Joint secrecy over the K-Transmitter Multiple Access Channel

0.49 MB·

by  Yanling Chen

Joint secrecy over the K-Transmitter Multiple
Access Channel
Yanling Chen∗, O. Ozan Koyluoglu† and A. J. Han Vinck∗
∗ Institute of Digital Signal Processing, University of Duisburg-Essen, Germany.
† Department of Electrical and Computer Engineering, The University of Arizona.
Email: {yanling.chen, han.vinck}@uni-due.de, [email protected].
coAmbmsturnacicta—tiTonhisovpearpaerKst-utrdainessmthiteteprrombulelmtipolef saecccuerses M1 Transmitter1 X1n
7 channel in the presence of an external eavesdropper, ... ... p(y,z|x1,x2,···,xK) Yn Receiver (Mˆ1,···,MˆK)
01 sleuabkjaegcet troataejforionmt stehcereccoyllceoctnisotnraoinftK(i.me.e,sisnafgoersmtaotioann MK TransmitterK XKn
2 etaabvelissdhrtohpepejorinist smecardeecyvaanchisiheivnagb)l.eArastearreegsiuolnt.,Twoethesis- Zn Eavesdropper ((M(1,(··(·,(M(K()
n
end, our results build upon two techniques in addition
a Fig. 1: K-transmitter DM-MAC with an eavesdropper.
to the standard information-theoretic methods. The
J
first is a generalization of Chia-El Gamal’s lemma on

2 entropy bound for a set of codewords given partial
2 information. The second is to utilize a compact repre- The rest of the paper is organized as follows. Section II
sentationofalistofsetsthat,togetherwithproperties introduces the system model and presents necessary def-
] of mutual information, leads to an efficient Fourier-
T initions; Section III gives the general conditional entropy
Motzkin elimination. These two approaches could also
I be of independent interests in other contexts. bound. The bound plays an important role in establishing
.
s the joint secrecy rate region that is given in Section III.
c
I. Introduction To enhance the flow of the paper, some details for the
[
Fourier-MotzkineliminationarerelegatedtoAppendixA.

1 Fortheproblemofreliablycommunicatingindependent
II. Preliminaries
v
messagesoveramultipleaccesschannel(MAC),Ahlswede
7
[1] first studied the 2-transmitter and 3-transmitter cases Consider a discrete memoryless MAC (DM-MAC) with
8
1 and determined the respective capacity regions; whilst K transmitters, one legitimate receiver, and one passive
6 Liao [2] considered the general K-transmitter MAC and eavesdropper, which is defined by p(y,z x1,x2, ,xK).
| ···
0 fully characterized its capacity region. The transmitter i, aims to send message mi, to the legiti-
1. Inspired by the pioneering works of Wyner [3] and mate receiver, where i = 1,2, ,K . Suppose that
∈K { ··· }
0 Csiszár and Körner [4] that studied the information the- xni is the channel input at transmitter i, and the channel
7 oretic secrecy of a point-to-point communication in the outputsatthelegitimatereceiverandeavesdropperareyn
1 presence of an external eavesdropper, a MAC with an andzn,respectively.Bythediscretememoryless natureof
:
v external eavesdropper was first introduced in [5]. In par- the channel (without any feedback), we have
i
X ticular, [5] focused on a degraded Gaussian MAC with n
Y
K-transmitters and established achievable rate regions p(yn,zn xn, ,xn)= p(y ,z x , ,x ). (1)
r | 1 ··· K i i| 1,i ··· K,i
a subjecttoapre-specifiedsecrecymeasure;whileadiscrete i=1
memoryless 2-transmitter MAC with an external eaves- A(2nR1,2nR2, ,2nRK,n)secrecycode nfortheDM-
dropperwasconsideredin[6].FurtherworksonMACwith ··· C
MAC consists of
an external eavesdropper include but not limited to [7]–
[10]. However, its secrecy capacity region, even for the 2- • K messagesetsM1,M2,··· ,MK,wheremi ∈Mi =
[1:2nRi] for i ;
transmitter case, still remains an open problem. ∈K
In this paper, we consider the secure communication • K encoders each assigning a codeword xni to message
m for i ; and
over a K-transmitter MAC subject to the joint secrecy i ∈K
• One decoder at the legitimate receiver that
constraint (i.e., information leakage rate from the collec-
declares an estimate of (m ,m , ,m ) say
tionofK messagestoaneavesdropperismadevanishing), 1 2 ··· K
(mˆ ,mˆ , ,mˆ ) or an error to the received
thechannelmodelofwhichisshowninFig.1.Asageneral 1 2 ··· K
sequence yn.
result, we establish a joint secrecy achievable rate region.
