- Research Article
- Open Access

# Secrecy Capacity of a Class of Broadcast Channels with an Eavesdropper

- Ersen Ekrem
^{1}and - Sennur Ulukus
^{1}Email author

**2009**:824235

https://doi.org/10.1155/2009/824235

Â© E. Ekrem and S. Ulukus. 2009

**Received:**1 December 2008**Accepted:**21 June 2009**Published:**6 August 2009

## Abstract

We study the security of communication between a single transmitter and many receivers in the presence of an eavesdropper for several special classes of broadcast channels. As the first model, we consider the degraded multireceiver wiretap channel where the legitimate receivers exhibit a degradedness order while the eavesdropper is more noisy with respect to all legitimate receivers. We establish the secrecy capacity region of this channel model. Secondly, we consider the parallel multireceiver wiretap channel with a less noisiness order in each subchannel, where this order is not necessarily the same for all subchannels, and hence the overall channel does not exhibit a less noisiness order. We establish the common message secrecy capacity and sum secrecy capacity of this channel. Thirdly, we study a class of parallel multireceiver wiretap channels with two subchannels, two users and an eavesdropper. For channels in this class, in the first (resp., second) subchannel, the second (resp., first) receiver is degraded with respect to the first (resp., second) receiver, while the eavesdropper is degraded with respect to both legitimate receivers in both subchannels. We determine the secrecy capacity region of this channel, and discuss its extensions to arbitrary numbers of users and subchannels. Finally, we focus on a variant of this previous channel model where the transmitter can use only one of the subchannels at any time. We characterize the secrecy capacity region of this channel as well.

## Keywords

- Markov Chain
- Channel Model
- Broadcast Channel
- Secrecy Capacity
- Wiretap Channel

## 1. Introduction

*degraded*wiretap channel. Later, his result was generalized to arbitrary,

*not necessarily degraded*, wiretap channels by Csiszar and Korner [2]. Recently, many multiuser channel models have been considered from a secrecy point of view [3â€“22]. One basic extension of the wiretap channel to the multiuser environment is

*secure broadcasting to many users*in the presence of an eavesdropper. In the most general form of this problem (see Figure 1), one transmitter wants to have confidential communication with an arbitrary number of users in a broadcast channel, while this communication is being eavesdropped by an external entity. Our goal is to understand the theoretical limits of secure broadcasting, that is, largest simultaneously achievable secure rates. Characterizing the secrecy capacity region of this channel model in its most general form is difficult, because the version of this problem without any secrecy constraints, is the broadcast channel with an arbitrary number of receivers, whose capacity region is open. Consequently, to have progress in understanding the limits of secure broadcasting, we resort to studying several special classes of channels, with increasing generality. The approach of studying special channel structures was also followed in the existing literature on secure broadcasting [8, 9].

In this paper, our approach will be two-fold: first, we will identify more general channel models than considered in [8, 9] and generalize the results in [8, 9] to those channel models, and secondly, we will consider somewhat more specialized channel models than in [8] and provide more comprehensive results. More precisely, our contributions in this paper are as follows.

(1) We consider the degraded multireceiver wiretap channel with an arbitrary number of users and one eavesdropper, where users are arranged according to a degradedness order, and each user has a less noisy channel with respect to the eavesdropper, see Figure 2. We find the secrecy capacity region when each user receives both an independent message and a common confidential message. Since degradedness implies less noisiness [2], this channel model contains the subclass of channel models where in addition to the degradedness order users exhibit, the eavesdropper is degraded with respect to all users. Consequently, our result can be specialized to the degraded multireceiver wiretap channel with an arbitrary number of users and a degraded eavesdropper, see Corollary 2 and also [23]. The two-user version of the degraded multireceiver wiretap channel was studied and the capacity region was found independently and concurrently in [9].

(2) We then focus on a class of parallel multireceiver wiretap channels with an arbitrary number of legitimate receivers and an eavesdropper, see Figure 3, where in each subchannel, for any given user, either the user's channel is less noisy with respect to the eavesdropper's channel, or vice versa. We establish the common message secrecy capacity of this channel, which is a generalization of the corresponding capacity result in [8] to a broader class of channels. Secondly, we study the scenario where each legitimate receiver wishes to receive an independent message for another subclass of parallel multireceiver wiretap channels. For channels belonging to this subclass, in each subchannel, there is a less noisiness order which is not necessarily the same for all subchannels. Consequently, this ordered class of channels is a subset of the class for which we establish the common message secrecy capacity. We find the sum secrecy capacity for this class, which is again a generalization of the corresponding result in [8] to a broader class of channels.

