The General Gaussian Multiple Access and Two-Way Wire-Tap Channels: Achievable Rates and Cooperative Jamming

0.44 MB·English

by  Ender Tekin

IEEETRANSACTIONSONINFORMATIONTHEORY 1
The General Gaussian Multiple Access and
Two-Way Wire-Tap Channels:
Achievable Rates and Cooperative Jamming
Ender Tekin, Student Member, IEEE and Aylin Yener, Member, IEEE
Abstract—The General Gaussian Multiple Access Wire-Tap A rigorous analysis of information theoretic secrecy was
8
Channel (GGMAC-WT) and the Gaussian Two-Way Wire-Tap first given by Shannonin [11].In this work, Shannonshowed
0
Channel(GTW-WT)areconsidered.IntheGGMAC-WT,multi-
0 that to achieve perfect secrecy in communications, which is
pleuserscommunicatewithanintendedreceiverinthepresence
2 equivalent to providing no information to an enemy crypt-
of an eavesdropper who receives their signals through another
n GMAC.IntheGTW-WT,twouserscommunicatewitheachother analyst, the conditional probability of the cryptogram given
a overacommonGaussianchannel,withaneavesdropperlistening a message must be independent of the actual transmitted
J through a GMAC. A secrecy measure that is suitable for this message. In other words, the a posteriori probability of a

5 multi-terminal environment is defined, and achievable secrecy message must be equivalent to its a priori probability.
2 rate regions are found for both channels. For both cases, the
In [12], Wyner applied this concept to the discrete memo-
powerallocationsmaximizingtheachievablesecrecysum-rateare
] determined.Itisseenthattheoptimumpolicymaypreventsome ryless channel. He defined the wire-tap channel, where there
T terminals from transmission in order to preserve the secrecy of is a wire-tapper who has access to a degraded version of the
I thesystem.Inspiredbythisconstruct,anewscheme,cooperative intended receiver’s signal. Using the normalized conditional
s. jamming, is proposed, where users who are prevented from entropy∆ofthetransmittedmessagegiventhereceivedsignal
c transmittingaccordingtothesecrecysum-ratemaximizingpower
atthewire-tapperasthesecrecymeasure,hefoundtheregion
[ allocation policy “jam” the eavesdropper, thereby helping the
remainingusers.Thisschemeisshowntoincreasetheachievable of all possible (R,∆) pairs, and the existence of a secrecy
2
v secrecy sum-rate. Overall, our results show that in multiple- capacity, Cs, the rate up to which it is possible to limit the
accessscenarios,userscanhelpeachothertocollectivelyachieve rateofinformationtransmittedtothewire-tappertoarbitrarily
2
positive secrecy rates. In other words, cooperation among users
1 small values.
can be invaluable for achieving secrecy for the system.
1 In [13], it was shown that for Wyner’s wire-tap chan-
2 Index Terms—Secrecy Capacity, Gaussian Multiple Access nel, it is possible to send several low-rate messages, each
0 Channel, Gaussian Two-Way Channel, Wire-Tap Channel, Con-
completely protected from the wire-tapper individually, and
7 fidential Messages
use the channel at close to capacity. However, if any of
0
/ the messages are available to the wire-tapper, the secrecy of
s
c I. INTRODUCTION the rest may also be compromised. Reference [14] extended
: Wyner’s results in [12] and Carleial and Hellman’s results in
v GAUSSIAN multiple-access channels and two-way chan-
[13] to Gaussian channels. The seminal work by Csisza´r and
i nelsaretwooftheearliestchannelsthatwereconsidered
X Ko¨rner, [15], generalized Wyner’s results to “less noisy” and
in the literature. The multiple-access channel capacity region
r “more capable” channels. Furthermore, it examined sending
a was determinedin [3],[4].The two-waychannelwas initially
commoninformationto both the receiverand the wire-tapper,
examined by Shannon, [5], where he found inner and outer
while maintaining the secrecy of some private information
bounds for the general two-way channel, and determined the
thatis communicatedto the intendedreceiveronly.Reference
capacity region for some special cases. In [6], it was shown
[16]suggestedthatthesecrecyconstraintdevelopedbyWyner
that the inner bound found by Shannon was not tight in
needed to be strengthened, since it constrains the rate of
general.ThecapacityregionoftheGaussiantwo-waychannel
information leaked to the wire-tapper, rather than the total
was found by Han in [7]. A related, somewhat more general
information, and the information of interest might be in this
case called two-user channels was studied in [8], [9]. For a
small amount.It was then shown that the results of [12], [15]
comprehensivereviewofthesechannels,thereaderisreferred
can be extended to “strong” secrecy constraints for discrete
to [10].
channels, where the limit is on the total leaked information
ratherthanjustthe rate,with nolossin achievablerates, [16].
Mauscript received February 16, 2007; revised September 30, 2007. This
work has been supported by NSF grant CCF-0514813 “Multiuser Wireless In the past two decades, commonrandomnesshas emerged
Security” and DARPA ITMANET Program grant W911NF-07-1-0028. This as a valuableresourcefor secretkey generation,[17],[18].In
work was presented in part in the 2006 Allerton Conference on Communi-
[17], it was shown that the existence of a “public” feedback
cations, Control, andComputing, [1],and2007International Symposiumon
Information Theory,[2]. channel can enable the two parties to be able to generate a
TheauthorsarewiththeDepartmentofElectricalEngineeringatthePenn- secretkeyevenwhenthewire-tapcapacityiszero.References
sylvania StateUniversity, University Park,PA16802(email: [email protected],
[19] and [20] examined the secret key capacity and common
[email protected]).
Digital ObjectIdentifier randomness capacity, for several channels. These results also
2 IEEETRANSACTIONSONINFORMATIONTHEORY
benefit from [16] to provide “strong” secret key capacities. referring to “weak” secrecy, where the rate of information
Maurer also examined the case of active adversaries, where leakedtotheadversaryislimited.Assuch,thiscanbethought
the wire-tapper has read/write access to the channel in [21]– of as “almost perfect secrecy”. We also find the sum-rate
[23].Thesecretkeygenerationproblemwasinvestigatedfrom maximizing power allocations for the general case, which is
a multi-party point of view in [24] and [25]. Notably, Csisza´r more interesting from a practical point of view. It is seen
and Narayan considered the case of multiple terminals where that as long as the users are not single-user decodable at
a number of terminals try to distill a secret key and a subset the eavesdropper, a secrecy-rate trade off is possible between
of these terminals can act as helper terminals to the rest in the users. Next, we show that a non-transmitting user can
[26], [27]. help increase the secrecy capacity for a transmitting user
Recently, severalnew models have emerged,examiningse- by effectively “jamming” the eavesdropper, and even enable
crecyforparallelchannels[28],[29],relaychannels[30],and secretcommunicationsthat would notbe possible in a single-
fading channels [31], [32]. Fading and parallel channels were userscenario.Wetermthisnewschemecooperativejamming.
