To Obtain or not to Obtain CSI in the Presence of Hybrid Adversary

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by  Y. Ozan Basciftci

To Obtain or not to Obtain CSI in the Presence of
Hybrid Adversary
Y. Ozan Basciftci C. Emre Koksal Fusun Ozguner
Dep. of Electrical & Computer Eng. Dep. of Electrical & Computer Eng. Dep. of Electrical & Computer Eng.
The Ohio State University The Ohio State University The Ohio State University
Columbus, Ohio, USA Columbus, Ohio, USA Columbus, Ohio, USA
Email: [email protected] Email: [email protected] Email: [email protected]
3 Abstract—We consider the wiretap channel model under the rate of reliable communication, but the adversary will have
1 presence of a hybrid, half duplex adversary that is capable of to make decisions under a higher uncertainty. In this paper,
0 eitherjammingoreavesdroppingatagiventime.Weanalyzedthe we study the tradeoffs involving obtaining CSI. We ask the
2 achievable rates under a variety of scenarios involving different
questions, should a legitimate pair obtain CSI and if so, what
methods for obtaining transmitter CSI. Each method provides
n a different grade of information, not only to the transmitter on should be their strategy?
a the main channel, but also to the adversary on all channels. To that end, we focus on the system depicted in Figure 1.
J
Our analysis shows that main CSI is more valuable for the We assume all three channels to be block fading. In each
1 adversary than the jamming CSI in both delay-limited and
block, based on the available CSI, the half-duplex adversary
3 ergodic scenarios. Similarly, in certain cases under the ergodic
can choose to do jamming at a fixed transmission power or
scenario, interestingly, no CSI may lead to higher achievable

] secrecy rates than with CSI. eavesdropping, but not both. Our objective is to maximize the
T
rate of reliable communication over the main channel, subject
I I. INTRODUCTION to full equivocation [4] (weak secrecy) at the adversary. The
.
s
Information theoretic security has received a significant adversary can follow an arbitrary strategy in its choice of
c
[ attention recently. One mainstream direction has been on the jamming vs. eavesdropping at any given block. In the case

wireless transmission of confidential messages from a source in which the receiver feeds back main CSI, it may do so
2
to a destination, in the presence of internal and/or external in two ways: directly by sending back the exact state of
v
9 eavesdroppers.Towardachievingthatgoal,thecommunicating the channel or by sending reverse pilots, trying to exploit
4 pair exploits the stochasticity and the asymmetry of wireless channel reciprocity (similar to [5]). While the former method
4 channels between the communicating pair and the eavesdrop- completely reveals the main CSI, it eliminates the possibility
6
pers. A stochastic encoder at the transmitter makes use of of the adversary to learn the jammer CSI. On the other hand,
.
1 the available channel state information (CSI) in a way for the while the pilot feedback successfully hides the main CSI, it
0 mutualinformationleakedtotheadversariesremainarbitrarily enables the adversary to estimate the jammer CSI. In terms
3
small. It is designed in a way that, even when the adversaries ofthesecrecyencodingstrategies,weaddressthepossibilities
1
have access to the full CSI of the main channel, i.e., between undertwogeneralscenarios:theergodicandthedelay-limited.
:
v the transmitter and the receiver as well as the eavesdropper In the former case, one message is encoded across infinitely
i
X channel, i.e., between the transmitter and itself, it still will many blocks and in the latter case, a separate message is
obtain an arbitrarily low rate of information on the message. encoded over each block, to be decoded immediately. Thus,
r
a Likewise, the adversary relies on CSI to make decisions. For in the delay-limited scenario, we also impose an additional
instance, a half-duplex hybrid adversary, capable of jamming probabilistic constraint on the decoding and secrecy outage
oreavesdroppingatagiventime(butnotbothsimultaneously) events.
decides between jamming vs. eavesdropping, based on the Inthedelaylimitedscenario,weshowthatbyrevealingthe
available CSI. main CSI, the receiver achieves a higher secrecy rate under
The assumption that the adversaries have full CSI of the outage constraint, compared to transmission with no CSI.
all channels is typical in the literature [1]-[3]. While this Furthermore, we show that main CSI is more valuable for
assumption leads to robust systems in terms of providing the adversary than the jamming CSIin both delay-limited and
security as it makes no assumptions on the adversaries, it can ergodicscenarios.Intheergodicscenario,weobservethatthe
be too conservative in some cases. For example, to obtain transmitter may not need the CSI to achieve higher secrecy
main CSI, an adversary relies on the same resource as the rates.
transmitter:feedbackfromthelegitimatereceiver.Hence,from There is a recent research interest on hybrid adversaris.
