Wiretap channel capacity: Secrecy criteria, strong converse, and phase change

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by  Eric Graves

Wiretap channel capacity: Secrecy criteria, strong
converse, and phase change
Eric Graves and Tan F. Wong
Abstract—This paper employs equal-image-size source parti- requirement is to view the problem as a binary hypothesis
tioningtechniquestoderivethecapacitiesofthegeneraldiscrete testing of the alternate hypothesis of M and Zn being in-
memoryless wiretap channel (DM-WTC) under four different dependent against the null hypothesis of M and Zn being
secrecy criteria. These criteria respectively specify requirements
7 on the expected values and tail probabilities of the differences, correlated.This is an interesting case in which we would like
1 in absolute value and in exponent, between the joint probability the false positive probability given by the likelihood ratio test
0 of the secret message and the eavesdropper’s observation and
n 2 tthheesceocrrrietseproiandreindgucperobbaacbkiltiotythifetshteayndwaerrdelienadkeapgeendaenndt.vSaorimateioonf PM,Zn(cid:18)(cid:26)(m,zn)∈M×Zn : PPMM(,mZn)P(mZn,z(znn)) ≥τ(cid:27)(cid:19)
a distance constraints that have been previously considered in the 1 as n
J literature.Thecapacitiesunderthesesecrecycriteriaarefoundto → →∞
5 bedifferentwhennon-vanishingerrorandsecrecytolerancesare where the decision threshold τ [0,1) serves as a measure
allowed.Basedonthesenewresults,weareabletoconcludethat ∈
2 of secrecy with τ 1 begin the most secret situation. Note
] tohnelysutrnodnegrctohnevtewrosesepcrroepceyrctyritgeerniaerbaallsyedhoonldcsofnosrtrtahieniDngMt-hWetTaCil that the log-likelih→ood log2 PPMM(,mZn)P(Zmn,z(znn)) may also be used
T probabilities. Under the secrecy criteria based on the expected in the hypothesis testing problem above.
I values, an interesting phase change phenomenon is observed as For every (m,zn) n, define
. the tolerance values vary. ∈M×Z
s
[c I. INTRODUCTION v(m,zn), 1− PPMM(,mZ)nP(Zmn,z(znn)) + if PM,Zn(m,zn)>0
1 The discrete memoryless wiretap channel (DM-WTC) (cid:12)(cid:12)0 (cid:12)(cid:12) if PM,Zn(m,zn)=0
(cid:12) (cid:12)
7v r(eXce,iPvYe,rZY|X,,aYnd×anZea)vecsodnrsoipstpseroZf.aAsmenedsseargeXM, aislteogbiteimseantet where |c|+ equals c if c>0 and 0 otherwise, and
734 rbeyliaZb.lyOfvreormnXusteos Yofatnhde DdiMsc-rWeetTlyC,agleatinfstne:avMesdr→oppXinng i(m,zn),(−0 log2 PPMM(,mZn)P(Zmn,z(znn)) iiff PPMM,,ZZnn((mm,,zznn))>=00.
0 and ϕn : n be the encoding and decoding functions
. respectiveYly em→plMoyed at X and Y, where = [1 : 2nR] is We see that all the secrecy requirements discussed above can
1 the message set and M is uniformlydistribuMted over . The be compactly specified in terms of the tail probabilities and
0
7 transmission reliability requirement is specified by M expected values of v(M,Zn) and i(M,Zn):
1
v: Pr{ϕn(Yn)6=M}≤ǫn (1) S1(δn):PM,Zn({v(M,Zn)>δn})→0
Xi wrehqeurieremǫnen∈ta(s0se,s1s)esdehnoowtemsuthche oenrreomr atoyleleraanrncea.bTouhteMsecfrreocmy SS23((δlnn))::PEMM,,ZZnn([vi((MM,,ZZnn))]=>klnPM),Zn0−PMPZnk1 ≤δn
r { } →
a Zn. This requirement is often quantified by measuring the S4(ln):EM,Zn[i(M,Zn)]=I(M;Zn) ln
≤
level of “independence” between M and Zn based on either
the variation distance wexhpeercetaδtinon∈w(.0r.,t.1]P,Mln,Z∈n.(N0,o∞te)t,haatndS2E(δMn,)Zna[n·]ddSen4(oltne)s tahree
1
P P P thevariationdistanceanddivergence(leakage)constraints,re-
M,Zn M Zn 1
2k − k spectively,whileS (δ )andS (l )correspondtothesecrecy
1 n 3 n
1
, PM,Zn(m,zn) PM(m)PZn(zn) requirementsspecifiedbythehypothesistestingproblemusing
2 | − | the likelihood and log-likelihood ratios, respectively.