examined together in [33], [34]. Broadcast and interference The GTW-WT is shown to be especially useful for secret
channels with confidential messages were considered in [35]. communications, as the multiple-access nature of the channel
References [36], [37] examined the multiple access channel hurts the eavesdropper without affecting the communication
with confidentialmessageswhere two transmitterstry to keep rate. This is due to the fact that the transmitted messages
their messages secret from each other while communicating of each user essentially help hide the other user’s secret
with a common receiver. In [36], an achievable region was messages,andreducetheextrarandomnessneededinwire-tap
foundin general, and the capacity regionwas foundfor some channels to confuse the eavesdropper.
special cases. MIMO channels were considered in [38], [39]. The rest of the paper is organized as follows: Section II
In [1], [40]–[42], we investigated multiple access channels describes the system model for the GGMAC-WT and GTW-
where transmitters communicate with an intended receiver in WT and the problem statement. Section III describes the
the presence of an external wire-tapper from whom the mes- general achievable rates for the GGMAC-WT and GTW-WT.
sages must be kept confidential. In [40]–[42], we considered Sections IV and V give the sum-secrecy rate maximizing
the case where the wire-tapper gets a degraded version of a power allocations, and the achievable rates with cooperative
GMACsignal, and definedtwo separatesecrecy measuresex- jamming. Section VI gives our numerical results followed by
tending Wyner’s measure to multi-user channels to reflect the our conclusions and future work in Section VII.
level of trust the network may have in each node. Achievable
rate regions were found for different secrecy constraints, and
II. SYSTEMMODELAND PROBLEM STATEMENT
it was shown that the secrecy sum-capacity can be achieved
We consider K users communicatingin the presence of an
using Gaussian inputs and stochastic encoders. In addition,
eavesdropper who has the same capabilities. Each transmitter
TDMA was shown to also achieve the secrecy sum-capacity.
k , 1,...,K has two messages, Ws which is
In this paper, we consider the General Gaussian Multiple ∈ K { } k
secret and Wo which is open2, from two sets of equally
Access Wire-Tap Channel (GGMAC-WT) and the Gaussian k
likely messages s = 1,...,Ms , o = 1,...,Mo .
Two-WayWire-TapChannel(GTW-WT),bothofwhichareof Wk { k} Wk { k}
Let W = (Ws,Wo), = s o, M = MsMo,
interest in wireless communicationsas they correspondto the k k k Wk Wk × Wk k k k
Wo = Wo , and Ws = Ws . The messages
ctraasnesmwihtteerres,asusicnhglaespihnyasincaaldc-hhoancnneeltwisourkti.lizWeed cboynmsidueltripalne areSencod{edku}skin∈gS(2nRk,n)Scodes{inkto}k{∈X˜Skn(Wk)}, where
external eavesdropper1 that receives the transmitters’ signals Rk = n1 log2Mk = n1 log2Mks+ n1 log2Mko =Rks +Rko. The
throughageneralGaussianmultipleaccesschannel(GGMAC) encoded messages {X˜k} = {X˜kn} are then transmitted. We
inbothsystemmodels.Weutilizeasuitablesecrecyconstraint assumethechannelparametersareuniversallyknown,andthat
whichisthenormalizedconditionalentropyofthetransmitted theeavesdropperalsohasknowledgeofthecodebooksandthe
secret messages given the eavesdropper’s signal, correspond- coding scheme. In other words, there is no shared secret. The
ing to the “collective secrecy” constraints used in [42]. We two channels we consider in this paper are described next.
show that satisfying this constraint implies the secrecy of
the messages for all users. In both scenarios, transmitters A. The General Gaussian Multiple-Access Wire-Tap Channel
are assumed to have one secret and one open message to
This is a scenario where the users communicate with a
transmit. This is different from [42] in that the secrecy rates
common base station in the presence of an eavesdropper,
are notconstrainedto be atleast a fixedportionof the overall
wherebothchannelsaremodeledasGaussianmultiple-access
rates. We find an achievable secrecy rate region, where users
channels as shown in Figure 1. The intended receiver and
can communicate with arbitrarily small probability of error
the wire-tapper receive Y˜ = Y˜n and Z˜ = Z˜n, respectively.
with the intended receiver under perfect secrecy from the
The receiver decodes Y˜ to get an estimate of the transmitted
eavesdropper, which corresponds to the result of [42] for the
messages,Wˆ s,Wˆ o.We wouldliketocommunicatewiththe
degraded case. We note that, in accordance with the recent K K
receiverwitharbitrarilylowprobabilityoferror,whilekeeping
literature, when we use the term perfect secrecy, we are
thewire-tapper(eavesdropper)ignorantofthesecretmessages,
1Even though we faithfully follow Wyner’s terminology in naming the
channels, admittedly in wireless system models, eavesdropper is a more 2Wewouldliketostressthatopenisnotthesameaspublic,i.e.,wedonot
appropriate termfortheadversary. imposeadecodability constraint fortheopenmessagesattheeavesdropper.
TEKINANDYENER:THEGENERALGAUSSIANMULTIPLEACCESSANDTWO-WAYWIRE-TAPCHANNELS 3
NM N1 N2
SOURCE1 W1s,W1o ENCODER1 X1 + Y RECEIVER SOURCE1 W1s,W1o REENCCEOIVDEERR11 Y√X1α11 + + √XYα222 ERNECCOEIDVEERR22 W2s,W2o SOURCE2
√h1
√h1 √h2
SOURCEK WKs,WKo ENCODERK XK √hK + Z EAVESDROPPER + Z EAVESDROPPER
NW NW
Fig.1. The standardized GMAC-WT system model. Fig.2. The standardized GTW-WTsystem model.