the perspective of the receiver, there is a tradeoff between In [7], the authors formulate the MIMO wiretap channel as
revealing CSI and keeping it secret: If the legitimate receiver a two player zero-sum game in which the payoff function
chooses not to reveal CSI, it will sacrifice some achievable is an achievable ergodic secrecy rate. The strategy of the
spectively, and P is the jamming power. Indicator function
j
G_m(i) I (i) = 1, if the adversary is in a jamming state in the
J
Main Channel i-th block; otherwise IJ(i) = 0. Channel gains, Gm(i),
G (i), and G (i) are defined to be the independent com-
e z
plex gains of transmitter-to-receiver channel, transmitter-to-

adversary channel, and adversary-to-receiver channel, respec-
G_z(i)
G_e(i) tively (as illustrated in Figure 1). Associated power gains
are denoted with H (i) = |G (i)|2, H (i) = |G (i)|2,
m m e e
and H (i) = |G (i)|2. We assume that channel reciprocity
z z
principle is valid for all channels, i.e., reverse channels and
forward channels have identical gains. We also assume that
jointprobabilitydensityfunctionofinstantaneouspowergains,
f (h), where H=[H (·)H (·)H (·)], is well defined and
H m e z
known by all entities.
Fig.1. SystemModel
B. Methods for Obtaining CSI
The legitimate pair may choose to obtain main CSI or
transmitter is to send the message in a full power or to
communicate without it. We call the latter strategy the no
utilize some of the available power to produce artificial noise.
CSI case. If they choose to obtain main CSI, at the beginning
The conditions under which pure Nash equilibrium exists are
of each time block, the transmitter sends training symbols to
studied. In [6], the authors consider fast fading main and
the legitimate receiver. In this paper, we ignore the overhead
eavesdropper channels and static jammer channel. Under this
associatedwiththistrainingprocess.Weassumethat,usingthe
channel configuration, they propose a novel encoding scheme
trainingsymbolssentatthebeginningofblocki,thelegitimate
which is called block-Markov Wyner secrey encoding. In [8],
receiverobtainsperfectknowledgeofG (i)andtheadversary
theauthorsintroduceapilotcontaminationattackinwhichthe m
obtains perfect knowledge of G (i).
adversary jams during the reverse training phase to prevent e
Once the receiver observes main CSI, it uses two possible
the transmitter from estimating the main CSI correctly. As a
methods for feeding back this information. The first one is
result, the transmitter incorrectly designs precoder which will
directly feeding back the observed channel state: the value of
increase the signal strength at the adversary that eavesdrops
G (i) is encoded at the receiver and sent to the transmitter
the main channel during the data transmission phase. m
in a feedback packet. Thus, we call this feedback method
The rest of this paper is organized as follows. In Section
the packet feedback. We assume that the legitimate receiver
II, we first describe the system model. We then explain the
and the adversary both decode this packet successfully and
channel model and CSI feedback models. At the end of the
learn G (i). The second method is using pilot based CSI
section, we explain the problem formulations. In Section III, m
feedback in which the receiver sends training symbols to the
we present the results for both delay limited and ergodic
transmitter. We call this second method the pilot feedback.
scenarios.InSectionIV,wepresentournumericalresultsand
We assume that by using these reverse training symbols, the
conclude the paper in Section V.
legitimatetransmitterobtainsperfectknowledgeofG (i)and
m
II. SYSTEMMODELANDPROBLEMFORMULATION the adversary obtains the perfect knowledge of Gz(i). Thus,
in the first method, the adversary obtains the knowledge of
A. Channel Model
G (i),butnotG (i),whereasthereverseistrueinthesecond
m z
In this paper, we focus on a block fading channel model.
method.
Time is divided into discrete blocks and there are N channel
uses in each block. Channel state is assumed to be constant C. Adversary Model
withinablockandvariesrandomlyfromoneblocktothenext. The goal of the adversary is to minimize the achieved
We assume all parties are half-duplex, thus the adversary can secrecyrate.Thestrategyspaceoftheadversaryineachblock
not jam and eavesdrop simultaneously. is binary: jamming or eavesdropping. The transmitter does
The observed signals at the legitimate receiver and the not observe the strategy of the adversary in any given block,
adversary in i-th block are as follows: whereas we assume that the adversary knows the strategy of
the transmitter a priori.
YN(i)=G (i)XN(i)+G (i)SN(i)I (i)+SN(i) (1)
m z j J m From one block to the next, the adversary chooses its
ZN(i)=Ge(i)XN(i)(1−IJ(i))+SeN(i) (2) strategy based on the transmitter’s strategy and the obtained
channel power gains1, i.e., h (i) for the no CSI case, h (i),
where XN(i) is the transmitted signal, P is the transmission e e
h (i) for the packet feedback case, and h (i), h (i) for the
power,YN(i)isthesignalreceivedbythelegitimatereceiver, m e z
pilot feedback case. We denote the vector of channel power
ZN(i) is the signal received by the adversary, SN(i), SN(i),
j m
and SN(i) are noise vectors distributed as complex Gaus-
e 1The realizations of the random variables are represented by lower case
sian, CN(0,PjIN×N), CN(0,IN×N), and CN(0,IN×N), re- lettersinthesequel.
gains observed by the adversary at the beginning of the ith strategies:
block with hA(i). 1. Encoding across blocks: We use this strategy in the
The secrecy level of a transmitted message is measured by no main CSI case. In this strategy, the NMR bits at the
m
the equivocation rate at the adversary. The equivocation rate output of the secrecy encoder is channel encoded via a single
at the adversary is defined as the entropy of the transmitted C(2NMRm,NM) Gaussian codebook and the NM symbols
message conditioned on the channel output and the available are transmitted over the channel over M blocks.
