Table Of ContentSecret Message Transmission by HARQ
with Multiple Encoding
Stefano Tomasin and Nicola Laurenti
Department of Information Engineering, University of Padova, ViaGradenigo 6/B, I-35131 Padova, Italy
{tomasin, nil}@dei.unipd.it
Abstract—Secure transmission between two agents, Alice and throughone-time-padschemes.Insteadofconsideringasimple
4 Bob, over block fading channels can be achieved similarly to retransmission approach, a hybrid automatic repeat request
1 conventionalhybridautomaticrepeatrequest(HARQ)byletting (HARQ)-like scheme is derived in [7] for secure communi-
0 Alice transmit multiple blocks, each containing an encoded
cations over a block-fading Gaussian channel. In this case,
2 version of the secret message, until Bob informs Alice about
successful decoding by a public error-free return channel. In a single codeword is generated and punctured versions of
n existing literature each block is a differently punctured version it are transmitted until the legitimate receiver decodes the
a
of a single codeword generated with a Wyner code that uses secretmessage.Forencoding,awiretapcodewithincremental
J
a common randomness for all blocks. In this paper instead we
redundancyisemployed,obtaininganincrementalredundancy
4 proposeamoregeneral approachwheremultiplecodewords are
1 generatedfromindependentrandomnesses.Theclassofchannels HARQ (IR-HARQ) scheme, and an outage formulation is
for which decodability and secrecy is ensured is characterized, considered. The approach of [7] is then extended in [8]
] with derivations for the existence of secret codes. We show in to a multiuser uplink scenario, with each user aiming at
R
particular that the classes are not a trivial subset (or superset) transmittingacombinationofpublicandconfidentialmessages
C of those of existing schemes, thushighlightingthe novelty of the
to a single base station,with the otherusersacting potentially
. proposedsolution.Theresultisfurtherconfirmedbyderivingthe
s as eavesdroppers. A suboptimal strategy of power allocation
average achievable secrecy throughput,thus takinginto account
c both decoding and secrecy outage. andschedulingtomaximizetheoverallnetworkutilityisthen
[
IndexTerms—Hybridautomaticrepeatrequest,PhysicalLayer derived.The HARQ secrecyscenarioisalso consideredin [9]
1 Security, Secret Message Transmission. witharatherdifferentapproach,wherestandardcodesareused
v with the addition of scrambling, but secrecy is expressed in
8 I. INTRODUCTION terms of the bit error probability at the eavesdropper. Lastly,
8
Physicallayersecrecyhasgainedalotofattentioninthelast thethroughputofHARQ withoutsecrecyconstraintshasbeen
0
3 fewyears,dueto itsability ofprovidinginformationtheoretic studied in [10], but the analysis does not fit immediately the
. unconditional security, thus adding security at the physical differentscenariowhereasecretmessagemustbetransmitted.
1
0 layer. Fromthe seminalworks[1]–[3] performancelimits and In this paper we consider a scenario similar to IR-HARQ
4 achievable rates have been derived in different scenarios for of [7], where transmissions occur on block fading channels.
1 thereliableyetsecrettransmissionofconfidentialinformation While for IR-HARQ at each retransmission a different punc-
:
v (see, e.g. [4] for a review). In particular, it has been shown turing of a single codeword of a Wyner code is used, we
i that diversity, in the form of fading (temporal diversity), proposetoencodethesecretmessagewithmultiplecodesand
X
multipath (frequency diversity) or multiple input multiple then send punctured versions of the multiple codewords until
r
a output (MIMO) (spatial diversity) is definitely beneficial to Bob decodes. The obtained solution is denoted secure HARQ
secret transmission. In fact, dimensions or instants in which (S-HARQ), and is a strict generalization of IR-HARQ. We
the legitimate receiver is at an advantage with respect to the provetheexistenceofcodesthatensurebothdecodabilityand
eavesdropper can be selected, even if the channels to both secrecy for all channel realizations that satisfy certain condi-
receivers have the same statistics. However, one of the main tions. While decodability and secrecy in IR-HARQ depend
obstacles to the effective implementation of such systems is on the average over all channel realization of the mutual
the need of knowing channel state information (CSI) towards informationbetweenthetransmittedandreceivedmessages,in
both the legitimate receiver and the eavesdropper at the time S-HARQ instead performance depends on multiple averages
of code design. overdifferentfadingblocks,thusprovidingadditionaldegrees
This drawback can be partly mitigated by the presence of of freedom1.