Ws. The signals at the intended receiver and the wire-tapper communicate the open and secret messages with arbitrarily
K
are given by low probability of error, while maintaining secrecy of the
secret messages. The signals at the intended receiver and the
K
Y˜ = hMX˜ +N˜ (1a) wiretapper are given by
k k M
XkK=1p Y˜1 =X˜1+ hM2X˜2+N˜1 (4a)
Z˜ = hWkX˜k+N˜W (1b) Y˜2 = hM1X˜p1+X˜2+N˜2 (4b)
Xk=1p Z˜ =phW1X˜1+ hW2X˜2+N˜W (4c)
whereN˜ ,N˜ aretheAWGN,X˜ isthetransmittedcodeword
M W k whereN˜ 0,σ2pandN˜ p 0,σ2 .Wealsoassume
of user k, and hM,hW are the channel gains of user k to the k ∼N k W ∼N W
k k the same power constraints given in (2) (with K = 2), and
intended receiver (main channel, M), and the eavesdropper (cid:0) (cid:1) (cid:0) (cid:1)
again use an equivalent standard form as illustrated in Figure
(wire-tapchannel,W),respectively.EachcomponentofN˜
0,σ2 andN˜ 0,σ2 .WealsoassumethefollowMin∼g 2:
N M W ∼N W
transmit power constraints: Y =√α X +X +N (5a)
(cid:0) (cid:1) (cid:0) (cid:1) 1 1 1 2 1
1 n X˜2 P˜¯ , k =1,...,K. (2) Y2 =X1+√α2X2+N2 (5b)
n ki ≤ k Z= h X + h X +N (5c)
i=1 1 1 2 2 W
X
Similar to the scaling transformationto obtain the standard where p p
fGoMrmACof-WthTe ibnytearnfereeqnucievaclehnatnnstealn,d[4ar3d],fowremc,a[n42r]e:present any • the codewords{X˜} are scaled to getX1 =rhσM122X˜1 and
K X = hM2X˜ ;
2 σ2 2
Y =kX=1Xk+NM (3a) • the marxim1um powers are scaled to get P¯1 = hσM122P˜¯1 and
K P¯ = hM2P˜¯ ;
Z= hkXk+NW (3b) • th2e traσn12sm2itters’ new channel gains are given by α1 =
where, for each k, kX=1p • thhM1σe2σ212wiarnedtapαp2e=r’shnM2σe1σ2w22;channel gains are given by h1 =
• the codewords are scaled to get Xk = hσMk2X˜k; hhMW1σσ222 and h2 = hhMW2σσ212;
• The new power constraints are P¯k = hσrMk2P˜¯kM; • th1e Wnoises are nor2maWlized by Nk = σ1k2N˜k, k =1,2 and
• The wiretapper’s new channel gains areMhk = hhMWkσσ2M2; NW = σ1W2N˜W.
k W
• The noises are normalized to get NM = Nσ˜2M and NW =
M C. Preliminary Definitions
N˜
W.
σ2
W In this section, we present some useful preliminary defini-
We can show that the eavesdropper gets a stochastically
tionsincludingthesecrecyconstraintwewilluse.Inparticular,
degraded version of the receiver’s signal if h =...=h
1 K ≡ the secrecy constraint we used is the “collective secrecy
h<1. We considered this special case in [41], [42].
constraint” we defined in [40], [42], and is suitable for the
multi-access nature of the systems of interest.
B. The Gaussian Two-Way Wire-Tap Channel
Definition 1 (Collective secrecy constraint): We use the
In this scenario, two transmitter/receiver pairs communi- normalized joint conditional entropy of the transmitted mes-
cate with each other over a common channel. Each receiver sages given the eavesdropper’s received signal as our secrecy
k = 1,2 gets Y˜k = Y˜kn and the eavesdropper gets Z˜ = Z˜n. constraint, i.e.,
Receiver k decodes Y˜ to get an estimate of the transmit- H(Ws Z)
k ∆ , S| (6)
ted messages of the other user. The users would like to S H(Ws)
S
4 IEEETRANSACTIONSONINFORMATIONTHEORY
for any set of users. For perfect secrecy of all positive secrecy rates and those who cannot. In addition, we
S ⊆ K
transmitted secret messages, we would like will say that a user is single-user decodableif its rate is such
that it can be decoded by treating the other user as noise.
H(Ws Z)
∆ = K| 1. (7) A user group is single-user decodable by the eavesdropper
K H(WKs) → if CM(P) CS˜W(P). Our achievable rates cannot guarantee
S ≤ S
Assume∆K 1 ǫforsomearbitrarilysmallǫasrequired. secrecy for such a group of users.
≥ −
Then,
H(Ws Z) H(Ws) ǫH(Ws) (8)
K| ≥ K − K III. ACHIEVABLE SECRECY RATEREGIONS
H(Ws Z) H(Ws)+H(Ws Ws)
S| ≥ ǫHS(Ws) HSc(|WsS Ws,Z) (9) A. The General Gaussian Multiple Access Wire-Tap Channel
− K − Sc| S
H(Ws) ǫH(Ws) (10) Inthissection,wepresentourmainresultsfortheGGMAC-
≥ S − K
∆ 1 ǫ′ (11) WT. We first define two separate regions and then give an
S
≥ − achievable region:
where ǫ′ , H(WKs)ǫ 0 as ǫ 0. If H(Ws) = 0, then Definition 3 (GGMAC-WT Superposition Region): Let
we define ∆SH(=WSs1). T→hus, the pe→rfect secrecy oSf the system Xk ∼ N (0,Pk) for all k. Then, the superposition region,
impliestheperfectsecrecyofanygroupofusers,guaranteeing MA-SUP, is given by
G
that when the system is secure, so is each individual user.
Definition 2 (Achievable rates): Let R = (Rs,Ro). The MA-SUP(P), R:
k k k G
rate vector R=(R ,...,R ) is said to be achievable if for
1 K (Rs +Rno) I(X ;Y X ),
anygivenǫ>0thereexistsacodeofsufficientlengthn such k k ≤ S | Sc ∀S ⊆K
that kX∈S
1 logMs Rs ǫ, k=1,...,K (12a) Rks ≤[I(XS;Y|XSc)−I(XS;Z)]+, ∀S ⊆K (19)
n k ≥ k− kX∈S o
1
logMo Ro ǫ, k=1,...,K (12b) which can be written as
n k ≥ k−
and
MA-SUP(P)= R:
1 G
P = P Wˆ =WW sent ǫ (12c) (cid:26)
e Kk=1Mk W∈×XKk=1W{k 6 | }≤ (Rks+Rko)≤ 21log 1+ k∈SPk , ∀S⊆K
is the aveQrage probability of error. In addition, we need kX∈S (cid:0) P (cid:1)
1
Rs log 1+ P
∆ 1 ǫ (12d) k ≤ 2 k
K ≥ − k∈S (cid:20) (cid:18) k∈S (cid:19)
X X
where ∆K denotes our secrecy constraint and is defined in log 1+ k∈ShkPk +, .