(m,znX)∈M×Zn
Clearly these secrecy requirements are related to each
or the divergence D(PM,Zn PMPZn) = I(M;Zn) between other. For example, S (δ ) implies S (δ ), and S (l ) im-
k 1 n 2 n 3 n
PM,Zn and PMPZn. Another way of quantifying the secrecy plies S (l ). Then, by Pinsker’s inequality, S (l ) implies
4 n 4 n
[e-mail]ensteaorfchEleLcatrbi,caAldaenldphCi,omMpDute2r0E78n3g,ineUe.rSin.Ag,. Sin2eq(cid:18)uqaliltny2,lnS2(cid:19)2(δnif) linmp∈lies(0S,1l(n2√2δ).n)Siimf iδlanrly by0.MAalrskoovb’ys
University ofFlorida,Gainesville, FL32611,[email protected] elementarylogarithm inequalities, S (δ ) imp→lies S 2δn if
T.F.WongwassupportedbytheNationalScienceFoundationunderGrant δ 0. Thus for vanishing toleranc1es,nS –S are e3sselnnt2ially
CCF-1320086. Eric Graves was supported by a National Research Council n → 1 4 (cid:0) (cid:1)
Research Associateship AwardatArmyResearch Lab. equivalent.
Special cases of these secrecy requirementshave been con- that lim (R ,ǫ ,η ) = (R,ǫ,η), for i 3,4 . We
n→∞ n n n
∈ { }
sideredintheliterature.Forexample,requiringǫ 0in(1), have instead chosen to present the “weak” versions of these
n
S (nr ) is the equivocation constraint originally→considered criteria, simply because their proofs trivially recover their
4 l
in [1]. Six secrecy requirements S –S are more recently “strong” counterparts. That is the ǫ-secrecy capacity under
1 6
considered in [2]. Setting ǫ 0, S is S (l ) for some S (l) orS (l), mustbe less than the ǫ-secrecycapacityunder
n 1 4 n 3 4
lln →00,,SS2 iissSS2((δln))ffoorrssoommeeδ→lnn→00,,Sa3nidsSS3(ilsn)Sfo(lrs)omfoer S3 nlogn2n and S4 nlogn2n -respectively. Yet, the secrecy
snom→e ln 4 0. 4 n n → 6 3 n cap(cid:16)acities u(cid:17)nder S3 (cid:16)nlogn2n (cid:17)and S4 nlogn2n can also be
Thenm→ajority of known secrecy capacity results under the achievedunderS3(ln(cid:16))andS(cid:17)4(ln)fors(cid:16)omeln (cid:17) 0,using[8,
→
abovesecrecy requirementsare for cases with vanishingerror Theorem 17.11].
tolerance, ǫ 0, and secrecy tolerance, l 0, ln 0, Write
n → n → n →
or δn 0. These results are nicely summarizedin [2], which C ,maxI(X;Y)
shows→thatthesecrecycapacitiesunderS1–S6 (seefootnote1) 1 PX
oftheDM-WTCareallgivenbymax I(U;Y) I(U;Z), C ,maxI(U;Y) I(U;Z)
PU,X − 2 PU,X −
❝ ❝
where U X Y,Z. Here we are mainly interested in
where the alphabet of U in the definition of C can be
cases where both the error tolerance ǫ and secrecy tolerance U 2
n assumed to have . Also restrict ǫ [0,1) and δ
δ , l or ln are non-vanishing, on which only a few partial |U| ≤ |X| ∈ ∈
n n n [0,1], and r [0, ). Then the following theorems give our
l
resultsexist.TheoldestsuchresultdatesbacktoWyner’sorig- ∈ ∞
main results regarding the secrecy capacities:
inalpaper[1],in whichthesecrecycapacityunderS (nr )of
4 l
the degraded DM-WTC (PY,Z|X = PZ|YPY|X) is calculated Theorem 1. The ǫ-secrecy capacity under S1(δ) of the DM-
for the case of ǫ 0. The ǫ-secrecy capacity under S (l ) WTC is given by
n 4 n
→
of the degraded DM-WTC is obtained in [3] for the case of
C if δ <1
ln 0. Thiscase hasalso beenextendedto the generalDM- C (δ), 2
WnT→C in [4] and [5]. The ǫ-secrecy capacity under S (δ) of 1 (C1 otherwise.