(3) We also investigate a class of parallel multireceiver wiretap channels with two subchannels, two users, and one eavesdropper, see Figure 4. For the channels in this class, there is a specific degradation order in each subchannel such that in the first (resp., second) subchannel the second (resp., first) user is degraded with respect to the first (resp., second) user, while the eavesdropper is degraded with respect to both users in both subchannels. This is the model of [8] for users and subchannels. This model is more restrictive compared to the one mentioned in the previous item. Our motivation to study this more special class is to provide a stronger and more comprehensive result. In particular, for this class, we determine the entire secrecy capacity region when each user receives both an independent message and a common message. In contrast, the work in [8] gives the common message secrecy capacity (when only a common message is transmitted) and sum secrecy capacity (when only independent messages are transmitted) of this class. We discuss the generalization of this result to arbitrary numbers of users and subchannels.

(4) We finally consider a variant of the previous channel model. In this model, we again have a parallel multireceiver wiretap channel with two subchannels, two users, and one eavesdropper, and the degradation order in each subchannel is exactly the same as in the previous item. However, in this case, the input and output alphabets of one subchannel are nonintersecting with the input and output alphabets of the other subchannel. Moreover, we can use only one of these subchannels at any time. We determine the secrecy capacity region of this channel when the transmitter sends both an independent message to each receiver and a common message to both receivers.

It is clear that all of the channel models we consider exhibit some kind of an ordered structure, where this ordered structure is in the form of degradedness in some channel models, and it is in the form of less noisiness in others. This common ordered structure in all channel models we considered implies that our achievability schemes and converse proofs use some common techniques. In particular, for achievability, we use stochastic encoding [2] in conjunction with superposition coding [24]; and for the converse proofs, we use outer bounding techniques in [1, 2], more specifically, the Csiszar-Korner identity, [2, Lemmaâ€‰7].

## 2. Degraded Multireceiver Wiretap Channels

We first consider the generalization of Wyner's degraded wiretap channel to the case with many legitimate receivers. In particular, the channel consists of a transmitter with an input alphabet , legitimate receivers with output alphabets and an eavesdropper with output alphabet . The transmitter sends a confidential message to each user, say to the th user, in addition to a common message, , which is to be delivered to all users. All messages are to be kept secret from the eavesdropper. The channel is assumed to be memoryless with a transition probability .

*the degraded multireceiver wiretap channel with a more noisy eavesdropper*. We note that this channel model contains the degraded multireceiver wiretap channel which is defined through the Markov chain:

because the Markov chain in (3) implies the less noisiness condition in (2).

where denotes any subset of . Hence, we consider only perfect secrecy rates. The secrecy capacity region is defined as the closure of all achievable rate tuples.

The secrecy capacity region of the degraded multireceiver wiretap channel with a more noisy eavesdropper is given by the following theorem whose proof is provided in Appendix .

Theorem 1.

Remark 1.

Theorem 1 implies that a modified version of superposition coding can achieve the boundary of the capacity region. The difference between the superposition coding scheme used to achieve (5) and the standard one in [24], which is used to achieve the capacity region of the degraded broadcast channel, is that the former uses stochastic encoding in each layer of the code to associate each message with many codewords. This controlled amount of redundancy prevents the eavesdropper from being able to decode the message.

As stated earlier, the degraded multireceiver wiretap channel with a more noisy eavesdropper contains the degraded multireceiver wiretap channel which requires the eavesdropper to be degraded with respect to all users as stated in (3). Thus, we can specialize our result in Theorem 1 to the degraded multireceiver wiretap channel as given in the following corollary.

Corollary 2.

We acknowledge an independent and concurrent work regarding the degraded multireceiver wiretap channel. The work in [9] considers the two-user case and establishes the secrecy capacity region as well.