CSI at the adversary. If the equivocation rate is equal to the
2. Block-by-block encoding: We use this strategy when
secrecyrate,themessageissaidtobetransmittedwithperfect
the main CSI is available. To utilize the main channel
secrecy. knowledge, the transmitter chooses some h∗ and encodes
z
the information using a Gaussian codebook with the rate
D. Encoding of Information (cid:16) (cid:17)
R(h (i))=log 1+ Phm(i) overblocki.Withthischoice,
In our system, we consider two levels of encoding. At m 1+Pjh∗z
the transmitted codeword will not be decoded successfully
the higher level, secrecy is realized Wyner code introduced
if the adversary is in the jamming state and h (i) > h∗.
in [4]. There, C (R ,R ,NM) represents a Wyner code of z z
s m s To handle this possibility, we use a plain ARQ strategy
size 2NMRm that bears a confidential message set Ws = in each block (similar to [9]). Transmissions that receive a
{1,2,...,2NMRs}, where NM is the codeword length in
negativeacknowledgement(NAK)areretransmitteduntilthey
number of bits. A message, w ∈ W is mapped to NMR
s s m are decoded successfully.
bits by a secrecy encoder [4] and these NMR bits are then
m
mappedtochannelencodedbitsatthelowerlevelofencoding
using a sequence of codes, Ci(2NR(hm(i)),N), one for each E. Problem Formulation
block i, 1 ≤ i ≤ M. Here, the code rate, R(h (i)), is
m
chosen based on the main CSI, obtained at the transmitter. One can notice that, in both the delay-limited and the
The sequence of codewords is denoted with xNM ∈ XNM, ergodicscenarios,weuseaconstantsecrecyrate2R .Thegoal
s
and the decoder, φ(·) maps the received sequence, YNM to ofthetransmitter istomaximizethesecrecy rate,R overthe
s
wˆ ∈W.Theaverageerrorprobabilityofthesequenceofcodes strategyspaceofsecrecyencodingandchannelencodingrates.
{C } is denoted with the associated sequence PNM. In this To that end, we consider the worst case scenario, in which
i e
paper, we focus on two scenarios as to how secrecy encoding the adversary perfectly tracks the strategy of the transmitter
and channel encoding are applied: delay limited and ergodic. in each block. Furthermore, since the transmitter does not
1) Delay-Limited Scenario: In the delay-limited scenario, know the real state of the adversary, the strategy pair of
the transmitter encodes a separate secret message, W (i) in the transmitter should satisfy the constraint for any arbitrary
s
eachblocki.Consequently,weuseaseparatesecrecyencoder strategy of the adversary. Thus, we choose the secrecy rate
C (i) = C (R (i),R ,N) for each block, where R (i) using:
s s m s m
is chosen to be identical to R(h (i)) to meet the channel R∗ = max minR (7)
m s s
rate. The channel encoder merely maps these NR (i) bits Rm(i),RsIJ(i)
m
to N Gaussian random variables, XN, forming a Gaussian for all i in both the delay-limited and the ergodic scenarios,
codebook Ci(2NR(hm(i),N). For the delay-limited scenario, subject to the following outage constraint in the delay-limited
wedefinethesecrecyoutageandconnectionoutageevents[3] scenario only:
as:
M
1 (cid:88)
I(XN;ZN|h (i)) lim I (i)I (i)+(1−I (i))I (i)I (i)≥α (8)
N e >R(hm(i))−Rs and (3) M→∞M i=1 J C J S C
I(XN;YN|hm(i)) <R(h (i)), (4) with probability 1, where IC(i) and IS(i) are indicator func-
N m tions that take on a value 0 in case of a connection and a
respectively, where XN ∼CN(0,PIN×N) secrecy outage, respectively. We evaluate Rs∗ under the no
CSI, packet feedback, and pilot feedback cases. Note that the
2) Ergodic Scenario: In the ergodic scenario, the transmit-
constraint enforces that the fraction of packets that are not in
ter has one secret message W and encodes it over M blocks
s
the both secrecy outage and the connection outage should be
using the C (R ,R ,NM) encoder. In the ergodic scenario,
s m s
larger than a threshold as the number of blocks, M, goes to
secrecyrateR issaidtobeachievableif,forany(cid:15)>0,there
s
infinity.
existssequenceofchannelcodes{C }forwhichthefollowing
i
are satisfied: Notethat,thereasonwhywefocusonthespecificencoding
strategies specified under delay limited and ergodic scenarios
PeNM ≤(cid:15) (5) in Section II-D is that, the solution, R∗, of the maximin
S
1 H(W |ZMN,hM)≥R −(cid:15) (6) problem stated in (7) is unknown [11].
MN s A s
forsufficientlylargeNandMandforanyhM ∈A suchthat
A M 2Forthedelay-limitedcase,itisconstantovertheentiresequenceofWyner
P[AM]=1.Here,weconsidertwopossiblechannelencoding codes.