a feedback channel, even with a limited rate and/or publicly
II. SYSTEMMODEL
accessible, which, contrary to the unconstrained transmis-
sion case, has been shown to increase secrecy capacity [5]. We consider the scenario of Fig. 1, where an agent Alice
The automatic repeat request (ARQ) mechanism, with its transmitsasecretmessageMtoanintendeddestinationagent,
intrinsic one-bit feedback is leveraged in [6] for the secure
generation of cryptographic keys, that can either be used 1Notation, throughout the paper EX[g(X)] denotes the expectation of
g(X)withrespecttoX,andI(X;Y)denotesthemutualinformationbetween
in traditional encryption systems or provide perfect secrecy X andY.
Alice Xn Yn Bob this end Alice encodes the secret message by using a random
k,m k,m
ENC. channel DEC. binning approach,independently(in the random message) for
Zn Eve each frame. The random message used at frame k to confuse
k,m
channel DEC. Eve is denoted as Md,k and bears nRd,k bits of information.
Therandommessageisindependentlygeneratedateachframe,
contains no secret information and may even be completely
Fig.1. SystemModel.
irrelevant for the three agents, since its purpose is only to
confuseEveaboutM.Wedenotethecollectionoftherandom
Bob,overtheAlice-Bobchannel.Athirdagent,Eve,overhears messages over the K frames as Me ={Md,1,...,Md,K}.
the message transmitted by Alice over an independent Alice- Encoding process: Let i∈ be the index of the (random)
Eve channel. secretmessageM in the set of2nRs messages. The encoding
Time is organized in consecutive slots of the same du- process works as follows: at frame k, Alice selects an index
ration, which are grouped into frames, each comprising M j ∈ {0,1,...,nR } randomly and uniformly, and gener-
k d,k
consecutive slots. The transmission of M spans in general ates the codeword XnM(i,j ), which is punctured into M
k k
many slots. At slot m of frame k, Alice transmits message codewords of n symbols Xn (i,j ),...,Xn (i,j ). Then,
Xkn,m of n symbols containing an encoded version of the thepuncturedcodewordXkn,mk,1(i,jkk)istransmk,iMttedatkthem-th
secret message. The secret message M bears nRs bits of slot of frame k. The set of transmitted messages over all K
information.Ateachslot,afterAlice’stransmission,Bobsends frames is identified by the (K +1)−ple (i,j ,...,j ).
1 K
a not acknowledge (NACK) message if he fails to correctly
Decodingstrategy: Weconsiderajointtypicalitydecoder
decodethe secretmessage.Whendecodingis successful,Bob
for both Bob and Eve. For the generic slot m of frame k,
sends an acknowledge (ACK) feedback to Alice, who stops
let Tn (XY) denote the set of all ǫ-jointly weakly typical
transmissions. After MK slots used for the transmission of k,m,ǫ
sequences ({Xn (i,j ),...,Xn (i,j )},{Yn ,...,Yn }).
the same secret message, Alice discards the secret message, 1,1 1 k,m k 1,1 k,m
Bob decides for message (ˆı,ˆ ,...,ˆ ) if {Xn(ˆı,ˆ )} is the
irrespectiveof Bob decodingoutcome.ACK/NACK messages 1 k ·,· k
only sequence taken from C that is ǫ-jointly typical with
are perfectly received by Eve. The procedure is similar to n
{Yn}. Otherwise an error is output.
