Feb 17, 2011 - indicating that over the multiple access channel (MAC) with an eavesdropper, secret communication can be enhanced with a ... H. Yagi is...

0 downloads 9 Views 202KB Size

No documents

Improved Rate-Equivocation Regions for Secure Cooperative Communication Ninoslav Marina, Member, IEEE, Hideki Yagi, Member, IEEE, and H. Vincent Poor, Fellow, IEEE

arXiv:1102.3500v1 [cs.IT] 17 Feb 2011

Abstract A simple four node network in which cooperation improves the information-theoretic secrecy is studied. The channel consists of two senders, a receiver, and an eavesdropper. One or both senders transmit confidential messages to the receiver, while the eavesdropper tries to decode the transmitted message. The main result is the derivation of a newly achievable rate-equivocation region that is shown to be larger than a rate-equivocation region derived by Lai and El Gamal for the relay-eavesdropper channel. When the rate of the helping interferer is zero, the new rate-equivocation region reduces to the capacity-equivocation region over the wire-tap channel, hence, the new achievability scheme can be seen as a generalization of a coding scheme proposed by Csiszár and Körner. This result can naturally be combined with a rate-equivocation region given by Tang et al. (for the interference assisted secret communication), yielding an even larger achievable rate-equivocation region. Index Terms Information-theoretic secrecy, wire-tap channel, eavesdropper channel, rate-equivocation region, secrecy capacity, perfect secrecy, physical layer security, cooperative communication.

I. I NTRODUCTION N this work we propose a scheme that increases the information theoretic secrecy in a simple cooperative communication network. The channel model includes a class of the wire-tap channels with a helping interferer introduced by Lai and El Gamal [1]. These authors considered several cooperation schemes over the relay-eavesdropper channel, in which the relay node helps to enhance the security level of communication between the sender and the receiver. The paper gives an interesting observation indicating that over the multiple access channel (MAC) with an eavesdropper, secret communication can be enhanced with a help of one of the two senders (called, the helping interferer or the helper). In addition, an achievable equivocation-rate region has been derived for this scheme. Subsequently, Tang et al. [2] have derived an improved rate-equivocation region using the fact that the receiver does not have to decode the sequence transmitted by a helper. One possibility is that the helper sends interference (dummy messages) in order to weaken the channel to the eavesdropper. When the rate of dummy messages of the helper is zero (there is no cooperation from the helper), the channel reduces to the (single-user) wire-tap channel introduced by Wyner [3], and generalized later by Csiszár and Körner [4]. In this reduced setting, however, the achievable rate-equivocation regions given by [1] and [2] do not coincide with the capacityequivocation region over the wire-tap channel, giving only its sub-region. When only perfect-secrecy is imposed (i.e., the eavesdropper is totally ignorant of the transmitted message), their results coincide with the secrecy-capacity of the wire-tap channel. Motivated by this fact, the first part of this paper gives a new achievable rate-equivocation region (i.e, an inner bound on the capacity-equivocation region) for the wire-tap channel with a helping interferer, showing that the new region is improved over the one given in [1]. When the rate of the helping interferer

I

N. Marina is supported by the European Commission under Marie Curie FP7 PEOPLE Programme, Grant #237669. H. Yagi is supported in part by MEXT under Grant-in-Aid for Young Scientists (B) No. 22760270, JST’s Special Coordination Funds for Promoting Science and Technology, and H. V. Poor is supported by the U.S. National Science Foundation under Grant CNS-09-05398. N. Marina is with Department of Electrical Engineering, Princeton University, Princeton, NJ 08544 USA (email: [email protected]). H. Yagi is with Center for Frontier Science and Engineering, The University of Electro-Communications, Chofu-shi, Tokyo 182-8585, Japan (email: [email protected]). H. V. Poor is with Department of Electrical Engineering, Princeton University, Princeton, NJ 08544 USA (email: [email protected]).

2

is zero, the new rate-equivocation region reduces to the capacity-equivocation region over the wire-tap channel, so the new achievability scheme can be seen as a generalization of the coding scheme given in [4]. Our result can naturally be combined with the additional rate-equivocation region given by [2], yielding an even larger rate-equivocation region. In the next section we present the previous results on the wire-tap channel with a helper. The main result of this work, that is the improved rate-equivocation region for the wire-tap channel with helping interferer, is presented in Section III, while in Section IV we derive an even larger rate-equivocation region. A note on the broadcast channel with confidential messages and the wire-tap channel with helping interferer is given in Section V. Section VI concludes the paper. II. P RELIMINARIES A. The Wire-Tap Channel with a Helping Interferer The cooperative channel considered in this paper is shown in Fig. 1 and consists of two senders, a receiver, and an eavesdropper, in which one sender transmits confidential messages to the receiver, and the eavesdropper tries to decode the transmitted message. The second sender plays the role of a “helper” to enhance the secrecy of communication. This model is referred to as the wire-tap channel with a (helping) interferer, and will be considered first. Let Xt be the channel input alphabet of sender t, W1 W2 Fig. 1.

f1 f2

X1N

YN Channel

X2N

p(y, z|x1 , x2 )

gr

ˆ1 W

ge

W1

ZN

A four node network of one sender, one receiver, one eavesdropper and one helper.

t = 1, 2, and let and Y and Z be the output alphabet of the receiver and the eavesdropper, respectively. We assume that all the alphabets are discrete and finite and the channel is memoryless, characterized by a conditional probability mass function (PMF) P (y, z|x1x2 ) for (x1 , x2 ) ∈ X1 × X2 and (y, z) ∈ Y × Z, i.e., xt , (xt1 , . . . , xtN ) ∈ XtN , y , (y1 , . . . , yN ) ∈ Y N and z , (z1 , . . . , yN ) ∈ Y N . Then, we have PN (y, z|x1 , x2 ) =

N Y

P (yn , zn |x1n , x2n )

n=1

where N denotes the number of channel uses. We assume that both of the receiver and the eavesdropper know P (y, z|x1, x2 ). Define Wt with t = 1, 2 as the set of integers {1, . . . , Mt } with Mt ≥ 1. Let w1 ∈ W1 be a uniformly distributed confidential message of sender 1. We also denote a random message of sender 2 by w2 ∈ W2 . Encoder t is a deterministic mapping denoted by ft : Wt → XtN .