III. RESULTS B. Ergodic Scenario
We first present a secrecy rate that is achievable under the
no CSI case.
A. Delay Limited Scenario
Theorem 2. The achieved secrecy under no CSI is:
The outage constraint in (8) is formulated as
RNoCSI =
s
M
1 (cid:88) (cid:20) (cid:20) (cid:18) PH (cid:19) (cid:21)(cid:21)+
C =Mli→m∞M i=1IJ(i)Ilog(cid:16)1+1+PPHjmH(zi()i)(cid:17)≥R(Hm(i)) E log 1+ 1+PjmHz −log(1+PHe) (12)
+(1−IJ(i))Ilog(1+PHe(i))≤R(Hm(i))−Rs The proof of (12) can be found at Section VII. Next, we
×I (9) presentanupperboundforthesecrecyratesachievedwiththe
log(1+PHm(i))≥R(Hm(i))
block-by-block encoding strategy.
Note that in (9), the secrecy outage event is represented with
log(1+Ph (i))>R(h (i))−R andtheconnectionoutage Theorem 3. Under the block-by-block encoding strategy, the
e m s
(cid:16) (cid:17)
events are given as log 1+ Phm(i) < R(h (i)) and achievable secrecy rate is upper bounded by
1+Pjhz(i) m
log(1+Phm(i))<R(hm(i)). (cid:20) (cid:18) PH (cid:19) (cid:21)+
R+ =E log 1+ m −log(1+PH )
s 1+P H e
Theorem 1. For the delay limited scenario, the solution of j z
(7), subject to constraint (9) leads to the following ordering ×P[Hz ≤h∗z] (13)
of the achievable rate with respect to the type of feedback:
Tofindthisupperbound,weemploythefollowingstrategy.
RNoCSI ≤RPacketfeedback ≤RPilotFeedback (10) When the transmitter receives a NAK signal, the transmitter
s s s
sendsanindependentgroupofbitsonthenextblockinsteadof
retransmittingthepreviouspacket[12].In[12],theauthorsuse
thisschemeforthesecretkeysharing.Thecrucialobservation
TheabovetheoremimpliesthatmainCSI,whichisobtained
isthatanupperboundfortheachievablesecretkeyrateisalso
with packet feedback is more valuable for the adversary than
anupperboundfortheachievablesecrecyrates.Thedetailsof
the jammer CSI, which is obtained with pilot feedback. We
theproofcanbefoundatSectionVIII.Notethatwiththeplain
nowgiveanoutlinefortheproof.Thedetailsoftheproofcan
ARQstrategydescribedinSectionII-D2,theachievablerateis
be found in Section VI.
identical to the expression provided in (13), with P[H ≤h∗]
The Outline of Proof of Theorem 1: The basic idea is z z
replacedwith(P[H ≤h∗])2,andH intheexpectationterm
to compare the feasible set of the problem (7) for three cases. z z z
replaced with h∗ as shown in [9].
The feasible set is defined as z
By comparing the upper bound given in Theorem 3, we
F ={(R ,R(·)):C ≥α w.p. 1, ∀A } (11) gain understanding on the performance of no CSI case. In
s p
particular, whenever the achievable rate with the no CSI case
where Ap is the set of channel power gains, hA(i), such that exceeds this bound, we know for sure that encoding across
theadversaryisinthejammingstateifhA(i)∈Ap.Theequiv- blocks (no CSI) is preferable over block-by-block encoding
alent form of (7) is as follows: Rs∗ =max(Rs,R(·))∈FRs. We with CSI. The main difference between block-by-block en-
observethatthesolutionto(7)isdirectlyrelatedtothesizeof coding and encoding across blocks is that, in the former, the
the feasible set. The strategy pair (Rs,R(·)) is the element of unsuccessfully received packets are discarded, whereas in the
thefeasibleset,F ifCmin =minApC[(Rs,R(·))]≥α w.p. 1. no CSI case, all the information received by the receiver is
We can write Cmin for the no CSI and packet feedback cases used to decode the message. The set of parameters for which
as follows: this is the case is illustrated in Section IV in an example.
