1 , U i → X i → Y 1 i → Z i (127) whic h is a consequence of the fact that the c hannel is memoryless and degraded. 24 Next, w e obtain an outer b ound for the sum ra t e of the second legitimate user’s public and conﬁden tial messages R p 2 + R s 2 as follows n ( R p 2 + R s 2 ) = H ( W p 2 , W s 2 ) (128) ≤ I ( W s 2 , W p 2 ; Y n 2 ) + nǫ n (129) = n X i =1 I ( W s 2 , W p 2 ; Y 2 i | Y i − 1 2 ) + nǫ n (130) = n X i =1 I ( W s 2 , W p 2 , Y i − 1 1 , Z n i +1 ; Y 2 i | Y i − 1 2 ) − I ( Y i − 1 1 , Z n i +1 ; Y 2 i | W s 2 , W p 2 , Y i − 1 2 ) + nǫ n (131) ≤ n X i =1 I ( W s 2 , W p 2 , Y i − 1 1 , Y i − 1 2 , Z n i +1 ; Y 2 i ) − I ( Y i − 1 1 , Z n i +1 ; Y 2 i | W s 2 , W p 2 , Y i − 1 2 ) + nǫ n (132) = n X i =1 I ( W s 2 , W p 2 , Y i − 1 1 , Z n i +1 ; Y 2 i ) − I ( Y i − 1 1 , Z n i +1 ; Y 2 i | W s 2 , W p 2 , Y i − 1 2 ) + nǫ n (133) = n X i =1 I ( U i ; Y 2 i ) − I ( Y i − 1 1 , Z n i +1 ; Y 2 i | W s 2 , W p 2 , Y i − 1 2 ) + nǫ n (134) ≤ n X i =1 I ( U i ; Y 2 i ) + nǫ n (135) where (133) comes from the fo llo wing Marko v c hain W s 2 , W p 2 , Z n i +1 , Y 2 i → Y i − 1 1 → Y i − 1 2 (136) whic h is a consequence of the fact that the c hannel is memoryless and degraded. 25 Finally , we obtain an outer b ound for the sum r a te R p 1 + R s 1 + R p 2 + R s 2 as follows n ( R p 1 + R s 1 + R p 2 + R s 2 ) = H ( W p 1 , W s 1 , W p 2 , W s 2 ) (137) = H ( W p 2 , W s 2 ) + H ( W p 1 , W s 1 | W p 2 , W s 2 ) (138) ≤ n X i =1 I ( U i ; Y 2 i ) − I ( Y i − 1 1 , Z n i +1 ; Y 2 i | Y i − 1 2 , W s 2 , W p 2 ) + nǫ n + H ( W p 1 , W s 1 | W p 2 , W s 2 ) (139) ≤ n X i =1 I ( U i ; Y 2 i ) − I ( Y i − 1 1 , Z n i +1 ; Y 2 i | Y i − 1 2 , W s 2 , W p 2 ) + I ( W p 1 , W s 1 ; Y n 1 | W p 2 , W s 2 ) + 2 ǫ n (140) = n X i =1 I ( U i ; Y 2 i ) − I ( Y i − 1 1 , Z n i +1 ; Y 2 i | Y i − 1 2 , W s 2 , W p 2 ) + I ( W p 1 , W s 1 ; Y 1 i | W p 2 , W s 2 , Y i − 1 1 ) + 2 nǫ n (141) ≤ n X i =1 I ( U i ; Y 2 i ) − I ( Y i − 1 1 , Z n i +1 ; Y 2 i | Y i − 1 2 , W s 2 , W p 2 ) + I ( W p 1 , W s 1 , Z n i +1 ; Y 1 i | W p 2 , W s 2 , Y i − 1 1 ) + 2 nǫ n (142) = n X i =1 I ( U i ; Y 2 i ) − I ( Y i − 1 1 , Z n i +1 ; Y 2 i | Y i − 1 2 , W s 2 , W p 2 ) + I ( Z n i +1 ; Y 1 i | W p 2 , W s 2 , Y i − 1 1 ) + I ( W p 1 , W s 1 ; Y 1 i | W p 2 , W s 2 , Y i − 1 1 , Z n i +1 ) + 2 nǫ n (143) = n X i =1 I ( U i ; Y 2 i ) − I ( Y i − 1 1 , Z n i +1 ; Y 2 i | Y i − 1 2 , W s 2 , W p 2 ) + I ( Z n i +1 ; Y 1 i | W p 2 , W s 2 , Y i − 1 1 ) + I ( W p 1 , W s 1 ; Y 1 i | U i ) + 2 nǫ n (144) = n X i =1 I ( U i ; Y 2 i ) − I ( Z n i +1 ; Y 2 i | Y i − 1 2 , W s 2 , W p 2 ) − I ( Y i − 1 1 ; Y 2 i | Y i − 1 2 , W s 2 , W p 2 , Z n i +1 ) + I ( Z n i +1 ; Y 1 i | W p 2 , W s 2 , Y i − 1 1 ) + I ( W p 1 , W s 1 ; Y 1 i | U i ) + 2 nǫ n (145) = n X i =1 I ( U i ; Y 2 i ) − I ( Y i − 1 2 ; Z i | W s 2 , W p 2 , Z n i +1 ) − I ( Y i − 1 1 ; Y 2 i | Y i − 1 2 , W s 2 , W p 2 , Z n i +1 ) + I ( Y i − 1 1 ; Z i | W p 2 , W s 2 , Z n i +1 ) + I ( W p 1 , W s 1 ; Y 1 i | U i ) + 2 nǫ n (146) = n X i =1 I ( U i ; Y 2 i ) − I ( Y i − 1 2 ; Z i | W s 2 , W p 2 , Z n i +1 ) − I ( Y i − 1 1 ; Y 2 i | Y i − 1 2 , W s 2 , W p 2 , Z n i +1 ) + I ( Y i − 1 2 , Y i − 1 1 ; Z i | W p 2 , W s 2 , Z n i +1 ) + I ( W p 1 , W s 1 ; Y 1 i | U i ) + 2 nǫ n (147) 26 = n X i =1 I ( U i ; Y 2 i ) − I ( Y i − 1 1 ; Y 2 i | Y i − 1 2 , W s 2 , W p 2 , Z n i +1 ) + I ( Y i − 1 1 ; Z i | W p 2 , W s 2 , Z n i +1 , Y i − 1 2 ) + I ( W p 1 , W s 1 ; Y 1 i | U i ) + 2 nǫ n (148) = n X i =1 I ( U i ; Y 2 i ) − I ( Y i − 1 1 ; Y 2 i , Z i | Y i − 1 2 , W s 2 , W p 2 , Z n i +1 ) + I ( Y i − 1 1 ; Z i | W p 2 , W s 2 , Z n i +1 , Y i − 1 2 ) + I ( W p 1 , W s 1 ; Y 1 i | U i ) + 2 nǫ n (149) = n X i =1 I ( U i ; Y 2 i ) − I ( Y i − 1 1 ; Y 2 i | Y i − 1 2 , W s 2 , W p 2 , Z n i +1 , Z i ) + I ( W p 1 , W s 1 ; Y 1 i | U i ) + 2 nǫ n (150) ≤ n X i =1 I ( U i ; Y 2 i ) + I ( W p 1 , W s 1 ; Y 1 i | U i ) + 2 nǫ n (151) ≤ n X i =1 I ( U i ; Y 2 i ) + I ( W p 1 , W s 1 , X i ; Y 1 i | U i ) + 2 nǫ n (152) = n X i =1 I ( U i ; Y 2 i ) + I ( X i ; Y 1 i | U i ) + 2 nǫ n (153) where (13 9) comes from (134), (146) is obtained by using Lemma 3, (147) is due to the follo wing Mark o v c hain Y i − 1 2 → Y i − 1 1 → W p 2 , W s 2 , Z n i +1 , Z i (154) whic h comes from the fact that the channel is degraded and memoryless, (149) is a conse- quence of the following Marko v chain W s 2 W p 2 Y i − 1 1 Y i − 1 2 Z n i +1 → Y 2 i → Z i (155) whic h also comes from the f act that the c hannel is degraded and memoryless , and (153) is due to the follow ing Mark ov c hain W s 1 W p 1 U i → X i → Y 1 i (156) whic h is a consequence of the fa ct that the c hannel is memoryless. The b ounds in (110), (126), (13 5) and (153) can b e single-letterized yielding the b ounds giv en in Theorem 2. B Pro of of Theorem 3 Here, w e ﬁrst consider a more general scenario than t he scenario intro duced in Section 2.3 . In t his more general scenario, the transmitter sends a pair of common public and conﬁdential messages to t he legitimate users in addition to a pair of public and conﬁden tial messages in tended to each legitimate user. Thus, in this case, the transmitter has the message tu- 27 ple ( W p 0 , W s 0 , W p 1 , W s 1 , W p 2 , W s 2 ), where the common public message W p 0 and the common conﬁden tial message W s 0 are sen t to b oth legitimate users, a nd a pair of public and conﬁ- den tial messages ( W pj , W sj ) are sen t to the j th legitimate user, j = 1 , 2 4 . Here also, there is no secrecy concern on the public messages W p 0 , W p 1 , W p 2 while the conﬁden tial messages W s 0 , W s 1 , W s 2 need to b e transmitted in p erfect secrecy , i.e., they need to satisfy the follo wing lim n →∞ 1 n I ( W s 0 , W s 1 , W s 2 ; Z n ) = 0 (157) The deﬁnition of a co de, achiev able rates, and t he capa city r egio n for this more general scenario can b e giv en similar to the deﬁnitions of a co de, ac hiev able r a tes, and the capacity region w e provid ed in Section 2. Here, w e prov ide the pro o f of Theorem 3 in tw o steps. In t he ﬁrst step, w e prov e an a c hiev able rate region for the more general scenario w e j ust in tro duced. In the second step, w e obta in the ac hiev a ble r a te region in Theorem 3 from the ach iev a ble ra t e r egion prov ed in the ﬁrst step b y using F ourier-Motzkin elimination in conjunction with the fact that since the common public and conﬁden tial messages W p 0 , W s 0 are deco ded b y b oth users, they can b e con v erted into public a nd conﬁden tial messages ( W p 1 , W s 1 , W p 2 , W s 2 ) of the legitimate users. B.1 P art I W e ﬁx t he join t distribution p ( q , u, v 1 , v 2 , x, y 1 , y 2 , z ) as follo ws p ( q , u ) p ( v 1 , v 2 , x | u ) p ( y 1 , y 2 , z | x ) (158) Next, we divide the common public message rate R p 0 in to t w o parts a s follows . R p 0 = ˜ R p 0 + ˜ ˜ R p 0 (159) In other words, w e divide the common public message W p 0 in to tw o