(1)

The receiver and the eavesdropper estimate the transmitted message from the received sequence y and z, with the decoding functions gr : Y N → W1 , and ge : Z N → W1 , respectively. Let Rt , t = 1, 2, be an information rate defined as Rt = log2 Mt /N. An (N, M1 , M2 , {ft , gt }t=1,2 ) code for the MAC with a helper consists of message sets V1 × V2 , encoding functions ft , and decoding functions gt with t = 1, 2. Provided that the transmitted message is w1 ∈ W1 ,

3 (N )

the decoder makes an error if gr (y) 6= w1 . The average probability of decoding error, denoted by Pe is 1 X Pe(N ) = Pr(gr (y) = 6 w1 | w1 sent). M1 w ∈W 1

,

1

The equivocation rate at the eavesdropper is defined as 1 H(W1 |Z N ). N The secrecy considered in this paper is defined as follows: Definition 1: A rate-equivocation pair (R1 , Re ) is said to be achievable if there exists a sequence of (N, M1 ) codes such that for every ǫ > 0, Re(N ) =

R1 ≥

log2 M1 − ǫ, N

Pe(N ) ≤ ǫ,

and Re(N ) ≥ Re − ǫ,

for all sufficiently large N. Definition 2: A perfect-secrecy rate R1 is said to be achievable if the rate-equivocation pair (R1 , R1 ) is achievable. The secrecy-capacity of the wire-tap channel with a helper is defined as the maximum of all achievable perfect-secrecy rates. Note that without sender 2, this channel model reduces the (single-user) wire-tap channel [3], [4]. Achievable rate-equivocation pairs, achievable perfect-secrecy rates, and the secrecy capacity for the wiretap channel are defined analogously. B. Known Achievable Rate-Equivocation Regions For the (single-user) wire-tap channel [3], [4], the following rate-equivocation region is the capacityequivocation region n [ (R1 , Re ) : 0 ≤ Re ≤ R1 , PQU PX1 |U PY Z|X

R1 ≤ I(U; Y ), o Re ≤ I(U; Y |Q) − I(U; Z|Q) ,

(2)

where Q and U are auxiliary random variables satisfying the Markov chain condition Q → U → X1 and the cardinality bounds |Q| ≤ |X1 | + 3 and |U| ≤ |X1 |2 + 4|X1 | + 3. Since we assume that all rates in this paper are always non-negative, if an upper-bound on Re happens to be negative, it means Re = 0. This rule will be applied throughout the paper when necessary. For the wire-tap channel with a helper, it was shown in [1, Theorem 3] that the following rateequivocation region is achievable: n [ (R1 , Re ) : R1 ≤ I(U1 ; Y |U2 ), co PU1 PU2 PX1 |U1 PX2 |U2 PY Z|X1 X2

0 ≤ Re ≤ R1 , Re ≤ I(U1 ; Y |U2 ) − min{I(U2 ; Y ), I(U2 ; Z)} o − I(U1 ; Z|U2 ) + min{I(U2 ; Y ), I(U2 ; Z|U1 )} , (3)

4

where co(S) denotes the convex hull of the set S, U1 and U2 are auxiliary random variables satisfying the Markov chain condition (U1 , U2 ) → (X1 , X2 ) → (Y, Z). We note that if I(U2 ; Y ) ≤ I(U2 ; Z), then the last inequality on Re becomes 0 ≤ Re ≤ I(U1 ; Y |U2 ) − I(U1 ; Z|U2 ), implying that the wire-tap channel with a helper becomes the ordinary wire-tap channel, i.e., there is no effect from the user cooperation. In this case, the region given by (2) reduces to a sub-region of the region given by (2). Note that the result of Tang et al. [2] implies that we might still have an advantage from the user cooperation in this case. For the wire-tap channel with a helper, it is known that the following perfect-secrecy rate is achievable [1, eq. (10)]: R1 = sup I(U1 ; Y |U2 ) − I(U1 ; Z|U2 ) + min{I(U2 ; Y ), I(U2 ; Z|U1 )} PU1 PU2 PX1 |U1 PX2 |U2

− min{I(U2 ; Y ), I(U2 ; Z)} where [x]+ denotes max{x, 0}.

+

,

(4)

III. I MPROVED R ATE -E QUIVOCATION R EGION In this section we show that it is possible to have a rate-equivocation region larger than the one given by (3). To that end we introduce an auxiliary random variable Q1 and we get the following improved region. Proposition 1: The following rate-equivocation region is achievable: [n C= (R1 , Re ) : 0 ≤ Re ≤ R1 , π

R1 ≤ R1′ + min{I(Q1 ; Y |U2 ), I(Q1 ; Z|U2 )}, n o ′ ′ ′ ′ Re ≤ max R1 + R2 − I(U1 ; Z|U2 Q1 ) − I(U2 ; Y |Q1 ), R1 + R2 − I(U1 U2 ; Z|Q1 ) (5) where π , PQ1 PU1 |Q1 PU2 PX1 |U1 PX2 |U2 PY Z|X1 X2 , R1′ , I(U1 ; Y |U2 Q1 ), and R2′ , min{I(U2 ; Y |Q1 ), I(U2 , Z|U1 )}. Q1 , U1 , and U2 are auxiliary random variables satisfying the following Markov chain conditions: Q1 → U1 → (U1 , U2 ) → (X1 , X2 )

X1 , and → (Y, Z).

(6)

As in [4], let the auxiliary random variable Q1 correspond to the sequence alphabet decoded by both the receiver and the eavesdropper, while letting U1 and U2 denote the sequence alphabets that can be decoded only by the receiver. First, we note that the constraint on Re in (5) can be re-written as I(U1 ; Y |U2 Q1 ) − I(U1 ; Z|U2 Q1 ), if I(U2 ; Y |Q1 ) ≤ I(U2 ; Z|Q1 ), Re ≤ I(U1 U2 ; Y |Q1 ) − I(U1 U2 ; Z|Q1 ), if I(U2 ; Z|Q1 ) ≤ I(U2 ; Y |Q1 ), ≤ I(U2 ; Z|U1 ), I(U ; Y |U Q ) − I(U ; Z|Q ), if I(U2 ; Z|U1 ) ≤ I(U2 ; Y |Q1 ). 1 2 1 1 1

5

It is straightforward that by setting Q1 = ∅, we have I(U1 ; Y |U2 Q1 ) + min{I(Q1 ; Y |U2 ), I(Q1 ; Z|U2 )} = I(U1 ; Y |U2 ), I(U1 ; Y |U2 Q1 ) − I(U1 ; Z|U2Q1 ) = I(U1 ; Y |U2 ) − I(U1 ; Z|U2 ). On the other hand, since I(U1 ; Y |U2 Q1 ) − I(U1 ; Z|U2 Q1 ) = I(U1 ; Y |U2 ) − I(U1 ; Z|U2 ) + I(Q1 ; Z|U2 ) − I(Q1 ; Y |U2 ) , we have sup PQ1 PU1 |Q1 PU2

I(U1 ; Y |U2 Q1 ) − I(U1 ; Z|U2 Q1 ) ≥ sup I(U1 ; Y |U2 ) − I(U1 ; Z|U2 ) .