CPacketFeedback = One can also write a general relationship between the
min
(cid:20) (cid:18) (cid:19)(cid:21) performance with packet feedback and pilot feedback:
PH
P R +log(1+PH )≤R(H )≤log 1+ m
s e m 1+P H Theorem4. Anysecrecyrateachievablewithpacketfeedback
j z
strategy is also achievable with pilot feedback strategy.
CNoCSI =
min
(cid:20) (cid:18) PH (cid:19)(cid:21) Theorem 4, shows that main CSI is more valuable for the
P R +log(1+PH )≤R≤log 1+ m
s e 1+P H adversary than the jamming CSI in the ergodic scenario, as
j z
was the case in the delay limited scenario. The proof can be
We can observe that FNoCSI ⊂ FPacketFeedback so we have
found at Section IX
RNoCSI ≤ RPacketfeedback. By assuming the adversary is full-
s s
duplex, we find a lower bound, Clowerbound for C at CSI IV. NUMERICALEVALUATION
min
feedbackcases.Then,weobserveClowerbound =CPacketFeedback Wefirstanalyzethedelay-limitedcase.Weassumethatboth
min
whichshowsCPacketFeedback ≤CPilotFeedback andthisconcludes main and eavesdropper channels are characterized by block
min min
the proof. Rayleigh fading, where the main channel and eavesdropper
channel power gains follow an exponential distribution with a
mean 10 and 1, respectively3. We also assume the jamming
channeldoesnotexperiencefading,wherepowergainisequal
to 1. The transmission power, P, and the jamming power,
P are identical and chosen to be 1. In Figure 2, we plot
j
the secrecy rate, R , as a function of the outage constraint
s
threshold, α under the no CSI, the packet feedback, and the
pilot feedback cases. The achievable rates in Figure 3 follow
the same ordering as given in Theorem (1).
Fig.3. TheregionwhereencodingacrossblockswithnoCSIoutperforms
block-by-blockencodingwithpacketfeedback.
delaylimitedscenariothat,thehighestsecrecyrateisachieved
with the pilot based feedback. Similarly, in the ergodic case,
we prove that the pilot-based CSI feedback outperforms the
packet-based CSI feedback, however interestingly, in certain
cases no CSI may lead to a higher achievable secrecy rates
than with CSI.
Fig.2. DelayLimitedScenario:ComparisonofCSIfeedbackmethodsunder VI. PROOFOFTHEOREM1
theoutageconstraint.
Feasible set for the problem (7) is defined to be
Next,wesimulatetheergodicscenarioandcomparethetwo
F ={(R ,R(·)):C ≥α, ∀A } (14)
strategies, encoding across blocks without CSI and block-by- s p
block encoding with packet CSI feedback. We used the same where A is the set of channel power gains such that the
p
power parameters as in the simulations for the delay-limited adversary is in the jamming state if h (i)∈A . We have the
A p
case. All three channels are assumed to be block Rayleigh- following lemma.
fading wtih E[H ]=1 We select the encoding parameter, h∗
such that P[H ≤z h∗]=0.75. z Lemma 5. If F (cid:54)= ∅, the solution to (7) is identical for all
z z
strategies of the adversary.
In Figure 3, we illustrate the region where the encoding
acrossblocksoutperformstheblock-by-blockencodingonthe Proof: The equivalent form of (7) is as follows:
(E[H ],E[H ]) space. The region to the left of the border,
e m
gthiveeennicnotdhiengploatcrcoosnstabinlosctkhserseestuolfts(Ein[Haeh],iEgh[eHrmse])crfeocrywrhaitceh. Rs∗ =(Rsm,Ra(·x))∈FRs (15)
As seen from (15), R∗ does not depend on strategy of the
The intuition behind this observation is that, when E[Hm] is s
adversary.
muchlargerthanE[H ],thepositiveoperatorinsidetheupper
e
bound, R+ loses its significance. However, the size of feasible set directly depends on the
CSIfeedbackscheme.From(15),wecanseethatthesolution
V. CONCLUSION to (7) is directly proportional with the size of the feasible set.
We consider the wiretap channel model under the presence In the rest of the proof, we will use this observation to show
of half duplex adversary that is capable of either jamming or the ordering.