parts as W p 0 = ( ˜ W p 0 , ˜ ˜ W p 0 ), where the rate of ˜ W p 0 is ˜ R p 0 , and the rate of ˜ ˜ W p 0 is ˜ ˜ R p 0 . W e use rate-splitting fo r the common public message b ecause due to [2], w e kno w tha t ra t e-splitting might enhance the achie v a ble public and conﬁden tial message rat e pairs ev en for the single legitimate user case. Co deb o ok generation: • Generate 2 n ˜ R p 0 length- n sequences q n through p ( q n ) = Q n i =1 p ( q i ), and index them as q n ( ˜ w p 0 ), where ˜ w p 0 ∈ { 1 , . . . , 2 n ˜ R p 0 } . • F or eac h q n ( ˜ w p 0 ) sequence, g enerate 2 n ( ˜ ˜ R p 0 + R s 0 +∆ 0 ) length- n sequences u n through 4 Another w ay to obtain the ac hiev a ble rate region in Theo rem 3 is to use rate- splitting for W p 1 , W s 1 , W p 2 , W s 2 as we mentioned ear lie r in Section 2.3. How ever, we chose to intro duce a pa ir of common public and conﬁdential messages here, b ecause the corres ponding scenario results in an achiev able scheme that encompa sses the one we can obtain by us ing rate-splitting. 28 p ( u n | q n ) = Q n i =1 p ( u i | q i ), and index them as u n ( ˜ w p 0 , ˜ ˜ w p 0 , w s 0 , d 0 ), where ˜ ˜ w p 0 ∈ { 1 , . . . , 2 n ˜ ˜ R p 0 } , w s 0 ∈ { 1 , . . . , 2 nR s 0 } , d 0 ∈ { 1 , . . . , 2 n ∆ 0 } . • F or eac h u n ( ˜ w p 0 , ˜ ˜ w p 0 , w s 0 , d 0 ) sequence, generate 2 n ( R p 1 + R s 1 +∆ 1 + L 1 ) length- n sequences v n 1 through p ( v n 1 | u n ) = Q n i =1 p ( v 1 i | u i ), and index them as v n 1 ( ˜ w p 0 , ˜ ˜ w p 0 , w s 0 , d 0 , w p 1 , w s 1 , d 1 , l 1 ), where w p 1 ∈ { 1 , . . . , 2 nR p 1 } , w s 1 ∈ { 1 , . . . , 2 nR s 1 } , d 1 ∈ { 1 , . . . , 2 n ∆ 1 } , l 1 ∈ { 1 , . . . , 2 nL 1 } . • F or eac h u n ( ˜ w p 0 , ˜ ˜ w p 0 , w s 0 , d 0 ) sequence, generate 2 n ( R p 2 + R s 2 +∆ 2 + L 2 ) length- n sequences v n 2 through p ( v n 2 | u n ) = Q n i =1 p ( v 2 i | u i ), and index them as v n 2 ( ˜ w p 0 , ˜ ˜ w p 0 , w s 0 , d 0 , w p 2 , w s 2 , d 2 , l 2 ), where w p 2 ∈ { 1 , . . . , 2 nR p 2 } , w s 2 ∈ { 1 , . . . , 2 nR s 2 } , d 2 ∈ { 1 , . . . , 2 n ∆ 2 } , l 2 ∈ { 1 , . . . , 2 nL 2 } . Enco ding: Assume W p 0 = w p 0 , W s 0 = w s 0 , W p 1 = w p 1 , W s 1 = w s 1 , W p 2 = w p 2 , W s 2 = w s 2 is the message to b e transmitted. Randomly pic k d 0 , d 1 , d 2 . Next, w e ﬁnd an ( l 1 , l 2 ) pair suc h t ha t the corr esp onding sequence tuple ( q n , u n , v n 1 , v n 2 ) is join tly ty pical. Due t o m utual co v ering lemma [2 2 ], if L 1 , L 2 satisfy L 1 + L 2 ≥ I ( V 1 ; V 2 | U ) (160) with high probabilit y , there exists at least one pair ( l 1 , l 2 ) suc h t hat the corr esp onding sequence tuple ( q n , u n , v n 1 , v n 2 ) is join tly t ypical. Deco ding: • The ﬁrst legitimate user deco des ( w p 0 , w s 0 , d 0 , w p 1 , w s 1 , d 1 ) in tw o steps. In the ﬁrst step, it deco des ( w p 0 , w s 0 , d 0 ) by lo oking f o r the unique ( q n , u n ) pair suc h that ( q n , u n , y n 1 ) is join tly t ypical. If the following conditio ns ar e satisﬁed, R p 0 + R s 0 + ∆ 0 ≤ I ( U ; Y 1 ) (161) ˜ ˜ R p 0 + R s 0 + ∆ 0 ≤ I ( U ; Y 1 | Q ) (162) the ﬁrst legitimate user can deco de ( w p 0 , w s 0 , d 0 ) with v anishingly small probability of error. In the second step, it deco des ( w s 1 , w p 1 , d 1 ) b y lo oking for the unique ( q n , u n , v n 1 ) tuple suc h that ( q n , u n , v n 1 , y n 1 ) is jointly typical. Assuming tha t ( w p 0 , w s 0 , d 0 ) is deco ded correctly in t he ﬁrst step, if the following conditio n is satisﬁed, R p 1 + R s 1 + ∆ 1 + L 1 ≤ I ( V 1 ; Y 1 | U ) (163) the ﬁrst legitimate user can deco de ( w p 1 , w s 1 , d 1 ) with v anishingly small probability of error. 29 • Similarly , the second legitimate user can deco de ( w p 0 , w s 0 , d 0 , w p 2 , w s 2 , d 2 ) with v anish- ingly small probabilit y of error if t he following conditions are satisﬁed. R p 0 + R s 0 + ∆ 0 ≤ I ( U ; Y 2 ) (1 64) ˜ ˜ R p 0 + R s 0 + ∆ 0 ≤ I ( U ; Y 2 | Q ) (165) R p 2 + R s 2 + ∆ 2 + L 2 ≤ I ( V 2 ; Y 2 | U ) (166) Equiv o cation computation: W e no w sho w that the prop osed co ding sc heme satisﬁes the p er- fect secrecy requiremen t on the conﬁdential messages giv en by (157). W e start as follo ws. H ( W s 0 , W s 1 , W s 2 | Z n ) ≥ H ( W s 0 , W s 1 , W s 2 | Z n , Q n ) (167) = H ( W s 0 , W s 1 , W s 2 , ˜ ˜ W p 0 , W p 1 , W p 2 , D 0 , D 1 , D 2 | Z n , Q n ) − H ( ˜ ˜ W p 0 , W p 1 , W p 2 , D 0 , D 1 , D 2 | Z n , Q n , W s 0 , W s 1 , W s 2 ) (168) = H ( W s 0 , W s 1 , W s 2 , ˜ ˜ W p 0 , W p 1 , W p 2 , D 0 , D 1 , D 2 | Q n ) − I ( W s 0 , W s 1 , W s 2 , ˜ ˜ W p 0 , W p 1 , W p 2 , D 0 , D 1 , D 2 ; Z n | Q n ) − H ( ˜ ˜ W p 0 , W p 1 , W p 2 , D 0 , D 1 , D 2 | Z n , Q n , W s 0 , W s 1 , W s 2 ) (169) = H ( W s 0 , W s 1 , W s 2 ) + H ( ˜ ˜ W p 0 , W p 1 , W p 2 , D 0 , D 1 , D 2 ) − I ( W s 0 , W s 1 , W s 2 , ˜ ˜ W p 0 , W p 1 , W p 2 , D 0 , D 1 , D 2 ; Z n | Q n ) − H ( ˜ ˜ W p 0 , W p 1 , W p 2 , D 0 , D 1 , D 2 | Z n , Q n , W s 0 , W s 1 , W s 2 ) (170) = H ( W s 0 , W s 1 , W s 2 ) + n ( ˜ ˜ R p 0 + R p 1 + R p 2 + ∆ 0 + ∆ 1 + ∆ 2 ) − I ( W s 0 , W s 1 , W s 2 , ˜ ˜ W p 0 , W p 1 , W p 2 , D 0 , D 1 , D 2 ; Z n | Q n ) − H ( ˜ ˜ W p 0 , W p 1 , W p 2 , D 0 , D 1 , D 2 | Z n , Q n , W s 0 , W s 1 , W s 2 ) (171) ≥ H ( W s 0 , W s 1 , W s 2 ) + n ( ˜ ˜ R p 0 + R p 1 + R p 2 + ∆ 0 + ∆ 1 + ∆ 2 ) − I ( W s 0 , W s 1 , W s 2 , ˜ ˜ W p 0 , W p 1 , W p 2 , D 0 , D 1 , D 2 , U n , V n 1 , V n 2 ; Z n | Q n ) − H ( ˜ ˜ W p 0 , W p 1 , W p 2 , D 0 , D 1 , D 2 | Z n , Q n , W s 0 , W s 1 , W s 2 ) (172) = H ( W s 0 , W s 1 , W s 2 ) + n ( ˜ ˜ R p 0 + R p 1 + R p 2 + ∆ 0 + ∆ 1 + ∆ 2 ) − I ( U n , V n 1 , V n 2 ; Z n | Q n ) − H ( ˜ ˜ W p 0 , W p 1 , W p 2 , D 0 , D 1 , D 2 | Z n , Q n , W s 0 , W s 1 , W s 2 ) (173) ≥ H ( W s 0 , W s 1 , W s 2 ) + n ( ˜ ˜ R p 0 + R p 1 + R p 2 + ∆ 0 + ∆ 1 + ∆ 2 ) − n ( I ( U, V 1 , V 2 ; Z | Q ) + γ 1 n ) − H ( ˜ ˜ W p 0 , W p 1 , W p 2 , D 0 , D 1 , D 2 | Z n , Q n , W s 0 , W s 1 , W s 2 ) (174) where (170)-(17 1) follo w fro m the facts that the messages W s 0 , W s 1 , W s 2 , ˜ ˜ W p 0 , W p 1 , W p 2 , D 0 , D 1 , 30 D 2 are indep enden t among themselv es, unifor mly distributed, and also ar e indep enden t of Q n , (173) stems f r om the fact that giv en the co dew ords ( Q n , U n , V n 1 , V n 2 ), ( W s 0 , W s 1 , W s 2 , ˜ ˜ W p 0 , W p 1 , W p 2 , D 0 , D 1 , D 2 ) and Z n are indep enden t, (174) comes fro m the fact tha t I ( U n , V n 1 , V n 2 ; Z n | Q n ) ≤ nI ( U, V 1 , V 2 ; Z | Q ) + nγ 1 n (175) where γ 1 n → 0 as n → ∞ . The b ound in (175) can b e sho wn by following the analysis in [23]. Next, w e consider the conditional entrop y term in (174). T o this end, w e introduce the following lemma. Lemma 5 We have H ( W p 1 , W p 2 , D 1 , D 2 | Z n , Q n , W s 0 , W s 1 , W s 2 , ˜ ˜ W p 0 , D 0 ) ≤ nγ 2 n (176) wher e γ 2 n → 0 as n → ∞ , if the fol lowin g c onditions a r e satisﬁe d. R p 1 + ∆ 1 + L 1 + R p 2 + ∆ 2 + L 2 ≤ I ( V 1 , V 2 ; Z | U ) + I ( V 1 ; V 2 | U ) (177) R p 1 + ∆ 1 + L 1 ≤ I ( V 1 ; Z, V 2 | U ) (178) R p 2 + ∆ 2 + L 2 ≤ I ( V 2 ; Z, V 1 | U ) (179) The pro of of Lemma 5 is giv en in App endix C. Using this lemma, w e ha v e the following corollary . Corollary 7 We ha ve H ( ˜ ˜ W p 0 , D 0 | Z n , Q n , W s 0 , W s 1 , W s 2 ) ≤ nγ 3 n (180) wher e γ 3 n → 0 as n → ∞ , if the fol lowin g c ondition is satisﬁe d. ˜ ˜ R p 0 + ∆ 0 ≤ I ( U ; Z | Q ) (181) No w, we set the rates ˜ ˜ R p 0 , ∆ 0 , R p 1 , ∆ 1 , L 1 , R p 2 , ∆ 2 , L 2 as follows. ˜ ˜ R p 0 + ∆ 0 = I ( U ; Z | Q ) − ǫ (182) L 1 + L 2 = I ( V 1 ; V 2 | U ) + ǫ 2 (183) R p 1 + ∆ 1 + R p 2 + ∆ 2 = I ( V 1 , V 2 ; Z | U ) − ǫ (184) R p 1 + ∆ 1 + L 1 < I ( V 1 ; Z, V 2 | U ) (185) R p 2 + ∆ 2 + L 2 < I ( V 2 ; Z, V 1 | U ) (186) In view of Lemma 5 and Corolla ry 7, the selections of ˜ ˜ R p 0 , ∆ 0 , R p 1 , ∆ 1 , L 1 , R p 2 , ∆ 2 , L 2 in 31 (182)-(186) imply that H ( ˜ ˜ W p 0 , W p 1 , W p 2 , D 0 , D 1 , D 2 | Z n , Q n , W s 0 , W s 1 , W s 2 ) = H ( ˜ ˜ W p 0 , D 0 | Z n , Q n , W s 0 , W s 1 , W s 2 ) + H ( W p 1 , W p 2 , D 1 , D 2 | Z n , Q n , W s 0 , W s 1 , W s 2 , ˜ ˜ W p 0 , D 0 ) (187) ≤ n ( γ 2 n + γ 3 n ) (188) using whic h in (174), we get H ( W s 0 , W s 1 , W s 2 | Z n ) ≥ H ( W s 0 , W s 1 , W s 2 ) + n ( ˜ ˜ R p 0 + R p 1 + R p 2 + ∆ 0 + ∆ 1 + ∆ 2 ) − n ( I ( U, V 1 , V 2 ; Z | Q ) + γ 1 n ) − n ( γ 2 n + γ 3 n ) (189) Moreo v er, using (1 82)-(184) in (189), w e ha ve H ( W s 0 , W s 1 , W s 2 | Z n ) ≥ H ( W s 0 , W s 1 , W s 2 ) − n 3 ǫ 2 − n ( γ 1 n + γ 2 n + γ 3 n ) (190) whic h implies tha t the prop osed co ding sc heme satisﬁes t he p erfect secrecy requiremen t on the conﬁden tial messages; completing t he equiv o cation computation. B.2 P art I I W e hav e sho wn that rate tuples ( R p 0 , R s 0 , R p 1 , R s 1 , R p 2 , R s 2 ) satisfying the fo llowing con- strain ts are ac hiev a ble. L 1 + L 2 = I ( V 1 ; V 2 | U ) (191) R p 0 + R s 0 + ∆ 0 ≤ min j =1 , 2 I ( U ; Y j ) (192) ˜ ˜ R p 0 + R s 0 + ∆ 0 ≤ min j =1 , 2 I ( U ; Y j | Q ) (193) R p 1 + R s 1 + ∆ 1 + L 1 ≤ I ( V 1 ; Y 1 | U ) (194) R p 2 + R s 2 + ∆ 2 + L 2 ≤ I ( V 2 ; Y 2 | U ) (195) ˜ ˜ R p 0 + ∆ 0 = I ( U ; Z | Q ) (196) R p 1 + ∆ 1 + R p 2 + ∆ 2 = I ( V 1 , V 2 ; Z | U ) (197) R p 1 + ∆ 1 + L 1 ≤ I ( V 1 ; Z, V 2 | U ) (198) R p 2 + ∆ 2 + L 2 ≤ I ( V 2 ; Z, V 1 | U ) (199) 32 W e eliminate ∆ 0 from the b ounds in (191)-(199) b y using F ourier-Motzkin elimination, whic h leads to the following region. L 1 + L 2 = I ( V 1 ; V 2 | U ) (200) R p 0 + R s 0 + I ( U ; Z | Q ) − ˜ ˜ R p 0 ≤ min j =1 , 2 I ( U ; Y j ) (201) R s 0 ≤ min j =1 , 2 I ( U ; Y j | Q ) − I ( U ; Z | Q ) (202) R p 1 + R s 1 + ∆ 1 + L 1 ≤ I ( V 1 ; Y 1 | U ) (203) R p 2 + R s 2 + ∆ 2 + L 2 ≤ I ( V 2 ; Y 2 | U ) (204) R p 1 + ∆ 1 + R p 2 + ∆ 2 = I ( V 1 , V 2 ; Z | U ) (205) R p 1 + ∆ 1 + L 1 ≤ I ( V 1 ; Z, V 2 | U ) (206) R p 2 + ∆ 2 + L 2 ≤ I ( V 2 ; Z, V 1 | U ) (207) 0 ≤ ˜ ˜ R p 0 (208) ˜ ˜ R p 0 ≤ R p 0 (209) ˜ ˜ R p 0 ≤ I ( U ; Z | Q ) (210) Next, w e eliminate ˜ ˜ R p 0 from the b ounds in (200)-(210) b y using F ourier-Motzkin elimination, whic h leads to the following region. L 1 + L 2 = I ( V 1 ; V 2 | U ) (211) R p 0 + R s 0 ≤ min j =1 , 2 I ( U ; Y j ) (212) R s 0 ≤ min j =1 , 2 I ( U ; Y j | Q ) − I ( U ; Z | Q ) (213) R p 1 + R s 1 + ∆ 1 + L 1 ≤ I ( V 1 ; Y 1 | U ) (214) R p 2 + R s 2 + ∆ 2 + L 2 ≤ I ( V 2 ; Y 2 | U ) (215) R p 1 + ∆ 1 + R p 2 + ∆ 2 = I ( V 1 , V 2 ; Z | U ) (216) R p 1 + ∆ 1 + L 1 ≤ I ( V 1 ; Z, V 2 | U ) (217) R p 2 + ∆ 2 + L 2 ≤ I ( V 2 ; Z, V 1 | U ) (218) (219) 33 Next, w e eliminate L 1 from the b ounds in (211)-(219) by using F our ier- Motzkin elimination, whic h leads to the following region. R p 0 + R s 0 ≤ min j =1 , 2 I ( U ; Y j ) (220) R s 0 ≤ min j =1 , 2 I ( U ; Y j | Q ) − I ( U ; Z | Q ) (221) R p 1 + R s 1 + ∆ 1 + I ( V 1 ; V 2 | U ) − L 2 ≤ I ( V 1 ; Y 1 | U ) (222) R p 2 + R s 2 + ∆ 2 + L 2 ≤ I ( V 2 ; Y 2 | U ) (223) R p 1 + ∆ 1 + R p 2 + ∆ 2 = I ( V 1 , V 2 ; Z | U ) (224) R p 1 + ∆ 1 + I ( V 1 ; V 2 | U ) − L 2 ≤ I ( V 1 ; Z, V 2 | U ) (225) R p 2 + ∆ 2 + L 2 ≤ I ( V 2 ; Z, V 1 | U ) (226) L 2 ≤ I ( V 1 ; V 2 | U ) (227) 0 ≤ L 2 (228) Next, w e eliminate L 2 from the b ounds in (220)-(228) by using F our ier- Motzkin elimination, whic h leads to the following region. R p 0 + R s 0 ≤ min j =1 , 2 I ( U ; Y j ) (229) R s 0 ≤ min j =1 , 2 I ( U ; Y j | Q ) − I ( U ; Z | Q ) (230) R s 1 ≤ I ( V 1 ; Y 1 | U ) − I ( V 1 ; Z | U ) (231) R s 2 ≤ I ( V 2 ; Y 2 | U ) − I ( V 2 ; Z | U ) (232) R s 1 + R s 2 ≤ I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) − I ( V 1 , V 2 ; Z | U ) (233 ) R p 1 + R s 1 + ∆ 1 ≤ I ( V 1 ; Y 1 | U ) (234) R p 2 + R s 2 + ∆ 2 ≤ I ( V 2 ; Y 2 | U ) (235) R p 1 + ∆ 1 ≤ I ( V 1 ; Z, V 2 | U ) (236) R p 2 + ∆ 2 ≤ I ( V 2 ; Z, V 1 | U ) (237) R p 1 + ∆ 1 + R p 2 + ∆ 2 = I ( V 1 , V 2 ; Z | U ) (238) 34 W e eliminate ∆ 1 from the b ounds in (229)-(238) b y using F ourier-Motzkin elimination, whic h leads to the following region. R p 0 + R s 0 ≤ min j =1 , 2 I ( U ; Y j ) (239) R s 0 ≤ min j =1 , 2 I ( U ; Y j | Q ) − I ( U ; Z | Q ) (240) R s 1 ≤ I ( V 1 ; Y 1 | U ) − I ( V 1 ; Z | U ) (241) R s 2 ≤ I ( V 2 ; Y 2 | U ) − I ( V 2 ; Z | U ) (242) R s 1 + R s 2 ≤ I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) − I ( V 1 , V 2 ; Z | U ) (243) R s 1 + I ( V 1 , V 2 ; Z | U ) − R p 2 − ∆ 2 ≤ I ( V 1 ; Y 1 | U ) (244 ) R p 2 + R s 2 + ∆ 2 ≤ I ( V 2 ; Y 2 | U ) (245 ) I ( V 1 , V 2 ; Z | U ) − R p 2 − ∆ 2 ≤ I ( V 1 ; Z, V 2 | U ) (246) R p 2 + ∆ 2 ≤ I ( V 2 ; Z, V 1 | U ) (247) 0 ≤ I ( V 1 , V 2 ; Z | U ) − R p 1 − R p 2 − ∆ 2 (248) 0 ≤ ∆ 2 (249) W e eliminate ∆ 2 from the b ounds in (239)-(249) b y using F ourier-Motzkin elimination, whic h leads to the following region. R p 0 + R s 0 ≤ min j =1 , 2 I ( U ; Y j ) (250) R s 0 ≤ min j =1 , 2 I ( U ; Y j | Q ) − I ( U ; Z | Q ) (251) R s 1 ≤ I ( V 1 ; Y 1 | U ) − I ( V 1 ; Z | U ) (252) R s 2 ≤ I ( V 2 ; Y 2 | U ) − I ( V 2 ; Z | U ) (253) R s 1 + R s 2 ≤ I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) − I ( V 1 , V 2 ; Z | U ) (254) R p 1 + R s 1 ≤ I ( V 1 ; Y 1 | U ) (255) R p 2 + R s 2 ≤ I ( V 2 ; Y 2 | U ) (256) R p 1 ≤ I ( V 1 ; Z, V 2 | U ) (257) R p 2 ≤ I ( V 2 ; Z, V 1 | U ) (258) R p 1 + R p 2 ≤ I ( V 1 , V 2 ; Z | U ) (259) W e note the fact that each legitimate user’s o wn conﬁden tial message rate R sj can b e given up in fav or of its public message rate R pj , j = 1 , 2, i.e., if the rate tuple ( R p 0 , R s 0 , R p 1 , R s 1 , R p 2 , R s 2 ) is ac hiev able, the rate tuple ( R p 0 , R s 0 , R p 1 + α 1 , R s 1 − α 1 , R p 2 + α 2 , R s 2 − α 2 ) is also ac hiev able for any non- negativ e ( α 1 , α 2 ) pairs satisfying α 1 ≤ R s 1 , α 2 ≤ R s 2 . Using this fa ct for t he 35 region given by (250)-(259), we obtain the follow ing region. R p 0 + R s 0 ≤ min j =1 , 2 I ( U ; Y j ) (260) R s 0 ≤ min j =1 , 2 I ( U ; Y j | Q ) − I ( U ; Z | Q ) (261) R s 1 + α 1 ≤ I ( V 1 ; Y 1 | U ) − I ( V 1 ; Z | U ) (262) R s 2 + α 2 ≤ I ( V 2 ; Y 2 | U ) − I ( V 2 ; Z | U ) (263) R s 1 + R s 2 + α 1 + α 2 ≤ I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) − I ( V 1 , V 2 ; Z | U ) (264) R p 1 + R s 1 ≤ I ( V 1 ; Y 1 | U ) (265) R p 2 + R s 2 ≤ I ( V 2 ; Y 2 | U ) (266) R p 1 − α 1 ≤ I ( V 1 ; Z, V 2 | U ) (267) R p 2 − α 2 ≤ I ( V 2 ; Z, V 1 | U ) (268) R p 1 + R p 2 − α 1 − α 2 ≤ I ( V 1 , V 2 ; Z | U ) (269) 0 ≤ α 1 (270) 0 ≤ α 2 (271) α 1 ≤ R p 1 (272) α 2 ≤ R p 2 (273) W e ﬁrst eliminate α 1 from the b ounds in (260)-(27 3) by using F ourier-Motzkin elimination, whic h leads to the following region. R p 0 + R s 0 ≤ min j =1 , 2 I ( U ; Y j ) (274) R s 0 ≤ min j =1 , 2 I ( U ; Y j | Q ) − I ( U ; Z | Q ) (275) R s 1 ≤ I ( V 1 ; Y 1 | U ) − I ( V 1 ; Z | U ) (276) R s 1 + R p 1 + R p 2 − α 2 ≤ I ( V 1 ; Y 1 | U ) − I ( V 1 ; Z | U ) + I ( V 1 , V 2 ; Z | U ) (277) R s 1 + R p 1 + R s 2 + α 2 ≤ I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 2 ; Z | U ) (278) R s 1 + R p 1 + R s 2 + R p 2 ≤ I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) (279) R s 1 + R s 2 + α 2 ≤ I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) − I ( V 1 , V 2 ; Z | U ) (280) R p 2 − α 2 ≤ I ( V 1 , V 2 ; Z | U ) (281) R s 2 + α 2 ≤ I ( V 2 ; Y 2 | U ) − I ( V 2 ; Z | U ) (282) R p 1 + R s 1 ≤ I ( V 1 ; Y 1 | U ) (283) R p 2 + R s 2 ≤ I ( V 2 ; Y 2 | U ) (284) R p 2 − α 2 ≤ I ( V 2 ; Z, V 1 | U ) (285) 0 ≤ α 2 (286) α 2 ≤ R p 2 (287) 36 Next, w e eliminate α 2 from the b ounds in (274 )-(287) by using F ourier-Motzkin elimination, whic h leads to the following region. R p 0 + R s 0 ≤ min j =1 , 2 I ( U ; Y j ) (288) R s 0 ≤ min j =1 , 2 I ( U ; Y j | Q ) − I ( U ; Z | Q ) (289) R s 1 ≤ I ( V 1 ; Y 1 | U ) − I ( V 1 ; Z | U ) (290) R s 2 ≤ I ( V 2 ; Y 2 | U ) − I ( V 2 ; Z | U ) (291) R s 1 + R s 2 ≤ I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) − I ( V 1 , V 2 ; Z | U ) (292) R p 1 + R s 1 ≤ I ( V 1 ; Y 1 | U ) (293) R p 2 + R s 2 ≤ I ( V 2 ; Y 2 | U ) (294) R s 1 + R p 1 + R s 2 ≤ I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 2 ; Z | U ) (295) R s 1 + R s 2 + R p 2 ≤ I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; Z | U ) (296) R s 1 + R p 1 + R s 2 + R p 2 ≤ I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) (297) W e note the fact that since the common public message is decoded by b oth legitimate users, its rate R p 0 can b e giv en up in fav or of eac h legitimate user’s o wn public message rate R pj , j = 1 , 2, i.e., if the ra te tuple ( R s 0 , R p 0 , R s 1 , R p 1 , R s 2 , R p 2 ) is ac hiev able, the rate tuple ( R p 0 − α − β , R s 0 , R p 1 + α, R s 1 , R p 2 + β , R s 2 ) is also ac hiev able for an y non-negat ive ( α , β ) pairs satisfying α + β ≤ R p 0 . Using this fact for the region give n b y (288)-(29 7 ), we obtain 37 the following region. α + β + R s 0 ≤ min j =1 , 2 I ( U ; Y j ) (298) R s 0 ≤ min j =1 , 2 I ( U ; Y j | Q ) − I ( U ; Z | Q ) (299) R s 1 ≤ I ( V 1 ; Y 1 | U ) − I ( V 1 ; Z | U ) (300) R s 2 ≤ I ( V 2 ; Y 2 | U ) − I ( V 2 ; Z | U ) (301) R s 1 + R s 2 ≤ I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) − I ( V 1 , V 2 ; Z | U ) (302) R p 1 − α + R s 1 ≤ I ( V 1 ; Y 1 | U ) (303) R p 2 − β + R s 2 ≤ I ( V 2 ; Y 2 | U ) (304) R s 1 + R p 1 − α + R s 2 ≤ I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 2 ; Z | U ) (305) R s 1 + R s 2 + R p 2 − β ≤ I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; Z | U ) (306) R s 1 + R p 1 − α + R s 2 + R p 2 − β ≤ I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) (307) 0 ≤ α (308) 0 ≤ β (309) α ≤ R p 1 (310) β ≤ R p 2 (311) 38 W e eliminate α from the b ounds in ( 2 98)-(311) b y using F ourier-Motzkin elimination, whic h leads to the following region. β + R s 0 ≤ min j =1 , 2 I ( U ; Y j ) (312) R s 0 ≤ min j =1 , 2 I ( U ; Y j | Q ) − I ( U ; Z | Q ) (313) R s 1 ≤ I ( V 1 ; Y 1 | U ) − I ( V 1 ; Z | U ) (314) R s 2 ≤ I ( V 2 ; Y 2 | U ) − I ( V 2 ; Z | U ) (315) R s 1 + R s 2 ≤ I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) − I ( V 1 , V 2 ; Z | U ) (316) R p 2 − β + R s 2 ≤ I ( V 2 ; Y 2 | U ) (317) R s 1 + R s 2 + R p 2 − β ≤ I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; Z | U ) (318) R s 1 + R s 2 + R p 2 − β ≤ I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) (319) R s 0 + β + R p 1 + R s 1 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) (320) β + R s 0 + R s 1 + R p 1 + R s 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 2 ; Z | U ) (321) R s 0 + R s 1 + R p 1 + R s 2 + R p 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) (322) 0 ≤ β (323) β ≤ R p 2 (324) 39 Next, w e eliminate β from the b ounds in (312)-(324) by using F ourier-Motzkin elimination, whic h leads to the following region. R s 0 ≤ min j =1 , 2 I ( U ; Y j | Q ) − I ( U ; Z | Q ) (325) R s 1 ≤ I ( V 1 ; Y 1 | U ) − I ( V 1 ; Z | U ) (326) R s 2 ≤ I ( V 2 ; Y 2 | U ) − I ( V 2 ; Z | U ) (327) R s 1 + R s 2 ≤ I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) − I ( V 1 , V 2 ; Z | U ) (328) R s 0 + R s 1 + R p 1 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) (329) R s 0 + R s 2 + R p 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 2 ; Y 2 | U ) (330) R s 0 + R s 1 + R p 1 + R s 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 2 ; Z | U ) (331) R s 0 + R s 1 + R s 2 + R p 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; Z | U ) (332) R s 0 + R s 1 + R p 1 + R s 2 + R p 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) (333) W e note the fact that since the common conﬁden tial message is deco ded b y b oth legitimate users, its rate can b e give n up in fa vor of eac h legitimate user’s ow n conﬁden tial message rate R sj , j = 1 , 2 , i.e., if the rate tuple ( R p 0 , R s 0 , R p 1 , R s 1 , R p 2 , R s 2 ) is ac hiev able, t he r a te tuple ( R p 0 , R s 0 − α − β , R p 1 , R s 1 + α, R p 2 , R s 2 + β ) is also ac hiev able f or any non-negativ e ( α, β ) pairs satisfying α + β ≤ R s 0 . Using this fact f or t he region giv en b y (325)-(333), w e 40 obtain the f ollo wing region. α + β ≤ min j =1 , 2 I ( U ; Y j | Q ) − I ( U ; Z | Q ) (334) R s 1 − α ≤ I ( V 1 ; Y 1 | U ) − I ( V 1 ; Z | U ) (335) R s 2 − β ≤ I ( V 2 ; Y 2 | U ) − I ( V 2 ; Z | U ) (336) R s 1 + R s 2 − α − β ≤ I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) − I ( V 1 , V 2 ; Z | U ) (337) β + R s 1 + R p 1 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) (338) α + R s 2 + R p 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 2 ; Y 2 | U ) (339) R s 1 + R p 1 + R s 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 2 ; Z | U ) (340) R s 1 + R s 2 + R p 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; Z | U ) (341) R s 1 + R p 1 + R s 2 + R p 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) (342) 0 ≤ α (343) 0 ≤ β (344) α ≤ R s 1 (345) β ≤ R s 2 (346) 41 W e eliminate α from the b ounds in ( 3 34)-(346) b y using F ourier-Motzkin elimination, whic h leads to the following