(7)

PU1 PU2

A similar derivation of (7) yields I(U1 U2 ; Y |Q1 ) − I(U1 U2 ; Z|Q1 ) ≥ sup I(U1 U2 ; Y ) − I(U1 U2 ; Z) , sup PU1 PU2

PQ1 PU1 |Q1 PU2

and sup PQ1 PU1 |Q1 PU2

I(U1 ; Y |U2 Q1 ) − I(U1 ; Z|Q1 ) ≥ sup I(U1 ; Y |U2 ) − I(U1 ; Z) . PU1 PU2

Hence, region C given by (5) is larger than or equal to the region given by (3). The random variable Q1 plays not only the role of convexification. The achievability of the region C will be shown in Appendix A. For the rate-equivocation region C, if I(U2 ; Y |Q1 ) ≤ I(U2 ; Z|Q1 ) for every PQ∗ 1 U1 U2 X1 X2 , PQ1 U1 PX1 |U1 PU2 X2 , the cooperation between sender 1 and sender 2 (the helper) has no effect, and the region is in a simpler form as the convex hull of n [ C˜ = (R1 , Re ) : 0 ≤ Re ≤ R1 , ∗ PQ

1 U 1 U 2 X1 X2

PY Z|X1 X2

R1 ≤ I(U1 ; Y |U2 ) o Re ≤ I(U1 ; Y |U2 Q1 ) − I(U1 ; Z|U2 Q1 ) . Although Tang et al. [2] give a larger region in this case, if I(U2 ; Y |U1 ) ≥ I(U2 ; Z|U1 ), then user cooperation does not take effect. In the following text, we denote the convex hull of C and C˜ by C ∗ and C˜∗ , respectively. When there is no helping interference, i.e., R2 = 0, then the region C˜ corresponds to the capacity-equivocation region for the ordinary wire-tap channel given by (2). Note that in the case R2 = 0, N the helper transmits a deterministic sequence uN 2 ∈ U2 , and both the receiver and the eavesdropper know this sequence. Therefore, the capacity-equivocation region is still characterized by U2 . When considering the perfect-secrecy rate, the auxiliary random variable Q1 introduced to derive a new rate-equivocation region has no impact. For the wire-tap channel with a helper, we can achieve the same perfect-secrecy rate as (4) derived in [1]. As the last result of this section we get the following theorem. Theorem 1: The rate-equivocation region C ∗ is achievable for the wire-tap channel with a helping interferer. Proof: From the above argument, by the coding scheme given in Appendix A, the region R1 (PQ∗ 1 X1 X2 ) is achievable for any given PQ∗ 1 X1 X2 , and hence R1 is achievable. By prefixing a conditional PMF PX1 |U1 PX2 |U2 , the region R2 , which is equivalent to C, is also achievable. The convex hull can be taken since we can time-share multiple input PMFs via the time-sharing principle [5]. ✷

6

IV. A N E VEN L ARGER R ATE -E QUIVOCATION R EGION We can combine the idea given in [2] with the achievable region C ∗ to get a larger achievable rateequivocation region. The key observation is that the receiver does not necessarily need to decode the dummy message W2 sent from the helper. For a fixed PQ∗ 1 U1 U2 X1 X2 , PQ1 U1 PX1 |U1 PU2 X2 ∈ P ∗ , let CA (PQ∗ 1 U1 U2 X1 X2 ) be defined as the rateequivocation region n ∗ CA (PQ1 U1 U2 X1 X2 ) = (R1 , Re ) : 0 ≤ Re ≤ R1 , R1 ≤ I(U1 ; Y |U2 Q1 ) + min{I(Q1 ; Y |U2 ), I(Q1 ; Z|U2 )}, n o ′ ′ Re ≤ max R3 − I(U1 ; Z|U2 Q1 ) − I(U2 ; Y |Q1 ), R3 − I(U1 U2 ; Z|Q1 ) (8) where R2′ = min{I(U2 ; Y |Q1 ), I(U2 , Z|U1 )}, R3′ , I(U1 ; Y |U2 Q1 ) + R2′ , and Q1 , U1 , and U2 are auxiliary random variables satisfying Markov chain conditions (6). Then the achievable rate-equivocation region C is expressed as [ CA (PQ∗ 1 U1 U2 X1 X2 ). (9) C= ∗ PQ

1 U 1 U 2 X1 X2

PY Z|X1 X2

We define another rate-equivocation region, for a fixed PQ∗ 1 U1 U2 X1 X2 ∈ P ∗ , as n CB (PQ∗ 1 U1 U2 X1 X2 ) = (R1 , Re ) : R1 ≤ I(U1 ; Y ), 0 ≤ Re ≤ R1 , o Re ≤ I(U1 ; Y |Q1 ) − I(U1 ; Z|Q1 ) . ˜ is given by the convex hull of Then, a new achievable rate-equivocation region, denoted by C, [ C˜ = CA (PQ∗ 1 U1 U2 X1 X2 ) ∪ CB (PQ∗ 1 U1 U2 X1 X2 ) .

(10)

∗ PQ 1 U 1 U 2 X1 X2

From equations (9) and (10), it is readily seen that in general we have C ∗ ⊆ C˜∗ where C˜∗ denotes the ˜ The region convex hull of C. CB (P ∗ ) \ (CA (P ∗) ∩ CB (P ∗ )) expresses an additional region to CA (P ∗ ) for a fixed P ∗ ∈ P ∗ , which is given by the observation in [2]. The rate-equivocation region C˜ can be seen as an extension of the result of [2] in the sense that we derive not only a perfect-secrecy rate but also a rate-equivocation region by introducing the auxiliary random variable Q1 . The key idea lies in the facts that: (i) The receiver and the eavesdropper can decode a partial message of W1 at the rate at most min{I(Q1 ; Y ), I(Q1 ; Z)},

and,

(ii) As for the other part of message, dummy message from the helper needs not be decoded, and can be treated as noise. Note that even though the region CB (PQ1 U1 U2 X1 X2 ) does not involve the rate R2 , user cooperation, i.e., interference by a helper, is necessary to achieve this region, and hence, the PMFs of random variables U2 and X2 are also included in the region. The achievability of the region C˜ is shown in Appendix B.

7

RZ

RZ

coop. B

coop. B

R2

R2 coop. A

RY RY R1

R1 I(X1 ; Y |X2 ) ≥ I(X1 ; Z|X2 )

I(X1 ; Y |X2 ) < I(X1 ; Z|X2)

Fig. 2. Pictorial representation for the equivocation gain when cooperation is used for the case (i) of Proposition 2 for the situations I(X1 ; Y |X2 ) ≥ I(X1 ; Z|X2 ) (left) and I(X1 ; Y |X2 ) ≥ I(X1 ; Z|X2 ) (right). Pentagons RY and RZ express an achievable region for the receiver’s MAC and the eavesdropper’s MAC, respectively. The cooperation scheme that achieves CA (P ∗ ) is labeled “coop. A” and the cooperation scheme that achieves CB (P ∗ ), is labeled “coop. B”.