2
the degraded DM-WTC is found in [6]. for all ǫ.
In this paper, we determine the secrecy capacities for the
Theorem 2. The ǫ-secrecy capacity under S (δ) of the DM-
generalDM-WTCundertheabovefoursecurityrequirements, 2
S –S ,withnon-vanishingtolerances.Alloftheseresultscan WTC is given by
1 4
be straightforwardly obtained using our recently developed
C if ǫ+δ <1
equal-image-size source partitioning techniques [4], [7]. Fur- C2(ǫ,δ), 2
ther,theǫ-secrecycapacityforeachofthesefourrequirements (C1 otherwise.
isunique.UnderS andS thestrongconversepropertyholds, Theorem3. Theǫ-secrecycapacityunderS (nr )oftheDM-
1 3 3 l
while it does not under S and S in general. In addition, WTC is given by
2 4
under S and S , the capacity can be broken into distinct
2 4 C (r ),C +min(r ,C C )
phases depending on the error tolerance. For instance, under 3 l 2 l 1− 2
S2 the capacity of the channel is either equal to the capacity for all ǫ.
of the channel with vanishing error, or the capacity of the
Theorem4. Theǫ-secrecycapacityunderS (nr )oftheDM-
channel with no secrecy requirement. We call this interesting 4 l
WTC is given by
phenomenon a phase change.
r
C (ǫ,r ),C +min l ,C C .
II. MAIN RESULTS 4 l 2 1 ǫ 1− 2
(cid:18) − (cid:19)
For i 1,2,3,4 , we call (fn,ϕn) a (n,R ,ǫ ,S (η ))- Theorems 1 and 4 state that the ǫ-secrecy capacities of the
n n i n
∈{ }
code if the domain of fn (i.e., ) is of cardinality 2nRn, DM-WTC under S (δ) and S (nr ) are invariantto the value
and the pair satisfy both (1) anMd Si(ηn). Further we say of ǫ [0,1) for al1l valid valu3es olf δ and rl, respectively. In
athcehiervaatebleerirforthesreecreexciysts(RaEsSe)q-tureipnlcee (oaf,b(n,c,)Rn∈,ǫnR,3Sii(sηnS))i-- aonthderS∈3w(onrrdls),.tAhelthsotruognhgicnovnavriearnsteopfrothpeeretryrohrotlodlseruanndceer,Sth1e(δǫ)-
caonddelsimsuch thaRt lim,ǫn→,η∞n(R=n,(ǫan,,bη,nc))=if(ia,b,c3),4if .iT∈h{e1n,2th}e, tsheecrelecaykacgapearcaittey run.dIenr sSp3e(cnifirlc), itshenoǫn-s-etrcirveicayllycadpeapceintydeunntdoenr
ǫR-sseuccrehcnyth→caa∞tptah(cid:0)ceitRnyEuSnn-dternirpt(cid:1)lhee(aRp,pǫr,oηp)riaisteSS-ia∈(·c)h{iisevtha}belme.aximum iSt3s(antrulr)atienscraetasCes,llitnheear(lnyoans-saecfruent)ctcioapnaociftyrloffrotmheCd2iscurnettiel
i 1
Note that for S and S , the above definition corresponds memoryless channel (DMC) ( ,P , ).
3 4 Y|X
to what is called “weak” secrecy in the literature [2]. If When the secrecy requiremeXntsare wYeakened to S (δ) and
2
“strong” secrecy is desired, the definition could be modi- S (nr ) from S (δ) and S (nr ) respectively, Theorems 2
4 l 1 3 l
fied to that the RES-triple (R,ǫ,η) is S -achievable when and 4 show that the strong conserve property no longer holds
i
there exists a sequence of (n,R ,ǫ ,S (η ))-codes such fortheDM-WTCastheǫ-secrecycapacitiesgenerallydepend
n n i n
onthevalueofǫ.UnderS (δ),theǫ-secrecycapacityremains and
2
at C as long as ǫ [0,1 δ). However, for ǫ [1 δ,1),
2 ∈ − ∈ − Ω (r), (m,zn) n :
the ǫ-secrecy capacity value experiences an abrupt phase n ∈M×Z
change,increasingtoC1 asifthereisnosecrecyrequirement. n PM,Zn(m,zn) 2n(r+λn)PM(m)PZn(zn) ,
Restricting to within either of the two value ranges, the ǫ- ≤
secrecy capacity under S2(δ) is invariant to ǫ. then o
r U=n0defrorSa4l(lnǫrl),[0th,e1)ǫ.-sNeocrteectyhactatphaicsitaylsroeminacilnusdeast Cth2ewcahseens PM,Zn(Ωn(r))≤PQn QSn(r) +µn.