region. β ≤ min j =1 , 2 I ( U ; Y j | Q ) − I ( U ; Z | Q ) (347) R s 1 + β ≤ min j =1 , 2 I ( U ; Y j | Q ) + I ( V 1 ; Y 1 | U ) − I ( U, V 1 ; Z | Q ) (348) R s 2 − β ≤ I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) − I ( V 1 , V 2 ; Z | U ) (349) R s 2 − β ≤ I ( V 2 ; Y 2 | U ) − I ( V 2 ; Z | U ) (350) R s 1 + R s 2 ≤ min j =1 , 2 I ( U ; Y j | Q ) + I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) − I ( U, V 1 , V 2 ; Z | Q ) (351) R s 2 + R p 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 2 ; Y 2 | U ) (352) β + R s 1 + R p 1 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) (353) R s 1 + R p 1 + R s 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 2 ; Z | U ) (354) R s 1 + R s 2 + R p 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; Z | U ) (355) R s 1 + R p 1 + R s 2 + R p 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) (356) R s 1 + 2 R s 2 + R p 2 − β ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) + 2 I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) − I ( V 1 , V 2 ; Z | U ) (357) 0 ≤ β (358) β ≤ R s 2 (359) 0 ≤ I ( V 1 ; Y 1 | U ) − I ( V 1 ; Z | U ) (360) 42 W e eliminate β from the b ounds in (347)-(360) by using F o urier-Motzkin elimination, whic h leads to the following region. R s 1 ≤ min j =1 , 2 I ( U ; Y j | Q ) + I ( V 1 ; Y 1 | U ) − I ( U, V 1 ; Z | Q ) (361) R s 2 ≤ min j =1 , 2 I ( U ; Y j | Q ) + I ( V 2 ; Y 2 | U ) − I ( U, V 2 ; Z | Q ) (362) R s 1 + R s 2 ≤ min j =1 , 2 I ( U ; Y j | Q ) + I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) − I ( U, V 1 , V 2 ; Z | Q ) (363) R s 1 + R p 1 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) (364) R s 2 + R p 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 2 ; Y 2 | U ) (365) R s 1 + R p 1 + R s 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 2 ; Z | U ) (366) R s 1 + R p 1 + R s 2 ≤ min j =1 , 2 I ( U ; Y j ) + 2 I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) − I ( V 1 , V 2 ; Z | U ) (367) R s 1 + R s 2 + R p 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; Z | U ) (368) R s 1 + R s 2 + R p 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) + 2 I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) − I ( V 1 , V 2 ; Z | U ) (369) R s 1 + R p 1 + R s 2 + R p 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) (370) 0 ≤ min j =1 , 2 I ( U ; Y j | Q ) − I ( U ; Z | Q ) (371) 0 ≤ I ( V 1 ; Y 1 | U ) − I ( V 1 ; Z | U ) (372) 0 ≤ I ( V 2 ; Y 2 | U ) − I ( V 2 ; Z | U ) (373) 0 ≤ I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) − I ( V 1 , V 2 ; Z | U ) (374) Finally , w e note that we can remov e the b ounds in (371)- (374) without enlarg ing the region giv en by (361)-(3 70), whic h will lea v e us with the desired achie v a ble rate region giv en in Theorem 3; completing the pro of. C Pro of of Lemma 5 Assume that the ea v esdropp er tries to deco de W p 1 , D 1 , L 1 , W p 2 , D 2 , L 2 b y using its kno wledge of ( W s 0 , W s 1 , W s 2 , W p 0 ). In par t icular, assume that, giv en ( W s 0 = w s 0 , W s 1 = w s 1 , W s 2 = w s 1 , W p 0 = w p 0 ), the ea v esdropp er tries to deco de W p 1 , D 1 , L 1 , W p 2 , D 2 , L 2 b y lo oking for the unique ( V n 1 , V n 2 ) suc h that ( q n , u n , v n 1 , v n 2 , z n ) is join tly t ypical. There are four p ossible error ev en ts: • E e 0 = { ( q n , u n , v n 1 , v n 2 , z n ) is not jointly t ypical f or the transmitted ( q n , u n , v n 1 , v n 2 ) } , 43 • E e i = { ( W p 1 , D 1 , L 1 ) 6 = (1 , 1 , 1) , ( W p 2 , D 2 , L 2 ) 6 = (1 , 1 , 1), and the corresp onding tuple ( q n , u n , v n 1 , v n 2 , z n ) is join tly t ypical } , • E e ii = { ( W p 1 , D 1 , L 1 ) = (1 , 1 , 1) , ( W p 2 , D 2 , L 2 ) 6 = (1 , 1 , 1), and the corresp onding tuple ( q n , u n , v n 1 , v n 2 , z n ) is join tly t ypical } , • E e iii = { ( W p 1 , D 1 , L 1 ) 6 = (1 , 1 , 1) , ( W p 2 , D 2 , L 2 ) = (1 , 1 , 1), and the corresp onding tuple ( q n , u n , v n 1 , v n 2 , z n ) is join tly t ypical } , Th us, the probability of deco ding error a t the eav esdropp er is giv en by Pr[ E e ] = Pr[ E e 0 ∪ E e i ∪ E e ii ∪ E e iii ] (375) ≤ Pr[ E e 0 ] + Pr[ E e i ] + Pr[ E e ii ] + Pr[ E e iii ] (376) ≤ ǫ 1 n + Pr[ E e i ] + Pr[ E e ii ] + Pr[ E e iii ] (377) where w e ﬁrst use the union b ound, a nd next the fact that Pr[ E e 0 ] ≤ ǫ 1 n for some ǫ 1 n satisfying ǫ 1 n → 0 as n → ∞ , which follows from the prop erties of the join tly typ ical sequences [13]. Next, we consider the ﬁrst term in (377) as follow s Pr[ E e i ] ≤ X ( w p 1 ,d 1 ,l 1 ) 6 =(1 , 1 , 1) ( w p 2 ,d 2 ,l 2 ) 6 =(1 , 1 , 1) Pr[( q n , u n , V n 1 , V n 2 , Z n ) ∈ A n ǫ ] (378) ≤ X ( w p 1 ,d 1 ,l 1 ) 6 =(1 , 1 , 1) ( w p 2 ,d 2 ,l 2 ) 6 =(1 , 1 , 1) X ( v n 1 ,v n 2 ,z n ) ∈A n ǫ p ( v n 1 | u n ) p ( v n 2 | u n ) p ( z n | u n ) (379) ≤ X ( w p 1 ,d 1 ,l 1 ) 6 =(1 , 1 , 1) ( w p 2 ,d 2 ,l 2 ) 6 =(1 , 1 , 1) X ( v n 1 ,v n 2 ,z n ) ∈A n ǫ 2 − n ( H ( V 1 | U ) − γ ǫ ) 2 − n ( H ( V 2 | U ) − γ ǫ ) 2 − n ( H ( Z | U ) − γ ǫ ) (380) = X ( w p 1 ,d 1 ,l 1 ) 6 =(1 , 1 , 1) ( w p 2 ,d 2 ,l 2 ) 6 =(1 , 1 , 1) |A n ǫ | 2 − n ( H ( V 1 | U )+ H ( V 2 | U )+ H ( Z | U ) − 3 γ ǫ ) (381) ≤ X ( w p 1 ,d 1 ,l 1 ) 6 =(1 , 1 , 1) ( w p 2 ,d 2 ,l 2 ) 6 =(1 , 1 , 1) 2 n ( H ( V 1 ,V 2 ,Z | U )+ γ ǫ ) 2 − n ( H ( V 1 | U )+ H ( V 2 | U )+ H ( Z | U ) − 3 γ ǫ ) (382) = X ( w p 1 ,d 1 ,l 1 ) 6 =(1 , 1 , 1) ( w p 2 ,d 2 ,l 2 ) 6 =(1 , 1 , 1) 2 − n ( I ( V 1 ,V 2 ; Z | U )+ I ( V 2 ; V 1 | U ) − 4 γ ǫ ) (383) ≤ 2 n ( R p 1 +∆ 1 + L 1 + R p 2 +∆ 2 + L 2 ) 2 − n ( I ( V 1 ,V 2 ; Z | U )+ I ( V 2 ; V 1 | U ) − 4 γ ǫ ) (384) where A n ǫ denotes the typical set, γ ǫ is a constan t whic h is a function of ǫ , and satisﬁes γ ǫ → 0 as ǫ → 0, (379) comes from the w a y w e generated the sequences ( q n , u n , v n 1 , v n 2 ), (380) stems f rom the prop erties of the ty pical sequences [13], and (382 ) comes fro m the b ounds on the size of the t ypical set A n ǫ [13]. Equation (384) implies that Pr[ E e i ] v anishes as n → ∞ if 44 the following condition is satisﬁed. R p 1 + ∆ 1 + L 1 + R p 2 + ∆ 2 + R p 2 < I ( V 1 , V 2 ; Z | U ) + I ( V 2 ; V 1 | U ) − 4 γ ǫ (385) Next, we consider Pr[ E e ii ] as follow s Pr[ E e ii ] ≤ X ( w p 2 ,d 2 ,l 2 ) 6 =(1 , 1 , 1) Pr[( q n , u n , v n 1 , V n 2 , Z n ) ∈ A n ǫ ] (386) ≤ X ( w p 2 ,d 2 ,l 2 ) 6 =(1 , 1 , 1) X ( v n 2 ,z n ) ∈A n ǫ p ( v n 2 | u n ) p ( z n | u n , v n 1 ) (387) ≤ X ( w p 2 ,d 2 ,l 2 ) 6 =(1 , 1 , 1) X ( v n 2 ,z n ) ∈A n ǫ 2 − n ( H ( V 2 | U ) − γ ǫ ) 2 − n ( H ( Z | U,V 1 ) − γ ǫ ) (388) = X ( w p 2 ,d 2 ,l 2 ) 6 =(1 , 1 , 1) |A n ǫ | 2 − n ( H ( V 2 | U ) − γ ǫ ) 2 − n ( H ( Z | U,V 1 ) − γ ǫ ) (389) ≤ X ( w p 2 ,d 2 ,l 2 ) 6 =(1 , 1 , 1) 2 n ( H ( V 2 ,Z | U,V 1 )+ γ ǫ ) 2 − n ( H ( V 2 | U ) − γ ǫ ) 2 − n ( H ( Z | U,V 1 ) − γ ǫ ) (390) = X ( w p 2 ,d 2 ,l 2 ) 6 =(1 , 1 , 1) 2 − n ( I ( V 2 ; Z,V 1 | U ) − 3 γ ǫ ) (391) ≤ 2 n ( R p 2 +∆ 2 + L 2 ) 2 − n ( I ( V 2 ; Z,V 1 | U ) − 3 γ ǫ ) (392) where (387) comes