When only the perfect-secrecy rate is concerned, the obtained rate-equivocation region is reduced to [ C˜′ = CA′ (π12 ) ∪ CB′ (π12 ) , (11) π12

where n CA′ (π12 ) , R1 :

R1 ≥ 0, R1 ≤ max I(U1 ; Y |U2 ) − I(U1 ; Z|U2 ) + R2′ − I(U2 ; Y ), o ′ I(U1 ; Y |U2 ) + R2 − I(U1 U2 ; Z)

and n o CB′ (π12 ) = R1 : 0 ≤ R1 ≤ [I(U1 ; Y ) − I(U1 ; Z)]+ . for a fixed input distribution π12 = PU1 X1 PU2 X2 . Then, the following perfect-secrecy rate is achievable: sup CA′ (π12 ) ∪ CB′ (π12 ) π12

which is the same as the one given in [2]. We next consider conditions under which we get an improvement to region CB (P ∗ ), i.e., CB (P ∗ ) \ (CA (P ∗ ) ∩ CB (P ∗ )) 6= ∅. We have the following proposition: Proposition 2: For a given P ∗ ∈ P ∗ , CB (P ∗ ) \ (CA (P ∗ ) ∩ CB (P ∗)) 6= ∅ if and only if either of the following two conditions is satisfied: I(U1 ; Y |Q) > I(U1 ; Z|Q) and 0 ≤ I(U2 ; Z|Q1 ) − I(U2 ; Y |Q1 ) ≤ I(U2 ; Z|U1 ) − I(U2 ; Y |U1 ), (ii) I(U1 ; Y |Q) > I(U1 ; Z|Q) and I(U2 ; Z|Q1 ) ≤ I(U2 ; Y |Q1 ) ≤ I(U2 ; Y |U1 ) ≤ I(U2 ; Z|U1 ). (i)

Proof: See Appendix C.

(12) (13) ✷

8

RZ

RZ

coop. B

R2

coop. B

R2 coop. A

RY

no coop. (R2 = 0)

RY

R1 I(X1 ; Y |X2 ) ≥ I(X1 ; Z|X2 )

R1 I(X1 ; Y |X2 ) < I(X1 ; Z|X2 )

Fig. 3. Pictorial representation for the equivocation gain when cooperation is used for the case (ii) in Proposition 2 for the situations I(X1 ; Y |X2 ) ≥ I(X1 ; Z|X2 ) (left) and I(X1 ; Y |X2 ) ≥ I(X1 ; Z|X2 ) (right). Pentagons RY and RZ express an achievable region for the receiver’s MAC and the eavesdropper’s MAC, respectively. The cooperation scheme that achieves CA (P ∗ ) is labeled “coop. A” and the cooperation scheme that achieves CB (P ∗ ), is labeled “coop. B”.

We illustrate both cases, in which CB (P ∗ ) is effective, in Figs. 2 and 3. For illustrative purpose, we consider rate-equivocation regions given by PX1 |Q1 PX2 . The actual region is obtained by prefixing PX1 |U1 PX2 |U2 as discussed in Appendix A. Fig. 2 describes the case that satisfies (12) in the following two situations: I(X1 ; Y |X2 ) ≥ I(X1 ; Z|X2) (left) and I(X1 ; Y |X2 ) ≥ I(X1 ; Z|X2 ) (right). Fig. 3 describes the case that satisfies that satisfies (13) in the following two situations: I(X1 ; Y |X2 ) ≥ I(X1 ; Z|X2 ) (left) and I(X1 ; Y |X2 ) ≥ I(X1 ; Z|X2 ) (right). In the figures “coop. A” denotes the cooperation scheme that achieves CA (P ∗ ), while “coop. B” the cooperation scheme that achieves CB (P ∗ ). Observe that in the right subfigures of both figures only cooperation scheme B gives positive equivocation, implying the usefulness of this cooperation scheme. V. A N OTE

B ROADCAST C HANNEL WITH C ONFIDENTIAL M ESSAGES AND A H ELPING I NTERFERER Since the effect of Q1 is not completely clear, one might doubt the true effect of Q1 . In this section, we discuss about the role of the introduced Q1 by comparing relationship between the broadcast channel with confidential messages (BCC) and the wire-tap channel with a helping interferer. We consider the following two items: (1) The constraint on R1 in the new achievable rate-equivocation region C involves the term ON THE

min{I(Q1 ; Y |U2 ), I(Q1 ; Z|U2 )}, whereas the capacity equivocation region for the ordinary wire-tap channel does not (c.f., (2)). (2) Although by introducing another auxiliary random variable Q1 we have a wider rate-equivocation region, this random variable gives no impact in terms of perfect-secrecy (i.e., (4)). Csiszár and Körner show in [4] that for the broadcast channel with confidential messages (BCC), the use of Q1 is essential. In the model of BCC (Fig. 4), there are two receivers, and the sender wishes to send public messages W0 of rate R0 to both receivers while public messages W1 of rate R1 is confidential to receiver 2. It is

9

Public Message

W0

Receiver

X1N

Wiretap Channel

Sender

W1

p(y, z|x1 )

Private Message

Fig. 4.

ˆ0 , W ˆ1 W

YN

Eve

Z

N

ˆ0 W

The broadcast channel with confidential messages (BCC).

known that the following region is the capacity-equivocation region for the BCC [4, Theorem 1] n CBCC = (R1 , Re , R0 ) : 0 ≤ R0 , 0 ≤ Re ≤ R1 , R0 + R1 ≤ I(U1 ; Y |Q1 ) + min{I(Q1 ; Y ), I(Q1 ; Z)}, Re ≤ I(U1 ; Y |Q1 ) − I(U1 ; Z|Q1 ), o R0 ≤ min{I(Q1 ; Y ), I(Q1 ; Z)}

(14)

where the random variables satisfy Q1 → U1 → X1 → (Y, Z).

(15)

From (14), the constraint on R0 + R1 also involves the term min{I(Q1 ; Y ), I(Q1 ; Z)} (c.f., above item 1). Furthermore, the random variable Q1 is essentially necessary because receiver 2 should estimate W0 from Z N reliably. Having this in mind, we can argue the BCC with a helping interferer as in Fig. 5, and we can achieve the following rate-equivocation region, that is the convex hull of [ CBCCH = CA′ (π ∗ ) ∪ CB′ (π ∗ ) . (16) π∗

where n CA′ (π ∗ ) , (R1 , Re , R0 ) :

0 ≤ R0 , 0 ≤ Re ≤ R1 , R0 + R1 ≤ I(U1 ; Y |U2 Q1 ) + min{I(Q1 ; Y |U2 ), I(Q1 ; Z|U2 )}, Re ≤ I(U1 ; Y |U2 Q1 ) − I(U1 ; Z|U2 Q1 ), o R0 ≤ min{I(Q1 ; Y |U2 ), I(Q1 ; Z|U2 )} ,

and n CB′ (π ∗ ) = (R1 , Re , R0 ) :

0 ≤ R0 , 0 ≤ Re ≤ R1 , R0 + R1 ≤ I(U1 ; Y |Q1 ) + min{I(Q1 ; Y ), I(Q1 ; Z)}, Re ≤ I(U1 ; Y |Q1 ) − I(U1 ; Z|Q1 ), o R0 ≤ min{I(Q1 ; Y ), I(Q1 ; Z)} .