l
of strong secrec∈y (S (l ) with l 0) and bounded leakage For proving achievability in Th(cid:0)eorems(cid:1)2 and 4, we will
4 n n
(S (l ) with l =l). Thus the stro→ng conversepropertyholds make use of the following lemma to simplify discussions:
4 n n
when rl = 0 as proven in [4] and [5]. For any fixed rl Lemma 6. For i 2,4 , if the RES-triple (R,0,η) is S -
∈ i
(0,C1 C2), the ǫ-secrecy capacity increases from C2 +rl achievable, then th∈e{RES-}triple (R,γ,(1 γ)η) is also S -
− i
to C and then levels off as ǫ increases in the range [0,1). −
1 achievable for any γ [0,1).
TheDM-WTCexhibitsa phasechangefromwherethestrong ∈
conversepropertyholdstowhereitdoesnot.Whenr C A. Proof of Theorem 1
l 1
≥ −
C , the ǫ-secrecy capacity value remains at C for all ǫ
2 1 (Direct) For any δ [0,1) and ǫ [0,1), the RES-
∈
[0,1), and the DM-WTC exhibits another phase change after triple (C ,ǫ,δ) being S∈-achievable follow∈s directly from [8,
2 1
which the strong converse property holds again.
Theorem 17.11], which in particular shows the RES-triple
(C ,0,0) is S -achievable.On the other hand, the RES-triple
III. PROOFS OF THEOREMS 2 1
(C ,ǫ,1)isS -achievablesinceC isthechannelcapacityfor
1 1 1
We prove the converses in Theorems 1–4 by employing
the DMC ( ,P , ), and δ =1 correspondsto no secrecy
Y|X
the following strong Fano’s inequality developed in [4] and X Y
constraint.
information stabilization result developed in [7]: (Converse) To prove that C (δ) is an upper bound on the
1
Strong Fano’s inequality. For any (fn,ϕn) of rate R that ǫ-secrecycapacityunderS1(δ), firstapplyLemma5toobtain
gives Pr ϕn(Yn) = M ǫ over the DM-WTC, there exist values τn, µn, and λn which converge to 0 as n increases,
scauabrradsenintdaolmi{tyiinsdaext mQo6nstthpao}tlyr≤naonmgieasloinvenr,aζnni→nde0x, asentdQanwinhdoesex ΩPsuMnc(h,0Z)ntha(anΩtdnP(Q0M)Sn,)(Z0n)(aΩ1snd(e0ρfi)n)nefd≤orinsPoLQmenme(cid:0)mQρnaSn(50.)W(cid:1)0e,+dalµuseno,thofaovSre1(tshδeat)st.