eavesdropping at a given time. We analyzed the achievable
Lemma 6. RNoCSI ≤RPacketFeedback
rates under a variety of scenarios involving different methods s s
for obtaining transmitter CSI. In particular, we considered no Proof: In the no feedback case, constraint term (9) is
CSI, CSI with packet based feedback, and CSI with pilot reduced to
based feedback. Each method provides a different grade of M
(cid:88)
ibnuftoramlsaotiotontnhoetaodnvlyerstoarytheontraanllsmchitatnerneolns.tWheemshaionwchfoarnntehle, C =Mli→m∞1/M i=1IJ(i)Ilog(cid:16)1+1+PPHjmH(zi()i)(cid:17)≥R
+(1−I (i))I I (16)
3Such a difference may occur in the cellular setting, when the receiver J log(1+PHe(i))≤R−Rs log(1+PHm(i))≥R
is a base station with many antennas or in a wireless LAN setting, where (cid:88)M
tehxeterrencaelivaderveirssaloryc.ated at a favorable position for reception, compared to an =Mli→m∞1/M i=1IHe(i)∈ApIlog(cid:16)1+1+PPHjmH(zi()i)(cid:17)≥R
+I I I (17) +I I I
He(i)(cid:54)∈Ap log(1+PHe(i))≤R−Rs log(1+PHm(i))≥R log(1+Phe)≤R(hm)−Rs log(1+Phm)≥R(hm) hm,he∈/Ap
(cid:20)
f (h )f (h )dh dh , w.p. 1 (26)
=E IHe∈ApIlog(cid:16)1+ PHm (cid:17)≥R He e Hm m e m
1+PjHz where (24) follows from the facet that I (i) =
(cid:3) J
+I I I , w.p. 1 (18)
(cid:90)He(cid:54)∈Ap(cid:20)log(1+PHe)≤R−Rs log(1+PHm)≥R IHm(i),He(i)∈Ap since the adversary knows both the
instantaneous power gain of the eavesdropper channel and
= E Ilog(cid:16)1+ PHm (cid:17)≥RIHe∈Ap the main channel. (25) follows from the strong law of large
he 1+PjHz
+ I I I |H =h (cid:3) numbers theorem and (26) follows from the independence of
log(1+PHe)≤R−Rs log(1+PHm)≥R He∈/Ap e e H , H , and H . A∗ set that minimizes C is as follows:
f (h )dh , w.p. 1 (19) e m z p
He e e
(cid:90) (cid:20) (cid:21) A∗ ={h ,h :log(1+Ph )≤R(h )−R }
p m e e m s
= heE Ilog(cid:16)1+1+PPHjmHz(cid:17)≥R Ihe∈Ap (cid:91){hm,he :log(1+Phm)≤R(hm)} (27)
+E[I ]I I
log(1+PHm)≥R log(1+Phe)≤R−Rs he∈/Ap When we combine (26) and (27), the minimized can be
fHe(he)dhe, w.p. 1 (20) CPacketFeedback written as
where (17) follows from the fact that we have IJ(i) = CPacketFeedback =
IpHowe(ei)r∈gAapinsoinfctehetheeavaedsvderorspapreyrochnalynnkenl,ow(1s8)thfoellionwstsanfrtoanmeothues P (cid:20)R +log(1+PH )≤R(H )≤log(cid:18)1+ PHm (cid:19)(cid:21).
s e m 1+P H
strong law of large numbers theorem and (20) follows from j z
the independence of H , H , and H .
e m z
Feasible set for the transmitter for the packet feedback case is
A∗p set that minimizes C is as follows: as follows
A∗p ={he :log(1+Phe)≤R−Rs} (21) FPacketFeedback ={(Rs,R(·)) :CPacketFeedback ≥α} (28)
The minimized C for a given rate pair (R ,R) is
s
It is easy to see that FNoCSI ⊂ FPacketFeedback then we have
CNoCSI = RNoCSI ≤RPacketFeedback.
s s
(cid:20) (cid:18) (cid:19)(cid:21)
PH
P R +log(1+PH )≤R≤log 1+ m
s e 1+P H Lemma 7. RPacketFeedback ≤RPilotFeedback
j z s s
Feasible set for the transmitter for the no CSI feedback case
Proof: We will show that for any given (R ,R(·)),
s
is as follows
CPacketFeedback isalowerboundfor(9)inthechannelfeedback
FNoCSI ={(R ,R) :CNoCSI ≥α} (22) cases. Constraint (9) can be written as