(7). We will call the set of all achievable rates, the secrecy- − 1+ h P ∀S⊆K
(cid:18) P k∈Sc k k(cid:19)(cid:21) (cid:27)
capacity region, and denote it MA for the GGMAC-WT, and (20)
TW for the GTW-WT, respectivCely. P
C
Before we state our results, we also define the following Definition 4 (GGMAC-WT TDMA Region): Let α be
k
notation which will be used extensively in the rest of this such that 0 α 1 for all k and K α ={ 1}. Let
≤ k ≤ k=1 k
paper: X (0,P /α ) for all k. Then, the TDMA region,
k k k
∼ N P
[ξ]+ ,max[ξ,0] (13) GMA-TDMA, is given by
1
CM(P), log 1+ P , (14) MA-TDMA(P,α), R:
S 2 k∈S k S ⊆K G
CSW(P), 12log(cid:0)1+Pk∈ShkP(cid:1)k , S ⊆K (15) RRkss +Rαko[I≤(XαkI;nY(XXk;Y)|XkIc()X, ;∀Zk∈XK )]+, k (21)
C˜W(P), 1log(cid:0)1+P k∈Shk(cid:1)Pk , (16) k ≤ k k | kc − k | kc ∀ ∈K
S 2 1+ h P S ⊆K o
(cid:18) P k∈Sc k k(cid:19) which is equivalent to
, P:0 P P¯ , k (17)
k Pk
P ≤ ≤ ∀
P¯ ,(cid:8)P¯1,...,P¯K (cid:9) (18) MA-TDMA(P,α)= R:
G
(cid:26)
andLawstelayk, wifeh(cid:8)iknf>orm1.aTllhyisc(cid:9)aisllathweakythofuisnedricsatrtionnggwifhehtkhe≤r th1e, Rks +Rko ≤ α2k log 1+ Pαk , ∀k∈K
intendedreceiverorthewiretapperisatamoreofanadvantage (cid:18) k(cid:19)
α P h P +
concerning that user, and is equivalent to stating whether the Rs k log 1+ k log 1+ k k , k .
single-user secrecy capacity of that user is positive or zero. k ≤ 2 (cid:20) (cid:18) αk(cid:19)− (cid:18) αk (cid:19)(cid:21) ∀ ∈K(cid:27)
(22)
We later extendthisconceptto referto userswhocanachieve
TEKINANDYENER:THEGENERALGAUSSIANMULTIPLEACCESSANDTWO-WAYWIRE-TAPCHANNELS 5
1
Remark 1: The superposition and TDMA regions can also Rs log 1+ P
be written as follows: k ≤ 2 k
k∈S (cid:20) (cid:18) k∈S (cid:19)
X X
GMA-SUP(P)= R: log 1+ k∈ShkPk +, (28)
− 1+ h P ∀S⊆K
(Rs +Rno) CM(P), (cid:18) P k∈Sc k k(cid:19)(cid:21)
k k ≤ S ∀S ⊆K
k∈S which we can also write as: P
X
+
Rks ≤ CSM(P)−C˜SW(P) , ∀S ⊆K (23) k∈S(Rks+Rko+Rkx)≤CSM, ∀S⊆K (29)
kX∈S h i o (Ro+Rx) CW, , with equality if = (30)
GMA-TDMA(P,α)=nR:P¯ PPkk∈∈SSRksk≤ CkSM−≤C˜SWS+,∀S∀⊆SK⊆K. S K (31)
Rks +Rko ≤αkCkM αk , ∀k ∈K PNote that if(cid:2)(31) is (cid:3)zero for a group of users, we cannot
(cid:18) k(cid:19) achieve secrecy for those users. When = , if the sum-
Rs α CM P¯k CW P¯k +, k (24) capacity of the main channel is less thaSn thaKt of the eaves-
k ≤ k k α − k α ∀ ∈K
(cid:20) (cid:18) k(cid:19) (cid:18) k(cid:19)(cid:21) o dropper channel, i.e., CKM ≤ CKW, secrecy is not possible for
in accordance with the definitions in (14)–(16). the system. Assume this quantity is positive. To ensure that
Theorem 1: The rate region given below is achievable for we can mutually satisfy both (31), (30), we can reclassify
the GGMAC-WT: some open messages as secret. Clearly, if we can guarantee
secrecy for a larger set of messages, secrecy is achieved for
MA =convex closure of the original messages. From the first set of conditionsin (25)
G
and the GMAC coding theorem, [44], with high probability
MA-SUP(P) MA-TDMA(P¯,α) . (25) the receiver can decode the codewords with low probability
P∈PG ! 0≤α≤1G ! of error. To show the secrecy condition in (12), first note
[ [ [
Σkαk=1 that, the coding scheme described is equivalent to each user
Proof: We first show that the superposition encoding k selecting one of Ms messages, and sending a uniformly
k
rate region given in (20) for a fixed power allocation is chosen codeword from among MoMx codewords for each.
k k
achievable. Consider the following coding scheme for rates Define X = K √h X , and we have
R MA-SUP(P) for some P : Σ k=1 k k
∈G ∈P
Superposition Encoding Scheme: For each user k, consider H(Ws Z)=PH(Ws) I(Ws;Z) (32)
K| K − K
the following scheme: =H(Ws) I(Ws;Z)+I(Ws;ZX ) (33)
1) Generate 3 codebooks Xs,Xo and Xx. Xs consists of K − K K | Σ
k k k k =H(Ws) h(Z)+h(ZWs)
Ms codewords,eachcomponentofwhichisdrawnfrom K − | K
Neack(h0,cλoskmPpkon−enεt).raCnoddoembolyokdrXawoknhafsroMmkoco(d0e,wλoorPdswiεth) =H(Ws) +Ih((XZ|X;ZΣ))+−Ih((XZ|W;ZKs,WXsΣ)) ((3345))
and Xx has Mx codewords with eachNcomponkenkt−ran- K − Σ Σ | K
domlykdrawn frkom (0,λxP ε) where ε is an arbi- where we used Ws XΣ Z, and thus we have
trarilysmallnumberNtoensukrekth−atthepowerconstraints h(ZWs,XΣ) = h(ZX→Σ) to get→(35). We will consider the
| |
two termsindividually.First, we have the trivialbounddueto
on the codewords are satisfied with high probability
and λs +λo + λx = 1. Define Rx = 1 logMx and channel capacity:
k k k k n k
Mt =MsMoMx.
k k k k I(X ;Z) nCW(P). (36)
2) To transmit message Wk = (Wks,Wko) ∈ Wks ×Wko, Σ ≤ K
user k finds the 2 codewords corresponding to compo-
Now write
nents of W and also uniformly chooses a codeword
k
Wx fromXx.User k thenaddsallthesecodewordsand I(X ;ZWs)=H(X Ws) H(X Ws,Z). (37)
k k Σ | K Σ| K − Σ| K
transmitsthe resultingcodeword,X , so thatit actually
k
transmits one of Mkt codewords. Let Rkt = n1 logMkt = Since user k independently sends one of MkoMkx codewords
Ro+Rs+Rx.Notethatsinceallcodewordsarechosen equally likely for each secret message,
k k k
uniformly, user k essentially transmits one of MoMx
k k K
ocovdereawllorradtseaotfrtaranndsommissfioorneiascRhtm. essage Wks, and its H(XΣ|WKs)=log (MkoMkx)! (38)
k k=1
Y
Specifically, we choose the rates to satisfy K
=n (Ro +Rx) (39)
1 k k
(Rks+Rko+Rkx)≤ 2log 1+ Pk , ∀S ⊆K (26) Xk=1 !