HARQ, except for the presence of Eve, thus we denote the ·,·
resulting scheme as S-HARQ. The code used at each frame is called frame code, while
The message received by Bob at slot m of frame k is HARQ code denotes the sequence of the frame codes. In
denoted by Yn , which is related to Xn by a given particular:
k,m k,m
transition probability distribution. Let us also define XknM = frame code: this is a subset Cn,k of 2n(Rs+Rd,k) words
[Xn ,...,Xn ] and YnM = [Yn ,...,Yn ]. Eve over- ofnM symbols,randomlychosen.Infact,foreachindexpair
k,1 k,M k k,1 k,M
hears the packet Zn at slot m of frame k, and let us (a,b), a ∈ {1,...,2nRs},b∈ {1,...,2nRd,k} we choose the
k,m
define ZnM = [Zn ,...,Zn ]. Both Bob and Eve use all word XnM(a,b) with independent symbols all drawn from a
k k,1 k,M k
previouslyreceivedpackets for decodingpurposes.Moreover, same distribution p (·);
Xk
Eve knows the encoding procedure followed by Alice. HARQ code: this is the set of 2n(Rs+PKk=1Rd,k)
We assume a block-fading channel, i.e., the channels do
codewords obtained by concatenating the K words
notchange within each slot, while theyvary from slot to slot.
XnKM(a,b ,...,b ) = [XnM(a,b ),...,XnM(a,b )]
We also assume that the number of channel uses n within 1 K 1 1 K K
and is denoted by C .
n
each slot is large enough so that we can use random cod-
We also denote the set of all possible codes that can be
ing arguments. For the Alice-Bob channel {p(Y |X )}
k,m k,m
generated for frame k as the ensemble C = {C }, and
denotes the symbol transition probability for slot m = n,k n,k
that of all possible HARQ codes as C = {C }. We assume
1,2,...,M, and frame k = 1,2,...,K. Similarly for the n n
that the actually selected code (as well as the ensemble) is
Alice-Eve channel we have {p(Z |X )}. Let p =
k,m k,m m,k
known to both Bob and Eve.
{p(Y |X ),p(Z |X )} denote a generic channel re-
k,m k,m k,m k,m
alization for slot m and frame k, while p = {p , m = Note that in the case of a single frame (K = 1) we
k m,k
1,2,...,M} is a set of channel realizations in frame k and obtainthescenarioconsideredin[7],whereasinglecodeword
is split into M parts that are sequentially transmitted until
p={p , m=1,2,...,M,k =1,2,...,K} (1)
m,k Bob decodes. On the other hand, when M = 1 we have
denotesagenericchannelrealizationforthewholesequenceof that a new codeword, generated by an independent random
frames. Fading implies that p is a random vector, and block message, is transmitted at each slot. Other cases (K > 1
fading statistics dictates the statistics of the vector. We also and M > 1) correspond to intermediate situations where
assume that CSI is not available to Alice before transmission. eachrandommessagespansmultipleslots,andmorethanone
random message may be used to confuse the eavesdropper
III. CODE CONSTRUCTION aboutthesame secretmessage,providedthatBobneedsmore
EncodingaimsatensuringbothdecodabilityofMbyBob, retransmissions. In the following we detail the general case
and secrecy, i.e., preventing information leakage to Eve. To for any value of K and M.
IV. DECODABILITY AND SECRECY CONDITIONS B. Secrecy codes ensemble characterization
The design of the S-HARQ code aims at ensuring that Todealwithsecrecy,wefirstdenotetheinformationleakage
a) Bob is able to decode the secret message with vanishing of the first M′ slots of the k-th frame to Eve when Alice uses
probability(decodability),and b)Eve getsvanishinginforma- code Cn,k over channel realization pk as
tion rate on the secret message (secrecy). Now we show that
L(C |p ,M′)=I(M;{Zn ,...,Zn }|C ,p ). (5)
asymptotically (n → ∞) for a given set P of channels, and n,k k k,1 k,M′ n,k k
for a given set of rates (Rs,Rd,1,...,Rd,K), there exists a Similarly, the information leakage for the transmission up to
S-HARQ code that provides both decodability and secrecy. slot M′ of frame K′ is defined as
Note that the code to be used is the same for all channels in
L(C |p,M′,K′)=
the set P. In the considered scenario, no CSI is available to n (6)
Alice, therefore if the channel is not in P we may have an I(M;Z1nM,...,ZKnM′−1,{ZKn′,1,...,ZKn′,M′}|Cn,p).