Here, Q1 , U1 , and U2 are auxiliary random variables satisfying the Markov chain conditions (6). Therefore, we think that in the case of the BCC with a helper, the use of Q1 is also essential. Furthermore, when R2 = 0, the above region reduces to the capacity-equivocation region for the BCC. Remark 1: We cannot directly use the achievability scheme from Appendix A, and the constraint on Re ∈ CA′ (P ∗ ), which is always achieved with R2 = 0, is smaller than that in CA (P ∗ ) given in (8) for the wire-tap channel with a helper. The main reason for this is that, by setting R2 ≤ min{I(U2 ; Y |Q1 ), I(U2 ; Z|U1 )}

10

W0 W1 W2

X1N

YN

Sender

Receiver Wiretap Channel

X2N Helper

p(y, z|x1 , x2 )

ZN Eve

ˆ0 , W ˆ1 W ˆ0 W

Dummy Message (interference) Fig. 5.

BC with confidential messages and with a helper.

as in Appendix A, then receiver 2 cannot always decode W2 (and equivalently, U2N ) correctly, and it cannot decode W0 accordingly for some R0 ≤ min{I(Q1 ; Y |U2 ), I(Q1 ; Z|U2 )}. If I(Q1 ; Y |U2 ) ≤ I(Q1 ; Z) is satisfied, then there is a possibility to have an advantage. On the other hand, in the case of the wire-tap channel with a helper, W2 needs not be decoded by the eavesdropper, so this problem does not occur. Despite the above remark, from (14) and (16), the cooperation by a helper gives a larger rate-equivocation region compared with the case of the ordinary BCC (with no helpers). This indicates that the cooperation has an effect even for the BCC case, and observation by Tang et al. [2] is also useful. VI. C ONCLUSION We have derived a new achievable rate-equivocation region for a class of wire-tap channels with a helping interferer, which has been shown to be larger than the rate-equivocation region given by [1]. Our result can naturally adopt the observation given by [2], yielding an even larger rate-equivocation region than the previously known regions. We also discussed about some relationship of our result with the capacity-equivocation over the broadcast channel with confidential messages in order to explain the role of the newly introduced random variable. A PPENDIX A ACHIEVABILITY OF THE N EW R EGION We shall show an achievability scheme for the region C via random coding. As in the wire-tap channel [4], we introduce rate splitting of R1 into R10 and R11 , where R10 denotes the rate of messages that can be decoded by both the receiver and the eavesdropper, and R11 denotes the rate of messages that can be decoded only by the receiver. First we define the following region: n [ R1 = (R1 , Re ) : R1 = R10 + R11 , 0 ≤ R10 , 0 ≤ Re ≤ R1 , PX1 Q1 PX2 PY Z|X1 X2

R10 ≤ min{I(Q1 ; Y |X2 ), I(Q1 ; Z|X2 )}, R11 ≤ I(X1 ; Y |X2 Q1 ), n Re ≤ max I(X1 ; Y |U2 Q1 ) − I(X1 ; Z|X2Q1 ), I(X1 ; Y |X2 Q1 ) − I(X1 X2 ; Z|Q1 ) o + min{I(X2 ; Y |Q1 ), I(X2 ; Z|X1 )} .

(17)

11

As discussed in [4], if R1 is achievable, the following region R2 is achievable by prefixing a conditional PMF PX1 |U1 PX2 |U2 : n [ R2 = (R1 , Re ) : R1 = R10 + R11 , 0 ≤ R10 , 0 ≤ Re ≤ R1 , PQ1 PU1 |Q1 PU2 PX1 |U1 PX2 |U2 PY Z|X1 X2

R10 ≤ min{I(Q1 ; Y |U2 ), I(Q1 ; Z|U2 )}, R11 ≤ I(U1 ; Y |U2 Q1 ), n Re ≤ max I(U1 ; Y |U2 Q1 ) − I(U1 ; Z|U2 Q1 ), I(U1 ; Y |U2 Q1 ) − I(U1 U2 ; Z|Q1 ) o + min{I(U2 ; Y |Q1 ), I(U2 ; Z|U1 )} . By using the relation R10 = R1 − R11 , Fourier-Motzkin elimination yields the region C given by (5). Hence, in terms of the achievability to the region C, it suffices to show that the rate-equivocation region R1 is achievable for every given PQ1 X1 PX2 . Next we show the achievability of R1 , given by (17), via random coding and the joint asymptotic equipartition property (AEP) [5]. We fix a joint PMF PQ∗ 1 X1 X2 , PQ1 X1 PX2 , and let the target region be denoted by R1 (PQ∗ 1 X1 X2 ). We consider two cases that will be called Case 1 and Case 2. A. Case 1: I(X1 ; Y |X2 Q1 ) ≤ I(X1 ; Z|X2 Q1 ) In this case, we need to consider only the case I(X2 ; Y |Q1 ) ≥ I(X2 ; Z|Q1 ), since otherwise the rateequivocation becomes zero because the first constraint on Re in (17) is apparently negative (i.e., it gives a trivial upper-bound on Re ). Then, the constraint on Re is expressed as Re ≤ [I(X1 ; Y |X2 Q1 ) + min{I(X2 ; Y |Q1 ), I(X2 ; Z|X1 )} − I(X1 X2 ; Z|Q1 )]+ .

(18)

1) Codebook generation: For a given PQ∗ 1 X1 X2 , we first generate 2N R10 independent and identically distributed(i.i.d.) sequences at random according to PQN1 (q) ,

N Y

PQ1 (qn ),

n=1

and index them as q(i), i ∈ [1, 2

N R10

], with

R10 ≤ min{I(Q1 ; Y |X2 ), I(Q1 ; Z|X2 )}.

(19)

When j1 ≤ j2 , [j1 , j2 ] denotes the set of all integers from j1 to j2 . For given q(i), i ∈ [1, 2N R10 ], we generate 2N R11 i.i.d. sequences at random according to PX1N |QN1 (x1 |q) ,

N Y

PX1 |Q1 (x1n |qn ),

n=1

and index them as x1 (i, b), b ∈ [1, 2N R ], with R ≤ I(X1 ; Y |X2 Q1 ). We also generate 2N R2 i.i.d. sequences at random according to PX2N (x2 ) , them as x2 (k), k ∈ [1, 2N R2 ], with

(20) QN

n=1

PX2 (x2n ), and index

R2 ≤ min{I(X2 ; Y |Q1 ), I(X2; Z|X1 )}.