k∈S (cid:18) k∈S (cid:19) =nCW(P). (40)
X X K
1
(Ro+Rx) log 1+ h P , ,
k k ≤ 2 k k ∀S⊆K We can also write
k∈S (cid:18) k∈S (cid:19)
X X
with equality if = (27) H(X Ws,Z) nδ (41)
S K Σ| K ≤ n
6 IEEETRANSACTIONSONINFORMATIONTHEORY
GMA−SUP, h=0.1, h= 0.3
0.8 GMA−TDMA,1 h=0.1,2 h=0.3 1.2 GMA−SUP(4,1.5)
1 2 GMA−SUP(4,4)
0.7 GMA−SUP, h1=0.3, h2=0.7 1 GMA−TDMA(4,4)
GMA−TDMA, h=0.3, h=0.7 GMA(4,4)
0.6 1 2 CMAC(4,4)
GMA−SUP, h=0.7, h=1.4
1 2 0.8
0.5 GMA−TDMA, h=0.7, h=1.4
1 2
R2 CMAC R2
0.4 0.6
0.3
0.4
0.2
0.2
0.1
00 0.2 0.4 0.6 0.8 1 1.2 00 0.2 0.4 0.6 0.8 1 1.2
R R
1 1
Fig.3. GGMAC-WTachievableregionsfordifferentchannelparam- Fig.4. GGMAC-WTachievable secrecy regionwhenP¯1 =4,P¯2 =
eters, GMA(P1 =4,P2 =2). 4,h1 =.1,h2 =.3.
where δ 0 as n since, with high probability, the
n → → ∞ combination of different superposition and TDMA regions.
eavesdroppercandecodeX givenWs dueto(30)andcode
Σ K Note also that the main extra condition for the superposition
generation. Using (36), (37), (40) and (41) in (35), we get
region is on the total extra randomness added. As a result,
H(Ws Z) H(Ws) nCW(P)+nCW(P) nδ (42) it is possible for “stronger” users to help “weak” users by
K| ≥ K − K K − n
=H(Ws) nδ . (43) contributingmoretothenecessaryextranumberofcodewords,
K − n which is the sum-capacity of the eavesdropper. Such a weak
Now, let us consider the TDMA region given in (22). This useronlyhastomakesurethatitisnotsingle-userdecodable,
region is obtained when users who can achieve single-user provided the stronger users are willing to sacrifice some of
secrecyuse a single-userwire-tapcodeas in [14]in a TDMA their own rate and generate more superfluous codewords. In
schedule, where the time-share of each user k is given by other words, we see that usersin a set are furtherprotected
a0ch≤ievαekse≤cre1cya,nid.e., hKka=v1inαgkh=k <1.1A, trtraannmsmitsittfeorrkαkwphoortcioann ufrnodmectohdeaebalve,escdormopppaerredbytothtehefacsitntghlaet-uuSsseerrscianses.etTShceaTreDaMlsAo
of the time when aPll other users are silent, using P¯k power, region, on the other hand, does not allow users to help each
αk
satisfying its average power constraint over the TDMA time- otherthisway.Assuch,onlyuserswhosechannelgainsallow
frame. This approach was used in [42] to achieve secrecy them to achieve secrecy on their own are allowed to transmit.
sum-capacity for individual constraints. When the channel is
degraded, i.e., hk = h for all k , then for collective For the special degraded case of h = ...= h , h 1,
∈ K 1 K
constraints the TDMA region is seen to be a subset of the the perfect secrecy rate region for Rs becomes the re≤gion
superposition region. However, this is not necessarily true for k
given by [42, Theorem 1] for δ = 1. We also observe that
thegeneralcase,andbytime-sharingbetweenthetwoschemes
even though there is a limit on the secrecy sum-rate achieved
we can generally achieve a larger achievable region, given in
by our scheme, it is possible to send open messages to the
(25).
intended receiver at rates such that the sum of the secrecy
We remark that it is possible to further divide the “open”
rate andopenrate forall usersis in the capacityregionofthe
messagestogetmoresetsof“private”messageswhicharealso
MACchanneltotheintendedreceiver.Eventhoughwecannot
perfectly secret, i.e., if we let o = ´s ´o, k, then as
longas weimposethesame resWtrikctionWsokn×R´Ws aks R∀s,we can send at capacity with secrecy, the codewords used to confuse
the eavesdropper may be used to communicate meaningful
achieveperfectsecrecyof W´ s, as in [14].However,this does
information to the intended receiver.
not mean that we have perfect secrecy at channel capacity, as
the secrecy sub-codes carry information about each other.
Observe that even for K = 2 users, a rate point in this
region is four dimensional, and hence cannot be accurately B. The Gaussian Two-Way Wire-Tap Channel
drawn. We can instead focus on the secrecy rate region, the
region of all achievable Rs. The sub-regions MA-SUP, MA-TDMA In this section, we present an achievable region for the
are shown for different channel gains in FiguGre 3 foGr fixed GTW-WT using a superposition coding similar to that used
transmit powers, and K = 2 users. Figure 4 represents how to achieve the region MA-SUP for the GGMAC-WT. We first
G
these regions change with different transmit powers when define
the channel gains are fixed. For the case shown, we need Definition 5 (GTW-WT Superposition region, TW(P)): Let
G
the convex hull operation, as the achievable region is a X (0,P ). Then, the GTW-WT superposition region,
k k
∼ N
TEKINANDYENER:THEGENERALGAUSSIANMULTIPLEACCESSANDTWO-WAYWIRE-TAPCHANNELS 7

0.8 0.8
GTW, h=0.1, h=0.3
1 2
0.7 0.7
GTW, h=0.3, h=0.7 GTW(2,1.5)
1 2
0.6 GTW, h=0.7, h=1.4 0.6 GTW(2,2)
1 2 GTW(4,1.5)
CTWC
0.5 0.5 GTW(4,2)
R2 R2 CTW(4,2)
0.4 0.4
0.3 0.3
0.2 0.2
0.1 0.1
0 0
0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2
R R
1 1
Fig.5. GTW-WTachievableregionsfordifferentchannelparameters, Fig. 6. GTW-WT achievable secrecy region when P¯1 = 4,P¯2 =
GTW(P1 =4,P2 =2). 2,h1 =.3,h2 =.7.
