Secrecy in Cooperative Relay Broadcast Channels

Secrecy in Co op erativ e Rela y Broadcast Channels ∗ Ersen Ekrem Senn ur Uluk u s Departmen t of Ele ctrical and Computer Enginee ring Univ ersit y of Maryland, College P ark, MD 20742 ersen@umd.e du ulukus@umd.e du No vem b er 2 , 2 018 Abstract W e in v estigate the eﬀects of user co op eration on the secrecy of broadcast channels b y considerin g a coop erativ e rela y broadcast c hannel. W e sho w th at user co op eration can increase the ac hiev able secrecy region. W e prop ose an ac hiev able sc heme that com bines Marton’s co ding scheme for broadcast channels and Cov er and El Gamal’s compress-and-forwa r d sc h eme for rela y c hann els. W e derive o u ter b ounds for the rate - equiv o cation region using auxiliary random v ariables for single-letteriza tion. Finally , w e consider a Gaussian c hann el and sho w that b oth users can ha ve p ositiv e secrecy rates, whic h is not p ossible f or scalar Gaussian b roadcast c hannels without co op eration. ∗ This work was supp or ted by NSF Gra nts CCF 0 4 -476 13, CCF 05-14 846, CNS 07 -1631 1 and CCF 07- 29127 , and presented in par t a t the Information Theory and Applications W orkshop, Sa n Diego, CA, January 2008 and t he IEE E In ternational Sympo sium on Information Theory , T oronto, Canada , July 2 008 [1]. 1 1 In tro ducti on The op en nature of wireless communic at ions facilitates co op eration by allowing users to ex- ploit the o v er-heard information to increase ac hiev able rates. Ho w ev er, the s ame open nature of wireless comm unications mak es it vulnerable to securit y attack s suc h as eav esdropping and jamming. In this pap er, w e in v estigate the interaction o f these t w o phenomena, namely co op eratio n and secrecy . In pa rticular, w e in v estigate the eﬀects of co op eration on secrecy . The eav esdropping a t t ac k w as ﬁrst studied fro m an information theoretic p oin t of view b y Wyner in [2], where he established the secrecy capacity for a sin gle-user de gr ade d wire- tap channe l. Later, Csiszar and Korner [3] studied the general, not necessarily degraded, single-user ea v esdropping c hannel, and found the secrecy capacit y . More recen tly multi-user v ersions o f the secrecy problem ha ve been considered for v arious c hannel mo dels. Refer- ences [4–8] consider multiple access channe ls (MA C), where in [4, 5] the ea ve sdropp er is an external en tit y , while in [6–8] the users in the MA C act as ea v esdropp ers on eac h other. Ref- erences [9 , 10] consider broadcast channe ls (BC) where b oth receiv ers wan t to ha v e secure comm unication with the transmitter; in here a s w ell, each receiv er of t he BC is an ea ve sdrop- p er for the other user. References [11–16] consider secrecy in r elay c hannels, where in [11 –13], the rela y is the ea v esdropp er, while in [14, 15] t here is an external ea ve sdropp er. In [16], the rela y helps the transmitter to impro v e its rate while it receiv es conﬁden tial mes sages that should b e ke pt hidden from the main receiv er. In a wirele ss medium, since all use rs receiv e a v ersion of all signals transmitted, they can co op era t e to improv e their communication rates. The simplest example o f a co op erativ e system is t he rela y c hannel [17] where the rela y helps increase the commun icatio n rate of a single-us er channel usin g its o v er-heard informa t io n. Multi-user v ersions of coop erativ e comm unication hav e b een studied mor e recen tly . In [18 ], a MA C is considered where b oth users o v er-hear a noisy v ersion of the signal transmitted b y the o t her user, and transmit in suc h a w a y to increase their achiev able ra t es. In [19–21], co op eration is done on t he receiv er side, where in a BC, one or b oth of the receiv ers transmit co op erative signals to improv e the ac hiev able ra t es of b oth users. Our goal is to study the eﬀects co op eratio n on the secrecy of multiple users where secrecy refers to sim ultaneous individual conﬁden tiality of all users. One of the simplest mo dels to study this in teraction is the co op erative rela y broadcast ch annel (CRBC), where there is a single transmitter and t w o receiv ers, and eac h receiv er w ould lik e to k eep its message secret from the other user; see F igures 1 and 2. In this mo del, in order to incorp orate the eﬀects of co op eratio n, there is either a single-sided (Figure 1) or double-sided (Figure 2) co op erative link b et wee n the users. F or clarit y of ideas and simplicit y of presen tation, for a ma jor pa r t of this pap er, w e will assume a CRBC with a single-side d co op eratio n link from the ﬁrst user to the second user. W e will in ve stigat e the eﬀects o f tw o-sided co op eration in Section 8. F o cusing on the single-side d CRBC, we note that if w e remo v e the co op eratio n link , our mo del reduces to the BC with conﬁden tial messages in [9, 10], and if w e set the rat e of the 2 User 2 User 1 CRBC Encoder X n 1 Y n 1 p ( y 1 , y 2 | x, x 1 ) Y n 2 ˆ W 1 ˆ W 2 X n ( W 1 , W 2 ) Figure 1: Co op erative rela y broadcast c hannel (CRBC) with single-sided co o p erativ e link. ﬁrst user t o zero, our mo del reduces to t he relay c hannel with conﬁden tial messages in [11–13], and if w e b oth set the ra te of the ﬁrst user to zero and remo v e t he co op eration link b etw een the users, our mo del reduces to the single-user ea v esdropp er c hannel in [2, 3]. Our mo del is the simplest mo del (except p erhaps for the “dual” mo del o f co op erating transmitters in a MA C with p er-user secrecy constrain ts [8]) that a llo ws us to study t he eﬀects of co op eration (or lac k there of ) of the ﬁrst user (the transmitting end of the co op erativ e link) on its ow n equiv o cation rate as well as on the equiv o cat io n rate o f the other user (receiving end of the co op erative link). Our motiv ation to study this problem can b e b est ex plained in a Gaussian example. Imagine a t w o - user Gaussian BC. This BC is degraded in one direction, hence b oth users cannot hav e p ositiv e secrecy rat es simultaneously [2, 9 , 10]. This has motiv ated [10 ] to use m ultiple an tennas at the transmitter in order to remo v e this degradedness in either of the directions and pro vide p ositiv e secrecy rates to b oth users sim ultaneously . W e wish to ac hiev e a similar eﬀect with a single t r a nsmitter an tenna, b y in tro ducing co o p eration from one user to the other. Imagine no w a G aussian CRBC [19, 20 ] as in Figure 1, where user 1 acts as a rela y fo r user 2’s message, i.e., that there is a co o p erativ e link from user 1 to user 2. Let us assume that in the underlying BC, user 1 has a b etter c hannel. Without the co op erative link, user 2 cannot ha v e secure communication with the transmitter. W e show that user 1 can transmit co op erative signals a nd impro ve the secrecy rate of user 2. Our main idea is that user 1 can use a compress-and-forward (CAF) based rela ying sc heme for the message of user 2, and increase user 2’s ra te to a lev el whic h is not deco dable b y user 1. This improv es user 2’s secrecy . No w, let us assume that in the underlying BC, user 1 ha s t he w orse channel. Without co op eration, user 1 cannot ha v e secure comm unication with the transmitter. W e sho w that user 1 can transmit a jamming signal in the co op erative c hannel ﬁrst to guaran tee a p ositiv e secrecy rate for it self assuming it has enough p o wer. This essen tially brings the system to the setting described in the previous case, and no w user 1 can send a co op erative signal to user 2 to help it ac hiev e a p ositiv e secrecy rate as well. In this pap er, w e prop ose an ac hiev able sc heme that com bines Mar t o n’s co ding sc heme for BCs [22 ] and Co v er and El G a mal’s CAF sc heme for rela y channe ls [17]. A similar ac hiev able 3 User 2 User 1 CRBC Encoder X n 1 Y n 1 ˆ W 1 ˆ W 2 X n ( W 1 , W 2 ) Y n 2 X n 2 p ( y 1 , y 2 | x, x 1 , x 2 ) Figure 2: Co op erativ e rela y bro a dcast channe l (CRBC) with a tw o-sided co o p eration link. sc heme has app eared in [23] whic h do es not consider a n y secrecy constrain ts, hence ours can b e view ed as a generalization of [23] to a secrec y conte xt. A similar achie v able sc heme a lso app eared in [11, 13], where CAF is a pplied to a relay channel to pro vide impro v ed secrecy for the main transmitter. A rela y c hannel can b e considered as a sp ecial case of the single-sided CRBC where the ra t e of t he ﬁrst user is set to zero. In this pap er, w e also dev elop a single-letter outer b ound on the rat e-equiv o cation region; w e accomplish singe-letterization by using to ols prop osed in [3 ], namely b y determining suitable auxiliary random v ariables. Besides this outer b ound, for the second user, that is b eing help ed in t he single-sided CRBC, w e dev elop another single-letter outer bo und whic h dep ends only on the channe l inputs and o utputs. T o visualize the eﬀects of co op eration on secrecy , w e consider a G aussian CRBC a nd sho w that b oth users can ha ve p ositiv e secrecy rates through user co op eration. T o obtain p o sitiv e secrecy rates for b oth users, we pro vide diﬀeren t assignmen ts for the auxiliary random v ari- ables app earing in the achie v able r a tes. Thes e auxiliary random v ariable assignmen ts hav e dirt y pap er co ding (DPC) interpretations [2 4]. In addition, we comb ine jamming and relay- ing to pro vide secrecy for b oth users when the relay ing user is w