In this paper, for a list of random variables W for
j
This work is supported in part by DFG Grant CH 601/2-1 and j , and a fixed , we denote WJ = Wi i .
∈K J ⊆K { | ∈J}
NSFawardCNS-1617335. Assume that the messages MK are uniformly distributed
over their corresponding message sets. Therefore, we have Lemma 6. Let R 0 for i , (cid:15) > 0, and
v,i
R = 1H(M ), for i . Denote the average prob- (Q,V , ,V ,Z) p(q≥) Q p(v q)∈p(zKv , ,v ).Let
i n i ∈ K 1 ··· K ∼ · i| · | 1 ··· K
ability of decoding error at the legitimate receiver as i∈K
(cid:26) (cid:27) Qn be a random sequence and each qn = (q(1), ,q(n))
Pen(Cn)=Pr S{Mi 6=Mˆi}|Cn .Definethejoint infor- distributed according to Qn p(q(t)). For i ··· , let
i∈K ∈ K
mn1Ia(tMionK;lZeank|aCgne).rTatheetroattehpeaeirav(eRs1d,rRop2p,·e·r·b,yRKR)L,iKs(sCanid) t=o Vquine(nlci)e,s litha∈t a[r1e c:on2dnitRivot,=ni]1a,llybe inadespeetndoefntrangidvoemn Qsen-
be achievable under the joint secrecy constraint, if there
and each vn = (v (1), ,v (n)) distributed accord-
esuxcishtsthaastequence of (2nR1,2nR2,··· ,2nRK,n) codes {Cn} ing to Qn pi(vi(t)q(t)i), a·n·d· leit be the codebook of
| C
(cid:0) t=1 (cid:1)
Pen(Cn)≤(cid:15)n, (2) oQf nV,nV,1n(fo1r),·i·· ,VKn(,2nwRitvh,K)an. LaerbtiLtriabreytphreorbaanbdiloitmy imndaesxs
RL,K(Cn)≤τn, (3) functiion. Then∈, ifKPr (Qn,Vn(L ), ,Vn(L ),Zn)
lim (cid:15)n =0 and lim τn =0. (4) n(Q,V , ,V ,Z) { 1 as1n 1 ··a·ndK K ∈
n→∞ n→∞ T(cid:15) 1 ··· K }→ →∞
X
deRfineictaiollntahnadt Klem=m{a1s.,2,··· ,K}. We have the following Rv,j ≥I(VJ;Z|Q), ∀J ⊆K (5)
j∈J
Definition 1. The indicator vector of a subset of set there exists a δ ((cid:15)) 0 as (cid:15) 0 and n , such
eKq,uadlentoot1edifbyi 1T, iasnad10×otKhervweicsteo,r,fowrit1h itsi i-tKhT.element th"at for n sufficniently→lar#ge, H(→L1,··· ,LK|Z→n,Q∞n,C) ≤
∈T ≤ ≤ n P Rv,j I(VK;Z Q) +nδn((cid:15)).
For instance, for K = 5, = 1,2,3,4,5 and = j∈K − |
K { } T
{1,3,5}, we have 1T =[1 0 1 0 1], and 1∅ =[0 0 0 0 0]. Proof: Given zn, let us define as the set of indices
Let 1 i t be a list of t subsets of . L
{Ti| ≤ ≤ } K (l1,l2, ,lK) such that
···
Definition 2. The presence vector of 1 i t is
Pt {Ti| ≤ ≤ } (qn,v1n(l1),··· ,vKn(lK),zn)∈T(cid:15)n(Q,V1,··· ,VK,Z).