So far we have determined the entire secrecy capacity region of the degraded multireceiver wiretap channel with a more noisy eavesdropper. This class of channels requires a certain degradation order among the legitimate receivers which may be viewed as being too restrictive from a practical point of view. Our goal is to consider progressively more general channel models. Toward that goal, in the following section, we consider channel models where the users are not ordered in a degradedness or noisiness order. However, the concepts of degradedness and noisiness are essential in proving capacity results. In the following section, we will consider multireceiver broadcast channels which are composed of independent subchannels. We will assume some noisiness properties in these subchannels in order to derive certain capacity results. However, even though the subchannels will have certain noisiness properties, the overall broadcast channel will not have any degradedness or noisiness properties.

## 3. Parallel Multireceiver Wiretap Channels

where is the input in the th subchannel where is the corresponding channel input alphabet, (resp., ) is the output in the th user's (resp., eavesdropper's) th subchannel where (resp., ) is the th user's (resp., eavesdropper's) th subchannel output alphabet.

We note that the parallel multireceiver wiretap channel can be regarded as an extension of the parallel wiretap channel [21, 22] to the case of multiple legitimate users. Though the work in [21, 22] establishes the secrecy capacity of the parallel wiretap channel for the most general case, for the parallel multireceiver wiretap channel, obtaining the secrecy capacity region for the most general case seems to be intractable for now. Thus, in this section, we investigate special classes of parallel multireceiver wiretap channels. These channel models contain the class of channel models studied in [8] as a special case. Similar to [8], our emphasis will be on the common message secrecy capacity and the sum secrecy capacity.

### 3.1. The Common Message Secrecy Capacity

We first consider the simplest possible scenario where the transmitter sends a common confidential message to all users. Despite its simplicity, the secrecy capacity of a common confidential message (hereafter will be called the common message secrecy capacity) in a general broadcast channel is unknown.

*reversely degraded*parallel channels. Here, we call them parallel degraded multireceiver wiretap channels to be consistent with the terminology used in the rest of the paper.) Although [8] established the common message secrecy capacity for this class of channels, in fact, their result is valid for the broader class in which we have either

valid for every and for any pair where , . Thus, it is sufficient to have a degradedness order between each user and the eavesdropper in any subchannel instead of the long Markov chain between all users and the eavesdropper as in (12).

for all
and any
pair where
,
. Hereafter, we call this channel *the parallel multireceiver wiretap channel with a more noisy eavesdropper*. Since the Markov chain in (12) implies either (15) or (16), the parallel multireceiver wiretap channel with a more noisy eavesdropper contains the parallel degraded multireceiver wiretap channel studied in [8].

The common message secrecy capacity is the supremum of all achievable secrecy rates.

The common message secrecy capacity of the parallel multireceiver wiretap channel with a more noisy eavesdropper is stated in the following theorem whose proof is given in Appendix .

Theorem 3.

where the maximization is over all distributions of the form .

Remark 2.

Theorem 3 implies that we should not use the subchannels in which there is no user that has a less noisy channel than the eavesdropper. Moreover, Theorem 3 shows that the use of independent inputs in each subchannel is sufficient to achieve the capacity, that is, inducing correlation between channel inputs of subchannels cannot provide any improvement. This is similar to the results of [25, 26] in the sense that the work in [25, 26] established the optimality of the use of independent inputs in each subchannel for the product of two degraded broadcast channels.

As stated earlier, the parallel multireceiver wiretap channel with a more noisy eavesdropper encompasses the parallel degraded multireceiver wiretap channel studied in [8]. Hence, we can specialize Theorem 3 to recover the common message secrecy capacity of the parallel degraded multireceiver wiretap channel established in [8]. This is stated in the following corollary whose proof can be carried out from Theorem 3 by noting the Markov chain .

Corollary 4.

where the maximization is over all distributions of the form .

### 3.2. The Sum Secrecy Capacity

where
is a permutation of
. We call this channel *the parallel multireceiver wiretap channel with a less noisiness order in each subchannel*. We note that this class of channels is a subset of the parallel multireceiver wiretap channel with a more noisy eavesdropper studied in Section 3.1, because of the additional ordering imposed between users' subchannels. We also note that the class of parallel degraded multireceiver wiretap channels with a degradedness order in each subchannel studied in [8] is not only a subset of parallel multireceiver wiretap channels with a more noisy eavesdropper studied in Section 3.1 but also a subset of parallel multireceiver wiretap channels with a less noisiness order in each subchannel studied in this section.