eak. Finally , we consider the CRBC with a tw o-sided co op eration link and pro vide an achiev able sc heme fo r this channe l. 2 The Ch ann el Mo de l and Deﬁnitions F rom here un til the b eginning o f Section 8, w e will fo cus on a single-sided CRBC, and refer to it simply as CRBC. The CRBC can b e view ed as a relay channel where t he transmitter sends messages b oth to the r ela y no de and the destination. Therefore, one of the users, user 1 in our case, in a CRBC b oth deco des its o wn message and also helps the other user. A CRBC consis ts of t w o message sets w 1 ∈ W 1 , w 2 ∈ W 2 , t w o input alphab ets, one at the transmitter x ∈ X a nd one at user 1 x 1 ∈ X 1 , and t wo output alphab ets y 1 ∈ Y 1 , y 2 ∈ Y 2 , where the former is fo r use r 1 a nd the latter is fo r user 2. The c hannel is a ssumed to b e memoryless a nd its transition probabilit y distribution is p ( y 1 , y 2 | x, x 1 ). A  2 nR 1 , 2 nR 2 , n  co de f o r this c hannel consists of tw o message sets as W 1 =  1 , . . . , 2 nR 1  4 and W 2 =  1 , . . . , 2 nR 2  , an enco der at the transmitter with mapping W 1 × W 2 → X n , a set of rela y functions at user 1, x 1 ,i = f i ( y 1 , 1 , . . . , y 1 ,i − 1 ) for 1 ≤ i ≤ n , t wo deco ders, one at eac h user with the mappings g 1 : Y n 1 → W 1 and g 2 : Y n 2 → W 2 . The proba- bilit y of error is deﬁned as P n e = max  P n e, 1 , P n e, 2  where P n e, 1 = Pr ( g 1 ( Y n 1 ) 6 = W 1 ) , P n e, 2 = Pr ( g 2 ( Y n 2 ) 6 = W 2 ). The secrecy of the users is measured b y the equiv o cation rates whic h are 1 n H ( W 1 | Y n 2 ) and 1 n H ( W 2 | Y n 1 , X n 1 ). Since user 1 has its o wn c hannel input, w e condition the en tropy rate of user 2 ’s messages on this c hannel input. A rate tuple ( R 1 , R 2 , R e, 1 , R e, 2 ) is said to be achie v able if there exists a  2 nR 1 , 2 nR 2 , n  co de with lim n →∞ P n e = 0 and lim n →∞ 1 n H ( W 1 | Y n 2 ) ≥ R e, 1 , lim n →∞ 1 n H ( W 2 | Y n 1 , X n 1 ) ≥ R e, 2 (1) 3 An Ac hi ev able Sc heme W e no w pro vide an achiev able sche me whic h com bines Marton’s co ding sc heme for BCs [22] and Cov er a nd El Gamal’s CAF sc heme for rela y channels [17]. A similar a c hiev able sche me has app eared in [23] without an y secrecy considerations. In this sc heme, user 1 sends a quan tized vers ion of its observ ation to user 2, which uses this informat ion to deco de it s own message. The corresp onding ac hiev able r a te-equiv o cation region is giv en b y the following theorem. Theorem 1 The rate tuples ( R 1 , R 2 , R e, 1 , R e, 2 ) satisfying R 1 ≤ I ( V 1 ; Y 1 | X 1 ) (2) R 2 ≤ I ( V 2 ; Y 2 , ˆ Y 1 | X 1 ) (3) R 1 + R 2 ≤ I ( V 1 ; Y 1 | X 1 ) + I ( V 2 ; Y 2 , ˆ Y 1 | X 1 ) − I ( V 1 ; V 2 ) (4) R e, 1 ≤ R 1 (5) R e, 1 ≤ h I ( V 1 ; Y 1 | X 1 ) − I ( V 1 ; Y 2 , ˆ Y 1 | V 2 , X 1 ) − I ( V 1 ; V 2 ) i + (6) R e, 2 ≤ R 2 (7) R e, 2 ≤ h I ( V 2 ; Y 2 , ˆ Y 1 | X 1 ) − I ( V 2 ; Y 1 | V 1 , X 1 ) − I ( V 1 ; V 2 ) i + (8) are a c hiev able for an y distribution of the form p ( v 1 , v 2 ) p ( x | v 1 , v 2 ) p ( x 1 ) p ( ˆ y 1 | x 1 , v 1 , y 1 ) p ( y 1 , y 2 | x, x 1 ) (9) sub ject to the constrain t I ( ˆ Y 1 ; Y 1 | X 1 , V 1 ) ≤ I ( ˆ Y 1 , X 1 ; Y 2 ) (10) This theorem is a sp ecial case of Theorem 4 and obtained from t he latter b y setting U = X 1 . Therefore, w e will omit the pro of o f Theorem 1 here and will provide the pro of of Theorem 4 5 in App endix D. In (6) and ( 8), ( x ) + is the p ositivit y op erato r, i.e., ( x ) + = max(0 , x ). Remark 1 W e note that b oth the form of the probability distribution in (9) a nd the con- strain t in (1 0) in Theorem 1 are somewhat diﬀeren t than those of the classic al CAF sc heme in [17]. First, we condition the distribution o f ˆ Y 1 on V 1 to preve nt the compressed v ersion of Y 1 to leak any additional information regarding user 1’s message on top of what user 2 already has through its own observ ation. The constraint in (10) also reﬂects this concern. Similar constrain ts on the distribution of ˆ Y 1 and on the compression rate hav e app eared in [23], where these mo diﬁcations a re not due to secrecy constrain ts con trary to here. In [23], these are imp osed to obtain higher rates for user 2 by removing user 1’s priv ate message from the compressed signal, whereas here, they a r e imp osed not to let ˆ Y 1 leak a n y additional informa- tion regarding user 1’s message. Moreov er, if w e let user 1 compress its observ atio n without erasing its o wn message fro m the observ at io n, i.e., if w e change the conditional distribution of ˆ Y 1 to p ( ˆ y 1 | x 1 , y 1 ), we can recov er the constrain t in [17] (see equations (29 )-(31) in [23]). Remark 2 If w e disable the assistance o f user 1 to us er 2 b y setting X 1 = ˆ Y 1 = φ , t he c hannel mo del reduces to the BC with secrec y constraints , and the a chiev able equiv o cation region becomes R B C e, 1 ≤ I ( V 1 ; Y 1 ) − I ( V 1 ; Y 2 | V 2 ) − I ( V 1 ; V 2 ) (11) R B C e, 2 ≤ I ( V 2 ; Y 2 ) − I ( V 2 ; Y 1 | V 1 ) − I ( V 1 ; V 2 ) (12) where w e require the Mar ko v c hain ( V 1 , V 2 ) → X → ( Y 1 , Y 2 ). This result w as deriv ed in [10]. Remark 3 If we disable b oth co op eration b etw een receiv ers b y setting X 1 = ˆ Y 1 = φ , and also t he conﬁden tial messages sen t to user 1 by setting V 1 = φ , the channel mo del r educes to the single-user eav esdropper channel, and the achie v able equiv o catio n rate for the second user b ecomes R e, 2 ≤ I ( V 2 ; Y 2 ) − I ( V 2 ; Y 1 ) (13) and t he Marko v chain V 2 → X → ( Y 1 , Y 2 ) is required by the probabilit y distribution in (9). This is exactly the secrecy capacit y o f the single-user eav esdropper c hannel give n in [3]. Remark 4 If w e disable the conﬁden tial messages sen t to use r 1 b y setting V 1 = φ , t he c hannel mo del reduces to a rela y c hannel with secrecy constrain ts, and the ac hiev able equiv- o cation rate for the second user b ecomes R e, 2 ≤ I ( V 2 ; Y 2 , ˆ Y 1 | X 1 ) − I ( V 2 ; Y 1 | X 1 ) (14) sub ject to I ( ˆ Y 1 ; Y 1 | X 1 ) ≤ I ( ˆ Y 1 , X 1 ; Y 2 ) (15) 6 and the corresp onding joint distribution reduces to p ( v 2 , x ) p ( x 1 ) p ( ˆ y 1 | x 1 , y 1 ) p ( y 1 , y 2 | x, x 1 ). F urther, if we make the p otentially sub optimal se lection of V 2 = X , the corresp onding ac hiev able secrecy rate and the constrain t coincide with their coun terpart s found in [11] for the relay c hannel. Remark 5 By comparing the equiv o cation rates of the users in (6) and (8) and the equiv- o cation rates of the users in the corresp onding BC given in (1 1) and (12), w e observ e that the equiv o cation rate of user 1 ma y decrease dep ending on the information con tained in ˆ Y 1 and t he equiv o cation rate of user 2 may increase depending on the c hannel conditions. Remark 6 W e will show in the next section, where w e dev elop outer b ounds for t he rate- equiv o cation region, that if the c hannel of user 2 is degraded with res p ect to the c hannel of user 1 then R e, 2 = 0 (see Remark 8), where degradedness is deﬁned thro ugh the Mark ov c hain X → ( X 1 , Y 1 ) → Y 2 . Here, w e sho w, as an intere sting ev aluation, that this ac hiev able sc heme cannot yield an y p ositive secrecy rates in this case, as exp ected. I ( V 2 ; Y 2 , ˆ Y 1 | X 1 ) − I ( V 2 ; Y 1 | V 1 , X 1 ) − I ( V 1 ; V 2 ) ≤ I ( V 2 ; Y 2 , ˆ Y 1 , V 1 | X 1 ) − I ( V 2 ; Y 1 | V 1 , X 1 ) − I ( V 1 ; V 2 ) (16) = I ( V 2 ; Y 2 , ˆ Y 1 | V 1 , X 1 ) + I ( V 2 ; V 1 | X 1 ) − I ( V 2 ; Y 1 | V 1 , X 1 ) − I ( V 1 ; V 2 ) (17) = I ( V 2 ; Y 2 , ˆ Y 1 | V 1 , X 1 ) − I ( V 2 ; Y 1 | V 1 , X 1 ) (18) ≤ I ( V 2 ; Y 2 , ˆ Y 1 , Y 1 | V 1 , X 1 ) − I ( V 2 ; Y 1 | V 1 , X 1 ) (19) = I ( V 2 ; Y 2 , Y 1 | V 1 , X 1 ) + I ( V 2 ; ˆ Y 1 | V 1 , X 1 , Y 1 , Y 2 ) − I ( V 2 ; Y 1 | V 1 , X 1 ) (20) = I ( V 2 ; Y 2 , Y 1 | V 1 , X 1 ) − I ( V 2 ; Y 1 | V 1 , X 1 ) (21) = I ( V 2 ; Y 2 | V 1 , X 1 , Y 1 ) (22) = 0 (23) where in (18), we used the fa ct that X 1 and ( V 1 , V 2 ) are indep enden t, i.e., I ( V 1 ; V 2 | X 1 ) = I ( V 1 ; V 2 ), in (21), w e used the Mark ov c hain ( V 2 , Y 2 ) → ( V 1 , X 1 , Y 1 ) → ˆ Y 1 whic h implies I ( V 2 ; ˆ Y 1 | V 1 , X 1 , Y 1 , Y 2 ) = 0, a nd in (23), w e used the Mark ov c hain ( V 1 , V 2 ) → X → ( X 1 , Y 1 ) → Y 2 whic h is due to the assumed degradedness. 4 An Oute r Boun d W e no w provid e an outer b ound fo r the rate- equiv o cation region. Our ﬁrst outer b ound in Theorem 2 uses auxiliary random v ariables. Ne xt, in Theorem 3, w e provide a simpler outer b ound for user 2 using only the c hannel inputs and outputs, without emplo ying any auxiliary random v ariables. 