defined to be t# = 1Ti, which counts the number of First we show that the expected size of this list, over all
i=1
presences of each element of over i 1 i t . randomly generated codebooks, is upper bounded by
K {T | ≤ ≤ }
Definition 3. A compact form of the element rearrange- 2K−1
X
ment for 1 i t is defined to be ∗ 1 i t , E( ) 1+ 2n[Ii+δ((cid:15))], (6)
{Ti| ≤ ≤ } {Tt,i| ≤ ≤ } |L| ≤
whereTt∗,icontainstheelementsthatpresentatleastitimes i=1
from all these t subsets, i.e., i ! where Ii =jP∈JiRv,j −I(VJi;Z|VJic,Q). Here, {Ji|i∈[1:
Tt∗,i = [ \ Tjk . 2Kc =− 1]} arfeorthie 2[1K:−2K1 n1o]n.-Nemotpetythsautbsets of K, and
{j1,···,ji}⊆[1:t] k=1 Ji K\Ji ∈ −
tiCmleaaxt,rlwy,heTrtie∗,sttt⊆mheaxTcto∗i,tsm−t1phae⊆clta·fr·og·rem⊆st weTltie∗,t1mh.oeAuntnttdoh,feTte#t∗,mi.p=Stoy∅{sTefott∗s,ri.|1i ≤> E(|L|)=P+r{(L1,···X,LK)∈L}Pr{(l1,··· ,lK)∈L},
≤ max} (l1,···,lK)6=(L1,···,LK)
Lemma 4. t# =iP=t11Ti =iP=t11Tt∗,i =tiPm=a1x1Tt∗,i. wthheeresou(rLc1e,s.···S,inLcKe)Parr{e(qnt,hve1n(tLr1u)e,··in·d,vicKnes(LKch)o,szenn) b∈y
witFhorins=tanc1e,,forK==13,,2K,={1=,2,32},3an.dW{Tei|h1a≤veit≤3=} Tto(cid:15)n1(Qa,sVn1,··· ,V.KA,sZ)fo}r→th1ea2sndn→term∞,,wtheec1asnt dteisrtmintgeunidshs
(2,2,1T)1, tma{x =} T2,2T3∗,1{={}1,2T,33},T{3∗,2 =}{1,2},T3∗,3 =#∅. (M2Kore−s1p)ecc→iafiscea∞sllya,ccfoorrdeiancghto ,thie v[a1lu:e2sKof (1l]1,,wl2e,·c·o·n,sliKde)r.
i
Lemma 5. Given two lists of sets 1 i t and J ∈ −
{T1i| ≤ ≤ 1} the following case:
1 i t ,elementsofwhicharet andt subsetsof
{seTt2i|,r≤espe≤cti2v}ely,iftheysharethesame1presen2cesequence • lj 6=Lj for j ∈Ji, and lj =Lj for j ∈Jic. P
K n Rv,j
t , then they also share the same compact form of the
# In this case, in total there are at most 2 j∈Ji
element rearrangement without empty sets.
possible (l , ,l ). By the joint typicality lemma,
1 K
···
III. Conditional entropy bound we can upper bound Pr (l , ,l ) by
1 K
{ ··· ∈L}
In this section, we give the conditional entropy bound, 2−nI(VJi;VJic,Z|Q)+nδ((cid:15)) (=a)2−nI(VJi;Z|VJic,Q)+nδ((cid:15)),
which is a generalization of Chia-El Gamal’s lemma [11,
Lemma 1] on entropy bound for a set of conditionally where (a) is due to the fact that VJi and VJic are
independent codewords given partial information. conditionally independent given Q.
Therefore, in this case, there are at most 2n[Ii+δ((cid:15))] Theorem 7. An achievable joint secrecy rate region of
number of (l ,l , ,l ) falling in the list . the K-transmitter DM-MAC with an external eavesdrop-
1 2 K
··· L
Summing up all the numbers of (l ,l , ,l ) falling in per is given by the union of non-negative rate pairs
1 2 K
the list over all these (2K 1) cases,·w·e· prove (6). (R1,R2, ,RK) that are defined by the followings:
L − ···
Furthermore, define the indicator variable E = 1 if X
(L1, ,LK) , and E =0 otherwise. We have Rj ≤I(VJ;Y|VJc,Q)−I(VJ;Z|Q), ∀J ⊆K
··· ∈L j∈J
H(L , ,L Zn,Qn, ) H(L , ,L ,E Zn,Qn, )
1 K 1 K where the union is over input probability distributions that
··· | C ≤ ··· | C
≤H(E)+H(L1,··· ,LK|Zn,Qn,E,C) factor as p(q) Q p(vi|q)p(xi|vi).