(21)

R′ , [R + R2 − I(X1 X2 ; Z|Q1 )]+

(22)

Let

12

express the rate that exceeds the eavesdropper’s ability to decode a sequence reliably. We also define ′ ′ W = [1, 2N R ], L = [1, 2N (R−R ) ], and B = W × L = [1, 2N R ]. Note that R′ ≤ R since R − R′ ≥ I(X1 X2 ; Z|Q1 ) − I(X2 ; Z|X1) = I(X1 ; Z|Q1 ). Hereafter, we assume R′ > 0 for simplicity. If this is not the case, no security level can be reached, and we achieve only (R1 , 0) such that R1 ≤ R which is still inside the rate-equivocation region R1 . We call this codebook generation and the encoding and decoding scheme described below Coding Scheme 1. 2) Encoding: For a given rate-equivocation pair (R10 , R11 , Re ) such that R1 = R10 +R11 and Re ≤ R1 , we consider the following encoding scheme: Assume that a secret message w1 = (w10 , w11 ) ∈ W1 with w10 ∈ W10 , [1, 2N R10 ] and w11 ∈ W11 , [1, 2N R11 ] is input to sender 1 and a random message w2 ∈ [1, 2N R2 ] is generated at sender 2. The encoding function for w11 at sender 1 operates in the following stochastic manner: ′ (i) If R11 > R′ , then we divide W11 into W and J , [1, 2N (R11 −R ) ] as W11 = W × J . Let g be ′ the partition that divides L into |J | = 2N (R11 −R ) subsets L′1 , . . . , L′2N(R11 −R′ ) with equal cardinalities 2N (R−R11 ) . The encoder determines (w, l) from w11 = (w, j) such that l is uniformly chosen from the partition L′j at random. In this case, there is a one-to-one correspondence between {(w, l)} and [1, 2N R ]. (ii) If R11 ≤ R′ , then the encoder obtains (w, l) by setting w , w11 and uniformly choosing l from L ′ at random. In this case, there is a one-to-one correspondence between {(w, l)} and [1, 2N (R11 +R−R ) ]. The transmitted sequence from sender 1 is x1 (i, b) with i = w10 and b = (w, l) ∈ [1, 2N R ]. Sender 2 transmits the sequence x2 (k) with k = w2 , where w2 ∈ [1, 2N R2 ] is uniformly selected. ˆ such that 3) Decoding:: Upon receiving y ∈ Y N , the receiver seeks a message pair (ˆi, k) ˆ y ∈ A(N ) q(ˆi), x2 (k), ǫ (N )

where Aǫ denotes the ǫ-jointly typical set [5] for any fixed ǫ > 0. If there does not exist or there are more than one such sequence, then the receiver declares a decoding error. Then, the receiver seeks a message ˆb = (w, ˆ ˆl) such that ˆ y ∈ A(N ) q(ˆi), x1 (ˆi, ˆb), x2 (k), ǫ

ˆ Having (ˆi, ˆb) such that ˆb = (w, for given (ˆi, k). ˆ ˆl), the receiver obtains the estimates of the transmitted message w1 = (w10 , w11 ) by setting wˆ10 , ˆi, w ˆ11 , w, ˆ g(ˆl) , if R11 > R′ , and wˆ10 , ˆi, w ˆ11 , w, ˆ if R11 ≤ R′ . 4) Analysis of Reliability: The average probability of decoding error for the receiver, denoted by provided that (i, b, k) is sent, is upper-bounded as

(N ) P e (i, b, k)

(N )

(N )

(N )

P e (i, b, k) ≤ P e,1 (i, k) + P e,2 (b|i, k), (N )

(23)

(N )

where P e,1 (i, k) and P e,2 (b|i, k) denote the probabilities of decoding error for the first step (estimation of (i, k)) and the second step (estimation of b given a true transmitted pair (i, k)), respectively. It is easily seen that the error probability of the first decoding step can be made arbitrarily small for all sufficiently large N by the AEP [5] since R10 and R2 satisfy (19), (21), and R10 + R2 ≤ min{I(Q1 ; Y ), I(Q1 ; Z)} + min{I(X2 ; Y |Q1 ), I(X2 ; Z|X1 )} ≤ I(Q1 X2 ; Y ).

(24)

Also, the error probability of the second decoding step can be made arbitrarily small for sufficiently large (N ) N by the AEP and (20), and so can the probability P e (i, b, k).

13 (N )

5) Analysis of Equivocation: The equivocation Re

=

1 H(W1 |Z N ) N

is lower-bounded by

1 H(W10 W11 |Z N ) N 1 ≥ H(W10 W11 |Z N W10 ) N 1 (25) = H(W11 |Z N W10 ), N where the inequality follows from the fact that conditioning does not increase the entropy. By a similar expansion for H(W11 |Z N W10 ) as in [1, eq. (45)], we obtain Re(N ) =

H(W11 |Z N W10 ) ≥ H(X1N X2N |W10 ) − I(X1N X2N ; Z N |W10 ) − H(X1N X2N |W10 W11 Z N ).

(26)

We shall consider bounding each term in (26). For the first term, we have H(X1N X2N |W10 ) = H(X1N |W10 ) + H(X2N ) and N N H(X1N |W10 ) = H(X1N |W10 QN 1 ) = H(X1 |Q1 ),

where the first equality is due to the fact that QN 1 is a deterministic function of W10 , while the last equality N follows from the Markov chain relationship W10 → QN 1 → X1 . Since the codewords are generated according to i.i.d. distributions, it follows that H(X1N |QN 1 ) = NH(X1 |Q1 ) ≥ NR,

(27)

H(X2N ) = NH(X2 ) ≥ NR2 .

(28)

and

It is sufficient that we directly replace these inequalities with NH(X1 |Q1 ) ≥ NI(X1 ; Y |X2 Q1 ) and NH(X2 ) ≥ N min{I(X2 ; Y |Q1 ), I(X2 ; Z|X1)}. For the second term in (26), we expand I(X1N X2N ; Z N |W10 ) = H(Z N |W10 ) − H(Z N |X1N X2N W10 ), (29) for which we have H(Z N |W10 ) = H(Z N |QN 1 ) = NH(Z|Q1 )

(30)

N due to the fact that QN 1 is a deterministic function of W10 , the Markov chain relationship W10 → Q1 → Z N , and an i.i.d. distribution for Z N given QN 1 . We also have

H(Z N |X1N X2N W10 ) = H(Z N |X1N X2N QN 1 ) = NH(Z|X1 X2 Q1 ).

(31)

It follows from (30) and (31) that (29) becomes I(X1N X2N ; Z N |W10 ) = NI(X1 X2 ; Z|Q1 ).