from the w ay w e generated sequences ( q n , u n , v n 1 , v n 2 ), (38 8) is due to the prop erties o f the t ypical sequences [13], and (390) comes from the b ounds o n the t ypical set A n ǫ [13]. Equation (392) implies that Pr[ E e ii ] → 0 as n → ∞ if the follo wing conditio n is satisﬁed. R p 2 + ∆ 2 + L 2 < I ( V 2 ; Z, V 1 | U ) − 3 γ ǫ (393) Similarly , w e can sho w that Pr[ E e iii ] → 0 as n → ∞ if the fo llo wing condition is satisﬁed. R p 1 + ∆ 1 + L 1 < I ( V 1 ; Z, V 2 | U ) − 3 γ ǫ (394) Th us, w e hav e sho wn tha t if the rat es ( R p 1 , ∆ 1 , L 1 , R p 2 , ∆ 2 , L 2 ) satisfy (385), (393 ), (394), the ea v esdropp er can deco de W p 1 , D 1 , L 1 , W p 2 , D 2 , L 2 b y using its knowledge of ( W s 0 , W s 1 , W s 2 , W p 0 ), i.e., Pr[ E e ] v anishes as n → ∞ . In view of this fact, using F ano’s lemma, we get H ( W p 1 , D 1 , L 1 , W p 2 , D 2 , L 2 | Z n , Q n , W s 0 , W s 1 , W s 2 , W p 0 , D 0 ) ≤ nγ 2 n (395) where γ 2 n → 0 as n → ∞ ; completing the pro of. 45 D Pro ofs of Theorems 5 and 6 D.1 Bac kground W e need some prop erties of the Fisher info rmation and the diﬀeren tial en tropy , whic h are pro vided next. Deﬁnition 1 ([7], Deﬁnition 3) L et ( U , X ) b e an arbitr arily c orr elate d length- n r an dom ve ctor p air with we l l-deﬁ ne d densities. T h e c onditional Fisher information matrix of X given U is deﬁne d as J ( X | U ) = E  ρ ( X | U ) ρ ( X | U ) ⊤  (396) wher e the ex p e ctation is over the joint density f ( u , x ) , and the c on ditional s c or e function ρ ( x | u ) is ρ ( x | u ) = ∇ log f ( x | u ) =  ∂ log f ( x | u ) ∂ x 1 . . . ∂ log f ( x | u ) ∂ x n  ⊤ (397) W e ﬁrst presen t the conditional fo rm of the Cramer-Rao inequalit y , whic h is prov ed in [7]. Lemma 6 ([7], Lemma 13) L et U , X b e arbitr aril y c orr elate d r ando m ve ctors w ith wel l- deﬁne d de nsities. L et the c onditional c ov a rianc e matrix of X b e Co v ( X | U ) ≻ 0 , then we have J ( X | U )  Co v ( X | U ) − 1 (398) which is satisﬁe d with e quality if ( U , X ) is joi n tly Gaussian with c onditional c o v arianc e matrix Cov( X | U ) . The follo wing lemma will b e used in the up coming pro of. The unconditional v ersion of this lemma, i.e., the case T = φ , is prov ed in L emma 6 of [7]. Lemma 7 ([7], Lemma 6) L et T , U , V 1 , V 2 b e r andom ve ctors such that ( T , U ) and ( V 1 , V 2 ) ar e indep endent. Mor e over, let V 1 , V 2 b e Gaussian r andom ve ctors with c ova rianc e matric es Σ 1 , Σ 2 such that 0 ≺ Σ 1  Σ 2 . The n , we have J − 1 ( U + V 2 | T ) − Σ 2  J − 1 ( U + V 1 | T ) − Σ 1 (399) The follow ing lemma address es the c hange of the Fisher informatio n w ith resp ect to conditioning. 46 Lemma 8 ([7], Lemma 17) L et ( V , U , X ) b e length- n r andom ve ctors with wel l-deﬁne d densities. Mor e ove r, assume that the p artial derivatives of f ( u | x , v ) with r esp e ct to x i , i = 1 , . . . , n exist and satisfy max 1 ≤ i ≤ n     ∂ f ( u | v , x ) ∂ x i     ≤ g ( u ) (400) for some inte gr able function g ( u ) . Then, if ( U , V , X ) satisfy the Markov chain U → V → X , we have J ( X | V )  J ( X | U ) (401) The following lemma will also b e used in the up coming pro of. Lemma 9 ([7], Lemma 8) L et K 1 , K 2 b e p ositive semi-deﬁnite matric es satisfying 0  K 1  K 2 , and f ( K ) b e a matrix-val ue d function such that f ( K )  0 for K 1  K  K 2 . Mor e over, f ( K ) is as sume d to b e gr adient of some sc alar ﬁeld. The n , we have Z K 2 K 1 f ( K ) d K ≥ 0 (402) The follow ing generalization of t he de Bruijn identit y [16, 24 ] is due to [17], where the unconditional form of this identit y , i.e., U = φ , is prov ed. Its generalization to this condi- tional f o rm for an arbitrary U is rather straightforw ard, and is given in Lemma 16 of [7]. Lemma 10 ([7], Lemma 16) L et ( U , X ) b e an arbi tr arily c o rr elate d r andom ve c tor p air with ﬁnite se c ond or der moments, and a lso b e indep endent of the r andom ve ctor N which is zer o-me an Gaussian w i th c ovari a nc e matrix Σ N ≻ 0 . Then, we h a ve ∇ Σ N h ( X + N | U ) = 1 2 J ( X + N | U ) (4 03) The fo llowing lemma is due to [25, 26] whic h low er b ounds the diﬀeren tial entrop y in terms of the Fisher information matrix. Lemma 11 ([25, 26]) L et ( U, X ) b e an ( n + 1) -dimensional r an dom ve ctor, w her e the c on- ditional Fisher i n formation matrix of X , c ondi tione d on U , exis ts. Then, we have h ( X | U ) ≥ 1 2 log | (2 π e ) J − 1 ( X | U ) | (404) W e also need t he following fact ab out strictly p ositiv e deﬁnite matrices. Lemma 12 L et A , B b e two p osi tive deﬁnite matric es, i.e ., A ≻ 0 , B ≻ 0 . If A  B , we have A − 1  B − 1 . 47 D.2 Pro ofs First, w e pr ov e Theorem 5 by sho wing that for a ny ( U, V , X ), there exists a Ga ussian ( U G , V G , X G ) whic h prov ides a larger region. Essen tially , this pro of will also yield a pro o f for Theorem 6 b ecause the outer b ound in Theorem 2 is deﬁned by the same inequalities that deﬁne the inner b ound given in Theorem 1 except the inequality in (1 1). Thus , we only pro vide the pro of of Theorem 5. First step: W e consider the b ound on R s 2 giv en in (8) as follows R s 2 ≤ I ( U ; Y 2 ) − I ( U ; Z ) (405) = [ h ( Y 2 ) − h ( Z )] + [ h ( Z | U ) − h ( Y 2 | U )] (406) First, we consider the ﬁrst term in (406) as follows h ( Z ) − h ( Y 2 ) = 1 2 Z Σ Z Σ 2 J ( X + N ) d Σ N (407) whic h fo llo ws from Lemma 10, and N is a Ga ussian random v ector with co v aria nce matrix Σ N satisfying Σ 2  Σ N  Σ Z . W e note t ha t due to Lemma 6, w e hav e J ( X + N )  (Cov( X ) + Σ N ) − 1  ( S + Σ N ) − 1 (408) where the second inequalit y comes from Lemma 12 and the fact that E  XX ⊤   S . Using this inequalit y in (407), we get h ( Z ) − h ( Y 2 ) ≥ 1 2 Z Σ Z Σ 2 ( S + Σ N ) − 1 d Σ N (409) = 1 2 log | S + Σ Z | | S + Σ 2 | (410) where (409) comes from Lemma 9. Next, we consider the second term in (406) as fo llo ws h ( Z | U ) − h ( Y 2 | U ) = 1 2 Z Σ Z Σ 2 J ( X + N | U ) d Σ N (411) whic h fo llo ws from Lemma 10, and N is a Ga ussian random v ector with co v aria nce matrix Σ N satisfying Σ 2  Σ N  Σ Z . Using Lemma 7, for any Σ N satisfying Σ 2  Σ N  Σ Z , w e ha v e J − 1 ( X + N 2 | U ) − Σ 2  J − 1 ( X + N | U ) − Σ N  J − 1 ( X + N Z | U ) − Σ Z (412) 48 Due to Lemma 12, the inequalities in (412) imply  J − 1 ( X + N Z | U ) − Σ Z + Σ N  − 1  J ( X + N | U )   J − 1 ( X + N 2 | U ) − Σ 2 + Σ N  − 1 (413) Using these inequalities in (411) in conjunction with Lemma 9, we get 1 2 log | J − 1 ( X + N Z | U ) | | J − 1 ( X + N Z ) − Σ Z + Σ 2 | ≤ h ( Z | U ) − h ( Y 2 | U ) ≤ 1 2 log | J − 1 ( X + N 2 | U ) − Σ 2 + Σ Z | | J − 1 ( X + N 2 ) | (414) whic h can b e expressed as f (0) ≤ h ( Z | U ) − h ( Y 2 | U ) ≤ f (1) (415) where f ( t ) is deﬁned as f ( t ) = 1 2 log | K 1 ( t ) + Σ Z | | K 1 ( t ) + Σ 2 | , 0 ≤ t ≤ 1 (4 16) and K 1 ( t ) is giv en by K 1 ( t ) = ( 1 − t )  J − 1 ( X + N Z | U ) − Σ Z  + t  J − 