(b) i∈K
1+H(L1, ,LK Zn,Qn,E =1, ) Proof: Fix p(q) and p(v q),p(x v ) for i . Gener-
≤ ··· | C i i i
+Pr{E =0}H(L1,··· ,LK|C), ate a random sequence qn, w|here p(q|n)= Qn ∈p(Kq(t)) with
where(b)followsfromthefactthatH(E)≤1sinceE isa each entry chosen as i.i.d. p(q). The sequetn=c1e qn is given
binary random variable; Pr E =1 1 and conditioning
{ }≤ to every node in the system.
does not increase the entropy. Codebook generation: For i , to construct code-
Since Pr{(qn,v1n(L1),··· ,vKn(LK),zn)∈T(cid:15)n(Q,V1,··· , book i, randomly generate 2n∈[RiK+Ri,r] i.i.d. sequences
VPKr{,(ZL)1},L2→,···1,LaKs) n∈/ L→} ca∞n,bethmenadePar{rbEitra=rily0s}ma=ll veainc(hmiw,Cimthi,rp)r,owbaitbhili(tmyip,(mvni,rq)n)∈=[1 Q:n2pn(Rvi](t×)q[1(t):).2nERvie,rr]y,
as n . Next, i| i |
→∞ t=1
H(L , ,L Zn,Qn,E =1, ) node in the network knows these codebooks. Denote the
1 ··· K| C overall codebook as .
(=c)HH((LL1,,··· ,,LLK|ZEn=,Q1n,,E, =)1,C,L,|L|) i rEanndcoodminlyg:aFnodruin∈ifCoKrm,ltyocsheonodsemsemssia,rge m[1i,:t2rnaRnis,mr]itatnedr
≤ X1 ··· K| L |L| findsvin(mi,mi,r).Then,giventhecodew∈ordvin(mi,mi,r),
= Pr =l H(L , ,L E =1, , =l) n
{|L| } 1 ··· K| L |L| it generates xn according to Pp(x (t)v (t)) and trans-
l∈supp(|L|) i i | i
t=1
(d) X mits this sequence to the channel.
≤ Pr{|L|=l}log2(l) Decoding: The legitimate receiver, upon receiving yn,
l∈supp(|L|) finds vn(mˆ ,mˆ ), vn(mˆ ,mˆ ), ... , vn(mˆ ,mˆ ) such
1 1 1,r 2 2 2,r K K K,r
(e) that (vn(mˆ ,mˆ ),vn(mˆ ,mˆ ), ,vn(mˆ ,mˆ ),yn)
=E(log ( )) log (E( )) 1 1 1,r 2 2 2,r ··· K K K,r
2 |L| ≤ 2 |L| is jointly typical.
(f) Analysis of the error probability of decoding: Consider
nmax 0, max I +K+nδ((cid:15))
i
≤ { i∈[1:2K−1] } the expected value of the error probability of decoding
  over the ensemble of random codes , i.e., P =E[P ( )].
e e
(g) X C C
≤n Rv,j −I(VK;Z|Q)+K+nδ((cid:15)), NreosetenttshtahtehrearnedComdleynogteensertahteedracnoddoembovoakritahbalteatdhhaetrreepto-
j∈K
the above scheme. From the decoding analysis for the
where(c)followsfromthefactthat and arefunctions
of the output Zn, given and Qn;L(d) is|Ld|ue to the fact multiple access channel, see, e.g., [13], Pe can be made
C approximately zero as n if
that, knowing E = 1, the sent indices (L ,L , ,L ) →∞
1 2 ··· K X
belong to the list L and the uncertainty is upper bounded [Rj +Rj,r]≤I(VJ;Y|VJc,Q), ∀J ⊆K. (7)
by the log cardinality of the list; (e) is by Jensen’s in- j∈J
equality;(f)isby(6)alon(cid:18)gwitha(cid:19)napplicationofthelog- Analysisofjointsecrecy:Forthejointsecrecyasdefined
sum-exp inequality: log2 xP∈X2x ≤ mx∈aXxx+Plog2(|X|); itnhi(s3e)n,dw,ewsehsohwowintthhaetfHol(loMwKinZgnth,QatnE, [)RL,Kn(CP)]R≤iτnn.τTno,
and (g) follows if the rates satisfies (5), i.e.: R | C ≥ −
v,j i∈K
j∈J ≥ as this implies I(MK;Zn ) I(MK;Zn,Qn ) nτn.
