How to Achieve Privacy in Bidirectional Relay Networks
aa r X i v : . [ c s . I T ] A p r How to Achieve Privacy in BidirectionalRelay Networks
Rafael F. Wyrembelski and Holger Boche
Lehrstuhl f¨ur Theoretische InformationstechnikTechnische Universit¨at M¨unchen, Germany
Abstract —Recent research developments show that the conceptof bidirectional relaying significantly improves the performance inwireless networks. This applies to three-node networks, where ahalf-duplex relay node establishes a bidirectional communicationbetween two other nodes using a decode-and-forward protocol. Inthis work we consider the scenario when in the broadcast phasethe relay transmits additional confidential information to onenode, which should be kept as secret as possible from the other,non-intended node. This is the bidirectional broadcast channelwith confidential messages for which we derive the capacity-equivocation region and the secrecy capacity region. The lattercharacterizes the communication scenario with perfect secrecy,where the confidential message is completely hidden from thenon-legitimated node.
I. I
NTRODUCTION
The use of relays is currently becoming more and moreattractive since they have the potential to significantly improvethe performance and coverage of wireless networks. Relaycommunication suffers from the fact that orthogonal resourcesare needed for transmission and reception. The inherent lossin spectral efficiency can be reduced if bidirectional commu-nication is considered [1, 2].Cellular system operators offer for several users differentservices simultaneously where some of them are subject tosecrecy constraints. Due to the nature of the wireless medium,a transmitted signal is received by the intended user but canalso be overheard by non-intended users. Consequently, asystem design that enables secure communication becomes animportant issue especially for confidential information, wherenon-legitimated receivers should be kept ignorant of it.In his seminal work [3] Wyner characterized the securecommunication problem for a single source-destination linkwith an eavesdropper, the so-called wiretap channel . In [4]Csisz´ar and K¨orner generalized this model and studied the broadcast channel with confidential messages . Recently, thesecure communication problem gained a lot of attention; fora current survey we refer, for example, to [5]. The multipleaccess channel with confidential messages is analyzed in [6],while [7] discusses the interference and broadcast channel. Se-cure communication in relay broadcast channels is addressed
The authors gratefully acknowledge the support of the TUM GraduateSchool / Faculty Graduate Center FGC-EI at Technische Universit¨at M¨unchen,Germany. The work of Holger Boche was partly supported by the GermanResearch Foundation (DFG) under Grant BO 1734/25-1.
R mR m R 21 p (a) MAC phase m c mR mm c R R 21 R c (b) BBC phaseFig. 1. Decode-and-forward bidirectional relaying. In the initial MAC phase,nodes 1 and 2 transmit their messages m and m with rates R and R tothe relay node. Then, in the BBC phase, the relay forwards the messages m and m and adds a confidential message m c for node 1 with rate R c to thecommunication which should be kept as secret as possible from node 2. in [8] and in two-way wiretap channels in [9].We consider bidirectional relaying in a three-node network,where a relay node establishes a bidirectional communicationbetween two nodes using a two-phase decode-and-forwardprotocol as shown in Figure 1. Here, our main concern ison enabling an additional confidential communication withinsuch a network. This differs from the wiretap scenario wherethe bidirectional communication itself should be secure fromeavesdroppers outside of the wireless network as, for example,studied from a signal processing point of view in [10, 11].In this work, we concentrate on the broadcast phase, wherethe relay has successfully decoded the two messages the nodeshave sent in the previous multiple access (MAC) phase. Thetask of the relay is then to transmit both messages and anadditional confidential message to one node, which should bekept as secret as possible from the other, non-legitimated node.For decoding, the receiving nodes can exploit the messagesthey have sent in the previous phase as side information sothat this channel differs from the classical broadcast channelwith confidential messages and is therefore called bidirectionalbroadcast channel (BBC) with confidential messages .For the BBC without confidential messages in [12, 13] itis shown that capacity is achieved by a single data stream,which combines both messages based on the network codingidea. Here, we address the problem of realizing additionalconfidential communication within a network that exploitsprinciples from network coding; hence, the optimal processingis by no means self-evident. Notation:
Discrete random variables are denoted non-italic capital lettersand their realizations and ranges by lower case letters and script letters,respectively; P ( · ) denotes the set of all probability distributions and A ( n ) ǫ ( · ) the set of (weakly) typical sequences, cf. for example [14]. I. B
IDIRECTIONAL B ROADCAST C HANNEL WITH C ONFIDENTIAL M ESSAGES
Let X and Y i , i = 1 , , be finite input and output sets. Thenfor input and output sequences x n ∈ X n and y ni ∈ Y ni , i =1 , , of length n , the discrete memoryless broadcast channel isgiven by W ⊗ n ( y n , y n | x n ) := Q nk =1 W ( y ,k , y ,k | x k ) . Sincewe do not allow any cooperation between the receiving nodes,it is sufficient to consider the marginal transition probabilities W ⊗ ni := Q nk =1 W i ( y i,k | x k ) , i = 1 , , only.In this work we consider the standard model with a blockcode of arbitrary but fixed length n . Let M i := { , ..., M ( n ) i } be the set of individual messages of node i , i = 1 , , which isalso known at the relay node. Further, M c := { , ..., M ( n ) c } is the set of confidential messages of the relay node. We usethe abbreviation M := M c × M × M .For the bidirectional broadcast (BBC) phase we assume thatthe relay has successfully decoded the individual messages m ∈ M from node 1 and m ∈ M from node 2 that itreceived in the previous multiple access phase (MAC) phase.Then the relay transmits both individual messages and anadditional confidential message m c ∈ M c to node 1, whichshould be kept as secret as possible from node 2. Definition 1: An ( n, M ( n ) c , M ( n )1 , M ( n )2 ) - code for the BBCwith confidential messages consists of one (stochastic) encoderat the relay node f : M c × M × M → X n and decoders at nodes 1 and 2 g : Y n × M → M c × M ∪ { } ,g : Y n × M → M ∪ { } , where the element in the definition of the decoders plays therole of an erasure symbol and is included for convenience.Since randomization may increase secrecy [4, 5], we allowthe encoder f to be stochastic. This means it is specified byconditional probabilities f ( x n | m ) with P x n ∈X n f ( x n | m ) =1 for each m = ( m c , m , m ) ∈ M . Here, f ( x n | m ) is theprobability that the message m ∈ M is encoded as x n ∈ X n .A code is measured by two performance criteria. First, alltransmitted messages have to be successfully decoded, i.e.,we want the average probability of a decoding error to besmall. In more detail, when the relay has sent the message m = ( m c , m , m ) , and nodes 1 and 2 have received y n and y n , the decoder at node 1 is in error if g ( y n , m ) = ( m c , m ) .Accordingly, the decoder at node 2 is in error if g ( y n , m ) = m . Then, the average probability of error at node i is given by µ ( n ) i := |M| P m ∈M λ i ( m ) , i = 1 , , where λ ( m ) denotesthe probability that node 1 decodes ( m c , m ) incorrectly if m = ( m c , m , m ) has been sent, and λ ( m ) the probabilitythat node 2 decodes m incorrectly.The second criterion is security. Similarly as in [3, 4] wecharacterize the secrecy level of the confidential message m c ∈ M c at node 2 by the concept of equivocation. Theequivocation H (M c | Y n , M ) describes the uncertainty of node2 about the confidential message M c having its own message M and the received sequence Y n as side information avail-able. Thus, the higher the equivocation, the more ignorant isnode 2 about the confidential message. Definition 2:
A rate-equivocation tuple ( R c , R e , R , R ) ∈ R is said to be achievable for the BBC with confidentialmessages if for any δ > there is an n ( δ ) ∈ N and a sequenceof ( n, M ( n ) c , M ( n )1 , M ( n )2 ) -codes such that for all n ≥ n ( δ ) wehave log M ( n ) c n ≥ R c − δ , log M ( n )2 n ≥ R − δ , log M ( n )1 n ≥ R − δ ,and n H (M c | Y n , M ) ≥ R e − δ (1)while µ ( n )1 , µ ( n )2 → as n → ∞ . The set of all achievablerate-equivocation tuples is the capacity-equivocation region ofthe BBC with confidential messages and is denoted by C BBC .If there is no additional confidential message for the relayto transmit, we have the classical BBC for which the capacity-achieving coding strategies are known [12, 13].