difference is that we choose the rates to satisfy
TW(P), is given by
G
1
Rs +Ro +Rx log(1+P ), k =1,2 (48)
k k k ≤ 2 k
TW(P)= R:
G 1
Rks +Rko ≤(cid:26)I(Xk;Y|Xkc), k =1,2 k∈S(Rko +Rkx)≤ 2log 1+k∈ShkPk!, ∀S⊆K
+ X X
Rs I(X ;Y X ) I(X ;Z) , (44) with equality if = (49)
k≤ k | kc − K ∀S⊆K S K
kX∈S (cid:20)kX∈S (cid:21) (cid:27) Rs 1 log(1+P )
k ≤ 2 k
which can be written as k∈S (cid:20)k∈S
X X
P +
log 1+ k∈S k , (50)
GTW(P)= R: − (cid:18) 1+P k∈ScPk(cid:19)(cid:21) ∀S⊆K
(cid:26)1 or equivalently P
Rs +Ro log(1+P ), k =1,2
k k ≤ 2 k Rs+Ro+Rx CM, k=1,2 (51)
1 k k k ≤ k
Rs log(1+P ) (Ro+Rx) CW, , with equality if = (52)
k ≤ 2 k k∈S k k ≤ S ∀S⊆K S K
kX∈S (cid:20)kX∈S P + Pk∈SRks ≤ k∈SCkM−CSW +, ∀S ⊆K. (53)
log 1+ k∈S k , . (45) asPsuming (53)(cid:2)Pis positive. Th(cid:3)e decodability of Ws from
− (cid:18) 1+P k∈ScPk(cid:19)(cid:21) ∀S⊆K(cid:27) Y ,Y comes from (51) and the capacity regionKof the
1 2
P Gaussian Two-Way Channel [7]. This gives the first set of
Remark 2: We can also write this region more compactly
termsintheachievableregion.Thekeyhereisthatsinceeach
as the following:
transmitter knows its own codeword, it can subtract its self-
interference from the received signal and get a clear channel.
TW(P)= R: Therefore, the Gaussian two-way channel decomposes into
G
(cid:26) two parallel channels.
Rs +Ro CM(P), k =1,2
k k ≤ k The second group of terms in (45), resulting from the
+
Rs CM(P) C˜W(P) , . (46) secrecy constraint, can be shown the same way as the proof
k ≤ k − S ∀S⊆K of Theorem 1, since Z has the same form for both channels.
k∈S (cid:20)k∈S (cid:21) (cid:27)
X X In other words, as far as the eavesdropper is concerned, the
Theorem 2: The rate region given below is achievable for channel is still a GMAC with K = 2 users. As such, we
the GTW-WT: need to send CW extra codewords in total, which need to be
K
shared by the two-terminals provided they are not single-user
convex closure of TW(P). (47) decodable.
G For different channel gains, the region of all Rs satisfying
P∈P
[
(45) is shown in Figure 5. Since we require four dimensions
Proof: The proofis verysimilar to the proofofTheorem foran accuratedepictionof the completerate region,we only
1. We use the same coding scheme as Theorem 1, the main focusonourmaininterest,i.e.,thesecrecyrateregion.Figure
8 IEEETRANSACTIONSONINFORMATIONTHEORY
6 shows the achievable secrecy rate region as a function of andwewouldliketofindthepowerallocationthatmaximizes
transmit powers. We note that higher powers always result in thisquantity.Statedformally,weareinterestedinthetransmit
alargerregion.We indicatetheconstraintontheoverallrates, powers that solve the following optimization problem:
correspondingtothecapacityregionoftheGaussianTwo-Way 1
Channel, by the dotted line. Note that the secrecy region has mP∈aPx2 log 1+ Kk=1Pk −log 1+ Kk=1hkPk
a structure similar to the GGMAC-WT with K =2. As far as h (cid:16) P (cid:17) (cid:16) 1 P (cid:17)i
the eavesdropperis concerned, there is no difference between = min logφK(P) (57)
P∈P 2
the two channels. However, since the main channel between
min φ (P) (58)
K
users decomposesinto two parallel channels, higher rates can ≡P∈P
be achievedbetweenthe legitimate terminals(users).Thus, in where
effect, each user’s transmitted codewords act as a secret key φ (P), 1+ k∈ShkPk, (59)
S
for the other user’s transmitted codewords, requiring fewer 1+ P S ⊆K
P k∈S k
extraneous codewords overall to confuse the eavesdropper,
and = yields (58). In obtaining(58), we simply used the
P
and a larger secrecy region. We note that a user may either S K
monotonicityofthelogfunction.Thesolutiontothisproblem
achieve secrecy or not, depending on whether it is single-
is given below:
user decodable or not. As a result, TDMA does not enlarge
Theorem 3: The secrecy sum-rate maximizing power allo-
the region, since each user can at least achieve their single- cation for MA-SUP satisfies P∗ = P¯ if k T and P∗ = 0 is
user secrecy rates. To see this, note that the constraint on the G k k ≤ k
k >T whereT 0,...,K issomelimitingusersatisfying
secrecy sum-rate can be written as: ∈{ }
1+ l h P¯
log(1+P1)+log(1+P2)−log(1+h1P1+h2P2) hT < 1+ k=l0 Pk¯ k ≤hT+1 (60)
=log(1+P ) log(1+h P ) P k=0 k
1 1 1
− h P andwedefineh0 ,0, P¯0 ,P0.Notethatthisallocationshows
+log(1+P2) log 1+ 2 2 (54) that only a subset of the strong users must be transmitting.