outage event, i.e., either Bob may not decode M or Eve may Then we start with the following lemma that establishes a
get some information on M. The outage probability Pout is relation between the information leakage of each frame and
theprobabilitythatanoutageeventoccurs.Fromthedefinition that of the transmission up to frame K′.
of set P we have a bound on P as
out
Lemma 2. Theinformationleakageoverallframesuptoslot
Pout ≤P[p∈/ P], (2) M′ of frame K′ is not larger than the sum of information
leakage for each frame, i.e.,
where P[·] denotes the probability operator. The characteriza-
tionofthesetP canthenguidethecodedesignanditsusage, K′−1
since we can obtain an estimate of the outage probability by L(Cn|p,M′,K′)≤ L(Cn,k|pk,M)+L(Cn,K′|pK′,M′).
assessing the probabilitythatthe channeloverwhichthecode k=1
X
is actually used is outside P. Proof: For the sake of a simpler notation we providethe
In order to characterize P we first derive conditionson the proof for K′ = K and M′ = M, the generalization being
realization p thatensure decodabilityby Bob on averageover straightforward.
a set of codes, then we derive conditions on p that ensure Since we use independent random binning in each trans-
secrecy with respect to Eve on average over a set of codes. mission, (ZnM,...,ZnM,C ) → (C ,M) → ZnM is a
1 k−1 n n,k k
Finally we characterize the set P over which a single code Markov chain.
provides both secrecy and decodability. By the chain rule for mutual information [11, eq. (2.62)]
From now on, for the sake of compactness we denote by we have
IkB,m(pk,m)=limn→∞ n1I(Xkn,m;Ykn,m|pk,m) the single letter I(M;ZnM,...,ZnM|C ,p)=
mutual information across the legitimate channel at slot m of 1 K n
frame k, and the analogous for the eavesdropper channel by K (7)
I(M;ZnM|ZnM,...,ZnM,C ,p )
IE (p )=lim 1I(Xn ;Zn |p ). k 1 k−1 n k
k,m k,m n→∞ n k,m k,m k,m k=1
X
A. Decodable codes ensemble characterization Each term in the sum can be upper bounded as
As decodabilityis concerned,we have the followingresult: I(M;ZnM|ZnM,...,ZnM,C ,p )=
k 1 k−1 n k
Lemma 1. Let (Mˆ,Mˆe) be the message decoded by the =H(ZknM|Z1nM,...,Zkn−M1,Cn,pk)−
ǫ-joint typicality decoder over K frames, and let the er- −H(ZnM|M,ZnM,...,ZnM,C ,p )
k 1 k−1 n k
ror probability associated with a given HARQ code C be
n ≤H(ZnM|M,C ,p )− (8)
P (C |p) = P[(M,M ) 6= (Mˆ,Mˆ )|C ,p] for a given k n,k k
e n e e n
channel realization p. For all K′ ≤ K and M′ ≤ M that −H(ZknM|M,Z1nM,...,Zkn−M1,Cn,pk)
satisfy =H(ZnM|C ,p )−H(ZnM|M,C ,p )
k n,k k k n,k k
K′ K′−1 M =I(M;ZknM|Cn,k,pk)=L(Cn,k|pk,M).
R +R < IB (p )−δ(ǫ) +
d,k s k,m k,m
Xk=1 kX=1 mX=1(cid:2) (cid:3) (3) Then we derive a bound on the information leakage at the
K′ M′ k-th frame by the following lemma.