7 Theorem 2 The ra te-equiv o cation region of the CRBC lies in t he union of the follo wing rate tuples 1 R 1 ≤ I ( V 1 ; Y 1 | X 1 ) (24) R 2 ≤ I ( V 2 ; Y 2 ) (25) R e, 1 ≤ min n ˜ R e, 1 , ¯ R e, 1 , R 1 o (26) R e, 2 ≤ min n ˜ R e, 2 , ¯ R e, 2 , R 2 o (27) where ˜ R e, 1 = I ( V 1 ; Y 1 | U ) − I ( V 1 ; Y 2 | U ) (28) ˜ R e, 2 = I ( V 2 ; Y 2 | U ) − I ( V 2 ; Y 1 | U ) (29) ¯ R e, 1 = I ( V 1 ; Y 1 | V 2 ) − I ( V 1 ; Y 2 | V 2 ) (30) ¯ R e, 2 = I ( V 2 ; Y 2 | V 1 ) − I ( V 2 ; Y 1 | V 1 ) (31) where the union is tak en o v er all joint distributions satisfying the Marko v chain U → ( V 1 , V 2 ) → ( X , X 1 , Y 1 ) → Y 2 (32) The pro of of this theorem is giv en in App endix A. Remark 7 The bounds o n the equiv o cation rates in Theorem 2 and those in [10], where the outer b ounds are for the equiv o catio n rates in a t w o- user BC with p er-user secrecy constrain ts a s in here, hav e the same expressions. The only diﬀerence b et wee n the tw o o uter b ounds is in the Marko v c hain o v er whic h the union is tak en. The Marko v c hain in (32) con tains the one in [10 ], which is U → ( V 1 , V 2 ) → X → ( Y 1 , Y 2 ) (33) whic h means that our outer bo und here ev aluates t o a larg er region than the one in [10]. This should be exp ected since the ac hiev able rat e- equiv o cation region here in our CRBC con tains the ac hiev able region in the BC. W e also prov ide a simpler outer b ound for the equiv o cation rate of user 2 whic h do es no t in v olve any auxiliary random v ariables. Theorem 3 The equiv o cation rate of user 2 is b ounded as follows R e, 2 ≤ max p ( x,x 1 ) I ( X ; Y 2 | X 1 , Y 1 ) (34) 1 Unfortunately , in the c onference version [1] o f this pap er, the outer b ound a pp e a red with some typo s. 8 The pro of of this theorem is giv en in App endix B. Remark 8 If the channe l is degraded, then the equiv o cation rate of user 2 is zero, since I ( X ; Y 2 | X 1 , Y 1 ) = 0 (35) whic h follow s from the Mark ov c hain X → ( X 1 , Y 1 ) → Y 2 whic h is a conse quence of the degradedness. Remark 9 W e generally exp ect the outer b ound in Theorem 3 to b e lo ose b ecause it essen- tially assumes that user 2 has a complete access to user 1’s observ ation 2 whereas, in reality , user 2 has only limited information ab out user 1’s observ ation, whic h it obtains through the coo p erativ e link. Ho w ev er, if the link from user 1 to user 2 is strong enough, user 1 ma y b e able to con v ey its observ a t io n to user 2 precisely in which case the outer b ound in Theorem 3 can b e close to the ac hiev a ble rate obtained via the CAF sc heme. F o r example, suc h a situation arises if the channel satisﬁes the follo wing Mark o v c hain X → ( X 1 , Y 2 ) → Y 1 (36) F or suc h c hannels, by selecting V 2 = X , V 1 = ˆ Y 1 = φ in the achiev able sc heme, w e get the follo wing equiv o cation rate for user 2 I ( X ; Y 2 | X 1 ) − I ( X ; Y 1 | X 1 ) = I ( X ; Y 2 , Y 1 | X 1 ) − I ( X ; Y 1 | X 1 ) = I ( X ; Y 2 | X 1 , Y 1 ) (37) where the ﬁrst equalit y is due to t he Mark ov chain in (36). Hence, the outer b ound in (34) giv es the secrecy capacit y for c ha nnels satisfying (36). Remark 10 Altho ug h w e a r e a ble to pro vide a simple outer b ound for the equiv o catio n rate of user 2, that dep ends only on the channel inputs and outputs, ﬁnding suc h a simple outer b ound for the equiv o cation rate of user 1 do es not seem to b e p ossible. One reason for this is that, use r 1 can use its observ atio n, i.e., Y 1 , for encoding its input, i.e., X 1 , and create correlation b etw een its channel inputs and outputs across time. Consequ ently , this correlation cannot b e accoun ted for without using auxiliary random v a r iables. Another reason will b e discuss ed in Remark 13. 2 In fact, this Sato-type [25] upp er-b ounding technique is used as a ﬁrst step (before intro ducing noise correla tion to tig ht en the upper b ound) in ﬁnding the secre c y capacity of the MIMO wiretap channel [2 6–29]. 9 5 An Example: Gaussian CRBC W e now pro vide an example t o sho w how t he prop osed ac hiev able sc heme can enlarge the secrecy regio n for a Ga ussian BC. The channe l outputs of a Gaussian CRBC are Y 1 = X + Z 1 (38) Y 2 = X + X 1 + Z 2 (39) where Z 1 ∼ N (0 , N 1 ) , Z 2 ∼ N (0 , N 2 ) and are indep enden t, E [ X 2 ] ≤ P , E [ X 2 1 ] ≤ aP . In this section, w e assume that N 2 > N 1 , i.e., user 1 has a stronger c hannel in the corresp o nding BC. Note that, in this case, if user 1 do es not help user 2, e.g., in the corresp onding BC, R e, 2 = 0. W e presen t tw o diﬀerent achiev able sc hemes for this channel where eac h o ne corresponds to a particular selection of the underlying random v aria bles in Theorem 1 satisfying the probabilit y distribution condition in (9). Prop osition 1 assigns indep enden t channe l inputs for eac h user, whereas Prop osition 2 uses a DPC sc heme. F or simplicit y , w e pro vide only the ac hiev able equiv o catio n regio n in the following prop ositions. Prop osition 1 The fo llo wing equiv o cation rates are ac hiev a ble for all α ∈ [0 , 1 ] R e, 1 ≤ 1 2 log  1 + αP ¯ αP + N 1  − 1 2 log  1 + αP N 2  (40) R e, 2 ≤ 1 2 log  1 + ¯ αP  1 αP + N 2 + 1 N 1 + N c  − 1 2 log  1 + ¯ αP N 1  (41) where ¯ α = 1 − α and N c is sub ject to N c ≥ N 2 ( ¯ αP + N 1 ) + P ( α ¯ αP + N 1 ) aP (42) Pro of: This a c hiev able region can b e obtained by selecting V 1 ∼ N (0 , α P ) , V 2 ∼ N (0 , ¯ αP ), X = V 1 + V 2 , X 1 ∼ N (0 , aP ), ˆ Y 1 = Y 1 − V 1 + Z c = V 2 + Z 1 + Z c and Z c ∼ N (0 , N c ), where V 1 , V 2 , X 1 and Z c are indep endent. The ra tes are found b y direc t calculatio n of the expressions in Theorem 1 using the ab o ve selection of random v a riables.  This ac hiev able region can b e enlarged b y in tro ducing correlation b etw een V 1 , V 2 . Since a join t encoding is p erformed at t he transmitter, one of the users’ signals can b e t r eated as a non-causally kno wn in terference, and DPC [24] can b e used. In the following prop osition, the tra nsmitter treats user 2’s signal a s a non- causally know n interfere nce. 10 Prop osition 2 The fo llo wing equiv o cation rates are ac hiev a ble for an y γ and all α ∈ [0 , 1] R e, 1 ≤ 1 2 log  1 + ( ¯ αγ + α ) 2 P ( α + γ 2 ¯ α ) N 1 + ( γ − 1) 2 α ¯ αP  − 1 2 log  1 + αP N 2  − 1 2 log  1 + γ 2 ¯ α α  (43) R e, 2 ≤ 1 2 log  1 + ¯ αP ( N 1 + N c ) + ¯ α (1 − γ ) 2 P ( αP + N 2 ) ( αP + N 2 )( N 1 + N c )  − 1 2 log  1 + α ¯ α ( γ − 1) 2 P ( α + γ 2 ¯ α ) N 1  − 1 2 log  1 + γ 2 ¯ α α  (44) where ¯ α = 1 − α and N c is sub ject to N c ≥ − η + p η 2 + 4 θ ω 2 θ (45) where θ = a ( α + ¯ αγ 2 ) P (46) η =  α + γ 2 ¯ α  P  aN 1 + (1 − γ ) 2 ¯ αP ( a + ¯ α )  − ( P + N 2 )  N 1 ( α + γ 2 ¯ α ) + α ¯ α ( γ − 1) 2 P  (47) ω =  ( P + N 2 )  (1 − γ ) 2 ¯ αP + N 1  − (1 − γ ) 2 ¯ α 2 P 2   N 1  α + γ 2 ¯ α  + P α ¯ α ( γ − 1) 2  (48) Pro of: These equiv o cation rates a r e obtained by applying DPC for user 1. Let the c hannel input o f the transmitter be X = U 1 + U 2 where U 1 ∼ N (0 , αP ) , U 2 ∼ N (0 , ¯ αP ) and are indep enden t. The auxiliary random v ariables are selected as V 2 = U 2 , V 1 = U 1 + γ U 2 , where for user 1, the signal of user 2 is treated as non-casually kno wn in terference at the transmitter. The c hannel output o f user 1 is compressed as ˆ Y 1 = Y 1 − V 1 + Z c = (1 − γ ) U 2 + Z 1 + Z c where Z c ∼ N (0 , N c ) is the compression noise. The c hannel input of user 1 is selected a s X 1 ∼ N (0 , aP ). Here, again, U 1 , U 2 , Z c and X 1 are all indep enden t. The rates are then found by direct calculation of the expressions in Theorem 1 using the ab ov e selection of random v ariables.  W e note that, in b oth of the prop ositions ab o ve , R e, 2 is a monotonically decreasing function of N c . Consequen tly , a c hiev able R e, 2 dep ends on the qualit y of the co op erativ e link b etw een the users. If this link gets b etter a llo wing user 1 to con ve y its observ at io n in a ﬁner form, user 2’s secrecy increases. F or illustrativ e purp oses, the rate regions giv en b y Prop ositions 1 and 2 a re ev a luated fo r the parameters P = 8 , N 1 = 1 , N 2 = 2 , a nd the corresp onding plots are giv en in Figures 3 and 4. Note that since N 2 > N 1 , if there w as no co op eratio n b et we en the users, us er 2 could not hav e a p ositiv e se crecy rate. W e observ e from these ﬁgures that, thanks to the co op eration of the users, b oth use rs enjoy p ositiv e secrecy ra tes. Ho w ev er, w e observ e that a p ositiv e secrecy fo r user 2 comes at t he exp ense of a decrease in the secrec y of user 1. 11 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 R e,1 (bits/channel use) R e,2 (bits/channel use) a = 5 a = 10 a = 50 a = 100 Figure 3: Achiev able equiv o catio n rate region f o r single - sided CRBC using Prop osition 1 where V 1 and V 2 are indep enden t. P = 8 , N 1 = 1 , N 2 = 2, i.e, user 2 has no secrecy rate in the underlying BC. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 R e,1 (bits/channel use) R e,2 (bits/channel use) a = 5 a = 10 a = 50 a = 100 Figure 4: Ac hiev able equiv o cat io n regio n for single-sided CRBC using Prop osition 2 where V 1 , V 2 are correlated, a dmitting a DPC interpre ta t io n. P = 8 , N 1 = 1 , N 2 = 2 , i.e., use r 2 has no secrecy rate in the underlying BC. 12 In particular, for b oth prop o sitions, maxim um secrecy rate for user 2 is ac hiev ed when user 1 do es not ha ve an y mess ag e itself and acts as a pure rela y for user 2. Similarly , user 1 ac hiev es the maxim um secrecy r a te when user 2 do es not ha ve an y message. F urthermore, we note that , for b oth ac hiev able sc hemes, as a → ∞ , t he equiv o catio n rate of user 2 approache s a limit. This is due to the fact that, as a → ∞ , the achiev able equiv o cation rates are limited b y the link b etw een the transmitter and user 1 . Moreov er, as a → ∞ , user 1 can send its observ ation to user 2 p erfectly . Th us, in this case, user 2 can b e assumed to hav e a channel output of ( Y 1 , Y 2 ), whic h makes the c ha nnel of user 1 degraded with resp ect to the ch a nnel of user 2. Conseque ntly , following the analysis carried out in Remark 9 , we exp ect the outer b ound in Theorem 3 to b ecome tigh t as a → ∞ , whic h is stated in the next coro llary . Corollary 1 As a → ∞ , the maxim um ac hiev able equiv o cation rate for user 2 b ecomes R e, 2 = 1 2 log  1 + P  1 N 1 + 1 N 2  − 1 2 log  1 + P N 1  (49) The pro of of this corollary is g iv en in App