s
M
We now show that FNoCSI ⊂ FPacketFeedback. For the packet (cid:88)(cid:2) (cid:3)
feedbackcase,channelencodingrateisafunctionofthemain C =Mli→m∞1/M 1−IHm(i),He(i),Hz(i)∈O (29)
i=1
CSI,H .Theadversaryknowstheinstantaneouspowergains
m
where
oftheeavesdroppingchannelandthemainchannel.Constraint
term in (9) is reducMed to O={hm(i),he(i),hz(i):IJ(i)Ilog(cid:16)1+1+PPhjmh(zi()i)(cid:17)≤R(hm(i)) =1}
(cid:88) (cid:91)
C =Mli→m∞1/M i=1IJ(i)Ilog(cid:16)1+1+PPHjmH(zi()i)(cid:17)≥R(Hm(i)) (cid:91){hm(i),he(i),hz(i):IE(i)Ilog(1+Phe(i))≥R(hm(i))−Rs =1}
{h (i),h (i),h (i):I (i)I =1}
+(1−IJ(i))Ilog(1+PHe(i))≤R(Hm(i))−Rs m e z E log(1+Phm(i))≤R(hm(i)) (30)
I (23)
log(1+PHm(i))≥R(Hm(i)) Constraint C decreases as the siz of set O increases. Let’s
M
=Mli→m∞1/M(cid:88)i=1I(Hm(i),He(i))∈ApIlog(cid:16)1+1+PPHjmH(zi()i)(cid:17)≥R(Hm(i)d)eOfiUnpepearnBouunpdper bound for O:
+I I
(cid:20) (Hm(i),He(i))(cid:54)∈Ap log(I1l+ogP(H1+e(Pi)H)≤mR(i()H)≥mR(i()H)−mR(is)) (24) (cid:91)={hm(i),he(i),hz(i):Ilog(cid:16)1+1+PPhjmH(zi()i)(cid:17)≤R(hm(i)) =1}
{h (i),h (i),h (i):I =1}
=E IHm,He∈ApIlog(cid:16)1+1+PPHjmHz(cid:17)≥R(Hm) (cid:91){hm(i),he(i),hz(i):Ilog(1+Phe(i))≥R(hm(i))−=Rs1} (31)
+I I I (cid:3), m e z log(1+Phm(i))≤R(hm(i))
Hm,He(cid:54)∈Ap log(1+PHe)≤R(Hm)−Rs log(1+PHm)≥R
where I (i) = I (i) = 1,∀i ∈ N. Then, we find a lower
w.p. 1 (25) J E
(cid:90) (cid:20) (cid:21) bound for C by putting OUpperBound in (29):
= hm,heE Ilog(cid:16)1+1+PPhjmHz(cid:17)≥R(hm) Ihm,he∈Ap CLowerBound
M
(cid:88) where (cid:15) = (cid:15) +(cid:15) . Here, (b) is satisfied for any (cid:15) > 0 and
=Mli→m∞i=1Ilog(cid:16)1+1+PPHjmH(zi()i)(cid:17)≥R(Hm(i))≥log(1+PHe(i))+Rs hMe ∈AM w1ith P2r[AM]=1 and M ≥M((cid:15)1) sinc1e
=CPacketCSI (32) 1 (cid:88)M
lim log(1+PH (i))=E[log(1+PH )]
Since CLowerBound = CPacketFeedback, we have FLowerBound = M→∞M e e
i=1
FPacketFeedback ⊂FPilotFeedback.
with probability 1, (c) follows from the Fano’s inequality.
Let’s define R (cid:44)R −R and PNM (cid:44)P[XNM (cid:54)=XˆNM]
e m s e
where XˆNM = g(ZNM,hM,W) is the estimation of the
e
codeword XNM.
VII. PROOFOFTHEOREM2
H(XNM|ZNM,W,hM) H(PNM)
e ≤PNMR + e (33)
NM e e NM
≤(cid:15) (34)
We employ the encoding across block strategy explained 2
(cid:104) (cid:16) (cid:17)(cid:105)
RirnemliaSbe<lcetiocEnom(cid:104)IlIom-gDu(cid:16)n,1icw+ahtieo1rn+PePHbjRmHymzj(cid:17)a(cid:105)m(cid:44),mthEinegalodagtveer1vsae+rryy1+PpbPrHleojvmHcekzn.tsN.tohtIeef HMfoer.retTh,heaensryeeqa(cid:15)us2eon>ncie0s,oth(f3ac4to)sdiinescssea(2RtiNsefiM=eRdIef(,oXRreNs)u,fafiZscNiMe|nHt→ley),h∞Pige.NhMN→an0d
that in no CSI case, the adversary only obtains h (i) so
e
equivocation rate is defined as H(W|ZNM,hMe ). Equivocation VIII. PROOFOFTHEOREM3
NM
analysis for the encoding across block is as follows.
We first show that if R is an achievable secrecy rate
s
H(W|ZNM,hM) under the packet feedback strategy, we have for any (cid:15) >
e 0, 1 H(W|ZNM,HM,HM)≥R −(cid:15),∀N ≥N((cid:15)),∀M ≥
=H(W,XNM|ZNM,hM)−H(XNM|ZNM,W,hM) NM m e s
e e M((cid:15)). Note that here, the message W is conditioned on
=H(XNM|ZNM,hM)+H(W|XNM,ZNM,hM) random vectors, HM and HM.