IB (p )−δ(ǫ) ,
k,m k,m
k=1m=1 Lemma 3. Foreachchannelrealizationpand(K′,M′)such
X X (cid:2) (cid:3) that M I(Xn ;Zn |p ) < R , k = 1,...,K′−1,
with δ(ǫ) > 0, then for each and n there exists a δ′(n) such m=1 k,m k,m k,m d,k
that δǫ′(n)−−−−→0 for each ǫ, and ǫ Mm=′P1I(XKn′,m;ZKn′,m|pK′,m)<Rd,K′, and for each n and
n→∞ ǫ we have a δ(ǫ) and a δ (n) such that δ (n)−−−−→0 and
ǫ ǫ
P n→∞
E [P (C |p)]≤δ (n). (4)
Cn e n ǫ 1
E L(C |p ,M) ≤δ(ǫ)+δ (n), k =1,...,K′−1
Proof: See the Appendix. Cn,k n n,k k ǫ
(cid:20) (cid:21)
1
ECn,K′ nL(Cn,K′|pK′,M′) ≤δ(ǫ)+δǫ(n). NfuonwctiobnysaPpp(l·y)ianngdtLh(e·)s,ewleicthtiorenfelreemncmeato[t4h,epragn.d1o4m] vtoaribaobtlhe
(cid:20) (cid:21) e
Proof: See the wiretap coding theorem [4, pg. 72]. Cn, we obtain a sequence of codes with vanishing error
By combining the two results we obtain the following probabilityandleakage.ByobservingthatbothPe(·)andL(·)
lemma. are non negative, we obtain (12) and (13).
We then have a single code sequence that provides both
Lemma 4. Foreachchannelrealizationpand(K′,M′)such
decodability and secrecy for all channels in the set P.
that
Remark 1: this result generalizes that of [7]: for that
M
codeconstructioninfact,sufficientconditionsforsecrecywere
IE (p )<R ∀k =1,2,...,K′−1
k,m k,m d,k ensured by a constrainton the sum of the mutual information
mX=1 (9) between Alice and Eve across slots of a single frame. In our
M′
scenario instead we need boundson each frameseparately, as
IEK,m(pK′,m)<Rd,K′ indicated by (9).
m=1
X
and for each n, we have a δ(ǫ) and a δ (n) such that V. SECURE CHANNEL SETS
ǫ
δǫ(n)−−−−→0 and SincethesetP isdefinedintermsofthemutualinformation
n→∞
oftheAlice-BobandAlice-Evechannels,wecanequivalently
1
E L(C |p,M′,K′) ≤K′δ(ǫ)+K′δ (n). (10) describe it by the set of mutual informations satisfying the
Cn n n ǫ
constraints, i.e., by the set
(cid:20) (cid:21)
Proof: Follows from Lemmas 3 and 2.
Q={{IB ,IE }:
k,m k,m
C. Characterization of the set P p∈P, and IB =IB (p ),IE =IE (p )}.
k,m k,m k,m k,m k,m k,m
Having derived sufficient conditions for decodability and (18)
secrecy for given channel realizations we are now ready to
From the results of the previous Section we have
derive conditions for both decodability and secrecy with the
same code. We now show that for the set K M
Q= Q(E)(K′,M′)∩Q(B)(K′,M′) , (19)
P ={p:∃(K′,M′) for which both (3) and (9) hold} (11) S S
K[′=1M[′=1h i
thereexistsasinglecode(sequence)thatprovidesbothsecrecy where Q(E)(K′,M′) indicates the set of channels for which
S
and decodability.
noinformationaboutthesecret messagehasleaked to Eveup
Theorem 1. For all n there exists a specific code C∗ with to the m-th slot of the k-th frame,
n
rates Rs and {Rd,k} such that, for all channels p∈P there Q(E)(K′,M′)= {IB ,IE }:
exists K′(p) and M′(p) such that S k,m k,m
M
(cid:8)
Pe(Cn∗|p)≤δǫ(n), L(Cn∗|p,M′(p),K′(p))≤δ(ǫ)+Kδǫ(n) IkE′,m′ ≤Rd,k′, for k′ =1,2,...,K′−1,