(32)

We now consider the third term in (26). Consider decoding of l given w1 = (w10 , w11 ) ∈ W1 by observing z ∈ Z N . For the case R11 > R′ , since this decoder knows j ∈ J , which is given by w11 = (w, j) ∈ W11 , and using the following inequalities 1 log2 |L′j | + R2 ≤ R − R′ + R2 = I(X1 X2 ; Z|Q1 ), N

14

and

1 log2 |L′j | ≤ R ≤ I(X1 ; Z|X2 Q1 ), (33) N the average probability of decoding error can be made arbitrarily small for sufficiently large N. Note that, in this case, R ≤ I(X1 ; Y |X2 Q1 ) ≤ I(X1 ; Z|X2 Q1 ). For the case R11 ≤ R′ , we also have 1 log2 |L| + R2 ≤ R − R′ + R2 = I(X1 X2 ; Z|Q1 ), N and 1 log2 |L| ≤ R ≤ I(X1 ; Z|X2Q1 ). (34) N Again, the average probability of decoding error can be made arbitrarily small with all sufficiently large N. Therefore, by Fano’s inequality [5], for any given ǫ′ > 0, we have 1 H(X1N X2N |W10 W11 Z N ) ≤ ǫ′ N for sufficiently large N. Substituting (28), (32), and (35) into (26) yields, for any given ǫ′ > 0,

(35)

Re(N ) ≥ R + R2 − I(X1 X2 ; Z|Q1 ) − ǫ′ for N sufficiently large. Since we can choose any pair of R and R2 subject to (20) and (21), there exist R and R2 such that, for any ǫ′ > 0, Re(N ) ≥ I(X1 ; Y |X2 Q1 ) + min{I(X2 ; Y |Q1 ), I(X2 ; Z|X1 )} − I(X1 X2 ; Z|Q1 ) − ǫ′ .

(36)

Hence, it follows from (22) and (36) that any equivocation Re satisfying (18) is achievable. B. Case 2: I(X1 ; Y |X2 Q1 ) > I(X1 ; Z|X2 Q1 ) In this case, if I(X2 ; Y |Q1 ) ≥ I(X2 ; Z|Q1 ), then the constraint on Re is given by (18). We can use Coding Scheme 1 discussed in Case 1 with a slight modification. We set + R′ = R + min{I(X2 ; Y |Q1 ), I(X2 ; Z|X1 )} − I(X1 X2 ; Z|Q1 ) ,

and we assume that R′ > 0, because no security level is obtained otherwise. Then, for the analysis of equivocation, the left hand side of (33) is bounded as 1 log2 |L′j | ≤ R − R′ = I(X1 X2 ; Z|Q1 ) − min{I(X2 ; Y |Q1 ), I(X2 ; Z|X1)}. N Since I(X2 ; Z|Q1 ) ≤ min{I(X2 ; Y |Q1 ), I(X2 ; Z|X1 )} and I(X2 ; Z|Q1 ) = H(X2 ) − H(X2 |Q1 Z) ≤ H(X2 ) − H(X2 |Q1 X1 Z) = I(X2 ; Z|X1 ) where the last equality follows from the Markov chain relationship Q1 → (X1 , Z) → X2 , we have (33) if R11 > R′ . From the same reasoning, we also have (34) if R11 ≤ R′ . Other arguments are quite similar to those for Case 1, and we can show that any rate-equivocation pair (R1 , R) ∈ R1 (PQ∗ 1 X1 X2 ) is achievable.

15

We then consider the case I(X2 ; Y |Q1 ) ≤ I(X2 ; Z|Q1 ). In this case, the constraint on Re in (17) is given by Re ≤ I(X1 ; Y |X2 Q1 ) − I(X1 ; Z|X2 Q1 ), which can be achieved by a similar coding/decoding scheme to Coding Scheme 1 by letting R ≤ I(X1 ; Y |X2 Q1 ), and R′ = [R1 − I(X1 ; Z|X2 Q1 )]+ . In this case, R2 can be arbitrarily set in the range 0 ≤ R2 ≤ I(X2 ; Y |Q1 ). We call this coding scheme Coding Scheme 2. To show the equivocation at the eavesdropper, note that 1 H(W10 W11 |Z N ) Re(N ) = N 1 H(W11 |Z N X2N W10 ). ≥ N Similarly to the derivation [1, eq. (49)], we obtain H(W11 |Z N X2N W10 ) ≥ H(X1N |W10 ) − I(X1N ; Z N |W10 X2N ) − H(X1N |W10 W11 Z N X2N ), in which the right hand side is lower-bounded by N(I(X1 ; Y |X2 Q1 ) − I(X1 ; Z|X2 Q1 ) − ǫ), for any given ǫ > 0, for all sufficiently large N. This completes the proof of the achievability to the region R1 (PQ∗ 1 X1 X2 ). A PPENDIX B A N ACHIEVABLE S CHEME FOR THE R EGION RB We give an achievability scheme for the region C˜ given in (10). Let π ∗ = PQ∗ 1 U1 U2 X1 X2 and RB (π ∗ ) be defined as n RB (π ∗ ) = (R1 , Re ) : R1 = R10 + R11 , 0 ≤ R10 , 0 ≤ Re ≤ R1 , R10 ≤ min{I(Q1 ; Y ), I(Q1 ; Z)}, R11 ≤ I(U1 ; Y |Q1 ), o Re ≤ I(U1 ; Y |Q1 ) − I(U1 ; Z|Q1 ) . By virtue of Fourier-Motzkin elimination, it is readily shown that [ [ CB (π ∗ ) = RB (π ∗ ), π∗

(37)

(38)

π∗

and from (9) and (10), C˜ =

[ CA (π ∗ ) ∪ RB (π ∗ ) π∗

= C∪

[

RB (π ∗ ).

π∗

The region C is achievable by the coding method given in Section III. Therefore, if we have an achievability scheme to achieve RB (PQ∗ 1 U1 U2 X1 X2 ) for any given PQ∗ 1 U1 U2 X1 X2 ∈ P ∗ , then the region C˜ is also achievable. We turn to showing an achievable scheme to the region RB (PQ∗ 1 U1 U2 X1 X2 ) for arbitrarily fixed π ∗ = PQ∗ 1 U1 U2 X1 X2 ∈ P ∗ . The description of a achievability scheme is a combination of the scheme in Appendix A and the scheme given in [2].

16

A PPENDIX C P ROOF OF P ROPOSITION 2 The condition I(U1 ; Y |Q) > I(U1 ; Z|Q) is necessary since otherwise there is no equivocation in CB (P ∗ ). As we have seen in (7), there are three cases for which the region CA (P ∗ ) is of different form. If I(U2 ; Y |Q1 ) ≤ I(U2 ; Z|Q1 ), then the constraint on Re ∈ CA (P ∗ ) is given by Re ≤ [I(U1 ; Y |U2 Q1 ) − I(U1 ; Z|U2 Q1 )]+ . Then the constraint on Re ∈ CB (P ∗ ) has an effect iff I(U1 ; Y |Q1 ) − I(U1 ; Z|Q1 ) ≥ I(U1 ; Y |U2 Q1 ) − I(U1 ; Z|U2 Q1 ).

(39)

First note that I(U1 ; Y |Q1 ) − I(U1 ; Z|Q1 ) − (I(U1 ; Y |U2 Q1 ) − I(U1 ; Z|U2 Q1 )) = I(U1 ; Z|U2 Q1 ) − I(U1 ; Z|Q1 ) − (I(U1 ; Y |U2 Q1 ) − I(U1 ; Y |Q1 )).