− 1+h P
(cid:18) 1 1(cid:19) Proof: We start with writing the Lagrangian to be mini-
log(1+P ) log(1+h P )
1 1 1 mized:
≥ −
+log(1+P ) log(1+h P ) (55)
2 2 2 K K
−
(P,µ)=φ (P) µ P + µ (P P¯ ) (61)
so that transmitting in the two-way channel always provides L K − 1k k 2k k− k
an advantage over the single-user channels. Xk=1 Xk=1
Equating the derivative of the Lagrangian to zero, we get
IV. MAXIMIZATIONOF SUM RATE ∂ (P∗,µ)
L =φ˙(j)(P∗) µ +µ =0 (62)
The achievable regions given in Theorems 1 and 2 depend ∂P∗ K − 1j 2j
j
on the transmit powers. We are, thus, naturally interested
where we define
in the power allocation P∗ that would maximize the total
h φ (P)
secrecy sum-rate. Recall that the standardized channel gain φ˙(j)(P), j − S (63)
for user k is hk = hhWkMσσM22, and that the higher hk is, the S 1+ k∈SPk
betterthecorrespondingkeaWvesdropperchannel.Withoutlossof for any set . P
S ⊆K
generality,assumethatusersareorderedintermsofincreasing Itiseasy tosee thatif hj >φK(P∗), thenµ1j >0,andwe
standardizedeavesdropperchannelgains,i.e.,h1 ... hK. have Pj∗ = P¯j. If hj < φK(P∗), then we similarly find that
Note that, we only need to concern ourselves w≤ith th≤e case Pj∗ =0. Finally, if hj =φK(P∗), then we also have
h1 < ... < hK, since we can combine users with the 1+ k∈K\jhkPk∗
same channelgains into one super-user. We can then split the hj = 1+ h P∗ (64)
resulting optimum power allocation for a super-user among Pk∈K\j k k
theactualconstitutingusersinanywaywe choose,sincethey and φK(P∗)= φK\j(P∗) doPes not depend on Pj, so we can
would all result in the same sum-rate. In addition, from a set P∗ = 0 with no effect on the secrecy sum-rate. Thus,
j
physical point of view, assuming that the channel parameters we have P∗ = P¯ if h < φ (P∗), and P∗ = 0 if h
j j j K j j ≥
are drawn according to a continuous distribution and then φ (P∗). Then, the optimal set of transmitters is of the form
K
fixed,theprobabilitythattwouserswouldhavethesameexact = 1,...,T since if a user T is transmitting, all users
T { }
standardized channel gain is zero. such that h < h must also be transmitting. We also note
k T
that φ (P∗) = φ (P¯). Let T be the last user satisfying this
K T
A. GGMAC-WT property, i.e. hT < φT(P¯) and hT+1 φT∪{T+1}(P¯). Note
≥
that
Wefirstexaminethesuperpositionregiongivenin(20).The
secrecy sum-rate achievablewith superpositioncoding for the 1+ T h P¯ 1+ T−1h P¯ +h P¯
h < k=1 k k = k=1 k k T T (65)
GGMAC-WT was given in Theorem 1 as T 1+ T P¯ 1+ T−1P¯ +P¯
P k=1 k P k=1 k T
RsMuA-mSUP=21"log 1+kK=1Pk!−log 1+kK=1hkPk!#+(56) hT−1 <hPT < 11++PTk=Tk−=1−111hPk¯Pk¯Pk =φT\{T}(P¯). (66)
X X
P
TEKINANDYENER:THEGENERALGAUSSIANMULTIPLEACCESSANDTWO-WAYWIRE-TAPCHANNELS 9
In other words, all sets = 1,...,S for S T also where
satisfy thisproperty,andareSviabl{ecandida}tesforth≤e optimal h 1+ k∈KhkPk
set of transmitting users. Therefore, we can claim that is ψ˙(j)(P), j − 1+Pj . (75)
T K P(1+P )
the optimum set of transmitting users, since from above we k∈K k
can iteratively see that φT(P¯)<φS(P¯) for all S <T. An argument similar toQthe one for the GGMAC-WT
Notethat,forthespecialcaseofK =2users,theoptimum establishes that if h > 1+ h P /(1+P ), or
j k∈K k k j
power allocation is equivalently if hj >1+hjc(cid:0)Pj∗c, tPhen Pj∗ =0(cid:1). When equality
(P1∗,P2∗)=((PP¯¯11,,0P¯)2,), iiff hh11 <<11,, hh22 <≥ 11++11++hhPP11¯¯PP11¯¯11 (67) ψwisKes(aaPtgis)afiicneadn,sebttheePsnejeψ˙n=K(jt)o(0Pn)oint=dteh0pisernecdgaasorend.lPeOsjsn. TotfohePcojo,nthsaeenrrdvehaapsnodswu,ecirhf,
(0,0), otherwise h < 1+ h P /(1+P ), then P∗ =P¯ .
j k∈K k k j j j
We also needto consider the TDMA region. In this case, Con(cid:0)siderPuser 1. If (cid:1)P1∗ = 0, and P2∗ > 0, this implies
the maximum achievable secrecy sum-rate is: that h2 < 1. Since h1 ≤ h2 < 1, we cannot have P1∗ = 0.
As a consequence of this contradiction, we see that P∗ = 0
max K αk log 1+ P¯k log 1+ hkP¯k . (68) whenever P1∗ =0. 2
Σ0≤kααk≤=11 kX=1 2 (cid:20) (cid:18) αk(cid:19)− (cid:18) αk (cid:19)(cid:21) P∗A.sWsuemweilPl1∗ha=ve PP¯1∗, =and0 icfonhsider1th+ehtwP¯o ;alatnerdnaPti∗ve=s fP¯or
This is a simple complex optimization problem that can if2h < 1 +h P¯ . 2These case2s ≥correspo1nd1to h <2 1 and2
2 2 1 1
easily be solved numerically. For the degraded case, we can h <1+h P¯ ,respectively.Thus,wehave(72)asthesecrecy
1 2 2
goebntaeirnal,awcelocsaendnofotrombtasionlusuticohn:aαsoklu=tion.HP¯kkoP¯wkevaser,initi[s42tr]i.viIanl sumRe-rmaaterkm3a:xiOmbizsienrgvepothwaetrthaleloscoaltuiotino.n in Theorem 4 has a
to note that users with h 1 should nPot be transmitting in structure similar to that in Theorem 3. In summary, it is
k
this scheme. The secrecy s≥um-rate is then the maximum of seen that as long as a user is not single-user decodable, it
the solutions given by the superposition and TDMA regions. should be transmitting with maximum power. Hence, when
both users can be made to be non-single-userdecodable,then
the maximum powers will provide the largest secrecy sum-
B. GTW-WT
rate. If this is not the case, then the user who is single-user
Now, we will examine the power allocation thatmaximizes
decodablecannottransmitwith non-zerosecrecyandwill just
the secrecy sum-rate given in Theorem 2 as
makethesecrecysum-rateconstrainttighterfortheremaining
1 + user by transmitting open messages.
RTW = log(1+P )+log(1+P ) log(1+h P +h P ) .
sum 2 1 2 − 1 1 2 2 Comparing (72) to (67), we see that the same form of
(69)
(cid:2) (cid:3) solutions is found, but the range of channel gains where
This problem is formally stated below:
transmission is possible is larger, showing that GTW-WT
1 allows secrecy even when the eavesdropper’s channel is not
max [log(1+P )+log(1+P ) log(1+h P +h P )]
1 2 1 1 2 2 very weak.
P∈P2 −
min ψ (P) (70)
K
≡P∈P V. SECRECY THROUGH COOPERATIVEJAMMING
where
1+ h P In the previous section, we found the secrecy sum-rate
ψ (P), k∈S k k (71)
S maximizingpowerallocations.ForboththeGGMAC-WTand
(1+P )
k∈PS k
GTW-WT, if the eavesdropperis not “disadvantaged enough”
and = yields (70). TQhe optimum power allocation is for some users, then these users’ transmit powers are set
S K
stated below:
to zero. We posit that such a user may be able to “help”
Theorem 4: The secrecy sum-rate maximizing power allo-
a transmitting user, since it can cause more harm to the
cation for the GTW-WT is given by
eavesdropper than to the intended receiver. We only consider
(P¯ ,P¯ ), if h 1+h P¯ , h <1+h P¯ thesuperpositionregion,sinceintheTDMAregionauserhas
1 2 1 2 2 2 1 1
≤ a dedicated time-slot, and hence does not affect the others.