I(VJ;Z), . This, along with previous remarks |C ≤ |C ≤
yields the d∀eJsir⊆ed Kinequality (by defining δn((cid:15)) to be the H(MK|Zn,Qn,C)=H(MK,Zn|Qn,C)−H(Zn|Qn,C)
arbitrary small term O((cid:15))+(K+1)/n). =H(MK,MK,r|Qn,C)+H(Zn|MK,MK,r,Qn,C)
IV. Achievable rate region H(Zn Qn, ) H(MK,r MK,Zn,Qn, )
− | C − | C
regIinonthoisf stehcetioKn-,twraensgmiviettaenr aDcMhi-eMvaAbCle jwoiitnht saencreexcyterrnatael (=a)H(MK,MK,r|Qn,C)+H(Zn|V1n,··· ,VKn,MK,MK,r,Qn,C)
eavesdropper. This result recovers the joint secrecy result −H(Zn|Qn,C)−H(MK,r|MK,Zn,Qn,C)
forK =2in[12,Theorem2],whichimproves[6,(8)]with (=b)nX[R +R ]+H(Zn Vn,Vn, ,Vn,Qn, )
channel prefixing as demonstrated in [12]. i i,r | 1 2 ··· K C
i∈K
−H(Zn|Qn,C)−H(MK,r|MK,Zn,Qn,C) – (cid:2)b+J(cid:3) is a (2K −1)×1 matrix with the i-th row
(≥c)nXi∈K"[Ri+Ri,r]−I(V1n,··· ,VKn;#Zn|Qn,C) – t(cid:2)toob−Jbb(cid:3)eeibbs+J−Jaii ==(2KII((VV−JJ1ii;;)YZ×||QV1J)mi;c,aQtr)i,x with the i-th row
−n XRi,r−I(VK;Z|Q)+εn (≥d)nXRi−nτn, Pr•op[e0rJti]esisoafb(+2Kan−d1b)−×arKegmivaetnriixnwLeitmhmzear8oienleAmpepnetns.dix
i∈K i∈K J J
A,whichplayacrucialroleinremovingtheredundantin-
where (a) follows from the fact that Vn, ,Vn are
1 ··· K quations so as to avoid the double exponential complexity
functions of (M ,M ), , (M ,M ), respectively,
1 1,r ··· K K,r of the Fourier-Motzkin elimination.
given Qn and ; (b) follows from the fact that
H(MK,MK,r|Qn,CC) = n P[Ri + Ri,r], and given Qn paNrtoictuelathr,atweA0h,aAv0e0 are both (2K+1−2)×K matrices. In
i∈K
aMnadrkCo,v(cMhaKin,M;(Kc),rf)ol→low(sVf1rno,mV2nL,e·m··m,aVK6n)(w→ithZRnv,ifo=rmRsi,ar A00 =(cid:20) [[11JJ]] (cid:21)=B⊗[1J], where B=(cid:20) 11 (cid:21). (10)
for i ) by requiring (5): − −
∈K
X Here is the Kronecker product. According to [15, Theo-
Rj,r I(VJ;Z Q), . (8) rem1⊗],wearelookingforaGsuchthat wA0x0 wbw
≥ | ∀J ⊆K
{ ≤ | ∈
j∈J G isequivalenttothefinalsystemoftheFourier-Motzkin
}
(d) is due to the fact that I(Vn, ,Vn;Zn Qn, ) elimination. Here, G is a base of essentially different
1 ··· K | C ≤
n[I(V , ,V ;Z Q)+ε ], the proof of which follows the minimal vectors of F that is the cone of non-negative
1 K n
proof of··[1·4, Lemm| a 3], and taking τ =2ε . solution w 0 of wA00 =0.