e e e m
−H(XNM|ZNM,W,hM) 1
e H(W|ZNM,HM,HM) (35)
≥H(XNM|ZNM,hM)+H(XNM|ZNM,W,hM) NM m e
e e (cid:90) 1
=H(XNM|hM)−I(XNM,ZNM|hM) = H(W|ZNM,hM,hM)f (hM,hM)dhM dhM
e +H(XNeM|ZNM,W,hMe ) (cid:90)AM NM M e HmM,HeM m e m e
(=a)MNRm−I(XNM,ZNM|hMe ) ≥ AM(Rs−(cid:15))fHmM,HeM(hMm,hMe )dhMm dhMe (36)
−H(XNM|ZNM,W,hMe ) =Rs−(cid:15) (37)
≥MNR −N(cid:88)M I(XN(i),ZN(i)|h (i)) where AM is the set defined in Section II-D2. Here, (36)
m e follows from the definition of achievability, and (37)follows
i=1 from the fact that P[A ] = 1. We consider the following
−H(XNM|ZNM,W,hM) M
e case for the rest of the proof. When the transmitter receives
M a NACK signal, on the next block, the transmitter sends
(cid:88)
≥MNR −N log(1+Ph (i))
m e an independent group of bits instead of retransmitting the
i=1 previous packet [12]. In [12], the authors use this scheme for
−H(XNM|ZNM,W,hMe ) thesecretkeysharing.Thecrucialobservationisthatanupper
where (a) follows from the fact that codeword XNM is boundforanachievablesecretkeyrateisalsoanupperbound
uniformly distributed over a set of size 2NMRm. We continue for achievable secrecy rates.
We define index set F that contains the indexes of blocks
with the following steps.
on which the transmitted codeword is successfully decoded.
H(W|ZNM,hMe ) Suppose that size of F is M(cid:48), n(cid:48) (cid:44)NM(cid:48), and n(cid:44)NM. We
NM now prove that 1H(W|Zn,HM,HM)≤R+ as n→∞.
(cid:80)M log(1+Ph (i)) H(XNM|ZNM,W,hM) n m e s
≥Rm− i=1 M e − NM e H(W|Zn,HmM,HeM)
(≥b)Rm−E[log(1+PHe)]−(cid:15)1 (≤a)H(W|Zn,HmM,HeM)
− H(XNM|ZNM,W,hMe ), −H(W|Zn,Yn,HmM,HzM,HeM)+nδn
NM (=b)H(W|Zn(cid:48),HM(cid:48),HM(cid:48))
(c) m e
≥ Rm−E[log(1+PHe)]−(cid:15)1−(cid:15)2 −H(W|Zn(cid:48),Yn(cid:48),HmM(cid:48),HzM(cid:48),HeM(cid:48))+nδn
=Rs−(cid:15), =I(W;Yn(cid:48),Hn(cid:48)|Zn(cid:48),HM(cid:48),HM(cid:48))+δ
z m e n
(cid:90)
(≤c)I(Xn(cid:48);Yn(cid:48),HzM(cid:48)|Zn(cid:48),HmM(cid:48),HeM(cid:48))+δn = P[Hm ∈AM(hMe )]fHeM(hMe )dhMe =1 (42)
hM
=I(Xn(cid:48);HM(cid:48)|Zn(cid:48),HM(cid:48),HM(cid:48)) e
z m e We can see that P[H ∈ A (hM)] = 1, ∀hM ∈ E such
+I(Xn(cid:48);Yn(cid:48)|Zn(cid:48),HmM(cid:48),HeM(cid:48),HzM(cid:48))+nδn that P[E] = 1. We nmow proMve tehe lemma wieth following
(=d)I(Xn(cid:48);Yn(cid:48)|Zn(cid:48),HM(cid:48),HM(cid:48),HM(cid:48)) inequalities.
m e z
M 1 (a) 1
(=e)(cid:88)I(XN(i);YN(i)|ZN(i),Hz(i),Hm(i),He(i))I(i) nH(W|Zn,hMe ) ≥ nH(W|Zn,hMe ,HmM) (43)
i=1 (cid:90) 1
= H(W|Zn,hM,hM)f (hM)dhM (44)
+nδn AM(hMe ) n e m HmM m m
(≤f)(cid:88)M NE(cid:20)log(cid:18)1+ PHm (cid:19)−log(1+PH )(cid:21)+I(i) (≥b)(cid:90) (R −(cid:15))f (hM)dhM (45)
i=1 1+PjHz e AM(hMe ) s HmM m m
+nδn (=c)Rs−(cid:15), ∀hMe ∈E with P[E]=1. (46)
where I(i) = IHz(i)≤h∗z. Here, (a) follows from Fano’s (a) follows from the fact that conditioning reduces the en-
inequality(b)followsfromtheindependentchoiceofthecode- tropy, (b) follows from the fact that since hM ∈ A (hM),
m M e
wordsymbolstransmittedineachblockthatdoesnotallowthe (hM,hM)∈A , and (c) follows from the fact that P[H ∈
m e M m
eavesdropper to benefit from the observations corresponding A (hM)]=1.
M e
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M
We define A (hM) = {hM : (hM,hM) ∈ A }.
M e m m e M
Since HM and HM are independent random vectors and
m e
P[(HM,HM) ∈ A ] = 1, we have P[HM ∈ A (hM)] =
m e M m M e
1,∀hM ∈A . To observe that
e M
(cid:90)
1= f (hM,hM)dhM dhM (40)
HM,HM m e m e
A m e
M
(cid:90) (cid:90)
= f (hM) f (hM)dhM dhM (41)
HM e HM m m e
hMe e hMmAM(hMe ) m