Theorem 1 ([12, 13]):
The capacity region of the BBC isthe set of all rate pairs ( R , R ) ∈ R satisfying R ≤ I (X; Y | Q) , R ≤ I (X; Y | Q) for random variables (Q , X , Y , Y ) ∈ Q × X × Y × Y andjoint probability distribution P Q ( q ) P X | Q ( x | q ) W ( y , y | x ) .The cardinality of the range of Q can be bounded by |Q| ≤ .Now, we focus our attention on the broadcast scenario with aconfidential message and present the main result of this work. Theorem 2:
The capacity-equivocation region C BBC of theBBC with confidential messages is a closed convex set of thoserate-equivocation tuples ( R c , R e , R , R ) ∈ R satisfying ≤ R e ≤ R c ,R e ≤ I (V; Y | U) − I (V; Y | U) ,R c + R i ≤ I (V; Y | U) + I (U; Y i ) , i = 1 , ,R i ≤ I (U; Y i ) , i = 1 , , for random variables (U , V , X , Y , Y ) ∈ U × V ×X × Y × Y and joint probability distribution P U ( u ) P V | U ( v | u ) P X | V ( x | v ) W ( y , y | x ) . Moreover, thecardinalities of the ranges of U and V can be bounded by |U| ≤ |X | + 3 , |V| ≤ |X | + 4 |X | + 3 . Remark 1:
While for the BBC without confidential mes-sages the auxiliary random variable Q only enables a time-sharing operation and carries no information, cf. Theorem 1,for the BBC with confidential messages we will see that theauxiliary random variable U carries the bidirectional informa-tion and V realizes an additional randomization.From Theorem 2 follows immediately the secrecy capacityregion C S BBC of the BBC with confidential messages whichis the set of rate triples ( R c , R , R ) ∈ R such that ( R c , R c , R , R ) ∈ C BBC . Corollary 1:
The secrecy capacity region C S BBC of the BBCwith confidential messages is the set of all rate triples ( R c , R , R ) ∈ R satisfying R c ≤ I (V; Y | U) − I (V; Y | U) ,R i ≤ I (U; Y i ) , i = 1 , , or random variables (U , V , X , Y , Y ) ∈ U × V ×X × Y × Y and joint probability distribution P U ( u ) P V | U ( v | u ) P X | V ( x | v ) W ( y , y | x ) .The capacity-equivocation region in Theorem 2 describesthe scenario where the confidential message is transmitted withrate R c at a certain secrecy level R e . Thereby, R e can beinterpreted as the amount of information of the confidentialmessage that can be kept secret from the non-legitimatednode. Therefore, Theorem 2 includes the case where thenon-legitimated node has some partial knowledge about theconfidential information, namely if R c > R e . The secrecycapacity region in Corollary 1 characterizes the scenario withperfect secrecy which is, of course, the practically morerelevant case. Since R c = R e , the confidential message canbe kept completely hidden from the non-legitimated node.III. S ECRECY -A CHIEVING C ODING S TRATEGY
In this section, we present a coding strategy that achievesthe desired rates with the required secrecy level and therewithprove the achievability part of the corresponding Theorem 2.
A. Codebook Design
A crucial part is the following Lemma 1 which ensuresthe existence of a suitable codebook with a specific structureconsisting of two layers.The first layer corresponds to a codebook suitable for theBBC with common messages [15] which means that thisset of codewords enables the relay to transmit (bidirectional)individual messages m ′ ∈ M ′ and m ′ ∈ M ′ to nodes 1 and2 as well as a common (multicast) message m ′ ∈ M ′ to bothnodes.Then, for each codeword there is a sub-codebook with aproduct structure similarly as in [4] for the classical broadcastchannel with confidential messages. The legitimate receiverfor the confidential message, i.e., node 1, can decode eachcodeword regardless to which column and row index it cor-responds. But the main idea behind such a codebook designis that the non-legitimated receiver, i.e., node 2, decodes thecolumn index of the transmitted codeword with the maximumrate its channel provides, and therefore is not able to decodethe remaining row index [5]. Lemma 1:
For any δ > let U → X → Y Y be a Markovchain of random variables and I (X; Y | U) > I (X; Y | U) . i) There exists a set of codewords u nm ′ ∈ U n , m ′ =( m ′ , m ′ , m ′ ) ∈ M ′ × M ′ × M ′ =: M ′ , with n (cid:0) log |M ′ | + log |M ′ | (cid:1) ≥ I (U; Y ) − δ, (2a) n (cid:0) log |M ′ | + log |M ′ | (cid:1) ≥ I (U; Y ) − δ, (2b)such that |M ′ | X m ′ ∈M ′ λ m ′ ,m ′ | m ′ ≤ ǫ ( n ) , (3a) |M ′ | X m ′ ∈M ′ λ m ′ ,m ′ | m ′ ≤ ǫ ( n ) , (3b)and ǫ ( n ) → as n → ∞ . Thereby, λ m ′ ,m ′ | m ′ denotesthe probability that node 1 decodes ( m ′ , m ′ ) ∈ M ′ × M ′ incorrectly if m ′ ∈ M ′ is given. The error event λ m ′ ,m ′ | m ′ for node 2 is defined accordingly. ii) For each u nm ′ ∈ U n there exists a set of (sub-)codewords x njlm ′ ∈ X n , j ∈ J , l ∈ L , m ′ ∈ M ′ , with n log |J | ≥ I (X; Y | U) − δ, (4a) n log |L| ≥ I (X; Y | U) − I (X; Y | U) − δ, (4b)such that |J ||L||M ′ | X j ∈J X l ∈L X m ′ ∈M ′ λ j,l | m ′ ≤ ǫ ( n ) , (5a) |J ||L||M ′ | X j ∈J X l ∈L X m ′ ∈M ′ λ j | l,m ′ ≤ ǫ ( n ) , (5b)and ǫ ( n ) → as n → ∞ . Here, λ j,l | m ′ is the probability thatnode 1 decodes j ∈ J or l ∈ L incorrectly if m ′ ∈ M ′ isknown. Similarly, λ j | l,m ′ is the probability that node 2 decodes j ∈ J incorrectly if m ′ ∈ M ′ and l ∈ L are given. Sketch of Proof:
Since the proof is based on the classicalbroadcast channel with confidential messages [4] and the BBCwith common messages [15] we only sketch the main ideas.For the first layer we generate |M ′ | codewords u nm ′ ∈ U n according to the distribution P U n ( u n ) = Q nk =1 P U ( u k ) anduse (weakly) typical sets A ( n ) ǫ (U , Y i ) , i = 1 , , for decodingat the receivers. Then, using random coding arguments, forthe BBC with common messages we know from [15] that (3)is satisfied if (2) is fulfilled proving the first part.To prove the second assertion, for each u nm ′ ∈ U n we generate |J ||L| codewords x njlm ′ ∈ X n according to P X n | U n ( x n | u n ) = Q nk =1 P X | U ( x k | u k ) and use typical sets A ( n ) ǫ (U , X , Y i ) , i = 1 , , for decoding at the receivers. Wenote that the structure of the sub-codewords is exactly thesame as for the classical broadcast channel with confidentialmessages [4, 5], where the latter assumes the average error cri-terion and uses random coding arguments as we do. Followingthe proof it is easy to show that (5) is satisfied if (4) is fulfilledproving the second part. B. Achievable Rate-Equivocation Region
Next, we use the codebook from Lemma 1 to constructsuitable encoder and decoders for the BBC with confidentialmessages.
Lemma 2:
Let U → X → Y Y and I (X; Y | U) >I (X; Y | U) . Using the codebook from Lemma 1 all rate-equivocation tuples ( R c , R e , R , R ) ∈ R satisfying ≤ R e = I (X; Y | U) − I (X; Y | U) ≤ R c , (6a) R c + R i ≤ I (X; Y | U) + I (U; Y i ) , i = 1 , , (6b) R i ≤ I (U; Y i ) , i = 1 , , (6c)are achievable for the BBC with confidential messages. Proof:
For given rate-equivocation tuple ( R c , R e , R , R ) ∈ R satisfying (6a)-(6c) we have to (U; Y ) I (U; Y ) I (X; Y | U) I (X; Y | U) LJM ′ M ′ M ′ R c ≥ I (X; Y | U) Fig. 2. The two bars visualize the available resources of both links. Each oneis split up into two parts: one designated for the bidirectional communication(gray) and one for the confidential message (white). Since R c ≥ I (X; Y | U ) ,some resources of the bidirectional communication have to be spent for theconfidential message as well (realized by a common message). construct message sets, encoders, and decoders with n log |M c | ≥ R c − δ, (7a) n log |M | ≥ R − δ, (7b) n log |M | ≥ R − δ, (7c)and further, cf. also (1), n H (M c | Y n , M ) ≥ I (X; Y | U) − I (X; Y | U) − δ. (8)The following construction is mainly based on the one forthe classical broadcast channel with confidential messages[4]. Thereby, we have to distinguish between two cases asvisualized in Figures 2 and 3.If R c ≥ I (X; Y | U) , cf. Figure 2, we construct the set ofconfidential messages as M c := J × L × M ′ where J and L are chosen as in Lemma 1 and M ′ is anarbitrary set of common messages such that (7a) is satisfied.The sets M = M ′ and M = M ′ are arbitrary such that(7b)-(7c) hold. Finally, we define the deterministic encoder f that maps the confidential message ( j, l, m ′ ) ∈ M c and theindividual messages m i ∈ M i , i = 1 , , into the codeword x njlm ′ ∈ X n with m ′ = ( m ′ , m ′ , m ′ ) and m ′ i = m i , i = 1 , . Remark 2:
Since R c ≥ I (X; Y | U) , a part of the confi-dential message must be transmitted as a common message.It is not possible to simply ”add” the remaining part to theindividual message for node 1, since this would require thatthis part of the confidential message is already available apriori as side information at node 2.If R c < I (X; Y | U) , cf. Figure 3, we set M c := K × L where K is an arbitrary set such that (7a) holds. Further, wedefine a mapping h : J → K that partitions J into subsets of”nearly equal size” [4], which means | h − ( k ) | ≤ | h − ( k ′ ) | , for all k, k ′ ∈ K . Moreover, since R c < I (X; Y | U) , there is no need fora set of common messages so that M ′ = ∅ . The sets M = M ′ and M = M ′ are arbitrary such that (7b)-(7c)hold. Finally, we define the stochastic encoder f that maps theconfidential message ( k, l ) ∈ M c and the individual messages m i ∈ M i , i = 1 , , into the codeword x njlm ′ ∈ X n with m ′ = (0 , m ′ , m ′ ) , where j is uniformly drawn from the set h − ( k ) ⊂ J and m ′ i = m i , i = 1 , . I (X; Y | U) I (X; Y | U) I (U; Y ) I (U; Y ) M ′ M ′ LJ R c < I (X; Y | U) K Fig. 3. Since R c < I (X; Y | U ) , there are more resources for the confidentialmessage available than needed. This allows the relay to enable a stochasticcoding strategy which exploits all the available resources by introducing amapping from J to K . Remark 3:
This time, the set J is not needed in total for theconfidential message. However, to force the non-legitimatedreceiver, i.e., node 2, to decode at its maximum rate, we definea stochastic encoder that spreads the confidential messagesover the whole set J .Up to now we defined message sets and the encoder. In bothcases the decoders are immediately determined by Lemma 1.To complete the proof it remains to show that the secrecylevel at node 2 fulfills (8). Proceeding exactly as in [4] wedefine the random variable X n with codewords x njlm ′ ∈ X n as realizations and M ′ = (M ′ , M ′ , M ′ ) as the third coordinateof the realization of X n . Then get for the equivocation H (M c | Y n , M ) ≥ H (X n | M ′ ) + H (Y n | X n ) − H (X n | M c , M ′ , Y n ) − H (Y n | M ′ ) . (9)Next, we bound all terms in (9) separately. We start with thefirst term and observe for given M ′ = m ′ that X n has |J ||L| possible values. Since X n is independently and uniformlydistributed, we have H (X n | M ′ ) = log |J | + log |L| . With thedefinition of J and L , cf. (4), we obtain n H (X n | M ′ ) → I (X; Y | U) . (10)For the second term in (9) we have n H (Y n | X n ) → H (Y | X) (11)as n → ∞ by the weak law of large numbers. If R c ≥ I (X; Y | U) , the third term in (9) vanishes. If R c < I (X; Y | U) , we define ϕ ( k, l, m ′ , y n ) := x nklm ′ if ( u nm ′ , x njlm ′ , y n ) ∈ A ( n ) ǫ (U , X , Y ) , h ( j ) = k , and arbitraryotherwise. Then we have P { X n = ϕ (M c , M ′ , Y n ) } ≤ ǫ ( n ) and therefore, by Fano’s lemma, cf. also [4, 5], n H (X n | M c , M ′ , Y n ) → (12)as n → ∞ . For the last term in (9) we define ˆ y n := y n if ( u nm ′ , y n ) ∈ A ( n ) ǫ (U , Y ) and arbitrary otherwise so that H (Y n | M ′ ) ≤ H (Y n | ˆY n ) + H ( ˆY n | M ′ ) . For the first term we have P { Y n = ˆY n } ≤ ǫ ( n ) by Fano’slemma, cf. [4, 5], so that it is negligible. Moreover, following[4, 5] it is easy to show that for the second term we have n H ( ˆY n | M ′ ) → H (Y | U) (13)which follows from the definition of the decoding sets A ( n ) ǫ (U , Y ) and the fact that the codewords are uniformlydistributed.inally, by substituting (10)-(13) into (9) we obtain (8)which establishes the desired secrecy level at node 2 andtherewith proves the lemma. C. Randomization and Convexity
Here, we complete the proof of achievability of Theorem 2.Since the argumentation is the same as for the classicalbroadcast channel with confidential messages [4], we onlysketch the main ideas.To obtain the whole region of Theorem 2, we proceedexactly as in [4] and introduce an auxiliary channel thatenables an additional randomization.