1 ( X + N 2 | U ) − Σ 2  (417) Since f ( t ) is con tinuous in t , due to the in termediate v alue theorem, there exists a t ∗ 1 suc h that 0 ≤ t ∗ 1 ≤ 1, and f ( t ∗ 1 ) = h ( Z | U ) − h ( Y 2 | U ) (418) = 1 2 log | K 1 + Σ Z | | K 1 + Σ 2 | (419) where K 1 = K 1 ( t ∗ 1 ). Since 0 ≤ t ∗ 1 ≤ 1, K 1 satisﬁes J − 1 ( X + N 2 | U ) − Σ 2  K 1  J − 1 ( X + N Z | U ) − Σ Z (420) in view of (417). Moreo v er, w e hav e K 1  J − 1 ( X + N Z | U ) − Σ Z (421)  J − 1 ( X + N Z ) − Σ Z (422)  Cov( X + N Z ) − Σ Z (423)  Cov( X ) (424)  S (425 ) where (42 2) is due to Lemma 8 and ( 4 23) comes from Lemma 6. Th us, in view of (42 0) and 49 (425), K 1 satisﬁes J − 1 ( X + N 2 | U ) − Σ 2  K 1  S (426) No w, using (41 0) and (419) in (406), we get the f o llo wing b ound on R s 2 R s 2 ≤ 1 2 log | S + Σ 2 | | K 1 + Σ 2 | − 1 2 log | S + Σ Z | | K 1 + Σ Z | (427) whic h completes the ﬁrst step of the pro of. Second step: W e consider the b ound on the conﬁdential message sum rate R s 1 + R s 2 giv en in (9) as follows R s 1 + R s 2 ≤ I ( U ; Y 2 ) + I ( X ; Y 1 | U ) − I ( X ; Z ) (428) = [ h ( Y 2 ) − h ( Z )] + [ h ( Y 1 | U ) − h ( Y 2 | U )] − 1 2 log | Σ 1 | | Σ Z | (429) ≤ 1 2 log | S + Σ 2 | | S + Σ Z | + [ h ( Y 1 | U ) − h ( Y 2 | U )] − 1 2 log | Σ 1 | | Σ Z | (430) where (430) comes from (410). Next, we consider the r emaining term in (430) as follows h ( Y 2 | U ) − h ( Y 1 | U ) = 1 2 Z Σ 2 Σ 1 J ( X + N | U ) d Σ N (431) whic h fo llo ws from Lemma 10, and N is a Ga ussian random v ector with co v aria nce matrix Σ N satisfying Σ 1  Σ N  Σ 2 . Due to Lemma 7, for any Gaussian random vec tor N with Σ N  Σ 2 , we ha v e J − 1 ( X + N | U ) − Σ N  J − 1 ( X + N 2 | U ) − Σ 2 (432)  K 1 (433) where (433) comes from (426). In view of Lemma 12, ( 4 33) implies J ( X + N | U )  ( K 1 + Σ N ) − 1 , Σ N  Σ 2 (434) Using (434) in (43 1) in conjunction with Lemma 9, w e hav e h ( Y 2 | U ) − h ( Y 1 | U ) ≥ 1 2 Z Σ 2 Σ 1 ( K 1 + Σ N ) − 1 d Σ N (435) = 1 2 log | K 1 + Σ 2 | | K 1 + Σ 1 | (436) 50 Using (436) in (43 0), w e get R s 1 + R s 2 ≤ 1 2 log | S + Σ 2 | | K 1 + Σ 2 | + 1 2 log | K 1 + Σ 1 | | Σ 1 | − 1 2 log | S + Σ Z | | Σ Z | (437) whic h completes the second step of the pro of. Third step: W e consider the b ound on R s 2 + R p 2 giv en in (10) as fo llo ws R p 2 + R s 2 ≤ I ( U ; Y 2 ) (438) = h ( Y 2 ) − h ( Y 2 | U ) (439) ≤ 1 2 log | (2 π e )( S + Σ 2 ) | − h ( Y 2 | U ) (440) where (4 40) comes from the maxim um en tropy t heorem [13 ]. Next, we consider the remaining term in (440). Using (41 9), we ha v e h ( Y 2 | U ) = h ( Z | U ) − 1 2 log | K 1 + Σ Z | | K 1 + Σ 2 | (441) ≥ 1 2 log | (2 π e ) J − 1 ( X + N Z | U ) | − 1 2 log | K 1 + Σ Z | | K 1 + Σ 2 | (442) ≥ 1 2 log | (2 π e )( K 1 + Σ Z ) | − 1 2 log | K 1 + Σ Z | | K 1 + Σ 2 | (443) = 1 2 log | (2 π e )( K 1 + Σ 2 ) | (444) where (442) is due to Lemma 1 1, a nd (44 3) comes fro m (421) and monotonicity of | · | in p ositiv e semi-deﬁnite ma t r ices. Using (44 4) in (44 0), w e get R p 2 + R s 2 ≤ 1 2 log | S + Σ 2 | | K 1 + Σ 2 | (445) whic h completes the t hird step of the pro of. 51 F ourth step: W e consider the b ound in (1 1) as follows R s 1 + R p 2 + R s 2 ≤ I ( U ; Y 2 ) + I ( X ; Y 1 | U ) − I ( X ; Z | U ) (446) = h ( Y 2 ) − h ( Y 2 | U ) + [ h ( Y 1 | U ) − h ( Z | U )] − 1 2 log | Σ 1 | | Σ Z | (447) ≤ 1 2 log | (2 π e )( S + Σ 2 ) | − h ( Y 2 | U ) + [ h ( Y 1 | U ) − h ( Z | U )] − 1 2 log | Σ 1 | | Σ Z | (448) ≤ 1 2 log | (2 π e )( S + Σ Z ) | − 1 2 log | (2 π e )( K 1 + Σ 2 ) | + [ h ( Y 1 | U ) − h ( Z | U )] − 1 2 log | Σ 1 | | Σ Z | (449) = 1 2 log | S + Σ 2 | | K 1 + Σ 2 | + [ h ( Y 1 | U ) − h ( Y 2 | U )] + [ h ( Y 2 | U ) − h ( Z | U )] − 1 2 log | Σ 1 | | Σ Z | (450) = 1 2 log | S + Σ 2 | | K 1 + Σ 2 | + [ h ( Y 1 | U ) − h ( Y 2 | U )] + 1 2 log | K 1 + Σ 2 | | K 1 + Σ Z | − 1 2 log | Σ 1 | | Σ Z | (451) ≤ 1 2 log | S + Σ 2 | | K 1 + Σ 2 | + 1 2 log | K 1 + Σ 1 | | K 1 + Σ 2 | + 1 2 log | K 1 + Σ 2 | | K 1 + Σ Z | − 1 2 log | Σ 1 | | Σ Z | (452) = 1 2 log | S + Σ 2 | | K 1 + Σ 2 | + 1 2 log | K 1 + Σ 1 | | Σ 1 | − 1 2 log | K 1 + Σ Z | | Σ Z | (453) where (448) comes from the maxim um entrop y theorem [13], (449 ) comes from (444), (45 1) is due to (41 9), a nd (452) comes fro m (4 36). Fifth step: W e consider the b ound in (12) as follows R p 1 + R s 1 + R p 2 + R s 2 ≤ I ( U ; Y 2 ) + I ( X ; Y 1 | U ) (454) = h ( Y 2 ) + [ h ( Y 1 | U ) − h ( Y 2 | U )] − 1 2 log | (2 π e ) Σ 1 | (455) ≤ 1 2 log | (2 π e )( S + Σ 2 ) | + [ h ( Y 1 | U ) − h ( Y 2 | U )] − 1 2 log | (2 π e ) Σ 1 | (456) ≤ 1 2 log | (2 π e )( S + Σ 2 ) | + 1 2 log | K 1 + Σ 1 | | K 1 + Σ 2 | − 1 2 log | (2 π e ) Σ 1 | (457) = 1 2 log | S + Σ 2 | | K 1 + Σ 2 | + 1 2 log | K 1 + Σ 1 | | Σ 1 | (458) where (456) comes from the maximum en trop y theorem [13], a nd (4 57) comes fro m (4 36). Hence, w e hav e show n that for an y feasible ( U, X ), there exists a G aussian ( U G , X G ) whic h yields a larger rate region. This completes the pro o f. E Pro of of Th e orem 7 W e now o btain an alternativ e rate region b y using the one given by (361)-(37 4). This alternativ e region is more a menable for ev aluation fo r the Gaussian MIMO c hannel. W e 52 note that the following region is included in the regio n g iven b y (361)-(37 4). R s 1 ≤ min j =1 , 2 I ( U ; Y j | Q ) + I ( V 1 ; Y 1 | U ) − I ( V 1 ; V 2 | U ) − I ( U ; Z | Q ) − I ( V 1 ; Z | U, V 2 ) (45 9) R s 2 ≤ min j =1 , 2 I ( U ; Y j | Q ) + I ( V 2 ; Y 2 | U ) − I ( U, V 2 ; Z | Q ) (460) R s 1 + R s 2 ≤ min j =1 , 2 I ( U ; Y j | Q ) + I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) − I ( U, V 1 , V 2 ; Z | Q ) (461 ) R s 1 + R p 1 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) − I ( V 1 ; V 2 | U ) (462) R s 2 + R p 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 2 ; Y 2 | U ) (463) R s 1 + R p 1 + R s 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) − I ( V 2 ; Z | U ) (464) R s 1 + R s 2 + R p 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) − I ( V 1 ; Z | U, V 2 ) (46 5) R s 1 + R p 1 + R s 2 + R p 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) (466) 0 ≤ min j =1 , 2 I ( U ; Y j | Q ) − I ( U ; Z | Q ) (467) 0 ≤ I ( V 1 ; Y 1 | U ) − I ( V 1 ; V 2 | U ) − I ( V 1 ; Z | U, V 2 ) ( 4 68) 0 ≤ I ( V 2 ; Y 2 | U ) − I ( V 2 ; Z | U ) (469) 53 W e note that w e can remo v e the constrain ts g iven b y (467)-(46 9) without enlarging t he region given by (459)-(466), which will lea v e us with the following region. R s 1 ≤ min j =1 , 2 I ( U ; Y j | Q ) + I ( V 1 ; Y 1 | U ) − I ( V 1 ; V 2 | U ) − I ( U ; Z | Q ) − I ( V 1 ; Z | U, V 2 ) (47 0) R s 2 ≤ min j =1 , 2 I ( U ; Y j | Q ) + I ( V 2 ; Y 2 | U ) − I ( U, V 2 ; Z | Q ) (471) R s 1 + R s 2 ≤ min j =1 , 2 I ( U ; Y j | Q ) + I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) − I ( U, V 1 , V 2 ; Z | Q ) (472 ) R s 1 + R p 1 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) − I ( V 1 ; V 2 | U ) (473) R s 2 + R p 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 2 ; Y 2 | U ) (474) R s 1 + R p 1 + R s 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) − I ( V 2 ; Z | U ) (475) R s 1 + R s 2 + R p 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) − I ( V 1 ; Z | U, V 2 ) (47 6) R s 1 + R p 1 + R s 2 + R p 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) (477) W e denote the region giv en by (470)-(4 77) b y R 21 . Similarly , the follo wing ac hiev able rate region can b e obtained as w ell. R s 1 ≤ min j =1 , 2 I ( U ; Y j | Q ) + I ( V 1 ; Y 1 | U ) − I ( U, V 1 ; Z | Q ) (478) R s 2 ≤ min j =1 , 2 I ( U ; Y j | Q ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) − I ( U ; Z | Q ) − I ( V 2 ; Z | U, V 1 ) (47 9) R s 1 + R s 2 ≤ min j =1 , 2 I ( U ; Y j | Q ) + I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) − I ( U, V 1 , V 2 ; Z | Q ) (480 ) R s 1 + R p 1 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) (481) R s 2 + R p 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) (482) R s 1 + R p 1 + R s 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) − I ( V 2 ; Z | U, V 1 ) (48 3) R s 1 + R s 2 + R p 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) − I ( V 1 ; Z | U ) (484) R s 1 + R p 1 + R s 2 + R p 2 ≤ min j =1 , 2 I ( U ; Y j ) + I ( V 1 ; Y 1 | U ) + I ( V 2 ; Y 2 | U ) − I ( V 1 ; V 2 | U ) (485) 54 whic h is denoted b y R 12 . Hence, we obtain the ac hiev able rate region R which is give n by R = con v ( R 12 ∪ R 21 ) (486 ) Next, we o utline an alternativ e metho d to o bta in the region R 21 . First, w e set L 1 , L 2 , R p 1 , R p 2 , ∆ 1 , ∆ 2 as follow s. L 1 = I ( V 1 ; V 2 | U ) (48 7) L 2 = 0 (488) R p 1 + ∆ 1 = I ( V 1 ; Z | U, V 2 ) (489) R p 2 + ∆ 2 = I ( V 2 ; Z | U ) (490) Using t he v alues of L 1 , L 2 , R p 1 + ∆ 1 , R p 2 + ∆ 2 giv en by (487)-(4 90) in (191)-(199), we hav e the following ac hiev able rate region. R p 0 + R s 0 + ∆ 0 ≤ min j =1 , 2 I ( U ; Y j ) (491) ˜ ˜ R p 0 + R s 0 + ∆ 0 ≤ min j =1 , 2 I ( U ; Y j | Q ) (492) R s 1 ≤ I ( V 1 ; Y 1 | U ) − I ( V 1 ; V 2 | U ) − I ( V 1 ; Z | U, V 2 ) (493) R s 2 ≤ I ( V 2 ; Y 2 | U ) − I ( V 2 ; Z | U ) (494) ˜ ˜ R p 0 + ∆ 0 = I ( U ; Z | Q ) (495) R p 1 + ∆ 1 = I ( V 1 ; Z | U, V 2 ) (496) R p 2 + ∆ 2 = I ( V 2 ; Z | U ) (497) Next, following the pro cedure in App endix B.2 , the achiev able rate region giv en b y (47 0)- (477), i.e., R 21 , can b e obtained by using the ac hiev able rate region giv en b y (4 9 1)-(497). Similarly , the other region R 12 can b e obtained as w ell. W e also note that this alternativ e deriv ation reve als that since w e select L 1 = I ( V 1 ; V 2 | U ) , L 2 = 0 to obta in the achiev able rate region R 21 , in this case, the transmitter ﬁrst enco des V n 2 , and then, next using the non-causal kno wledge o f V n 2 , enco des V n 1 , i.e., uses Gelfand- Pinsk er enco ding for V n 2 . Next, w e obtain an a c hiev able rate region for the Ga ussian MIMO m ulti-receiv er wiretap c hannel with public and conﬁden tial messages. W e provide this ach iev a ble rate region b y ev aluating the regions R 12 and R 21 with a sp eciﬁc c hoice of Q, U, V 1 , V 2 , X . In particular, t o ev aluate R 21 , we use the fo llowing selection fo r Q, U, V 1 , V 2 , X : • Q is selected as a zero-mean Gaussian random v ector with co v ariance matrix S − K 0 − K 1 − K 2 , where K 0 , K 1 , K 2 are p o sitiv e semi-deﬁnite matrices satisfying K 0 + K 1 + K 2  S , • U is selected a s U = Q + Q ′ , where Q ′ is a zero-mean Ga ussian ra ndom v ector with 55 co v ariance matrix K 0 , and is indep enden t of Q , • V 2 is selected as V 2 = U + U 2 , where U 2 is a zero-mean Gaussian ra ndo m v ector with co v ariance matrix K 2 , and is indep enden t of Q, Q ′ , • V 1 is selected as V 1 = U 1 + A U 2 + U , where U 1 is a zero-mean G a ussian ra ndom v ector with co v ariance matrix K 1 , and is indep enden t o f Q, Q ′ , U 2 . The enco ding matrix A is giv en by A = K 1 [ K 1 + Σ 1 ] − 1 , • X is selected as X = Q + Q ′ + U 2 + U 1 . W e note that w e use dirt y-pap er co ding [19] to enco de V 1 , whic h leads t o the following. I ( V 1 ; Y 1 | U ) − I ( V 1 ; V 2 | U ) = 1 2 log | K 1 + Σ 1 | | Σ 1 | (498) The other m utual information terms in the region R 21 can b e computed straigh tforw ardly , 56 whic h leads to the following region. R s 1 ≤ min j =1 , 2 1 2 log | K 0 + K 1 + K 2 + Σ j | | K 1 + K 2 + Σ j | + 1 2 log | K 1 + Σ 1 | | Σ 1 | − 1 2 log | K 0 + K 1 + K 2 + Σ Z | | K 1 + K 2 + Σ Z | − 1 2 log | K 1 + Σ Z | | Σ Z | (499) R s 2 ≤ min j =1 , 2 1 2 log | K 0 + K 1 + K 2 + Σ j | | K 1 + K 2 + Σ j | + 1 2 log | K 1 + K 2 + Σ 2 | | K 1 + Σ 2 | − 1 2 log | K 0 + K 1 + K 2 + Σ Z | | K 1 + Σ Z | (500) R s 1 + R s 2 ≤ min j =1 , 2 1 2 log | K 0 + K 1 + K 2 + Σ j | | K 1 + K 2 + Σ j | + 1 2 log | K 1 + K 2 + Σ 2 | | K 1 + Σ 2 | + 1 2 log | K 1 + Σ 1 | | Σ 1 | − 1 2 log | K 0 + K 1 + K 2 + Σ Z | | Σ Z | (501) R s 1 + R p 1 ≤ min j =1 , 2 1 2 log | S + Σ j | | K 1 + K 2 + Σ j | + 1 2 log | K 1 + Σ 1 | | Σ 1 | (502) R s 2 + R p 2 ≤ min j =1 , 2 1 2 log | S + Σ j | | K 1 + K 2 + Σ j | + 1 2 log | K 1 + K 2 + Σ 2 | | K 1 + Σ 2 | (503) R s 1 + R p 1 + R s 2 ≤ min j =1 , 2 1 2 log | S + Σ j | | K 1 + K 2 + Σ j | + 1 2 log | K 1 + Σ 1 | | Σ 1 | + 1 2 log | K 1 + K 2 + Σ 2 | | K 1 + Σ 2 | − 1 2 log | K 1 + K 2 + Σ Z | | K 1 + Σ Z | (504) R s 1 + R s 2 + R p 2 ≤ min j =1 , 2 1 2 log | S + Σ j | | K 1 + K 2 + Σ j | + 1 2 log | K 1 + K 2 + Σ 2 | | K 1 + Σ 2 | + 1 2 log | K 1 + Σ 1 | | Σ 1 | − 1 2 log | K 1 + Σ Z | | Σ Z | (505) R s 1 + R p 1 + R s 2 + R p 2 ≤ min j =1 , 2 1 2 log | S + Σ j | | K 1 + K 2 + Σ j | + 1 2 log | K 1 + K 2 + Σ 2 | | K 1 + Σ 2 | + 1 2 log | K 1 + Σ 1 | | Σ 1 | (506) W e denote the region g iv en in (499) -(506) by R 21 ( K 0 , K 1 , K 2 ). Similarly , w e can ev alu- ate the region R 12 to obta in another ac hiev able ra te region R 12 ( K 0 , K 1 , K 2 ) for the Gaus- sian MIMO multi-receiv er wiretap c hannel, where R 12 ( K 0 , K 1 , K 2 ) can b e obtained from R 21 ( K 0 , K 1 , K 2 ) by sw apping the subscripts 1 and 2. References [1] A. Wyner. The wire-tap c hannel. Bel l System T e c h n ic al Journal , 54(8):13 55–1387, Jan. 1975. [2] I. Csiszar a nd J. Korner. Broadcast channels with conﬁdential messages. IEEE T r ans. 57 Inf. The ory , IT-24(3):33 9–348, May 1978. [3] G. Bag herik a ram, A. S. Motahar i, a nd A. K. Khandani. The secrecy ra te regio n of the broadcast c hannel. In 46 th Annual Al lerton Conf . Commun., Contr. and Comp ut. , Sep. 2008. Also a v ailable at [a rXiv:0806.4200]. [4] E. Ekrem and S. Ulukus. On secure bro adcasting. In 42th Asilomar Co nf. Signals , Syst. and Comp. , Oct. 2008. [5] E. Ekrem and S. Ulukus. Secrecy capacit y of a class of broa dcast c hannels with an ea v es- dropp er. 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Multi-receiver Wiretap Channel with Public and Confidential Messages

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