n n
≥
Joint secrecy achievable rate region: We summarize the In fact, for our system defined by (9), we find
requirements in order to guarantee a reliable communica- (cid:2) (cid:3)
G=D I , where D= 1 1 . (11)
tion under the joint secrecy constraint as follows: ⊗ 2K−1
• the non-negativity for rates; Here In is an n n identity matrix. In the following, we
×
• the conditions for a reliable communication, i.e., (7); show that G (as defined in (11)) is a base of essentially
• the conditions for the joint secrecy, i.e., (8). different minimal vectors of F. Note that the system
That is, we have the following system of inequalities: defined by wA0x0 wbw G is:
{ ≤ | ∈ }
X[Rj +Rj,r]≤I(VJ;Y|VJc,Q), for all J ⊆K, GA0x0 ≤Gb ⇔ [1J]x0 ≤(cid:2)b+J −b−J(cid:3). (12)
j∈J Firstly, it is clear that w G is positive, i.e., w 0.
X ∈ ≥
Rj,r I(VJ;Z Q), for all . Secondly, we show that wA00 = 0 for any w G. Note
≥ | J ⊆K ∈
j∈J thatA00 =B [1J]by(10).Therefore,ifw (B [1J])=0,
⊗ · ⊗
To obtain the desired region of R i , the variables we have w A00 =0 as well. It is easy to verify that
i ·
{ | ∈ K}
Rofi{Ri0,rf|oir∈iK}araeretonboteienlicmluidneadtehde.r(eT,hsienrcaettehceoynwstirlalinnotst G·(B⊗[1J])=(D⊗I2K−1)·(B⊗[1J])(=a)(DB)⊗[1J]=[0J],
≥ ∈K
be involved in the Fourier-Motzkin elimination. But they where (a) is due to the mixed-product property of the
will be included in the final derived region.) Kronecker product: (A B)(C D)=(AC) (BD).
Fourier-Motzkin Elimination: A matrix notation of the Furthermore, by defin⊗ition of⊗G in (11), ea⊗ch row of G
above system of inequalities can be written as follows:
is essentially different from the others. And, they are all
" [1 ] #  R1  " [1 ] #  R1,r   (cid:2)b+(cid:3)  minimal according to [15, Theorem 2].
J . J . J
· .. + · .. ≤ (cid:2) (cid:3) , Most importantly, we show in the following that any
| [0{Jz] } | R{Kz } | −{[1zJ] } | R{Kz,r } | −{bz−J } odtuhceer opnolsyitirveeduvnecdtaonrts icnesqautiastfyioinngs.cF·oAr 00co=nve0n,iewniclle,prwoe-
A0 x0 A00 x00 b (9) denote c = [c+ c−], with c+ = [c+1,··· ,c+2K−1] and
where c− = [c−, ,c− ]. Then, c A00 = 0 is equivalent to
1 ··· 2K−1 ·
•• tSo1hJninelycisesettJthhKee=n(1wo∅×nhi-ineKcmthropidndtedyufiicncceaihsttiooooirncnelvysiescrogtefoidvrJuenonfdinnatehneDetdeisnfituoenbqistubeioatentJico1on)nos;-f, vcc−i+ecR)·teo[n1cruJamol]lfb=Jtehrica,−otafn·fpod[1rrJeths1]eu.n≤scce+isi o>≤f J02iK(.oLr−ect−i1,>1J0i)iisndtihceatiensdci+icat(oorr
sidered. Let , , be the list of 2K 1
different non{-eJm1p·t·y· sJu2bKse−t1s}of . We have − Jc+ ={|J1,·{·z· ,J}1,··· ,|Ji,·{·z· ,J}i,··· ,|J2K−1,·{·z· ,J2K−}1};
{J1,··· ,J2K−1}; K J ∈ c+1 c+i c+2K−1
• [bC1eoJrt]rheiesspaionnd(d2icKinagt−olyr1,v)e×ctKor m1Jait,riwxh,ewreith1t≤heii≤-th2Krow−t1o. Jc− ={|J1,·c{·−1z· ,J}1,··· ,|Ji,·c{·−iz· ,J}i,··· ,|J2K−1,c−2·{K·z·−,1J2K−}1}.