The sum secrecy capacity is defined to be the supremum of all achievable sum secrecy rates.

The sum secrecy capacity for the class of parallel multireceiver wiretap channels with a less noisiness order in each subchannel studied in this section is stated in the following theorem whose proof is given in Appendix .

Theorem 5.

for all and any .

Remark 3.

Theorem 5 implies that the sum secrecy capacity is achieved by sending information only to the strongest user in each subchannel. As in Theorem 3, here also, the use of independent inputs for each subchannel is capacity-achieving, which is again reminiscent of the result in [25, 26] about the optimality of the use of independent inputs in each subchannel for the product of two degraded broadcast channels.

As mentioned earlier, since the class of parallel multireceiver wiretap channels with a less noisiness order in each subchannel contains the class of parallel degraded multireceiver wiretap channels studied in [8], Theorem 5 can be specialized to give the sum secrecy capacity of the latter class of channels as well. This result was originally obtained in [8]. This is stated in the following corollary. Since the proof of this corollary is similar to the proof of Corollary 4, we omit its proof.

Corollary 6.

for all input distributions on and any .

So far, we have considered special classes of parallel multireceiver wiretap channels for specific scenarios and obtained results similar to [8], only for broader classes of channels. In particular, in Section 3.1, we focused on the transmission of a common message, whereas in Section 3.2, we focused on the sum secrecy capacity when only independent messages are transmitted to all users. In the subsequent sections, we will specialize our channel model, but we will develop stronger and more comprehensive results. In particular, we will let the transmitter send both common and independent messages, and we will characterize the entire secrecy capacity region.

## 4. Parallel Degraded Multireceiver Wiretap Channels

We consider a special class of parallel degraded multireceiver wiretap channels with two subchannels, two users, and one eavesdropper. We consider the most general scenario where each user receives both an independent message and a common message. All messages are to be kept secret from the eavesdropper.

If we ignore the eavesdropper by setting , this channel model reduces to the broadcast channel that was studied in [25, 26].

where denotes any subset of . The secrecy capacity region is the closure of all achievable secrecy rate tuples.

The secrecy capacity region of this parallel degraded multireceiver wiretap channel is characterized by the following theorem whose proof is given in Appendix .

Theorem 7.

where the union is over all distributions of the form .

Remark 4.

If we let the encoder use an arbitrary joint distribution instead of the ones that satisfy , this would not enlarge the region given in Theorem 7, because all rate expressions in Theorem 7 depend on either or but not on the joint distribution .

Remark 5.

The capacity-achieving scheme uses either superposition coding in both subchannels or superposition coding in one of the subchannels, and a dedicated transmission in the other one. We again note that this superposition coding is different from the standard one [24] in the sense that it associates each message with many codewords by using stochastic encoding at each layer of the code due to secrecy concerns.

Remark 6.

If we set , we recover the capacity region of the underlying broadcast channel [26].

Remark 7.

where the union is over all . This region can be obtained through either Corollary 2 or Theorem 7 (by setting and eliminating redundant bounds) implying the consistency of the results.

Next, we consider the scenario where the transmitter does not send a common message, and find the secrecy capacity region.

Corollary 8.

where the union is over all distributions of the form .

Proof.

Since the common message rate can be exchanged with any user's independent message rate, we set where . Plugging these expressions into the rates in Theorem 7 and using Fourier-Moztkin elimination, we get the region given in the corollary.

Remark 8.

If we disable the eavesdropper by setting , we recover the capacity region of the underlying broadcast channel without a common message, which was found originally in [25].

At this point, one may ask whether the results of this section can be extended to arbitrary numbers of users and parallel subchannels. Once we have Theorem 7, the extension of the results to an arbitrary number of parallel subchannels is rather straightforward. Let us consider the parallel degraded multireceiver wiretap channel with subchannels, and in each subchannel, we have either the following Markov chain:

for any . We define the set of indices (resp., ) as those where for every (resp., ), the Markov chain in (33) (resp., in (34)) is satisfied. Then, using Theorem 7, we obtain the secrecy capacity region of the channel with two users and subchannels as given in the following theorem which is proved in Appendix .