Thus S (δ) and≥Lemm−a 5 together imply→that
1 1
R , q :R I(M;Yn Q =q )+ζ
Qn (cid:26) n ∈Qn ≤ n | n n n(cid:27) PQn QSn(0) ≥PM,Zn(Ωn(0))−µn ≥1−ρn−µn. (2)
satisfying P ( R) 1 ǫ ζ . Butthe(cid:0)nthestr(cid:1)ongFano’sinequalityand(2)togethergivethe
Qn Qn ≥ − − n
existence of a q such that
Information stabilization. For the (fn,ϕn) pair, random n ∈Qn
iinnddeexx Qsunb,seatndiZndex setsQatnisfaybinogveP,ther(e eZx)ist ξ1n →ξ0:a1ndan R≤ n1I(M;Yn|Qn =qn)+ζn (3)
1) PZZnn:|QPnZQ(nZnˆ|Qn⊆n(q(znQn)n||qqnn))≥=.ξ1n−2−ξHnQ,(ZnwnhQ|eQrnne=Zq≥ˆnn)(q,−n),n (cid:8)zn ∈ n1I(M;Zn|Qn =qn)≤τn (4)
2) tPhere (ex˜is(tqs )qa) M˜1(qn)ξ , an⊆d P M(cid:9)(msqati)sfy=.ing sfoinrcelarPgeQnenQouRngh∩nQSna(n0d)ǫ ≥ [10,−1)ǫ. −Coζmnb−ininρgn −(3)µannd>(40)
M|Qn M n | n ≥ − n M|Qn | n ξn (cid:0) (cid:1) ∈
3) P2−H(M|Qn(=˜qnn)(mfo,rqea)cmh,mq ∈) M˜(qn),1and ξ where walilthǫ[8, L[0e,m1)m.aO1n7.t1h2e],oitthfeorllhoawnsd,thwathRen≤δC=2+1,ζnth+e ηstnrofnogr
Zn|M,Qn Z n | n ≥ − n . ∈
2Z˜−nH(m(Z,nq|Mn),Q,n=qnz)n,∈ Zn : PZn|M,Qn(zn|m,qn) =ξn Fasanion’sthienesqtuaanlditayrd(is.etr.,on(g3))cognivveersseRar≤gumCe1n+t foζnr tfhoer DalMl ǫC,
(cid:8) ( ,P , ).
for each qn ∈QZn. (cid:9) X Y|X Y
B. Proof of Theorem 2
Obtained through the information stabilization result in the
appendix, the following lemma will also be needed: (Direct) The RES-triple (C ,ǫ,δ) is S -achievable, once
2 2
again, by [8, Theorem 17.11], for ǫ+δ < 1. For ǫ+δ 1,
Lemma 5. For any r ≥ 0, there exist τn → 0, µn → 0, and the RES-triple (C ,ǫ,1 ǫ) is S -achievable by Lemm≥a 6,
λn →0 satisfying nλn →∞ such that by defining since the RES-trip1le (C −,0,1) is S2 -achievable.
1 2
S(r), q : 1I(M;Zn Q =q ) r+τ (Converse) On the other hand, to prove that C2(ǫ,δ) is an
Qn n ∈Qn n | n n ≤ n upper bound on the ǫ-secrecy capacity under S (δ), observe
(cid:26) (cid:27) 2
that S (δ) implies
. 2
(cid:12)(cid:12)n11lFoogr2aanny−nonn1-nleogga2tibvne(cid:12)(cid:12)λ≤nλ→n.0,an>0,andbn>0,an=λn bn means δ ≥kPM,Zn −PMPZnk1
≥ PM,Zn(m,zn)−PM(m)PZn(zn) together with (8) implies that there must be a qn ∈QRn such
≥(m,zn)∈MX×Zn\Ωn(0)PM,Zn(m,zn) 1−2−nλn that n1I(M;Zn|Qn =qn)≤ rl1+ αnǫlogζ2nn. (9)
(m,zn)∈MX×Zn\Ωn(0) (cid:0) (cid:1) Again by the strong Fano’s inequality−, for−this q we also
= 1−2−nλn [1−PM,Zn(Ωn(0))]. (5) have(3).Hencecombining(3)and(8)with[8,Lemmna17.12],
Thus (cid:2)combining L(cid:3)emma 5 and (5) gives it follows that
r + αlog n
P S(0) 1 δ 2−nλn µ . R C + l n 2 +ζ .
Qn Qn ≥ − − − n ≤ 2 1 ǫ ζ n
n
− −
As a result, if ǫ(cid:0)+δ <(cid:1)1, then there must exist a q
n n The strong Fano’s inequality alone also gives the bound R
∈ Q
such that (3) and (4) are simultaneously satisfied since ≤
C +ζ . Thus we have
1 n
PQn(QRn ∩QSn(0))≥1−ǫ−δ−ζn−2−nλn −µn >0 R min C + rl+ αnlog2n +ζ , C +ζ .
≤ 2 1 ǫ ζ n 1 n
forallsufficientlylargen. Henceagainby[8, Lemma17.12], (cid:26) − − n (cid:27)
we have R C2 +ζn +τn. If ǫ+δ 1, then the strong IV. CONCLUSIONS
≤ ≥
Fano’s inequality (i.e., (3)) gives R C +ζ .