(12) m′=1
X
and M′
lim P (C∗|p)=0, lim 1L(C∗|p)≤δ(ǫ). (13) IKE′,m′ ≤Rd,K′ (20)
n→∞ e n n→∞n n mX′=1
Proof: From the definition of P and lemma 3 we imme- while Q(B)(K′,M′) indicates the set of channels for which
diately have S
the secret message is decodable by Bob within the m-th slot
E [E [P (C |p)]]≤δ (n), (14) of the k-th frame,
p∈P Cn e n ǫ
while from Lemma 4 (and by the fact that K′(p) < K) we Q(B)(K′,M′)= {IB ,IE }:
S k,m k,m
also have K′−1 M (cid:8) +
E E 1L(C |p,M′(p),K′(p)) ≤Kδ(ǫ)+Kδ (n). IkB′,m′ −Rd,k′
p∈P Cn n n ǫ k′=1 "m′=1 #
(cid:20) (cid:20) (cid:21)(cid:21) (15) X X +
M′
Texhpeenc,tastiimonilsarolyvertothtehechaapnpnreolacshetoafnd[7t]hewceodceasn (sswinacpe tthhee + IKB′,m′ −Rd,K′ ≥Rs (21)
integrands are non negative and finite), obtaining mX′=1
where [x]+ = 0 if x < 0 and [x]+ = x if x > 0. Condition
E [E [P (C |p)]]≤δ (n) (16)
Cn p∈P e n ǫ (21) follows by applying Lemma 3 to the Alice-Bob channel,
E E 1L(C |p,M′(p),K′(p)) ≤Kδ(ǫ)+Kδ (n). as we observe that if Mm=1IkB,m ≤Rd,k, Bob will not make
Cn p∈P n n ǫ use of the signal received in frame k to decode the secret
(cid:20) (cid:20) (cid:21)(cid:21) (17) message. P
I(B) =I(B) I(E) =I(E)
2,1 1,2 2,1 1,2
RIR
d
R
12s
RIR
d
R +R
d,2 s
R
d,2
I(B) I(E)
1,1 1,1
Rd,1+Rs R12sRdIR Rd,1 RdIR
Fig. 2. R12s = Rd,1+Rd,2+Rs Bob’s decoding region Q(SB)(2,1) Fig. 3. Eve’s failure region Q(SE)(2,1) for S-HARQ with K = 2 and
for S-HARQ with K = 2 and M = 1 (dashed area), and Q(B)(1,2) of M =1(dashedarea)andQ(B)(1,2)ofIR-HARQ(grayarea).
S S
IR-HARQ(grayarea).
1
A. Outage Analysis
0.8
A bound on the reliability outage probability for the whole
transmission is then
0.6
P ≤P {IB ,IE }6∈Q(B)(K,M) , (22) F
o k,m k,m D
C
while the probabilihtythat decodinghappensexacitly at the m- 0.4
the slot of the k-th frame is bounded as M=1, K=6
P (k,m)≥ 0.2 M=2, K=3
D
M=3, K=2
P {IB ,IE }∈Q(B)(k,m)\Q(B)(k,m−1) m>1
k,m k,m
. 0
Ph{IkB,m,IkE,m}∈Q(B)(k,1)\Q(B)(k−1,M)i m=1 −0.2 0 0.2 ∆0T.4[bit/s/0H.6z] 0.8 1 1.2
h i (23)
Fig. 4. CDF of the difference between the throughputs of S-HARQ and
Assuming that the Alice-Bob and Alice-Eve channels are IR-HARQforvarious values ofM,overaRicefadingwiretap channel.
independent, the secrecy outage probability up to slot m of
frame k is bounded by
BothAlice-BobandAlice-Evechannelsareblock-fading,with
P (k,m)≤P {IB ,IE }6∈Q(E)(k,m) . (24) independentRicechannelsateachslot:theRicefactoris0dB
s k,m k,m
forbothandtheaveragesignaltonoiseratio(SNR)is4dBfor
h i
VI. NUMERICAL RESULTS the Alice-Bob channel and 5dB, for the Alice-Eve channel.
We first provide some insight into the performance of the The performance of the proposed approach is assessed over
proposedsolutionbyconsideringatransmissionwithonlytwo block fading channels, by considering the achievable secret
framesK =2andoneslotperframeM =1.Forgivenvalues throughput, i.e.