endix C. 6 Join t Jamming and Rela ying The prop osed ac hiev ability sc heme and its application to Ga ussian CRBC sho w us that user co op eration can enlarge the secrecy region. How ev er, this achie v abilit y sche me and the Ga ussian example provide us with o nly a limited picture of what can b e ac hiev ed. In particular, the a chiev ability sc heme pr o p osed in Section 3 is designed with the co op erating user (user 1) b eing the stronger of the tw o users in mind. Next, we w ant to explore what can b e done when the co op erating user (user 1) is the we a ker of the t w o users. In this case, without the co o p erativ e link, us er 1 cannot ha ve a p ositive secrecy ra te. Therefore, the ﬁrst question t o ask is, whether user 1 can ha ve a p ositiv e secrecy rate b y utilizing the co op erative link. The answ er to this question is p ositive if user 1 uses the co op erative link to send a jamming signal to user 2. Ho w eve r, a more intere sting question is whether b o t h users can ac hiev e p o sitiv e secrecy sim ultaneously . The f o llo wing theorem pro vides an achiev able sc heme, where user 1 p erforms a comb inat io n of jamming and relaying, to provide b oth users with p ositiv e secrecy ra tes. 13 Theorem 4 The rate quadruples ( R 1 , R 2 , R e, 1 , R e, 2 ) satisfying R 1 ≤ I ( V 1 ; Y 1 | X 1 ) (50) R 2 ≤ I ( V 2 ; Y 2 , ˆ Y 1 | U ) (51) R 1 + R 2 ≤ I ( V 1 ; Y 1 | X 1 ) + I ( V 2 ; Y 2 , ˆ Y 1 | U ) − I ( V 1 ; V 2 ) (52) R e, 1 ≤ R 1 (53) R e, 1 ≤ h I ( V 1 ; Y 1 | X 1 ) − I ( V 1 ; Y 2 , ˆ Y 1 | V 2 , U ) − I ( V 1 ; V 2 ) i + (54) R e, 2 ≤ R 2 (55) R e, 2 ≤ h I ( V 2 ; Y 2 , ˆ Y 1 | U ) − I ( V 2 ; Y 1 | V 1 , X 1 ) − I ( V 1 ; V 2 ) i + (56) are a c hiev able for an y distribution of the for m p ( v 1 , v 2 ) p ( x | v 1 , v 2 ) p ( u ) p ( x 1 | u ) p ( ˆ y 1 | u, v 1 , y 1 ) p ( y 1 , y 2 | x, x 1 ) (57) sub ject to the follo wing constrain t I ( ˆ Y 1 ; Y 1 | X 1 , V 1 , U ) ≤ I ( ˆ Y 1 , U ; Y 2 ) (58) The pro of of this theorem is giv en in App endix D. Remark 11 In Theorem 4, U denotes the actual help signal, while the c hannel input X 1 , whic h is correlated with U , ma y include an additional jamming attack. The intuition b ehind this ach iev able sc heme is that, although user 2 should b e able to deco de U , it cannot deco de the en tire X 1 . Therefore, since user 2 cannot deco de and eliminate X 1 from Y 2 , its c hannel b ecomes an attac ked one, where deco ding V 1 ma y b e imp ossible. Therefore, in this sc heme, user 1 ﬁrst att a c ks user 2 to mak e it s c hannel w orse by asso ciating U with many X 1 s ( hence, it conf uses user 2) , and then helps it to impro v e its secre cy ra te. Remark 12 W e note that this achie v able sc heme is reminiscen t of “co op erativ e jamming” [30]. In [30], the fo cus is on a t wo user MAC with an external eav esdropp er, where one of the users attack s bo th the legitimate receiv er and the ea v esdropp er, with the hop e that it h urts the ea v esdropp er more than it h urts the legitimate receiv er, a nd improv es the secrecy of the legitimate receiv er. In con tra st, in our work, the rela y (user 1) attac ks user 2 to improv e its o wn secrec y . 7 Gaussian E xample Revis ited Consider ag ain the Gaussian CRBC, no w with N 1 > N 2 . T he sche me prop osed in Theorem 4 w orks a s follow s: user 1 divides X 1 in to tw o pa rts. The ﬁrst part carries the noise and the second part carries the bin index of ˆ Y 1 . Although Theorem 4 is v alid for all cases, assume here 14 that user 1 has large enough p o wer. Then, the ﬁrst part makes user 2 ’s channe l noisier than user 1’s channel. This brings the situation to the case studied in Section 5. Consequen tly , w e can now ha v e a p ositiv e secrecy ra te for user 1, and also provide a p ositiv e secrecy rate to user 2, by sending a compressed v ersion of Y 1 to it, as in Section 5. Prop osition 3 The follo wing equiv o catio n ra tes a r e ac hiev able for all ( α , β ) ∈ [0 , 1 ] × [0 , 1] R e, 1 ≤ 1 2 log  1 + αP ¯ αP + N 1  − 1 2 log  1 + αP a ¯ β P + N 2  (59) R e, 2 ≤ 1 2 log  1 + ¯ αP  1 N 1 + N c + 1 αP + N 2 + a ¯ β P  − 1 2 log  1 + ¯ αP N 1  (60) where ¯ α = 1 − α, ¯ β = 1 − β , a nd N c is sub ject t o N c ≥ ¯ αP ( α P + N 2 + a ¯ β P ) + N 1 ( P + N 2 + a ¯ β P ) aβ P (61) Pro of: This ac hiev a ble region is obtained by selecting the ra ndom v ariables in Theorem 4 as X = V 1 + V 2 where V 1 ∼ N (0 , α P ) , V 2 ∼ N (0 , ¯ αP ), X 1 = U + Z j where U ∼ N (0 , aβ P ) , Z j ∼ N (0 , a ¯ β P ), ˆ Y 1 = Y 1 − V 1 + Z c = V 2 + Z 1 + Z c where Z c ∼ N (0 , N c ). Moreo ve r, V 1 , V 2 , U, Z j , Z c are all indep enden t. Here, Z j serv es as the jamming signal, and U serv es as the help er signal. User 1 ﬁrst j ams user 2 a nd mak es its c hannel noisier than its own b y using Z j and then helps user 2 through sending a compressed v ersion of its o bserv a tion b y using U . The rates are then found by direct calculation o f the expressions in Theorem 4 using the ab o ve selection of random v aria bles.  Moreo v er, as in Section 5, w e can use DPC based sc hemes in this case also. The following prop osition c haracterizes the DPC sc heme for Theorem 4. Prop osition 4 The follow ing equiv o cation r a tes are ac hiev able for an y γ and fo r all ( α, β ) ∈ [0 , 1] × [0 , 1] R e, 1 ≤ 1 2 log  1 + ( ¯ αγ + α ) 2 P ( α + γ 2 ¯ α ) N 1 + ( γ − 1) 2 α ¯ αP  − 1 2 log  1 + αP ( a ¯ β P + N 2 )  − 1 2 log  1 + γ 2 ¯ α α  (62) R e, 2 ≤ 1 2 log  1 + ¯ αP ( N 1 + N c ) + ¯ α (1 − γ ) 2 P ( αP + a ¯ β P + N 2 ) ( αP + a ¯ β P + N 2 )( N 1 + N c )  − 1 2 log  1 + α ¯ α ( γ − 1) 2 P ( α + γ 2 ¯ α ) N 1  − 1 2 log  1 + γ 2 ¯ α α  (63) where ¯ α = 1 − α, ¯ β = 1 − β and N c is sub ject to N c ≥ − η + p η 2 + 4 θ ω 2 θ (64) 15 where θ = aβ ( α + ¯ αγ 2 ) P (65) η =  α + γ 2 ¯ α  P  aβ N 1 + (1 − γ ) 2 ¯ αP ( aβ + ¯ α )  − ( P + a ¯ β P + N 2 )  N 1 ( α + γ 2 ¯ α ) + α ¯ α ( γ − 1) 2 P  (66) ω =  ( P + a ¯ β + N 2 )  (1 − γ ) 2 ¯ α P + N 1  − (1 − γ ) 2 ¯ α 2 P 2   N 1  α + γ 2 ¯ α  + P α ¯ α ( γ − 1) 2  (67) Pro of: All r andom v aria ble selections are the same as in Prop osition 2 except for X 1 , U . Here, w e c ho o se X 1 = Z j + U a nd U ∼ N (0 , aβ P ) , Z j ∼ N (0 , a ¯ β P ). U, Z j are indep enden t.  W e ﬁrst note that Prop ositions 3, 4 reduce to Prop ositions 1, 2, resp ectiv ely , by simply selecting β = 0, i.e., no j amming. W e prov ide a num erical example in Figures 5, 6 for P = 8 , N 1 = 2 , N 2 = 1. Since N 1 > N 2 , a positive secrecy rate for user 1 w ould not b e p ossible if the co op erat ive link did not exist. Ho w ev er, if user 1 has enough p o we r to ma ke user 2’s c hannel noisier b y injecting G a ussian no ise to it, user 1 can provide secrecy for itself. F or user 1 to ha ve p ositive secrecy , we need a ≥ N 1 − N 2 P (68) Otherwise, user 1 cannot ha v e p ositive secrecy by using stra t egies emplo ye d in Prop ositions 3, 4. In addition, con tra r y to Section 5, we observ e f rom Figures 5 and 6 t ha t here DPC based sc hemes do not pro vide an y gain with resp ect t o the indep enden t selection of V 1 , V 2 . F urthermore, w e also apply Prop ositions 3 and 4 to the case where user 1 is stronger than user 2 b y selecting the noise v ariances as N 1 = 1 , N 2 = 2 as in Section 5 to sho w that prop ositions presen t ed in this section cov er the ones in Section 5. W e provide t he corresp onding graphs in Figures 7 and 8. Comparing Fig ures 3 (resp. 4) and 7 (resp. 8), w e observ e that ev en though the maxim um secrecy ra t e of user 2 remains the same, the maxim um secrec y rate of user 1 is improv ed signiﬁcan tly . This improv emen t comes, b ecause t hr o ugh Pro p ositions 3 and 4, user 1 jams the receiv er of user 2. Next, w e examine Fig ures 3 and 7 in more detail. In Figure 3, for instance when a = 100, the largest R e, 2 , which is ab out 0.2 5 bits/c hannel use, is obtained when R e, 1 = 0. This corresp onds to the case where user 1 ’s rate and secrecy rate are set to zero. In this case, user 1 serv es a s a pure relay for user 2. The secrecy rate w e obtain a t this extreme is the same as [11 , 13]. A t the ot her extreme, the lar gest R e, 1 , whic h is ab out 0.42 bits/c hannel use, is obtained when R e, 2 = 0. In this case, user 2 is just an ea v esdropp er in a single-user c hannel from the transmitter to user 1. The secrecy rat e we obtain at this extreme is the same as [2, 3, 31 ]. Moreov er, as w e see from Figure 3, whenev er user 1 helps user 2 to hav e p ositiv e secrecy , it needs to deviate fro m t his extrem e p oin t. Th us, user 2’s p ositiv e sec recy rates 16 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 R e,1 (bits/channel use) R e,2 (bits/channel use) a = 5 a = 10 a = 50 a = 100 Figure 5: Ac hiev able equiv o cation rate region using Prop osition 3 wh ere user 1 jams a nd rela ys, and V 1 , V 2 are independen t. P = 8 , N 1 = 2 , N 2 = 1, i.e., user 1 cannot ha ve a n y p ositiv e secrecy in the underlying BC. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 R e,1 (bits/channel use) R e,2 (bits/channel use) a = 5 a = 10 a = 50 a = 100 Figure 6: Ac hiev able equiv o cation rate region using Prop osition 4 wh ere user 1 jams a nd rela ys, and V 1 , V 2 are correlated, admitting a DPC in terpretatio n. P = 8 , N 1 = 2 , N 2 = 1, i.e., user 1 cannot hav e an y p ositiv e secrecy in the underlying BC. 17 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 R e,1 (bits/channel use) R e,2 (bits/channel use) a = 5 a = 10 a = 50 a = 100 Figure 7: Ac hiev able equiv o cation rate region using Prop osition 3 wh ere user 1 jams a nd rela ys, and V 1 , V 2 are indep enden t. P = 8 , N 1 = 1 , N 2 = 2, i.e., user 1’s channe l is stronger than user 2. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 R e,1 (bits/channel use) R e,2 (bits/channel use) a = 5 a = 10 a = 50 a = 100 Figure 8: Ac hiev able equiv o cation rate region using Prop osition 4 wh ere user 1 jams a nd rela ys, and V 1 , V 2 are correlated, admitting a DPC in terpretatio n. P = 8 , N 1 = 1 , N 2 = 2, i.e., user 1’s c hannel is stronger than user 2. 18 come a t the exp ense of a decrease in user 1’s secrecy ra te. If w e consider Fig ure 7, the larg est R e, 2 is the same as that in Figure 3 , whic h is ag ain ac hiev ed when R e, 1 = 0, i.e., when user 1 acts as a pure rela y for user 2. Ho w ev er, in Figure 7, user 1’s maxim um secrecy r a te increases dramatically due to its ja mming capabilities in Prop osition 3. In Fig ure 7, user 1 achiev es its maxim um secrecy ra te, whic h is ab out 1.58 bits/c hannel use, when it uses all of its p o we r for jamming user 2’s receiv er and when the rate o f user 2 is set to zero. W e note tha t t his rate is larg er than that is ac hiev a ble in the corr esp o nding single-user ea ves dropp er c hannel from the transmitter to user 1, while user 2 is an eav esdropp er. W e observ e from Figure 7 t ha t when user 1 is able to jam and rela y j oin tly , it can prov ide secrecy for user 2 while its o wn secrecy ra te is still larger than that of the corr esp o nding single-user eav esdropp er c hannel. Th us, a s opp osed to the case where it can only relay , i.e., Prop osition 1, b oth users enjoy secrecy in Prop osition 3 , while user 1 do es no t hav e to compromise fro m its own secrecy r ate that is ac hiev able in the underlying ea v esdropp er c hannel. Remark 13 W e are no w ready to disc uss wh y we could not ﬁnd an o ut er b ound f o r the equiv o cation ra t e of user 1 that relies only on the c hannel inputs and outputs. T o understand this, w e ﬁrst examine the outer b ound w e f ound on the equiv o catio n rate of user 2 in Theorem 3. This outer bo und is obtained by giving the en tire observ atio n of user 1 to user 2 (i.e., N c = 0). Hence, this is the b est p ossible scenario as far as the channe l of user 2 is concerned, and thus , it yields an outer b ound. How ev er, a similar approac h cannot w ork for user 1, because altho ug h user 1 can hav e access to the observ a t ion of user 2, user 1 still has additional freedom (and opp ortunities) to increase its o wn secrecy rat e b y sending jamming signals ov er the co o p erativ e link, as show n in this section. This is the main reason wh y we could not ﬁnd a simple outer b ound for user 1’s secrec y rate using only the channel inputs/outputs. 8 Tw o-side d Co o p e ration In this section, w e provide an achie v able sche me f o r CRBC with tw o-sided co op era t io n. In this case, eac h user can act as a relay for the other o ne; see Figure 2 . The corresp onding c hannel consists of tw o message sets w 1 ∈ W 1 , w 2 ∈ W 2 , three input alphab ets, one at the transmitter x ∈ X , one at user 1 x 1 ∈ X 1 and one at user 2 x 2 ∈ X 2 . The c hannel consists of t w o output a lpha b ets denoted b y y 1 ∈ Y 1 , y 2 ∈ Y 2 at the tw o users. The c hannel is assumed to b e memoryless and its transition probabilit y distribution is p ( y 1 , y 2 | x, x 1 , x 2 ). A  2 nR 1 , 2 nR 2 , n  co de for this channel consists of t w o message set as W 1 =  1 , . . . , 2 nR 1  and W 2 =  1 , . . . , 2 nR 2  , an enco der at the transmitter whic h maps eac h pa ir ( w 1 , w 2 ) ∈ ( W 1 × W 2 ) to a co dew ord x n ∈ X n , a set of rela y functions at user 1, x 1 ,i = f 1 ,i ( y 1 , 1 , . . . , y 1 ,i − 1 ) , 1 ≤ i ≤ n, and a set of relay functions at user 2, x 2 ,i = f 2 ,i ( y 2 , 1 , . . . , y 2 ,i − 1 ) , 1 ≤ i ≤ n, t wo deco ders, one at user 1 and one at user 2 with the mappings g 1 : Y n 1 → W 1 , g 2 : Y n 2 → W 2 . 19 Deﬁnitions for the error probability fo r this t w o- sided case a r e the same as in the single - sided case. The secrecy of the users is again measured b y the equiv o cation rat es whic h a re 1 n H ( W 1 | Y n 2 , X n 2 ) and 1 n H ( W 2 | Y n 1 , X n 1 ). In this case, since user 2 has a c hannel input also, we condition t he en tropy rate of user 1’s messages on this c hannel input. A rate tuple ( R 1 , R 2 , R e, 1 , R e, 2 ) is said to be achie v able if there exists a  2 nR 1 , 2 nR 2 , n  co de with lim n →∞ P n e = 0 , and lim n →∞ 1 n H ( W 1 | Y n 2 , X n 2 ) ≥ R e, 1 , lim n →∞ 1 n H ( W 2 | Y n 1 , X n 1 ) ≥ R e, 2 (69) The f ollo wing theorem characterize s an achie v able region f o r this c hannel mo del. Theorem 5 The rate tuples ( R 1 , R 2 , R e, 1 , R e, 2 ) satisfying R 1 ≤ I ( V 1 ; Y 1 , ˆ Y 2 | X 1 , U 2 ) (70) R 2 ≤ I ( V 2 ; Y 2 , ˆ Y 1 | X 2 , U 1 ) (71) R 1 + R 2 ≤ I ( V 1 ; Y 1 , ˆ Y 2 | X 1 , U 2 ) + I ( V 2 ; Y 2 , ˆ Y 1 | X 2 , U 1 ) − I ( V 1 ; V 2 ) (72) R e, 1 ≤ R 1 (73) R e, 1 ≤ h I ( V 1 ; Y 1 , ˆ Y 2 | X 1 , U 2 ) − I ( V 1 ; Y 2 , ˆ Y 1 | V 2 , X 2 , U 1 ) − I ( V 1 ; V 2 ) i + (74) R e, 2 ≤ R 2 (75) R e, 2 ≤ h I ( V 2 ; Y 2 , ˆ Y 1 | X 2 , U 1 ) − I ( V 2 ; Y 1 , ˆ Y 2 | V 1 , X 1 , U 2 ) − I ( V 1 ; V 2 ) i + (76) are a c hiev able for an y distribution of the for m p ( v 1 , v 2 ) p ( x | v 1 , v 2 ) p ( u 1 , x 1 ) p ( ˆ y 1 | u 1 , y 1 ) p ( u 2 , x 2 ) p ( ˆ y 2 | u 2 , y 2 ) p ( y 1 , y 2 | x, x 1 , x 2 ) (77) sub ject to the follo wing constrain ts I ( ˆ Y 1 ; Y 1 | U 1 , X 1 , U 2 ) ≤ I ( ˆ Y 1 , U 1 ; Y 2 | X 2 ) (78) I ( ˆ Y 2 ; Y 2 | U 2 , X 2 , U 1 ) ≤ I ( ˆ Y 2 , U 2 ; Y 1 | X 1 ) (79) The pro of of this theorem is giv en in App endix E. Con trary to the previous a chiev able sc hemes giv en in Theorem 1 and 4, here users do not compress their observ ations af t er erasing their co dew ords from the observ a t io ns; this is why w e did not condition ˆ Y 1 (resp. ˆ Y 2 ) on V 1 (resp. V 2 ) in ( 7 7). In fact, they cannot remo v e their o wn co dew or ds from their observ at io ns b ecause each us er emplo ys a sliding-windo w type deco ding sc heme, i.e., they should wait until the next blo ck to deco de their o wn co dew ords, whereas compression should b e p erfo rmed righ t after the reception of the previous blo c k, at whic h time they hav e not y et deco ded their own messages. How eve r, w e no te that this ac hiev able sc heme also provides o pp ortunities for jamming as did the ac hiev able sc heme pro vided in Section 6. 20 9 Gaussian E xample for Two-sided Co o p eration The channel outputs of a Gaussian CRBC with tw o -sided co op eration are Y 1 = X + X 2 + Z 1 (80) Y 2 = X + X 1 + Z 2 (81) where Z 1 ∼ N (0 , N 1 ) , Z 2 ∼ N (0 , N 2 ) and are indep enden t, E [ X 2 ] ≤ P , E [ X 2 1 ] ≤ a 1 P , E [ X 2 2 ] ≤ a 2 P . W e presen t the following prop osition whic h c haracterizes an achiev able equiv o cation re- gion. Prop osition 5 The fo llo wing equiv o cation rates are ac hiev a ble for all ( α, β 1 , β 2 ) ∈ [0 , 1] 3 R e, 1 ≤ 1 2 log  1 + αP ( N 1 + a 2 ¯ β 2 P + N 2 + N c, 2 ) ¯ αP ( N 1 + a 2 ¯ β 2 P + N 2 + N c, 2 ) + ( N 1 + a 2 ¯ β 2 P )( N 2 + N c, 2 )  − 1 2 log  1 + α P  1 a 1 ¯ β 1 P + N 2 + 1 N 1 + N c, 1  (82) R e, 2 ≤ 1 2 log  1 + ¯ αP ( N 2 + a 1 ¯ β 1 P + N 2 + N c, 1 ) αP ( N 2 + a 1 ¯ β 1 P + N 1 + N c, 1 ) + ( N 2 + a 1 ¯ β 1 P )( N 1 + N c, 1 )  − 1 2 log  1 + α P  1 a 2 ¯ β 2 P + N 1 + 1 N 2 + N c, 2  (83) where ¯ α = 1 − α, ¯ β 1 = 1 − β 1 , ¯ β 2 = 1 − β 2 , and N c, 1 , N c, 2 are sub ject to N c, 1 ≥ − b 11 + p b 2 11 + 4 a 11 c 11 2 a 11 (84) N c, 2 ≥ − b 22 + p b 2 22 + 4 a 22 c 22 2 a 22 (85) and a 11 = a 1 β 1 P (86) b 11 = P  P + a 1 β 1 ( P + N 1 )  − ( P + N 1 + a 2 ¯ β 2 P )( P + N 2 + a 1 ¯ β 1 P ) (87) c 11 = ( P + N 1 + a 2 ¯ β 2 P )  P N 1 + ( P + N 1 )( N 2 + a 1 ¯ β 1 P )  (88) a 22 = a 2 β 2 P (89) b 22 = P  P + a 2 β 2 ( P + N 2 )  − ( P + N 1 + a 2 ¯ β 2 P )( P + N 2 + a 1 ¯ β 1 P ) (90) c 22 = ( P + N 2 + a 1 ¯ β 1 P )  P N 2 + ( P + N 2 )( N 1 + a 2 ¯ β 2 P )  (91) Pro of: This ac hiev able region is o btained by selecting X = V 1 + V 2 where V 1 ∼ N (0 , α P ), V 2 ∼ N (0 , ¯ αP ) and are independen t, X i = U i + ˜ Z i where U i ∼ N (0 , a i β i P ), ˜ Z i ∼ N (0 , a i ¯ β i P ) , 21 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 R e,1 (bits/channel use) R e,2 (bits/channel use) a 1 = a 2 = 5 a 1 = a 2 = 10 a 1 = a 2 = 50 a 1 = a 2 = 100 Figure 9: Ac hiev able equiv o cation ra te region using Prop osition 5 where eac h user can join tly jam and relay . P = 8 , N 1 = 1 , N 2 = 2, i.e., user 2 cannot ha v e an y p ositiv e secrecy in the underlying BC. i = 1 , 2 and independen t, and ˆ Y i = Y i + Z c,i where Z c,i ∼ N (0 , N c,i ) , i = 1 , 2 a nd are indep enden t of all other random v ariables. Direct calculation o f rates in Theorem 5 with these ra ndom v ariable selections yields the ach iev able region.  A n umerical example is given in F igure 9 for the case P = 8 , N 1 = 1 , N 2 = 2. Comparing Figure 9 with Figures 7 and 8, w e observ e that user 2’s secrecy rate improv es signiﬁcantly b ecause now user 2 can jam user 1 to impro ve its own secrecy rate. W e also o bserv e that user 1’s secrecy rat e impro v es as w ell, compared to Section 7. The increase in user 1’s secre cy in this tw o-sided case is due to the f a ct that user 2 no w a cts as a relay fo r user 1. How ev er, when user 1 j a ms user 2 using all of its p o we r, it limits the help that comes from user 2, hence Theorem 5 prov ides o nly a mo dest se crecy rat e increase f o r use r 1 on top of what Theorem 4 already provides . 