1 n Employing the recently developed techniques of equal-
≤
image-size partitioning, we obtained the ǫ-secrecy capacities
C. Proof of Theorem 3
under S (δ), S (δ), S (nr ), and S (nr ) of the DM-WTC
(Direct)The RES-triple ((C +min(r ,C C ),ǫ,r) is 1 2 3 l 4 l
S achievabledueto[8,Theore2m17.13],wl hic1h−sho2wsthatlthe fornon-vanishingǫ, δ, and rl. The secrecycriteria considered
3 include the standard leakage and variation distance secrecy
RES-triple ((C +min(r ,C C ),0,r) is S achievable.
(Converse) O2n the othler h1a−nd, 2to provle that3C (r ) is an constraints often employed in the literature. Our new results
3 l show that both the capacity value and the strong converse
upper bound on the ǫ-secrecy capacity under S (nr ) of the
3 l property of the DM-WTC are in fact dependent on the
DM-WTC, we notethat Lemma5 and S (nr ) directlyimply
3 l secrecy criterion adopted. We conjecture that the interesting
PQn QSn(rl) ≥PM,Zn(Ωn(rl))−µn ≥1−ρn−µn (6) phasechangephenomenonobservedincaseswherethestrong
conserve property does not hold is commonplace in many
for som(cid:0)e ρ (cid:1)0. Thusas beforethe strongFano’sinequality
n other multi-terminal DMCs.
→
and(6)togethergivetheexistenceofaq satisfying(3)
n n
∈Q
and APPENDIX
1
I(M;Zn Q =q ) r +τ (7) A. Proof of Lemma 5
n n k n
n | ≤
We need the following lemma to prove Lemma 5:
since P R S(r ) 1 ǫ ζ ρ µ > 0.
Qn Qn ∩Qn l ≥ − − n − n − n
Hencebyapplying[8,Lemma17.12]tothecombinationof(3) Lemma7. LetQ bearandomindexrangingover ,whose
n n
(cid:0) (cid:1) Q
and (7), we have R < C +r +ζ +τ . The strong Fano’s cardinality is at most polynomialin n, and V be any discrete
2 l n n
inequalityalso gives the boundR≤C1+ζn. Thuswe obtain random variable distributed over V. Then there exist λn →0
R≤min{C2+rl+ζn+τn,C1+ζn} for all ǫ. and ξn′ →0 such that nλn →∞ and
.
D. Proof of Theorem 4 P (v,q ) :P (v q )= P (v)
V,Qn n ∈V ×Qn V|Qn | n λn V
(Direct) First note the RES-triple 1 ξ′.
≥ −(cid:0)(cid:8)n (cid:9)(cid:1)
(C2+min 1r−lǫ,C1−C2 ,0,1r−lǫ is S4 achievable Note that λn and ξn′ both depend only on the polynomial
(cid:16)due to [8,(cid:16) Theorem 1(cid:17)7.13]. H(cid:17)ence the RES-triple cardinality bound on .
n
(C +min rl ,C C ,ǫ,r is S -achievable by Q
2 1−ǫ 1− 2 l 4 Proof: Let α > 0 be such that nα. First
(cid:16)Lemma 6. (cid:16) (cid:17) (cid:17) write = (v,q ) :P (v|Qqn)|>≤n2αP (v)
(Converse) To prove C4(ǫ,rl) upper-bounds the ǫ-secrecy and A= (v,q )n ∈V ×Q:nP V|Q(nv q|)n<n−2αPV(v) .
capacity under S4(nrl) of the DM-WTC, notice that S4(nrl) ThenB (cid:8) n ∈V ×Qn V|Qn | n V (cid:9)
implies (cid:8) (cid:9)
.
r 1I(M;Zn) 1I(M;Zn Q ) αlog n PV,Qn (v,qn)∈V ×Qn :PV|Qn(v|qn)=λn PV(v)
l ≥ n ≥ n | n − n 2 ≥1−(cid:0)(cid:8)PV,Qn(A∪B) (cid:9)(cid:1)(10)
1 α
≥ nI(M;Zn|Qn =qn)PQn(qn)− n log2n where λn = 2nαlog2n. Thus the lemma is verified by (10) if
qnX∈QRn we canshowthatPV,Qn(A∪B)→0.In particular,we doso
1 α by bounding P ( ) n−α and P ( ) n−2α, and
min I(M;Zn Q =q )P ( R) log n (8) V,Qn A ≤ V,Qn B ≤
≥qn∈QRn n | n n Qn Qn − n 2 setting ξn′ =n−α+n−2α.