Denote n+ = 2KP−1c+ and n− = 2KP−1c−. Simply vector c that is other than the rows of G (as defined in
i i (11)) such that c A00 = 0, therefore, the final system of
rep−re,sen−ti,ng ,Jc+−i==,1we{Jha1+v,eJ2+,··· ,Jn++i}=1and Jc− = (9)Aiss eaquciovnaclelunstiotno·,th(e12o)nees(t1a2b)l.ishes the resulting joint
{J1 J2 ··· Jn−}
secrecy region.
n+ n−
X X
c+·[1J]= i=11Ji+, c−·[1J]= i=11Ji−; (13) ProperAtpiepsenodfixb+Aand b−
T T
c+ (cid:2)b+(cid:3)=Xn+ b+ , c− (cid:2)b−(cid:3)=Xn− b− . (14) Recall that b+T =I(VT;Y|VTc,Q) and b−T =I(VT;Z|Q)
· J J+ · J J− as defined in (9). We have the following lemma:
i i
i=1 i=1
tNhoattebthotaht cJ+c+·[a1nJd] J=c−c−sh·a[1reJ]thaendsa(m13e)ptroegseetnhceervimecptolyr LofeKm,maand8i.tsGcoivmenpa{ctTif|o1rm≤oif≤thet}eleams eantlisrteaorfratngseumbseentst
c# = c+ · [1J] (according to Definition 2). Denote the {Tt∗,i|1≤i≤t}, we have
largest element of c to be c .
# max t t t t
comLeptac{tJfio⊕r|m1s≤ofith≤e enl+em}eanntdre{aJrri(cid:9)an|1ge≤meint≤(dne−fi}nitbieonthoef Xb+Ti ≥Xb+Tt∗,i and Xb−Ti ≤Xb−Tt∗,i.
i=1 i=1 i=1 i=1
which is given in Definition 3) for Jc+ and Jc−, respec- To prove Lemma 8, we need the following two lemmas.
tively. Then, according to Lemma 5, they also share the
same compact form of the element rearrangement without Lemma 9. Given 1 i t 1 as a list of t 1
i
empty sets, i.e., ⊕ 1 i c . That is, subsets of , and lis{tT | 1≤ i≤ t−(w}ith one more sub−set
{Ji | ≤ ≤ max} includedK), let ∗{Ti1| ≤i ≤t} 1 and ∗ 1 i
⊕ = (cid:9), for 1 i c ; (15) Tt {Tt−1,i| ≤ ≤ − } {Tt,i| ≤ ≤
Ji Ji ≤ ≤ max t be their compact forms of the element rearrangement,
Ji⊕ =Jj(cid:9) =∅, for cmax <i≤n+ & cmax <j ≤n−. r}espectively. We have
Since the element arrangement does not change the pres-  ∗ i=1
ence vector, we have by Lemma 4: ∗ = Tt∗−1,1S∪(cid:0)Tt ∗ T (cid:1) 1<i<t .
c# =c+·[1J]=cXmax1Ji⊕. (16) PTtr,oiof:ThTTett∗−−p11r,,oit−o1f∩fToTtr−t1i,i−=1 1,Tttis straii=ghttforward by
i=1 definitionsof ∗ and ∗.For1<i<t,theproofisgiven
Now if we sum up the rows of (12) with respect to Tt,1 Tt,t
insteps(19)-(21)(atthetopofthenextpage),where(19)
{Ji⊕|1≤i≤cmax}, we obtain the following inequation: is by the definition of Tt∗,i; (20) is due to the distributive
cXim=a1x1Ji⊕x0 ≤cXim=a1xhb+Ji⊕ −b−Ji⊕i. (17) Llaewms mofath1e0s.etFsoarn∀dT1(2,T1)2 i⊆s bKy, twheehdaevfienition of Tt∗−1,i.
1) b+ +b+ b+ +b+ ;
O(9n),twheeoobtthaeirnhtahnedf,olilfowwiengapipnleyqucattioont:he original system 2) bT−T11 +bT−T22 ≥≤bT−T11∩∩TT22 +bT−T11∪∪TT22.