10 Concl u sions In this pap er, w e inv estigated the eﬀects of co op eration on secrecy . W e show ed that user co op eratio n can increase secrecy , i.e., ev en an untruste d party can help. An imp ortant p o int to observ e though is tha t whether co op eratio n can improv e secrecy or not dep ends o n the co op eratio n metho d employ ed. F or instance, ev en though a deco de-a nd- forw ard (DA F ) based co op eratio n sc heme can increase the ra te, it cannot impro ve secrecy , b ecause in this case the co op erating part y , whic h is also the eav esdropp er, needs to deco de the message it forw ar ds. 22 Ho w ev er, in CAF, we do not require the co o p erating party to deco de the message. In fact, in CAF, the co op erat ing part y helps increase the rate of the main transmitter to lev els whic h it itse lf cannot dec o de, hence impro ving the secrecy of the main transmitter-receiv er pair against itself. App endices A Pro o f of Theo rem 2 Here w e pro ve the outer b ound o n the capacit y-equiv o cation region of the CRBC given in Theorem 2 whic h closely follow s the con v erse giv en in [3] and the outer b o und in [10]. First, deﬁne the follow ing random v ariables U i = Y i − 1 1 Y n 2 ,i +1 (92) V 1 ,i = W 1 U i (93) V 2 ,i = W 2 U i (94) whic h satisfy t he follow ing Mark ov c hain U i → ( V 1 ,i , V 2 ,i ) → ( X i , X 1 ,i , Y 1 ,i ) → Y 2 ,i (95) but do not satisfy the f o llo wing one U i → ( V 1 ,i , V 2 ,i ) → ( X i , X 1 ,i ) → ( Y 1 ,i , Y 2 ,i ) (96) b ecause of the enco ding function emplo ye d at user 1 whic h can generate correlation b etw een Y 1 ,i and  Y n 1 ,i +1 , Y n 2 ,i +1  through X 1 ,i +1 that cannot b e resolv ed by conditioning on ( X i , X 1 ,i ). F or a similar discussion, the reader can refer to [19]. 23 W e start with the ac hiev able rate of user 1. nR 1 = H ( W 1 ) = I ( W 1 ; Y n 1 ) + H ( W 1 | Y n 1 ) (97) ≤ I ( W 1 ; Y n 1 ) + ǫ n (98) = n X i =1 I ( W 1 ; Y 1 ,i | Y i − 1 1 ) + ǫ n (99) = n X i =1 H ( W 1 | Y i − 1 1 ) − H ( W 1 | Y i − 1 1 , Y 1 ,i ) + ǫ n (100) = n X i =1 H ( W 1 | Y i − 1 1 , X 1 ,i ) − H ( W 1 | Y i − 1 1 , Y 1 ,i ) + ǫ n (101) ≤ n X i =1 H ( W 1 | Y i − 1 1 , X 1 ,i ) − H ( W 1 | Y i − 1 1 , Y 1 ,i , X 1 ,i ) + ǫ n (102) = n X i =1 I ( W 1 ; Y 1 ,i | Y i − 1 1 , X 1 ,i ) + ǫ n (103) ≤ n X i =1 H ( Y 1 ,i | X 1 ,i ) − H ( Y 1 ,i | Y i − 1 1 , X 1 ,i , W 1 ) + ǫ n (104) ≤ n X i =1 H ( Y 1 ,i | X 1 ,i ) − H ( Y 1 ,i | Y i − 1 1 , X 1 ,i , W 1 , Y n 2 ,i +1 ) + ǫ n (105) = n X i =1 I ( V 1 ,i ; Y 1 ,i | X 1 ,i ) + ǫ n (106) where (98) is due to F ano’s lemma, (101) follow s fro m the Marko v c hain W 1 → Y i − 1 1 → X 1 ,i , (102), (104) and (105) are due to the fact that conditioning cannot increase en t r o p y , and (106) follows from the deﬁnition of V 1 ,i in (93). Similarly , for the achie v able ra te of user 2, w e ha v e nR 2 ≤ I ( W 2 ; Y n 2 ) + ǫ n (107) = n X i =1 I ( W 2 ; Y 2 ,i | Y n 2 ,i +1 ) + ǫ n (108) = n X i =1 H ( Y 2 ,i | Y n 2 ,i +1 ) − H ( Y 2 ,i | Y n 2 ,i +1 , W 2 ) + ǫ n (109) ≤ n X i =1 H ( Y 2 ,i ) − H ( Y 2 ,i | Y n 2 ,i +1 , W 2 , Y i − 1 1 ) + ǫ n (110) ≤ n X i =1 I ( V 2 ,i ; Y 2 ,i ) + ǫ n (111) where (107) is due to F ano’s lemma, (110) is due to the fact that conditioning cannot increase en tropy , and (1 11) follows from the deﬁnition of V 2 ,i giv en in ( 9 4). 24 W e now deriv e the outer b ounds on the equiv o cation rates. W e start with user 1. nR e, 1 = H ( W 1 | Y n 2 ) = H ( W 1 ) − I ( W 1 ; Y n 2 ) (112) = I ( W 1 ; Y n 1 ) − I ( W 1 ; Y n 2 ) + H ( W 1 | Y n 1 ) (113) ≤ I ( W 1 ; Y n 1 ) − I ( W 1 ; Y n 2 ) + ǫ n (114) = n X i =1 I ( W 1 ; Y 1 ,i | Y i − 1 1 ) − I ( W 1 ; Y 2 ,i | Y n 2 ,i +1 ) + ǫ n (115) = n X i =1 I ( W 1 , Y n 2 ,i +1 ; Y 1 ,i | Y i − 1 1 ) − I ( Y n 2 ,i +1 ; Y 1 ,i | Y i − 1 1 , W 1 ) − I ( W 1 , Y i − 1 1 ; Y 2 ,i | Y n 2 ,i +1 ) + I ( Y i − 1 1 ; Y 2 ,i | Y n 2 ,i +1 , W 1 ) + ǫ n (116) where (1 14) is due to F ano’s lemma. Using [3] n X i =1 I ( Y n 2 ,i +1 ; Y 1 ,i | Y i − 1 1 , W 1 ) = n X i =1 I ( Y i − 1 1 ; Y 2 ,i | Y n 2 ,i +1 , W 1 ) (117) in (1 16), we obtain nR e, 1 ≤ n X i =1 I ( W 1 , Y n 2 ,i +1 ; Y 1 ,i | Y i − 1 1 ) − I ( W 1 , Y i − 1 1 ; Y 2 ,i | Y n 2 ,i +1 ) + ǫ n (118) = n X i =1 I ( W 1 ; Y 1 ,i | Y i − 1 1 , Y n 2 ,i +1 ) + I ( Y n 2 ,i +1 ; Y 1 ,i | Y i − 1 1 ) − I ( W 1 ; Y 2 ,i | Y n 2 ,i +1 , , Y i − 1 1 ) − I ( Y i − 1 1 ; Y 2 ,i | Y n 2 ,i +1 ) + ǫ n (119) No w, using [3 ] n X i =1 I ( Y n 2 ,i +1 ; Y 1 ,i | Y i − 1 1 ) = n X i =1 I ( Y i − 1 1 ; Y 2 ,i | Y n 2 ,i +1 ) (120) in (1 19), we obtain nR e, 1 ≤ n X i =1 I ( W 1 ; Y 1 ,i | Y i − 1 1 , Y n 2 ,i +1 ) − I ( W 1 ; Y 2 ,i | Y n 2 ,i +1 , Y i − 1 1 ) + ǫ n (121) = n X i =1 I ( W 1 ; Y 1 ,i | U i ) − I ( W 1 ; Y 2 ,i | U i ) + ǫ n (122) = n X i =1 I ( W 1 , U i ; Y 1 ,i | U i ) − I ( W 1 , U i ; Y 2 ,i | U i ) + ǫ n (123) = n X i =1 I ( V 1 ,i ; Y 1 ,i | U i ) − I ( V 1 ,i ; Y 2 ,i | U i ) + ǫ n (124) where (122) and (124) follow f rom the deﬁnitions of U i and V 1 ,i giv en in (92) and (93), 25 resp ectiv ely . Similarly , w e can use the preceding tec hnique for user 2’s equiv o catio n rate as w ell after noting that nR e, 2 ≤ H ( W 2 | Y n 1 , X n 1 ) ≤ H ( W 2 | Y n 1 ) (125) whic h leads to nR e, 2 ≤ n X i =1 I ( V 2 ,i ; Y 2 ,i | U i ) − I ( V 2 ,i ; Y 1 ,i | U i ) + ǫ n (126) The o ther b ounds o n the equiv o cation rates can b e deriv ed as follows. nR e, 1 = H ( W 1 | Y n 2 ) ≤ H ( W 1 , W 2 | Y n 2 ) (127) = H ( W 1 | W 2 , Y n 2 ) + H ( W 2 | Y n 2 ) (128) ≤ H ( W 1 | W 2 , Y n 2 ) + ǫ n (129) = I ( W 1 ; Y n 1 | W 2 ) − I ( W 1 ; Y n 2 | W 2 ) + H ( W 1 | W 2 , Y n 1 ) + ǫ n (130) ≤ I ( W 1 ; Y n 1 | W 2 ) − I ( W 1 ; Y n 2 | W 2 ) + ǫ ′ n (131) = n X i =1 I ( W 1 ; Y 1 ,i | W 2 , Y i − 1 1 ) − I ( W 1 ; Y 2 ,i | W 2 , Y n 2 ,i +1 ) + ǫ ′ n (132) = n X i =1 I ( W 1 , Y n 2 ,i +1 ; Y 1 ,i | W 2 , Y i − 1 1 ) − I ( W 1 , Y i − 1 1 ; Y 2 ,i | W 2 , Y n 2 ,i +1 ) + ǫ ′ n (133) = n X i =1 I ( W 1 ; Y 1 ,i | W 2 , Y i − 1 1 , Y n 2 ,i +1 ) − I ( W 1 ; Y 2 ,i | W 2 , Y n 2 ,i +1 , Y i − 1 1 ) + ǫ ′ n (134) = n X i =1 I ( W 1 ; Y 1 ,i | W 2 , U i ) − I ( W 1 ; Y 2 ,i | W 2 , U i ) + ǫ ′ n (135) = n X i =1 I ( W 1 , U i ; Y 1 ,i | W 2 , U i ) − I ( W 1 , U i ; Y 2 ,i | W 2 , U i ) + ǫ ′ n (136) = n X i =1 I ( V 1 ,i ; Y 1 ,i | V 2 ,i ) − I ( V 1 ,i ; Y 2 ,i | V 2 ,i ) + ǫ ′ n (137) where (129) and (13 1) are due to F ano’s lemma, and (133) and (134 ) a r e due to the follow ing iden tities [3] n X i =1 I ( Y n 2 ,i +1 ; Y 1 ,i | W 1 , W 2 , Y i − 1 1 ) = n X i =1 I ( Y i − 1 1 ; Y 2 ,i | W 1 , W 2 , Y n 2 ,i +1 ) (138) n X i =1 I ( Y n 2 ,i +1 ; Y 1 ,i | W 2 , Y i − 1 1 ) = n X i =1 I ( Y i − 1 1 ; Y 2 ,i | W 2 , Y n 2 ,i +1 ) (139) resp ectiv ely . Finally , (135) and (137) follow from the deﬁnitions of U i , V 1 ,i and V 2 ,i giv en 26 in (92), (93) and (94), resp ectiv ely . Similarly , w e can use this tec hnique to b ound user 2’s equiv o cation rate after noting that H ( W 2 | Y n 1 , X n 1 ) ≤ H ( W 2 | Y n 1 ), whic h leads to nR e, 2 ≤ H ( W 2 | Y n 1 , X n 1 ) ≤ H ( W 2 | Y n 1 ) ≤ n X i =1 I ( V 2 ,i ; Y 2 ,i | V 1 ,i ) − I ( V 2 ,i ; Y 2 ,i | V 1 ,i ) + ǫ ′ n (140) T o expres s the outer b ounds obtained ab o ve in a single-letter form, we deﬁne U = J U J , V 1 = V 1 ,J , V 2 = V 2 ,J , X = X J , X 1 = X 1 ,J , Y 1 = Y 1 ,J , Y 2 = Y 2 ,J where J is a random v ariable which is uniformly distributed o ve r { 1 , . . . , n } . Using these new deﬁnitions, w e can reac h the single-letter expressions giv en in Theorem 2, hence completing the pro of. B Pro of o f The o rem 3 The pro of is as follows. R e, 2 ≤ H ( W 2 | Y n 1 , X n 1 ) ≤ I ( W 2 ; Y n 2 | X n 1 ) − I ( W 2 ; Y n 1 | X n 1 ) + H ( W 2 | Y n 2 , X n 1 ) (141) ≤ I ( W 2 ; Y n 2 | X n 1 ) − I ( W 2 ; Y n 1 | X n 1 ) + ǫ n (142) ≤ I ( W 2 ; Y n 2 | X n 1 , Y n 1 ) + ǫ n (143) ≤ I ( X n , W 2 ; Y n 2 | X n 1 , Y n 1 ) + ǫ n (144) = I ( X n ; Y n 2 | X n 1 , Y n 1 ) + ǫ n (145) = n X i =1 I ( X n ; Y 2 ,i | X n 1 , Y n 1 , Y i − 1 2 ) + ǫ n (146) ≤ n X i =1 H ( Y 2 ,i | X 1 ,i , Y 1 ,i ) − H ( Y 2 ,i | X n 1 , Y n 1 , Y i − 1 2 , X n ) + ǫ n (147) = n X i =1 H ( Y 2 ,i | X 1 ,i , Y 1 ,i ) − H ( Y 2 ,i | X 1 ,i , Y 1 ,i , X i ) + ǫ n (148) = n X i =1 I ( X i ; Y 2 ,i | X 1 ,i , Y 1 ,i ) + ǫ n (149) where (142) is due to F ano’s lemma, (145) fo llows from the fact that giv en X n , W 2 is indep enden t of a ll other random v ariables, (147) is due to the f act that conditioning cannot increase entrop y , a nd (148) fo llo ws from the Mark ov c hains ( Y 1 ,i , Y 2 ,i ) → ( X i , X 1 ,i ) → ( Y i − 1 1 , Y i − 1 2 , X i − 1 , X i − 1 1 ) (150) Y 2 ,i → ( X i , X 1 ,i , Y 1 ,i ) → ( Y n 1 ,i +1 , X n i +1 , X n 1 ,i +1 ) (151) Th us, a fter deﬁning an indep enden t random v a riable J , tha t is uniformly distributed ov er { 1 , . . . , n } , and X = X J , X 1 = X 1 ,J , Y 1 = Y 1 ,J , Y 2 = Y 2 ,J , w e can obta in the single-letter expression in Theorem 3, completing the pro of. 27 C Pro of o f C orollary 1 In Prop ositions 1 and 2, if we take a → ∞ , then the secrecy rate in (49) can b e sho wn to b e ac hiev able. As a notational r emark, H ( · ) denotes the diﬀeren tial entrop y in this section. W e now compute an outer b ound for R e, 2 using Theorem 3, R e, 2 ≤ I ( X ; Y 2 | X 1 , Y 1 ) (152) = H ( Y 2 | X 1 , Y 1 ) − H ( Z 2 | Z 1 ) (153) ≤ H ( X + Z 2 | Y 1 ) − H ( Z 2 ) (154) ≤ H ( X + Z 2 − α Y 1 ) − 1 2 log(2 π eN 2 ) (155) ≤ 1 2 log(2 π e ) E  ( X + Z 2 − αY 1 ) 2  − 1 2 log(2 π eN 2 ) (156) ≤ 1 2 log  (1 − α ) 2 P + α 2 N 1 + N 2  − 1 2 log( N 2 ) (157) where