Lemma 3:
Let U → V → X → Y Y and I (V; Y | U) > I (V; Y | U) . Then all rate-equivocation tuples ( R c , R e , R , R ) ∈ R satisfying ≤ R e ≤ I (V; Y | U) − I (V; Y | U) ≤ R c , (14a) R c + R i ≤ I (V; Y | U) + I (U; Y i ) , i = 1 , , (14b) R i ≤ I (U , Y i ) , i = 1 , , (14c)are achievable for the BBC with confidential messages. Thecorresponding rate region is denoted by R . Sketch of Proof:
The prefixing realized by the randomvariable V is exactly the same as in [4, Lemma 4].Moreover, it is obvious that if the rate-equivocation tuple ( R c , R e , R , R ) is achievable, than each rate-equivocationtuple ( R c , R ′ e , R , R ) with ≤ R ′ e ≤ R e is also achievable.Consequently, we can further replace the equality in (6a) byan inequality in (14a). Lemma 4:
The rate region R is convex. Sketch of Proof:
Exactly as in [4, Lemma 5] it is easyto show that any linear combination of two rate tuples in R is contained in R which proves the convexity.It remains to show that R describes the same rate region asthe one specified by Theorem 2. Lemma 5:
The rate region R equals the capacity region C BBC of the BBC with confidential messages.
Proof:
It is obvious that
R ⊆ C
BBC holds. To show thereversed inclusion, i.e., C BBC ⊆ R , let ( R c , R e , R , R ) ∈C BBC be any rate-equivocation tuple. For this, we construct asin [4] the maximal achievable confidential and equivocationrates that are possible for given individual rates R and R as R ∗ c := I (V; Y | U) + min (cid:8) I (U; Y ) − R , I (U; Y ) − R (cid:9) ,R ∗ e := I (V; Y | U) − I (V; Y | U) . Then we have R e ≤ R ∗ e , R ∗ e ≤ R c ≤ R ∗ c , and therewithalso ( R ∗ c , R ∗ e , R , R ) ∈ R . Now, from the definition of R follows that the rate-equivocation tuples ( R ∗ c , R ∗ e , R , R ) , ( R ∗ c , , R , R ) , and (0 , , R , R ) belong to R as well. Fi-nally, from the convexity of R , cf. Lemma 4, follows that ( R c , R e , R , R ) ∈ R which proves the lemma.To complete the proof of achievability it remains to boundthe cardinalities of the ranges of U and V . Since the boundsof the cardinalities depend only the structure of the randomvariables, the result follows immediately from [4, Appendix]where the same bounds are established for the classical broad-cast channel with confidential messages. D. Weak Converse
Already the coding strategy indicates that, basically, ideasfrom the BBC [12, 15] and from the classical broadcast chan-nel with confidential messages [4] are exploited. Based on thisobservation it is straightforward to establish the weak conversefor the BBC with confidential messages by extending theconverse of the classical broadcast channel with confidentialmessages [4] using standard arguments for the BBC [12, 15].IV. D
ISCUSSION
In this work, our focus was on privacy in bidirectional relaynetworks, where additionally to the two bidirectional messagesthe relay node transmits a confidential message to one ofthe nodes, which should be kept as secret as possible fromthe other, non-legitimated node. For this scenario we charac-terized the corresponding capacity-equivocation and secrecycapacity regions in detail. This scenario is completely differentfrom the bidirectional broadcast wiretap channel, where thebidirectional communication itself should be kept secret fromeavesdroppers outside of the bidirectional relay network [10,11]. This is an interesting and important topic for itself.R
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