Theorem 9.

where the union is over all distributions of the form .

We are now left with the question whether these results can be generalized to an arbitrary number of users. If we consider the parallel degraded multireceiver wiretap channel with more than two subchannels and an arbitrary number of users, the secrecy capacity region for the scenario where each user receives a common message in addition to an independent message does not seem to be characterizable. Our intuition comes from the fact that, as of now, the capacity region of the corresponding broadcast channel without secrecy constraints is unknown [27]. However, if we consider the scenario where each user receives only an independent message, that is, there is no common message, then the secrecy capacity region may be found, because the capacity region of the corresponding broadcast channel without secrecy constraints can be established [27], although there is no explicit expression for it in literature. We expect this particular generalization to be rather straightforward, and do not pursue it here.

## 5. Sum of Degraded Multireceiver Wiretap Channels

where , and . Thus, if the transmitter chooses to use its first alphabet, that is, , the second user (resp. eavesdropper) receives a degraded version of user 1's (resp., user 2's) observation. However, if the transmitter uses its second alphabet, that is, , the first user (resp. eavesdropper) receives a degraded version of user 2's (resp. user 1's) observation. Consequently, the overall channel is not degraded from any user's perspective, however, it is degraded from eavesdropper's perspective.

where denotes any subset of . The secrecy capacity region is the closure of all achievable secrecy rate tuples.

The secrecy capacity region of this channel is given in the following theorem which is proved in Appendix .

Theorem 10.

where the union is over all and distributions of the form .

Remark 9.

This channel model is similar to the parallel degraded multireceiver wiretap channel of the previous section in the sense that it can be viewed to consist of two parallel subchannels, however, now the transmitter cannot use both subchannels simultaneously. Instead, it should invoke a time-sharing approach between these two so-called parallel subchannels ( reflects this concern). Moreover, superposition coding scheme again achieves the boundary of the secrecy capacity region, however, it differs from the standard one [24] in the sense that it needs to be modified to incorporate secrecy constraints, that is, it needs to use stochastic encoding to associate each message with multiple codewords.

Remark 10.

An interesting point about the secrecy capacity region is that if we drop the secrecy constraints by setting , we are unable to recover the capacity region of the corresponding broadcast channel that was found in [26]. After setting , we note that each expression in Theorem 10 and its counterpart describing the capacity region [26] differ by exactly . The reason for this is as follows. Here, not only denotes the time-sharing variable but also carries an additional information, that is, the change of the channel that is in use is part of the information transmission. However, since the eavesdropper can also decode these messages, the term , which is the amount of information that can be transmitted via changes of the channel in use, disappears in the secrecy capacity region.

## 6. Conclusions

In this paper, we studied secure broadcasting to many users in the presence of an eavesdropper. Characterizing the secrecy capacity region of this channel in its most general form seems to be intractable for now, since the version of this problem without any secrecy constraints is the broadcast channel with an arbitrary number of receivers, whose capacity region is open. Consequently, we took the approach of considering special classes of channels. In particular, we considered degraded multireceiver wiretap channels, parallel multireceiver wiretap channels with a more noisy eavesdropper, parallel multireceiver wiretap channels with less noisiness orderings in each subchannel, and parallel degraded multireceiver wiretap channels. For each channel model, we obtained either partial characterization of the secrecy capacity region or the entire region.

## Appendices

### A. Proof of Theorem 1

First, we show achievability, then provide the converse.

#### A.1. Achievability

- (i)
Generate length- sequences through and index them as where , and .

- (ii)
For each , where , generate length- sequences through and index them as where and .

- (iii)
Finally, for each , generate length- sequences through and index them as where and .

- (iv)

ncoding

Assume the messages to be transmitted are . Then, the encoder randomly picks a set and sends .

Deco ing

Equivocation C ulation

We now calculate the equivocation of the code described above. To that end, we first introduce the following lemma which states that a code satisfying the sum rate secrecy constraint fulfills all other secrecy constraints.

Lemma 11.

where denotes any subset of .

Proof.

which is a consequence of the fact that message sets are uniformly and independently distributed.

where each term will be treated separately. Since given , can take values uniformly, the first term is

The second term in (A.19) is

#### A.2. Converse

The first term on the right-hand side of (A.33) is bounded as follows:

which is a consequence of the fact that the legitimate receivers exhibit a degradation order.