in ( 1 54), w e used the fact that conditioning cannot increase entrop y a nd that H ( Z 2 | Z 1 ) = H ( Z 2 ) due to the indep endence of Z 1 and Z 2 . Equation (15 5) is a gain due to the f a ct that conditioning cannot increase en tropy , (156) comes from the fact that Gaussian distribution maximizes en t r op y sub ject to a p o we r constraint, and (157) is obtained b y using the p ow er constrain t on X . Finally , w e note that ( 1 57) is a v alid outer b ound fo r ev ery α and if we select α as α = P P + N 1 (158) w e get (49), completing the pro o f. D Pro of o f The o rem 4 The transmitter uses the join t encoding sc heme of Marton [22] and user 1 uses a CAF sc heme [1 7]. User 2 employ s list deco ding to ﬁnd whic h ˆ Y 1 is sen t. Let A n ǫ ( V 1 ) and A n ǫ ( V 2 ) denote the sets of strongly ty pical i.i.d. length- n sequences of v 1 and v 2 , resp ectiv ely . Let A n ǫ ( V 1 | v 2 ) (resp. A n ǫ ( V 2 | v 1 )) denote the set o f length- n seque nces V 1 (resp. V 2 ) that are join tly ty pical with v 2 (resp. v 1 ). F urthermore, let S n ǫ ( v 1 ) (resp. S n ǫ ( v 2 )) denote the set of v 1 (resp. v 2 ) seque nces for whic h A n ǫ ( V 2 | v 1 ) (resp. A n ǫ ( V 1 | v 2 )) are non-empt y . Fix the probabilit y distribution as p ( v 1 , v 2 ) p ( x | v 1 , v 2 ) p ( u, x 1 ) p ( ˆ y 1 | u, v 1 , y 1 ) (159) Co deb o ok st ructure: 28 1. Select 2 nR ( V i ) v i sequence s thr o ugh p ( v i ) =    1 || S n ǫ ( v i ) || , if v i ∈ S n ǫ ( v i ) 0 , ot herwise (160) in a n i.i.d. manner and index them as v i ( w i , ˜ w i , l i ) where w i ∈  1 , . . . , 2 nR i  , ˜ w i ∈ { 1 , . . . , 2 n ˜ R i } and l i ∈  1 , . . . , 2 nL i  for i = 1 , 2. R i , ˜ R i , L i and R ( V i ) are related through R ( V i ) = R i + ˜ R i + L i , i = 1 , 2 (161) F urthermore, w e set L 1 + L 2 = I ( V 1 ; V 2 ) + ǫ (162) to ensure that for g iven pairs ( w 1 , ˜ w 1 ) and ( w 2 , ˜ w 2 ), w e can ﬁnd a jointly t ypical pair ( v 1 ( w 1 , ˜ w 1 , l 1 ) , v 2 ( w 2 , ˜ w 2 , l 2 )) for some l 1 , l 2 . 2. F or each ( w 1 , w 2 ), the transmitter randomly picks ( ˜ w 1 , ˜ w 2 ) and ﬁnds a pa ir ( v 1 ( w 1 , ˜ w 1 , l 1 ) , v 2 ( w 2 , ˜ w 2 , l 2 )) that is join tly t ypical. Such a pair exists with high probabilit y due to (162). Then, giv en this pair of ( v 1 , v 2 ), the transmitter generates its c hannel inputs through Q n i =1 p ( x i | v 1 ,i , v 2 ,i ). 3. User 1 generates 2 nR 0 length- n sequences u through p ( u ) = Q n i =1 p ( u i ) and lab els them as u ( s i ) where s i ∈ { 1 , . . . , 2 nR 0 } . 4. F or each u ( s i ), us er 1 generates 2 n ˆ R length- n sequences ˆ y 1 through p ( ˆ y 1 | u ) = Q n i =1 p ( ˆ y 1 ,i | u i ) and indexes them as ˆ y 1 ( z i | s i ) where z i ∈ { 1 , . . . , 2 n ˆ R } . 5. F or each u ( s i ), user 1 generates 2 nR ′ 0 length- n sequences x 1 through p ( x 1 | u ) = Q n i =1 p ( x 1 ,i | u i ) and indexes them as x 1 ( t i | s i ) where t i ∈ { 1 , . . . , 2 nR ′ 0 } . P artit ioning: • P artitio n 2 n ˆ R in to cells S s i where s i ∈ { 1 , . . . , 2 nR 0 } . Enco ding: The tra nsmitter sends x corresp onding t o the pair ( w 1 , w 2 ). Us er 1 (rela y) sends x 1 ( t i | s i ) if the estimate of y 1 ( i − 1), i.e., ˆ z i − 1 , falls in to S s i and t i is c hosen randomly f rom { 1 , . . . , 2 nR ′ 0 } . The use of many x 1 ( t i | s i ) for a ctual help signal u ( s i ) aims t o confuse user 2 and to decrease its deco ding capability . Deco ding: 29 a. Deco ding at user 1: 1. User 1 se eks a unique t ypical pair of ( y 1 ( i ) , v 1 ( w 1 ,i , ˜ w 1 ,i , l i ) , x 1 ( t i | s i )) whic h can b e ac hiev ed with v anishingly small error probability if R ( V 1 ) ≤ I ( V 1 ; Y 1 | X 1 ) (163) 2. User 1 decides that z i is receiv ed if there exists a jointly typic al pair ( ˆ y 1 ( z i | s i ) , y 1 ( i ) , v 1 ( w 1 ,i , ˜ w 1 ,i , l i ) , x 1 ( t i | s i )) whic h can b e g uaran teed to o ccur if ˆ R ≥ I ( ˆ Y 1 ; Y 1 | U, X 1 , V 1 ) (164) b. Deco ding at user 2: 1. User 2 seeks a unique join tly t ypical pair of ( y 2 ( i ) , u ( s i )) whic h can b e fo und with v anishingly small error probability if R 0 ≤ I ( U ; Y 2 ) (165) 2. User 2 employs list deco ding t o deco de ˆ y 1 ( z i − 1 | s i − 1 ). It ﬁrst calculates its ambiguit y set as L ( ˆ y 1 ( z i − 1 | ˆ s i − 1 )) = { ˆ y 1 ( z i − 1 | ˆ s i − 1 ) : ( ˆ y 1 ( z i − 1 | ˆ s i − 1 ) , y 2 ( i − 1)) is join tly t ypical } (166) and ta k es its in tersection with S ˆ s i whic h results in a unique and correct in tersection p oin t if ˆ R ≤ I ( ˆ Y 1 ; Y 2 | U ) + R 0 ≤ I ( ˆ Y 1 , U ; Y 2 ) (167) Equations (164) and (167) lead to the compression constraint in (58). 3. User 2 decides that v 2 ( w 2 ,i − 1 , ˜ w 2 ,i − 1 , l 2 ,i − 1 ) is receiv ed if there exists a unique jointly t ypical pair ( v 2 ( w 2 ,i − 1 , ˜ w 2 ,i − 1 , l 2 ,i − 1 ) , y 2 ( i − 1) , ˆ y 1 ( ˆ z i − 1 | ˆ s i − 1 )), whic h can b e found with v anishingly small error probability if R ( V 2 ) ≤ I ( V 2 ; Y 2 , ˆ Y 1 | U ) (168) Equiv o cation computation: W e no w sho w t ha t R e, 1 and R e, 2 satisfying (53)-(54) and (55)-(56) are ac hiev able with the co ding sc heme presen ted. T o this end, w e treat sev eral p ossible cases separately . First, 30 assume that R 1 ≥ I ( V 1 ; Y 1 | X 1 ) − I ( V 1 ; Y 2 , ˆ Y 1 | V 2 , U ) − I ( V 1 ; V 2 ) (169) R 2 ≥ I ( V 2 ; Y 2 , ˆ Y 1 | U ) − I ( V 2 ; Y 1 | V 1 , X 1 ) − I ( V 1 ; V 2 ) (170) F or t his case, w e select the tot a l nu mber of co dew ords, i.e., R ( V i ) , i = 1 , 2, as R ( V 1 ) = I ( V 1 ; Y 1 | X 1 ) (171) R ( V 2 ) = I ( V 2 ; Y 2 , ˆ Y 1 | U ) (172) With this selection, we hav e ˜ R 1 + L 1 ≤ I ( V 1 ; Y 2 , ˆ Y 1 | V 2 , U ) + I ( V 1 ; V 2 ) (173) ˜ R 2 + L 2 ≤ I ( V 2 ; Y 1 | V 1 , X 1 ) + I ( V 1 ; V 2 ) (174) W e start with user 1’s equiv o cation rate, H ( W 1 | Y n 2 ) ≥ H ( W 1 | Y n 2 , V n 2 , U n , ˆ Y n 1 ) (175) = H ( W 1 , Y n 2 , V n 2 , ˆ Y n 1 | U n ) − H ( Y n 2 , V n 2 , ˆ Y n 1 | U n ) (176) = H ( V n 1 , W 1 , Y n 2 , V n 2 , ˆ Y n 1 | U n ) − H ( V n 1 | W 1 , Y n 2 , V n 2 , ˆ Y n 1 , U n ) − H ( Y n 2 , V n 2 , ˆ Y n 1 | U n ) (177) = H ( V n 1 | U n ) + H ( W 1 , Y n 2 , V n 2 , ˆ Y n 1 | U n , V n 1 ) − H ( V n 1 | W 1 , Y n 2 , V n 2 , ˆ Y n 1 , U n ) − H ( Y n 2 , V n 2 , ˆ Y n 1 | U n ) (178) ≥ H ( V n 1 | U n ) − I ( V n 1 ; Y n 2 , V n 2 , ˆ Y n 1 | U n ) − H ( V n 1 | W 1 , Y n 2 , V n 2 , ˆ Y n 1 , U n ) (179) where each term will b e treated separately . First term is H ( V n 1 | U n ) = H ( V n 1 ) = nR ( V 1 ) = nI ( V 1 ; Y 1 | X 1 ) (180) where the ﬁrst equalit y is due to the indep endence of U n and V n 1 . The second equality follo ws fr om the fact that V n 1 can tak e 2 nR ( V 1 ) v alues with equal pro babilit y . The third equalit y comes fr o m our selection in (171). The second term of (1 79) can b e b ounded as I ( V n 1 ; Y n 2 , V n 2 , ˆ Y n 1 | U n ) ≤ nI ( V 1 ; Y 2 , V 2 , ˆ Y 1 | U ) + nǫ n (181) using t he approach devised in Lemma 3 o f [10 ]. T o b ound t he last term in (179 ), we assume that user 2 is tr ying to deco de V n 1 giv en the side information W 1 = w 1 . Since V n 1 can tak e less than 2 n ( I ( V 1 ; Y 2 , ˆ Y 1 | U,V 2 )+ I ( V 1 ; V 2 )) v alues (see (173)) giv en W 1 = w 1 , user 2 can deco de V n 1 with v anishingly small error probabilit y as long as W 1 = w 1 is given . Consequen tly , the use 31 of F ano’s lemma yields H ( V n 1 | W 1 , Y n 2 , V n 2 , ˆ Y n 1 , U n ) ≤ ǫ n (182) Plugging (180), (181) and (182) in to (179), w e get H ( W 1 | Y n 2 ) ≥ nI ( V 1 ; Y 1 | X 1 ) − nI ( V 1 ; Y 2 , ˆ Y 1 , V 2 | U ) − nǫ n (183) = nI ( V 1 ; Y 1 | X 1 ) − nI ( V 1 ; Y 2 , ˆ Y 1 | V 2 , U ) − nI ( V 1 ; V 2 ) − nǫ n (184) where (184) f o llo ws from the indep endence of ( V 1 , V 2 ) a nd U , i.e., I ( V 1 ; V 2 | U ) = I ( V 1 ; V 2 ). Similarly , w e can b ound equiv o cation of user 2 as follow s, H ( W 2 | Y n 1 , X n 1 ) ≥ H ( W 2 | Y n 1 , X n 1 , V n 1 ) (185) = H ( W 2 , Y n 1 , V n 1 | X n 1 ) − H ( Y n 1 , V n 1 | X n 1 ) (186) = H ( W 2 , V n 2 , Y n 1 , V n 1 | X n 1 ) − H ( V n 2 | W 2 , Y n 1 , V n 1 , X n 1 ) − H ( Y n 1 , V n 1 | X n 1 ) (187) = H ( V n 2 | X n 1 ) + H ( W 2 , Y n 1 , V n 1 | X n 1 , V n 2 ) − H ( V n 2 | W 2 , Y n 1 , V n 1 , X n 1 ) − H ( Y n 1 , V n 1 | X n 1 ) (188) ≥ H ( V n 2 | X n 1 ) − I ( V n 2 ; Y n 1 , V n 1 | X n 1 ) − H ( V n 2 | W 2 , Y n 1 , V n 1 , X n 1 ) (189) where the ﬁrst t erm is H ( V n 2 | X n 1 ) = H ( V n 2 ) = nR ( V 2 ) = nI ( V 2 ; Y 2 , ˆ Y 1 | U ) (190) where the ﬁrst equality is due t o the indep endence of V n 2 and X n 1 , the second equality comes from the f a ct that V n 2 can tak e 2 nR ( V 2 ) v alues with equal probability and the last equalit y is a consequence of o ur c hoice in (1 7 2). The second term of (189) can b e b ounded as I ( V n 2 ; Y n 1 , V n 1 | X n 1 ) ≤ nI ( V 2 ; Y 1 , V 1 | X 1 ) + nǫ n (191) follo wing the approac h of Lemma 3 of [10]. T o b ound the la st term of (189 ), w e assume that user 1 is trying to deco de V n 2 giv en the side information W 2 = w 2 . Since V n 2 can take at most 2 n ( I ( V 2 ; Y 1 | V 1 ,X 1 )+ I ( V 2 ; V 1 )) v alues (see (174)) giv en W 2 = w 2 , user 1 can deco de V n 2 with v anishingly small error probability a s long as this side information is a v ailable. Cons equen tly , the use of F ano’s lemma yields H ( V n 2 | W 2 , Y n 1 , V n 1 , X n 1 ) ≤ ǫ n (192) 32 Plugging (190), (191) and (192) in to (189), w e get H ( W 2 | Y n 1 , X n 1 ) ≥ nI ( V 2 ; Y 2 , ˆ Y 1 | U ) − nI ( V 2 ; Y 1 , V 1 | X 1 ) − nǫ n (193) = nI ( V 2 ; Y 2 , ˆ Y 1 | U ) − nI ( V 2 ; Y 1 | V 1 , X 1 ) − nI ( V 1 ; V 2 ) − nǫ n (194) where (1 94) follows f r om t he independence of ( V 1 , V 2 ) and X 1 , i.e., I ( V 1 ; V 2 | X 1 ) = I ( V 1 ; V 2 ). W e hav e completed the equiv o catio n calculation for the case describ ed b y (169)-(170). The pro ofs of o ther cases in volv e no diﬀeren t arguments b esides decreasing the total num b er co dew ords in (171)-(172). F or example, if R 1 ≤ I ( V 1 ; Y 1 | X 1 ) − I ( V 1 ; Y 2 , ˆ Y 1 | V 2 , U ) − I ( V 1 ; V 2 ) (195) then we select the total n um b er of co dew ords for user 1 a s R ( V 1 ) = R 1 + I ( V 1 ; Y 2 , ˆ Y 1 | V 2 , U ) + I ( V 1 ; V 2 ) (196) whic h is equiv alent to sa ying that ˜ R 1 + L 1 = I ( V 1 ; Y 2 , ˆ Y 1 | V 2 , U ) + I ( V 1 ; V 2 ) (197) In this case, following the steps from (175) to (179), w e can b o und the equiv o cation of user 1 as follo ws, H ( W 1 | Y n 2 ) ≥ H ( V n 1 | U n ) − I ( V n 1 ; Y n 2 , V n 2 , ˆ Y n 1 | U n ) − H ( V n 1 | W 1 , Y n 2 , V n 2 , ˆ Y n 1 , U n ) (198) where the ﬁrst t erm is now H ( V n 1 | U n ) = H ( V n 1 ) = nR ( V 1 ) = n ( R 1 + I ( V 1 ; Y 2 , ˆ Y 1 | V 2 , U ) + I ( V 1 ; V 2 )) (199) where the ﬁrst equalit y is due to the independence of V n 1 and U n , the second eq uality is due to the fact that V n 1 can tak e at most 2 nR ( V 1 ) v alues with equal pro ba bilit y and the la st equalit y is a consequence of our choice in (19 6). An upp er b ound o n the second term w as already obtained in (181). The third term can a lso b e shown to deca y to zero as n go es to inﬁnit y considering the case that user 2 is deco ding V n 1 using side informatio n W 1 = w 1 . Since V n 1 can take 2 n ( I ( V 1 ; Y 2 , ˆ Y 1 | V 2 ,U )+ I ( V 1 ; V 2 )) v alues given W 1 = w 1 , user 2 can deco de V n 2 with v anishingly small error probability a s long as this side information is av ailable. Therefore, the use of F ano’s lemma implies H ( V n 1 | W 1 , Y n 2 , V n 2 , ˆ Y n 1 , U n ) ≤ ǫ n (200) 33 Plugging (181),(199), (200) in to (198), we get H ( W 1 | Y n 2 ) ≥ n ( R 1 + I ( V 1 ; Y 2 , ˆ Y 1 | V 2 , U ) + I ( V 1 ; V 2 )) − I ( V 1 ; Y 2 , V 2 , ˆ Y 1 | U ) − nǫ n (201) = nR 1 − nǫ n (202) where w e used the fact that U and ( V 1 , V 2 ) are indep enden t, i.e., I ( V 1 ; V 2 | U ) = I ( V 1 ; V 2 ). Th e other cases leading to diﬀeren t equiv o catio n rates can b e prov ed similarly , hence omitted. E Pro o f of Theo rem 5 Fix the probability distribution as p ( v 1 , v 2 ) p ( x | v 1 , v 2 ) p ( u 1 , x 1 ) p ( ˆ y 1 | u 1 , y 1 ) p ( u 2 , x 2 ) p ( ˆ y 2 | u 2 , y 2 ) (203) Co deb o ok st ructure: 1. Select 2 nR ( V i ) v i sequence s thr o ugh p ( v i ) =    1 || S n ǫ ( v i ) || , if v i ∈ S n ǫ ( v i ) 0 , ot herwise (204) in a n i.i.d. manner and index them as v i ( w i , ˜ w i , l i ) where w i ∈  1 , . . . , 2 nR i  , ˜ w i ∈ { 1 , . . . , 2 n ˜ R i } and l i ∈  1 , . . . , 2 nL i  for i = 1 , 2. R i , ˜ R i , L i and R ( V i ) are related through R ( V i ) = R i + ˜ R i + L i , i = 1 , 2 (205) F urthermore, w e set L 1 + L 2 = I ( V 1 ; V 2 ) + ǫ (206) to ensure that for g iven pairs ( w 1 , ˜ w 1 ) and ( w 2 , ˜ w 2 ), w e can ﬁnd a jointly t ypical pair ( v 1 ( w 1 , ˜ w 1 , l 1 ) , v 2 ( w 2 , ˜ w 2 , l 2 )) for some l 1 , l 2 . 2. F or each ( w 1 , w 2 ), the transmitter randomly picks ( ˜ w 1 , ˜ w 2 ) and ﬁnds a pa ir ( v 1 ( w 1 , ˜ w 1 , l 1 ) , v 2 ( w 2 , ˜ w 2 , l 2 )) that is join tly t ypical. Such a pair exists with high probabilit y due to (206). Then, giv en this pair of ( v 1 , v 2 ), the transmitter generates its c hannel inputs through Q n i =1 p ( x i | v 1 ,i , v 2 ,i ). 3. User j generates 2 nR 0 ,j length- n sequences u j through p ( u j ) = Q n i =1 p ( u j,i ) and lab els them as u j ( s j,i ) where s j,i ∈ { 1 , . . . , 2 nR 0 ,j } where j = 1 , 2. 34 4. F or eac h u j ( s j,i ), user j generates 2 n ˆ R j length- n sequen ces ˆ y j through p ( ˆ y j | u j ) = Q n i =1 p ( ˆ y j,i | u j,i ) and indexes them as ˆ y j ( z j,i | s j,i ) where z j,i ∈ { 1 , . . . , 2 n ˆ R j } , j = 1 , 2. 5. F or eac h u j ( s j,i ), use r j generates 2 nR ′ 0 ,j length- n sequence s x j through p ( x j | u j ) = Q n i =1 p ( x j,i | u j,i ) and indexes them as x j ( t j,i | s j,i ) where t j,i ∈ { 1 , . . . , 2 nR ′ 0 ,j } , j = 1 , 2. P artit ioning: • P artitio n 2 n ˆ R j in to cells S s j,i where s j,i ∈ { 1 , . . . , 2 nR 0 ,j } , j = 1 , 2. Enco ding: The tra nsmitter sends x corresp o nding to the pair ( w 1 , w 2 ). User j sends x j ( t j,i | s j,i ) if the estimate of y j ( i − 1), i.e., ˆ z j,i − 1 , falls into S s j,i and t j,i is c hosen randomly from { 1 , . . . , 2 nR ′ 0 ,j } . The use o f man y x j ( t j,i | s j,i ) for actual help signal u j ( s j,i ) aims to confuse the other user and to decrease its decoding capabilit y . Deco ding: W e only consider deco ding at user 1. Final expressions regarding user 2 will follow due to symmetry . 1. User 1 seeks a unique jointly t ypical pair of ( y 1 ( i ) , u 2 ( s 2 ,i )) whic h can b e found with v anishingly small error probability if R 0 , 2 ≤ I ( U 2 ; Y 1 | X 1 ) (207) 2. User 1 decides on ˆ y 1 ( z 1 ,i | s 1 ,i ) b y lo oking for a jointly ty pical pair ( ˆ y 1 ( z 1 ,i | s 1 ,i ) , y 1 ( i ) , u 2 ( s 2 ,i ) , x 1 ( t 1 ,i | s 1 ,i )) whic h can b e ensured to exist if ˆ R 1 ≥ I ( ˆ Y 1 ; Y 1 | U 1 , U 2 , X 1 ) (208) 3. User 1 emplo ys list deco ding to deco de ˆ y 2 ( z 2 ,i − 1 | s 2 ,i − 1 ). It ﬁrst calculates its amb iguity set as L ( ˆ y 2 ( z 2 ,i − 1 | ˆ s 2 ,i − 1 )) = { ˆ y 2 ( z 2 ,i − 1 | ˆ s 2 ,i − 1 ) : ( ˆ y 2 ( z 2 ,i − 1 | ˆ s 2 ,i − 1 ) , y 1 ( i − 1)) is join tly t ypical } (209) and then tak es its interse ction with S ˆ s 2 ,i whic h results in a unique and correct in ter- section po in t if ˆ R 2 ≤ I ( ˆ Y 2 ; Y 1 | U 2 , X 1 ) + R 0 , 2 ≤ I ( ˆ Y 2 , U 2 ; Y 1 | X 1 ) (210) 4. User 1 decides that v 1 ( w 1 ,i − 1 , ˜ w 1 ,i − 1 , l 1 ,i − 1 ) is receiv ed if there exists a unique join tly t yp- ical pa ir ( v 1 ( w 1 ,i − 1 , ˜ w 1 ,i − 1 , l 1 ,i − 1 ) , y 1 ( i − 1) , ˆ y 2 ( ˆ z 2 ,i − 1 | ˆ s 2 ,i − 1 )) whic h can b e found with v anishingly small error probability if R ( V 1 ) ≤ I ( V 1 ; Y 1 , ˆ Y 2 | X 1 , U 2 ) (211) 35 Equiv o cation computation: Similar t o the previous pro o f s, w e treat eac h case separately . D ue t o symmetry , w e only consider user 1. If the rate of user 1 is suc h that R 1 ≥ I ( V 1 ; Y 1 , ˆ Y 2 | X 1 , U 2 ) − I ( V 1 ; Y 2 , ˆ Y 1 | X 2 , V 2 , U 1 ) − I ( V 1 ; V 2 ) (212) then we select the total n um b er of co dew ords as R ( V 1 ) = I ( V 1 ; Y 1 , ˆ Y 2 | X 1 , U 2 ) (213) whic h implies that ˜ R 1 + L 1 ≤ I ( V 1 ; Y 2 , ˆ Y 1 | X 2 , V 2 , U 1 ) + I ( V 1 ; V 2 ) (214) The equiv o cation rate can b e b ounded a s follows, H ( W 1 | Y n 2 , X n 2 ) ≥ H ( W 1 | Y n 2 , X n 2 , ˆ Y n 1 , V n 2 , U n 1 ) (215) = H ( W 1 , Y n 2 , ˆ Y n 1 , V n 2 | X n 2 , U n 1 ) − H ( Y n 2 , ˆ Y n 1 , V n 2 | X n 2 , U n 1 ) (216) = H ( W 1 , V n 1 , Y n 2 , ˆ Y n 1 , V n 2 | X n 2 , U n 1 ) − H ( V n 1 | W 1 , Y n 2 , ˆ Y n 1 , V n 2 , X n 2 , U n 1 ) − H ( Y n 2 , ˆ Y n 1 , V n 2 | X n 2 , U n 1 ) (217) = H ( V n 1 | X n 2 , U n 1 ) + H ( W 1 , Y n 2 , ˆ Y n 1 , V n 2 | X n 2 , U n 1 , V n 1 ) − H ( V n 1 | W 1 , Y n 2 , ˆ Y n 1 , V n 2 , X n 2 , U n 1 ) − H ( Y n 2 , ˆ Y n 1 , V n 2 | X n 2 , U n 1 ) (218) ≥ H ( V n 1 | X n 2 , U n 1 ) − I ( V n 1 ; Y n 2 , ˆ Y n 1 , V n 2 | X n 2 , U n 1 ) − H ( V n 1 | W 1 , Y n 2 , ˆ Y n 1 , V n 2 , X n 2 , U n 1 ) (219) W e treat eac h term in (219) separately . The ﬁrst term is H ( V n 1 | X n 2 , U n 1 ) = H ( V n 1 ) = nR ( V 1 ) = nI ( V 1 ; Y 1 , ˆ Y 2 | X 1 , U 2 ) (220) where the ﬁrst equality is due to the indep endence of V n 1 and ( X n 2 , U n 1 ), the second equality follo ws from the fact that V n 1 can tak e 2 nR ( V 1 ) v alues with equal probabilit y and the last equalit y is due to o ur c hoice in (213). The second term of (219) can b e b ounded as I ( V n 1 ; Y n 2 , ˆ Y n 1 , V n 2 | X n 2 , U n 1 ) ≤ nI ( V 1 ; Y 2 , ˆ Y 1 , V 2 | X 2 , U 1 ) + nǫ n (221) follo wing Lemma 3 of [10]. T o b ound the la st term of (219), we consider the case that user 2 is trying to decode V n 1 giv en the side information W 1 = w 1 . Since V n 1 can tak e 2 n ( I ( V 1 ; Y 2 , ˆ Y 1 | X 2 ,V 2 ,U 1 )+ I ( V 1 ; V 2 )) v alues at most, use r 2 can deco de V n 1 with v anishingly small error probability as long as this side informat io n is a v ailable. Hence, the use of F ano’s 36 lemma yields H ( V n 1 | W 1 , Y n 2 , ˆ Y n 1 , V n 2 , X n 2 , U n 1 ) ≤ ǫ n (222) Plugging (220), (221), (222) into (219), w e get H ( W 1 | Y n 2 , X n 2 ) ≥ nI ( V 1 ; Y 1 , ˆ Y 2 | X 1 , U 2 ) − nI ( V 1 ; Y 2 , ˆ Y 1 , V 2 | X 2 , U 1 ) − nǫ n (223) = nI ( V 1 ; Y 1 , ˆ Y 2 | X 1 , U 2 ) − nI ( V 1 ; Y 2 , ˆ Y 1 | X 2 , V 2 , U 1 ) − nI ( V 1 ; V 2 ) − nǫ n (224) where (224) follows from the indep endence of ( X 2 , U 1 ) and ( V 1 , V 2 ), i.e., I ( V 1 ; V 2 | X 2 , U 1 ) = I ( V 1 ; V 2 ). F or t he other case, i.e., if the rate o f user 1 is such that R 1 ≤ I ( V 1 ; Y 1 , ˆ Y 2 | X 1 , U 2 ) − I ( V 1 ; Y 2 , ˆ Y 1 | X 2 , V 2 , U 1 ) − I ( V 1 ; V 2 ) (225) w e select the total num b er of co dew ords as R ( V 1 ) = R 1 + I ( V 1 ; Y 2 , ˆ Y 1 | X 2 , V 2 , U 1 ) + I ( V 1 ; V 2 ) (226) and f ollo wing the same lines o f computation, w e can sho w that H ( W 1 | Y n 2 , X n 2 ) ≥ nR 1 − nǫ n (227) completing the pro of. References [1] E. 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Secrecy in Cooperative Relay Broadcast Channels

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