To bound P ( ), note that for all (v,q ) ,
wherenα isthecardinalityboundon n.Butfromthestrong V,Qn A n ∈A
Fano’s inequality, we have P ( RQ) 1 ǫ ζ . This P (q ) n−2α (11)
Qn Qn ≥ − − n Qn n ≤
since since (m,zn,q ) Ξ . Thus Lemma 5 results from (16) by
n n
∈
settingτ =4λ +2ξ andµ =3ξ′ +4ξ ,becausewehave
P (v) P (v q )P (q ) n2αP (v)P (q ). n n n n n n
V ≥ V|Qn | n Qn n ≥ V Qn n from (14)
Then the upper bound on P ( ) follows from (11) as
below: V,Qn A PM,Zn(Ωn(r))
P ( )= P (v q )P (q ) ≤PM,Zn,Qn(Ξn∩Γn∩Ωn(r)×Qn)+3ξn′ +4ξn
V,Qn A (v,Xqn)∈A V|Qn | n Qn n ≤PQn QSn(r) +3ξn′ +4ξn.
P (v q )n−2α n−α. B. Proof of L(cid:0)emma(cid:1)6
≤ V|Qn | n ≤
(v,Xqn)∈A (n,FRor ,(1i γ)ǫ∈+γ,S{2(,(41}, γ)wl e))-cocdaen (fˆnco,nϕsntr)u,ctgivena
The upper bound on PV,Qn(B) follows similarly in that that thnere e−xistsna (n,Rni,ǫn,−Si(lnn))-code (fn,ϕn). Whence
P ( )= P (v q )P (q ) the lemma follows by the definition of the RES-triples.
V,Qn B V|Qn | n Qn n Letting Mˆ be a random variable distributed identical, but
(v,Xqn)∈B independent, to M. The new encoder, fˆn, is constructed by
P (v)P (q )n−2α n−2α.
≤ V Qn n ≤ setting it equal to f(M) with probability 1 γ and to f(Mˆ)
(v,Xqn)∈B with probability γ. While the new decoder ϕ−ˆn =ϕn.
Clearly, an error will likely occur if fˆ(M) is set equal to
Apply Lemma 7 three times with V = M, V = Zn, and f(Mˆ). On the other hand, the probability of error will revert
V =(M,Zn), respectively. Writing to that of (fn,ϕn) if fˆ(M) is set equal to f(M). Thus the
probability of error for (fˆn,ϕˆn) is at most (1 γ)ǫ +γ.
Γn , (m,zn,qn)∈M×Zn×. Qn : Letting PZn,M be the joint distribution o−f Znn,M for
n PPMM,|ZQnn|(Qmn|(qmn),z=.nλ|qnnP)M=λ(mn P),Ma,nZdn(m,zn), fPiˆnnduacasen(dd1bP−yfγn).P,BwZunet,Mctahne+nw,γrfPiotZerntthhPeeMjvo,aiwnritahdtiilioestnrtihdbeuistmtiaoannrcgoeif,nZalns,rMemafoinr
. M Zn
where λ is obPtaZinne|Qdni(nznL|eqmn)m=aλ7n, PwZenh(azvne)o k(1−γ)PZn,M +γPZnPM −PZnPMk1
n =(1 γ) PZn,M PZnPM 1.
− k − k
PM,Zn,Qn(Γn)≥1−3ξn′. (12) And for divergence
Next define
D((1 γ)PZn,M +γPZnPM PZnPM)
Ξn , (m,zn,qn)∈M×Zn×Qn :qn ∈QZn, ≤(−1−γ)D(PZn,M||PZnP|M| )+γD(PZnPM||PZnPM)
n m ˜(q ), and zn ˆn(q ) ˜n(m,q ) =(1 γ)D(PZn,M PZnPM).
n n n − ||
∈M ∈Z ∩Z
o
with the corresponding Z, ˜(q ), ˆn(q ), and ˜n(m,q )
Qn M n Z n Z n
as given in the information stabilization result summarized in REFERENCES
Section III. Similar to before,
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Combining (12) and (13) gives
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∈ ∩
memoryless channels,” ArXiv e-prints, Dec. 2015. Available at
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≤
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n n n n
≥ n | −