We now bound the terms of the summation in (A.33) for . Let us use the shorthand notation, , then

which follows from the fact that each user's channel is less noisy with respect to the eavesdropper's channel. Finally, we bound the following term where we again use the shorthand notation ,

where and , and this concludes the converse.

### B. Proof of Theorem 3

Achievability of these rates follows from [8, Propositionâ€‰2]. We provide the converse. First let us define the following random variables:

where ,â€‰ . Start with the definition

where (B.6) and (B.8) are due the following identities:

respectively, which are due to [2, Lemmaâ€‰7]. Now, we will bound each summand in (B.8) separately. First, define the following variables:

Hence, the summand in (B.8) can be written as follows:

where (B.16) and (B.18) follow from the following identities:

respectively, which are again due to [2, Lemmaâ€‰7]. Now, define the set of subchannels, say , in which the th user is less noisy with respect to the eavesdropper. Thus, the summands in (B.18) for are negative and by dropping them, we can bound (B.18) as follows:

Moreover, for , we have

where both are due to the fact that for , in this subchannel the th user is less noisy with respect to the eavesdropper. Therefore, adding (B.21) and (B.22) to each summand in (B.20), we get the following bound:

where an equality follows from the following Markov chain:

which is a consequence of the facts that channel is memoryless and subchannels are independent. Finally, using (B.24) in (B.8), we get

which completes the proof.

### C. Proof of Theorem 5

We first introduce the following lemma.

Lemma 12.

Proof.

because is the observation of the strongest user in the th subchannel, that is, its channel is less noisy with respect to all other users in the th subchannel. This concludes the proof of the lemma.

where the first inequality follows from the fact that conditioning cannot increase entropy.

We now start the converse proof:

for all and any . Thus, we can further bound (C.9) as follows:

which is a consequence of the facts that channel is memoryless, and the subchannels are independent.

### D. Proofs of Theorems 7 and 9

#### D.1. Proof of Theorem 7

We prove Theorem 7 in two parts, first achievability and then converse. Throughout the proof, we use the shorthand notations , , .

#### D.1.1. Achievability

- (i)
- (ii)
- (iii)

We now show the achievability of these regions separately. Start with the first region.

Proposition 13.

The region defined by (D.1) is achievable.

Proof.

- (i)
Split the private message rate of user 1 as .

- (ii)
Generate length- sequences through and index them as where and .

- (iii)
Generate length- sequences through and index them as where , and .

- (iv)
For each , generate length- sequences through and index them as where , .

- (v)

E coding

If is the message to be transmitted, then the receiver randomly picks and sends the corresponding codewords through each channel.

D oding

where we used the degradedness of the channel. Thus, we only need to show that this coding scheme satisfies the secrecy constraints.

Equivocation Computation

which completes the proof.

Achievability of the region defined by (D.2) follows due to symmetry. We now show the achievability of the region defined by (D.3).

Proposition 14.

The region described by (D.3) is achievable.

Proof.

- (i)
Generate length- sequences through and index them as where , .

- (ii)
For each , generate length- sequences through and index them as where , .

- (iii)
Generate length- sequences through and index them as where , .

- (iv)
For each , generate length- sequences through and index them as where , .

- (v)

En oding

Assume that the messages to be transmitted are . Then, after randomly picking the tuple , corresponding codewords are sent.

ecoding

After plugging the values of given by (D.22) into (D.29)â€“(D.31), one can recover the region described by (D.3) using the degradedness of the channel.

Equivocation Calculation

which concludes the proof.