(P∗,P∗)= (P¯ ,0), if h <1, h 1+h P¯ (72)
1 2  1 1 2≥ 1 1 We will next show that this type of cooperative behavior is
(0,0), otherwise
indeed useful, notably exploiting the fact that the established
Proof: The Lagrangian is, achievablesecrecysum-rateisadifferenceofthesum-capacity
expressionsfortheintendedchannel(s)andtheeavesdropper’s
2 2
(P,µ)=ψ (P) µ P + µ (P P¯ ). (73) channel. As a result, reducing the latter more than the former
K 1k k 2k k k
L − − actually results in an increase in the achievable secrecy sum-
k=1 k=1
X X rate.
Equating the derivative of the Lagrangian to zero for user
Formally, the scheme we are considering implies partition-
j, we get
ingthesetofusers, intoasetoftransmittingusers, anda
∂L(∂PP∗∗,µ) =ψ˙K(j)(P∗)−µ1j +µ2j =0 (74) istettroafnjsammimtsinXgusersKT(cP=∗IK,0−)Tin.sItfeaadusoefrckoidsejwamormdsi.TngIn,ththeins
j k ∼ N k
10 IEEETRANSACTIONSONINFORMATIONTHEORY
case, we can show that we can achieve higher secrecy rates allocation is of the form
when the “weaker” users are jamming. We also show that
1,...,T,T +1,...,J 1, J ,J +1,...,K
the GTW-WT, has an additional advantage compared to the
{ − }
GGMAC-WT, that is the fact that the receiver already knows P∗=P¯ P∗=0 PJ∗ P∗=P¯
the jamming sequence. As such, this scheme only harms the
| {ztra}nsm|itting,i.e.{,∈zT } |{zja}mm|ing,i.e{.,z∈Tc }
eavesdropper and not the intended receivers, achieving an
even higher secrecy sum-rate. Once again, without loss of with | {z } | {z }
generality, we consider h < ... < h . In addition, we will +
1 K c + c2 4c c
assume that a user can either take the action of transmitting P∗ = min P¯ ,− 2 2− 1 3 (81)
J J 2c
its information or jamming the eavesdropper, but not both. It " ( p 1 )#
is readily shown in Section V-A below that we do not lose
and
any generality by doing so, and that splitting the power of a
user between the two actions is suboptimal from the secrecy c =h h P∗ h P∗ (82)
sum-rate maximization point of view. 1 J J k − k k
(cid:18) k∈T k∈T (cid:19)
X X
c =h 2+ h P∗+ h P∗ P∗
2 J k k k k k
A. GGMAC-WT (cid:18) k∈XK\J k∈XTc\J (cid:19)kX∈T
The problem is formally presented below: h 2+ P∗+ P∗ h P∗ (83)
− J k k k k
T⊆mKa,Px∈P 21(cid:20)log(cid:18)1+ 1+Pk∈kT∈TPckPk(cid:19) c3= 1+ (cid:18) hkkP∈XKk∗\J 1+ k∈XTc\JhkP(cid:19)k∗kX∈T Pk∗
log 1+P k∈T hkPk (76) (cid:18) k∈XK\J (cid:19)(cid:18) k∈XTc\J (cid:19)kX∈T
− (cid:18) 1+P k∈TchkPk(cid:19)(cid:21) −hJ 1+ Pk∗ 1+ Pk∗ hkPk∗ (84)
min φK(P) P (77) (cid:18) k∈XK\J (cid:19)(cid:18) k∈XTc\J (cid:19)kX∈T
≡T⊆K,P∈P φTc(P) whenever the positive real root exists, and 0 otherwise.
Proof: We first solve the subproblem of finding the
where we recall that φ (P) is given by (59), such that
S
optimal power allocation for a set of given transmitters, .
1+ h P T
φ (P)= k∈K k k (78) The solution to this will also giveus insight into the structure
K 1+P k∈KPk of the optimal set of transmitters, T∗. We start with writing
1+ h P the Lagrangian:
φ (P)= Pk∈Tc k k. (79)
Tc
1+P k∈TcPk φ (P) 2 2
(P,µ)= K µ P + µ (P P¯ ). (85)
To see that a user should not bPe splitting its power among L φTc(P) − 1k k 2k k− k
k=1 k=1
X X
jammingandtransmitting,itissufficienttonotethatregardless
The derivative of the Lagrangian depends on the user:
of how a user splits its power, φ (P) will be the same, and
K
the user only affects φTc(P). Assume the optimum solution ∂ (P∗,µ)
is such that user j splits its power, so j and j c. 0= L∂P∗
∈ T ∈ T j
Then, it is easy to see that if h < φ (P∗), the sum-rate is
j Tc φ˙(j)(P∗)
increased when that user uses its jamming power to transmit, = φTKc(P∗) −µ1j +µ2j, if j∈T (86)
and when hj >φTc(P∗), the sum-rate is increased when the φ˙K(j)(P∗)φTc(P∗)−φK(P∗)φ˙T(jc)(P∗) +µ , if j c
user uses its transmit power to jam. When hj = φTc(P∗),  φ2Tc(P∗) 2j ∈T
tshaemne,reagnadrdwleesscaonf ahsoswumitesupsoewrejreiisthseprlittr,atnhsemsiutsmo-rrajtaemiss.the sinCceonasiudseerraju∈seTrcjsatisfi.eTshPej∗sa>me0,aritgummuesntthaasveinµt1hje=su0m. -
Note that we must have φK(P) ≤ φTc(P) to have a non- rate maximization pr∈ooTf leads to Pj∗ = P¯j if hj < φK(P∗)
ozevreorsneoctrejacmymsuimng-.raTteh,isansdchφeTmce(Pca)n>be1sthoohwanvetoanacahdiveavnetathgee aWnedcPa1n∗ w=ri0teif(8h6j) a≥s φK(P∗). Now examine a user j ∈ Tc.
following secrecy sum-rate:
ρ (P∗)
Theorem 5: The secrecy sum-rate using cooperative jam- j +µ =0 (87)
2 2 2j
ming is 1+ k∈KPk 1+ k∈TchkPk
whe(cid:0)re P (cid:1) (cid:0) P (cid:1)
1 P∗
RSUP-MA-CJ = log 1+ k∈T k
sum 2 (cid:18) 1+P k∈TcPk∗(cid:19) ρj(P), hj 1+ Pk 1+ Pk hkPk
1log P1+ k∈T hkPk∗ (80) − (cid:18) kX∈K (cid:19)(cid:18) kX∈Tc (cid:19)kX∈T
− 2 1+ h P∗
(cid:18) P k∈Tc k k(cid:19) + 1+ h P 1+ h P P .(88)
k k k k k
where is the set of transmitters andPthe optimum power (cid:18) k∈K (cid:19)(cid:18) k∈Tc (cid:19)k∈T
X X X
T