cA0x0 ≤cb ⇒ c+[1J]x0 ≤c+(cid:2)b+J(cid:3)−c−(cid:2)b−J(cid:3) proofParopopfl:ieWsetoonb−ly+givbe−t.he proof for b+T1+b+T2. A similar
n+ n− T1 T2
(⇒a) c#x0 ≤Xi=1b+Ji+ −Xi=1b−Ji−, (18) b+T1 +b+T2 (=a)I(VT1;Y|VT1c,Q)+I(VT2;Y|VT2c,Q)
(b)
where (a) is due to (16) and (14). = I(VT1∩T2,VT1∩T2c;Y|VT1c,Q)+I(VT2;Y|VT2c,Q)
Note that the LHS of (18) is the same as the LHS of
(c)
(17) (according to (16)). However, we can show that = I(VT1∩T2c;Y|VT1c,Q)+I(VT1∩T2;Y|V(T1∩T2)c,Q)
n+ n− n+ n− +I(VT2;Y|VT2c,Q)
X X (b)X X
RHS of (18)= b+ b− b+ b− (g)
i=1 Ji+ −i=1 Ji− ≥ i=1 Ji⊕ −i=1 Ji(cid:9) ≥ I(VT1∩T2;Y|V(T1∩T2)c,Q)+I(VT1∩T2c;Y|VT1c∩T2c,Q)
(=c)cXim=a1xhb+Ji⊕ −b−Ji⊕i=RHS of (17), +(=e)II((VVTT21;∩YT|2V;TY1∩|VT(2Tc1,∩VTT21)cc∩,TQ2)c,+QI)(VT1∪T2;Y|V(T1∪T2)c,Q),
where (b) is according to Lemma 8 (as given in Appendix where (a) is by the definition of b+; (b) is by the fact
T
A)and(c)isdueto(15)andb+ =b− =0.Thatis,(18)is that = ( ) ( c); (c) is by the chain
∅ ∅ T1 T1∩T2 ∪ T1∩T2
redundantsinceitisalreadyimpliedby(17),whichcanbe rule of the mutual information; (d) is by the facts that
derived as a linear combinatory with positive coefficients T2c = (T1∩T2c)∪(T1c∩T2c) and I(VT1∩T2c;Y|VT1c,Q) =
of inequations of (12). Since this applies to any positive I(VT1∩T2c;VT1c∩T2,Y|VT1c∩T2c,Q) ≥ I(VT1∩T2c;Y|VT1c∩T2c,
   
! ! ! !
[ \i  [ \i [ [ i\−1 \ 
Tt∗,i= Tjk = Tjk Tjk Tt (19)
   
{j1,···,ji}⊆[1:t] k=1 {j1,···,ji}⊆[1:t−1] k=1 {j1,···,ji−1}⊆[1:t−1] k=1
 !  ! 
 [ \i [ [ i\−1 \ 
= Tjk  Tjk  Tt (20)
   
{j1,···,ji}⊆[1:t−1] k=1 {j1,···,ji−1}⊆[1:t−1] k=1
[(cid:16) \ (cid:17)
=Tt∗−1,i Tt∗−1,i−1 Tt , (21)
Qin)d,epwehnidchenhtogldivsensinQce; VanT1d∩T(2ec), VisT1dc∩uTe2 taondthVeTf1ac∩ctTs2cthaaret Le∗mma10afonrd2thefjacttthat1T(t∗−si1n,jce∩(cid:0)T∗t∗−1,j−1∩∗Tt(cid:1)=
= ( c) and c c =( )c. bTyt−d1,ejfi∩niTttion); st≤ep (c≤) i−s by applyTint−g1L,je⊆mmTta−91,.j−In1
T1N∪oTw2weTg2iv∪e tTh1e∩pTro2of of LTem1 m∩aT82 as foTl1lo∪wsT.2 particular, (c ) is by tjhe fact that ∗ = ∗ S ;
truFeirbsyt,iwndeuschtoiown.that the statement iP=t1b+Ti ≥ iP=t1b+Tt∗,i is s(cid:0)ottehp∗er(citn−t1e)rmise1bd(cid:1)yia=tthees∗ftaecpfotsrtah1rae<tbTjyt∗,<tth=te.TTftat,∗−1ct1,tt−hT1att−T1TT,t1∗−t;1a,jTn∪td
Tt−1,j−1∩Tt Tt,j
• For t = 1, we have by definition T1∗,1 = T1. Thus, A similar proof can be applied to show that the state-
• bF+To1r=t=b+T21∗,,1tahnedsttahteemsteantetmisentrtuiesbtryuLeefmormta=110.. ment for iP=t1b−Ti is also true. This concludes our proof.
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