#### D.1.2. Converse

We remark that although and are correlated, at the end of the proof, it will turn out that selection of them as independent will yield the same region. We start with the common message rate:

where (D.47) is due to Fano's lemma, the equality in (D.48) is due to the fact that the eavesdropper's channel is degraded with respect to the first user's channel. We bound each term in (D.50) separately. First term is

both of which are due to the fact that subchannels are independent, memoryless, and degraded. We now consider the second term in (D.50),

We now bound the sum rates to conclude the converse:

where (D.80) follows from Fano's lemma, (D.81) is due to the fact that the eavesdropper's channel is degraded with respect to both users' channels, (D.83) is obtained by adding and subtracting from the first term of (D.82). Now, we proceed as follows:

Adding to (D.87), we get

which is a consequence of the degradation orders that subchannels exhibit. Thus, (D.91) can be expressed as

which is due to the degradedness of the channel. Moreover, the second term in (D.97) is zero as we show in what follows:

which can be further bounded as follows:

Finally, (D.106) is due to our previous result in (D.75). We keep bounding terms in (D.84):

which is a consequence of the fact that each subchannel is memoryless. Thus, we only need to bound in (D.84) to reach the outer bound for the sum secrecy rate:

So far, we derived outer bounds, (D.62), (D.63), (D.77), (D.78), (D.124), (D.125), on the capacity region which match the achievable region provided. The only difference can be on the joint distribution that they need to satisfy. However, the outer bounds depend on either or but not on the joint distribution . Hence, for the outer bound, it is sufficient to consider the joint distributions having the form . Thus, the outer bounds derived and the achievable region coincide yielding the capacity region.

### D.2. Proof of Theorem 9

#### D.2.1. Achievability

for .

which implies that random variable tuples are mutually independent. Using this fact, one can reach the expressions given in Theorem 9.

#### D.2.2. Converse

where each term will be treated separately. The first term can be bounded as follows:

Thus, using these new auxiliary random variables in (D.134), we get

We now bound the second term in (D.131) as follows:

where, for the second term we already obtained, an outer bound given in (D.149). We now bound the first term:

We now bound the sum secrecy rate. We first borrow the following outer bound from the converse proof of Theorem 7:

where, for the first and third terms, we already obtained outer bounds given in (D.156) and (D.149), respectively. We now bound the second term as follows:

which is due to the facts that channel is memoryless and subchannels are independent. Thus, plugging (D.149), (D.156), and (D.167) into (D.161), we get

Finally, we note that all outer bounds depend on the distributions but not on any joint distributions of the tuples implying that selection of the pairs to be mutually independent is optimum.

### E. Proof of Theorem 10

We prove Theorem 10 in two parts; first, we show achievability, and then we prove the converse.

#### E.1. Achievability

- (i)
- (ii)
- (iii)

To show the achievability of each surface, we first introduce a codebook structure.

Codebook Structure

- (i)
Generate length- sequences through and index them as where , and .

- (ii)
For each , generate length- sequences through and index them as where , .

- (iii)
Generate length- sequences through and index them as where , and .

- (iv)
For each , generate length- sequences through and index them as where , .

- (v)
and .

- (vi)

Enc ding

When the transmitted messages are ,â€‰ , we randomly pick , and send corresponding codewords.

Decodin

Thus, the third surface can be achieved with vanishingly small error probability. As of now, we showed that all rates in the so-called capacity region are achievable with vanishingly small error probability, however we did not claim anything about the secrecy conditions which will be considered next.

Equivocation Calcula

which completes the achievability part of the proof.

#### E.2. Converse

Similar to the converse of Theorem 7, here again, and can be arbitrarily correlated. However, at the end of converse, it will be clear that selection of them as independent would yield the same region. Start with the common message rate:

where (E.35) is due to Fano's lemma, (E.36) is due to the fact that the eavesdropper's channel is degraded with respect to the first user's channel. Once we obtain (E.38), using the analysis carried out in the proof of Theorem 7, we can obtain the following bounds:

We now consider the sum of common and independent message rates:

Finally, we derive the outer bounds for the sum secrecy rate:

So far, we derived outer bounds on the secrecy capacity region which match the achievable region. Hence, to claim that this is indeed the capacity region, we need to show that computing the outer bounds over all distributions of the form yields the same region which we would obtain by computing over all . Since all the expressions involved in the outer bounds depend on either or but not on the joint distribution , this argument follows, establishing the secrecy capacity region.

## Declarations

### Acknowledgments

This work was supported by NSF Grants CCF 04-47613, CCF 05-14846, CNS 07-16311, and CCF 07-29127, and was presented in part at the 42nd Asilomar Conference on Signals, Systems and Computers, Pacific Grove, Calif, USA, October 2008 [23].

## Authorsâ€™ Affiliations

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