of R , R and R , Fig. 2 shows as a dashed area the set
d,1 d,2 s K M
P (k,m)[1−P (k,m)]R
Q(B)(2,1) with K = 2 and M = 1. We also show in gray T = max D s s , (25)
theSset Q(B)(1,2) with K =1 frame and M =2 slot, that is {Rd,k}k=1m=1 M(k−1)+m
S X X
when the IR-HARQ scheme is used with a random rate RIR whichis the average(overthe numberof slots) of the secrecy
d
over the same channel. We observe that the shape of the two ratedividedbythe numberof slotsneededfordetection.Asa
areas are different. Similarly, for given values of Rd,1, Rd,2 reference value we consider the achievable secret throughput
andR , Fig. 3 showsas a dashedarea theset Q(E)(2,1)with oftheIR-HARQschemeT(IR) andwefocusontheadditional
s S
K = 2 and M = 1. We also show in gray the performance secret throughputdefined as ∆ =T(S)−T(IR). Using (22),
T
of IR-HARQ. Also in this case we observe that the shape of (23) and (24) with equalitiesin (25) we obtain a lower bound
the two areas are different. We conclude that S-HARQ is a on the achievable secret throughput. Note that we resort to
non-trivial extension of IR-HARQ. theboundssinceasexactperformanceisnotknown,although
Inordertofurtherconfirmthisconclusioninamoregeneral the difference of the lower bounds ∆ may not in general
T
setting, we have considered S-HARQ with a total of 6 slots. be a bound of the actual difference. For various values of
M, Fig. 4 shows the CDF of ∆ . Moreover, the total of 6 We can split E as the union
T (1,j1,...,jK):{∃k:jk6=1} i,j1,...,jK
slots can be split into the following frame configurations: a) of the events where an error occurs in at least one frame. Let
S
K = 1, M = 6, b) K = 2, M = 3, c) K = 3, M = 2, d) S be the set of k for which error occurs, then we have
Kthe=IR6-,HMAR=Q 1sy.sRteemca.llFrtohmat Kthe=fig1u,rMe, w=e6noctoerrtehsaptofnodrsatlol P Ei,jk6=1,k∈S,ju=1,u∈K\S ≤2−n(Pk∈SI(XkMn;YkMn|pk)−δ(ǫ))
(31)
M <6, we have ∆T >0 with non-zeroprobability,therefore wi(cid:2)th K = {1,2,...,K}(cid:3), and we have 2nRd,k|S| − 1 of
therearecaseswhenthediversityprovidedbydifferentframes
these events. Hence, the most restrictive condition on R
within the same total number of slots, yields a strictly higher d,k
to havevanishingerrorprobabilityiswhenonlyoneof theK
throughput than choosing K = 1. Moreover, we observe that
messages is different from 1 and in this case we have
by varying M the distribution of ∆ changes, thus leaving
T
spaceforoptimizationofthesystem,tobeconsideredinfuture P[E1,j1,...,jK]≤2−n(I(XknM;YknM|pk)−δ(ǫ)), ∃!k :jk 6=1
studies. (32)
VII. CONCLUSIONS and we have 2nRd,k −1 of such events.
Event E , i 6= 1 occurs when F , i 6= 1 occurs
We have proposed a secret message transmission scheme i,j1,...,jK i,j,k
for all k, in which case from (30) we have
overblockfadingchannelswithafeedbackfromthelegitimate
receiver with no CSI. Numerical results have highlighted the P[E1,j1,...,jK]≤2−n(PKk=1I(XknM;YknM|pk)−δ(ǫ)), ∃!k :jk 6=1
non-trivial relation with existing schemes and the fact that (33)
for some channel conditions the proposed solution providesa and we have (2nRs −1)2PKk=1Rd,k of such events.
higher available secret throughput. From (27) we have
ACKNOWLEDGMENT ECn[Pe(Cn|p)]≤
This work was supported in part by the MIUR project K
ESCAPADE (Grant RBFR105NLC) under the “FIRB-Futuro δǫ(n)+ (2nRd,k −1)2−n(I(XknM;ZknM|pk)−δ(ǫ))+
in Ricerca 2010” funding program. Xk=1
2PKk=1Rd,k(2nRs −1)2−nPKk=1(I(XknM;YknM|pk)−δ(ǫ)).
APPENDIX
Lastly, observing that
In this Appendix we provide the proof of Lemma 1.
For the sake of a simpler notation we provide the proof M
for K′ = K and M′ = M, the generalization being I(XknM;YknM|pk)= I(Xkn,m;Ykn,m|pk,m), (34)
straightforward. m=1
X
WithoutrestrictionwesupposeM=i=1.Indicatingwith we conclude the proof.
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Y