(40)

Since I(U1 ; Z|U2 Q1 ) − I(U1 ; Z|Q1 ) = I(U1 U2 ; Z|Q1 ) − I(U1 ; Z|Q1 ) − I(U2 ; Z|Q1 ) = I(U2 ; Z|U1 ) − I(U2 ; Z|Q1 ) and also I(U1 ; Y |U2 Q1 ) − I(U1 ; Y |Q1 ) = I(U2 ; Y |U1 ) − I(U2 ; Y |Q1 ), then (40) becomes I(U1 ; Y |Q1 ) − I(U1 ; Z|Q1 ) − (I(U1 ; Y |U2 Q1 ) − I(U1 ; Z|U2 Q1 )) = I(U2 ; Z|U1 ) − I(U2 ; Z|Q1 ) − (I(U2 ; Y |U1 ) − I(U2 ; Y |Q1 ))).

(41)

Therefore, (39) holds iff I(U2 ; Z|U1 ) − I(U2 ; Z|Q1 ) ≥ I(U2 ; Y |U1 ) − I(U2 ; Y |Q1 ), leading to (12). If I(U2 ; Z|Q1 ) ≤ I(U2 ; Y |Q1 ) ≤ I(U2 ; Z|U1 ), then the constraint on Re ∈ CA (P ∗ ) is given by Re ≤ [I(U1 U2 ; Y |Q1 ) − I(U1 U2 ; Z|Q1 )]+ . Then the constraint on Re ∈ CB (P ∗ ) has an effect iff I(U1 ; Y |Q1 ) − I(U1 ; Z|Q1 ) ≥ I(U1 U2 ; Y |Q1 ) − I(U1 U2 ; Z|Q1 ).

(42)

We note that I(U1 ; Y |Q1 ) − I(U1 ; Z|Q1 ) − (I(U1 U2 ; Y |Q1 ) − I(U1 U2 ; Z|Q1 )) = I(U2 ; Z|U1 ) − I(U2 ; Y |U1 ).

(43)

Therefore, (42) holds iff (13) holds. If I(U2 ; Z|U1 ) ≤ I(U2 ; Y |Q1 ), then the constraint on Re ∈ CA (P ∗ ) is given by Re ≤ [I(U1 U2 ; Y |Q1 ) − I(U1 ; Z|Q1 )]+ . In this case, since it always holds I(U1 ; Y |Q1 ) − I(U1 ; Z|Q1 ) ≤ I(U1 U2 ; Y |Q1 ) − I(U1 ; Z|Q1 ), the constraint on Re ∈ CB (P ∗) has no effect.

(44) ✷

17

A PPENDIX D T HE W IRE -TAP C HANNEL WITH A D EAF -I NTERFERER In wireless network settings, sender 2 (the helper) in the wire-tap channel with a helper can observe a noisy sequence of the transmitted sequence X1N from sender 1. Let Y1N denote the sequence observed by sender 2. For some security systems, it is desired to avoid leaking information about W1 to sender 2, which motivates the introduction of another type of the wire-tap channel with a helper, called the wire-tap channel with a deaf-helper (a deaf-interferer) [1]. The wire-tap channel with a deaf-helper looks like the relay-eavesdropper channel, in which a relay node observes Y1N and helps to increase the rate of W1 or the equivocation at the eavesdropper. Note that in this channel model, the relay node might (partially) decode the message W1 for the cooperation. On the other hand, the scenario of the wire-tap channel with a deaf-helper describes the setting in which sender 1 with secret messages does not fully trust the other sender (the helper) but still wishes to get help from the user cooperation. As in [1], we assume that sender 2 is not malicious, and willing to help the communication from sender 1 to the receiver. Since sender 2 ”forwards” a dummy sequence instead of forwarding a (partial) message of sender 1, the cooperation scheme is called a noise-forwarding (NF) strategy. In this setting, a rate-equivocation region is defined by introducing an additional security constraint as follows: Definition 3: A rate-equivocation pair (R1 , Re ) is said to be achievable if there exists a sequence of (N, M1 ) codes such that for every ǫ > 0, log2 M1 − ǫ, N ≤ ǫ, 1 H(W1 |Z N ) ≥ Re − ǫ, , N 1 H(W1 |Y1N X2N ) ≥ Re − ǫ , N

R1 ≥ Pe(N ) Re(N ) Rs(N )

for all sufficiently large N. We conjecture that the convex hull of the following rate-equivocation region is achievable n [ CDH = (R1 , Re ) : 0 ≤ Re ≤ R1 , PQ1 PU1 |Q1 PU2 PX1 |U1 PX2 |U2 PY Z|X1 X2

R1 ≤ I(U1 ; Y |U2 Q1 ) + min{I(Q1 ; Y |U2 ), I(Q1 ; Z|U2 )}, n o ′ ′ Re ≤ max R3 − I(U1 ; Z|U2 Q1 ) − I(U2 ; Y |Q1 ), R3 − I(U1 U2 ; Z|Q1 ) o + Re ≤ [I(U1 ; Y |U2 Q1 ) − I(U1 ; Y1 |U2 Q1 )] where R2′ = min{I(U2 ; Y |Q1 ), I(U2 , Z|U1 )}, R3′ = I(U1 ; Y |U2 Q1 ) + R2′ , and Q1 , U1 and U2 are auxiliary random variables satisfying the Markov chain conditions given by (6). As for the perfect-secrecy rate, the above achievable rate-equivocation region reduces to the following result, which is the same as that given in [1, Theorem 6]. Theorem 2: The perfect-secrecy rate for the wire-tap channel with a deaf-helper, given by R1 =

sup P U 1 X1 P U 2 X2

min{Re,1 , Re,2 },

18

where Re,1 , max I(U1 ; Y |U2 ) − I(U1 ; Z|U2 ) + R2′ − I(U2 ; Y |), I(U1 ; Y |U2 ) + R2′ − I(U1 U2 ; Z) , and Re,2 , [I(U1 ; Y |U2 ) + R2′ − I(U1 ; Y1|U2 )]+ , is achievable. R EFERENCES [1] L. Lai and H. E. Gamal, “The relay-eavesdropper channel: Cooperation for secrecy,” IEEE Trans. Inform. Theory, vol. 54, no. 9, pp. 4005–4019, Sep. 2008. [2] X. Tang, R. Liu, P. Spasojevi´c, and H. V. Poor, “Interference assisted secret communication,” in Proc. IEEE Inform. Theory Workshop, Porto, Portugal, 2008. [3] A. Wyner, “The wire-tap channel,” Bell Syst. Tech. J., vol. 54, no. 8, pp. 1355–1387, Jan. 1975. [4] I. Csiszár and J. Korner, “Broadcast channels with confidential messages,” IEEE Trans. Inform. Theory, vol. 24, no. 3, pp. 339–348, May 1978. [5] T. M. Cover and J. A. Thomas, Elements of Information Theory. New York, NY: John Wiley & Sons, 1991.